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Study of Elastic Modulus and Yield Strength of Polymer Thin Films Using Atomic Force Microscopy Binyang Du,† Ophelia K. C. Tsui,*,‡ Qingling Zhang,† and Tianbai He*,† State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun, Jilin, 130022, China, and Department of Physics, Hong Kong University of Sciences & Technology, Clear Water Bay, Kowloon, Hong Kong Received October 12, 2000. In Final Form: February 13, 2001 Nanometer-scale elastic moduli and yield strengths of polycarbonate (PC) and polystyrene (PS) thin films were measured with atomic force microscopy (AFM) indentation measurements. By analysis of the AFM indentation force curves with the method by Oliver and Pharr, Young’s moduli of PC and PS thin films could be obtained as 2.2 ( 0.1 and 2.6 ( 0.1 GPa, respectively, which agree well with the literature values. By fitting Johnson’s conical spherical cavity model to the measured plastic zone sizes, we obtained yield strengths of 141.2 MPa for PC thin films and 178.7 MPa for PS thin films, which are ∼2 times the values expected from the literature. We propose that it is due to the AFM indentation being asymmetric, which was not accounted for in Johnson’s model. A correction factor, , of ∼0.72 was introduced to rescale the plastic zone size, whereupon good agreement between theory and experiment was achieved.
Introduction There is increasing interest to study the physical properties of thin polymer films because of their widespread use in a large number of modern technologies. They include thermal or protective coatings, dielectrics, nonlinear optics, and numerous other fields.1 In these applications, thermal properties of polymer thin films such as the glass transition temperature, Tg, and thermal expansion coefficient are important parameters to take into account if operations at elevated temperatures are desired.2,3 The successful long-term performance and reliability of a thin film when used under these conditions is usually limited by their mechanical properties. There is thus increasing need to develop convenient and reliable characterization techniques to determine the mechanical characteristics of thin polymer films, such as elasticity, plasticity, friction, and so on. Evaluation of mechanical properties of thin films can be carried out by first producing indents or scratches with an indenter, followed by ex situ study of the resultant deformation profile with optical or electron microscopy and a surface profiler.4-6 With this approach, elastic modulus, hardness, and yield strength of thin films have been determined. However, the transfer of specimen from the indentation machine to the microscope or surface profiler inevitably results in a nonnegligible amount of time delay, which is not desirable for studies of polymers, which are viscoelastic. Therefore, recovery or relaxation of the indentation may take place * To whom correspondence should be addressed. E-mail addresses:
[email protected] and
[email protected]. † Changchun Institute of Applied Chemistry. ‡ Hong Kong University of Sciences & Technology. (1) Feng, Y. P.; Sinha, S. K.; Deckman, H. W.; Hastings, J. B.; Siddons, D. P. Phys. Rev. Lett. 1993, 71, 537-540. (2) Murata, H.; Merritt, C. D.; Inada, H.; Shirota, Y.; Zakya, H. K. Appl. Phys. Lett. 1999, 21, 3252-3254. (3) Grohens, Y.; Brogly, M.; Labbe, C.; David, M.-O.; Schultz, J. Langmuir 1998, 14, 2929-2932. (4) Gerberich, W. W.; Kramer, D. E.; Tymiak, N. I.; Volinsky, A. A.; Bahr, D. F.; Kriese, M. D. Acta Mater. 1999, 47, 4115-4123. (5) Bahr D. F.; Gerberich, W. W. Metal. Mater. Trans. A 1996, 27A, 3793-3800. (6) Briscoe B. J.; Sebastian, K. S. Proc. R. Soc. London, Sect. A 1996, 452, 439-457.
during the transfer, resulting in added uncertainty in any mechanical parameter thus determined, notably the mechanical yield strength. Furthermore, cutting-edge technology often involves thin films with thickness on the nanometer scale. These applications will require evaluation techniques with resolution of similar or smaller length scales. With the invention of atomic force microscopes (AFMs),7 it is now possible to carry out characterization of surface morphology and various kinds of physical properties of thin film samples over local areas from ∼1002 µm2 down to ∼1 nm2.8-12 By use of AFM in the method described above for the mechanical evaluation of polymer films, the study of an indent can be carried out in situ, with a resolution that is typically better than a nanometer. To perform an indentation experiment with an AFM, the tip is forced to penetrate into the sample surface until the desired deflection has been reached. Then the topography of the depression is measured almost immediately by using the same AFM cantilever/tip at a very light load. It must be noted that some “recovery” of the surface indent may take place during the 5-10 s elapse time between the act of indentation and the subsequent in situ imaging of the indent.13 This time delay is much smaller compared to that needed in ex situ measurements, in cases where the relaxation time of the polymer is comparable to the elapse time of 5-10 s notably near the glass transition temperature of the polymer. AFM techniques that make use of the response of a sample to an indentation force have been employed for the study of various nanomechanical properties of polymer single crystals and amorphous polymer thin films, including elasticity,14-16 hardness,17 (7) Binng, G.; Quate, C. F.; Gerber, C. Phys. Rev. Lett. 1986, 56, 930-933. (8) Magonov, S. N.; Reneker, D. H. Annu. Rev. Mater. Sci. 1997, 27, 175-222. (9) Radmacher, M.; Tillmann, R. W.; Fritz, M.; Gaub, H. E. Science 1992, 257, 1900-1905. (10) Domke, J.; Radmacher, M. Langmuir 1998, 14, 3320-3325. (11) Capella, G.; Dietler, G. Surf. Sci. 1999, 34, 1-104. (12) Fretigny, C.; Basire, C. J. Appl. Phys. 1997, 82, 43-48. (13) Sangwal K.; Gorostiza, P.; Servat, J.; Sanz, F. J. Mater. Res. 1999, 14, 3973-3982. (14) Du, B.; Liu, J.; Zhang, Q.; He, T. Polymer 2001, 42, 5901-5907.
10.1021/la001434a CCC: $20.00 © 2001 American Chemical Society Published on Web 04/26/2001
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and viscoelasticity.18,19 In this work, we demonstrate the use of the AFM indentation technique to study the elasticity, plasticity, and yield strength of thin polycarbonate (PC) and polystyrene (PS) films at the nanometer scale. The elastic moduli were deduced directly from the indentation force curves, using the method by Oliver and Pharr.20,21 The yield strength was determined by fitting Johnson’s conical spherical cavity model to the plastic zone size deduced from sectional analyses of the indent. A correction factor, , was introduced to make Johnson’s cavity model suitable for the analysis of realistic AFM indents, which are typically asymmetric.
diamond tip used in this experiment, β ∼ 1.034, Et ) 1141 GPa, and νt ) 0.07.20 To account for the effect of nonroundness of our three-sided pyramidal tip, the projected contact area, A should be corrected to23
A ) 3(31/2)(tan2 R)(h0 + ∆hT)2
where h0 is the depth of the depression that would be formed in the sample if the indenter were a flat punch. (For conical indenters such as the one used here, the indentation depth is ∼0.72h0.20) The correction factor, ∆hT, is given by
∆hT ) (R/8) cot2 R
Experimental Section AFM indentation tests were performed in a Digital Instruments NanoScope IIIa multimode scanning probe microscope (Digital Instruments, Santa Barbara, CA) with a diamond tip mounted on a stainless steel cantilever with nominal spring constant 186 N/m (Nanosensors). The diamond tip is a threesided pyramid with an apex angle of about 60°, yielding an included half-angle, R, of ∼21.6° (manufacturer’s specification). The cantilever sensitivity was determined through measurements with a sapphire sample before and after the indentation tests as described elsewhere.15,22 Before experiments with the AFM, the equipment was carefully calibrated to minimize any measurement uncertainties. Unless otherwise stated, a z scan rate of 4 Hz and a lateral compensation angle of 25° were used to eliminate effects due to creeps in the piezoelectric scanner and lateral motion of the AFM tip during the indentation process. Proper choice of the z scan rate and lateral compensation angle can be found elsewhere.23 Samples. The samples used were bisphenol A-polycarbonate (Mw ) 32 000 Da, Mw/Mn ) 2.1, Tg ) 150 °C) and polystyrene (Mw ) 250 000 Da, Mw/Mn ) 7.1, Tg ) 100 °C) purchased from Aldrich Chemical Co., Milwaukee, WI. Thin film samples with thickness ∼10 µm were prepared by casting 2 wt % solutions of the polymers in THF onto cleaned silicon substrates. To completely remove the residue solvent, the cast films were annealed in a vacuum oven for several days at 120 °C. No sign of dewetting could be found in the samples even upon prolonged annealing at 120 °C. The surface roughness, Rq was found to be less than 1 nm from 10 µm × 10 µm AFM topographical images of the samples. Data Analysis of the Nanoindentation. The detailed experimental procedure and data analysis for nanomechanical characterization by AFM indentation can be found in a previous work by us.14 In brief, an AFM indentation curve acquired from a sample can be analyzed to give the sample’s elastic modulus, Es, using the method of Oliver and Pharr20
[
Es ) (1 - νs2)
]
2 -1
A1/2 1 - νt 2 β 1/2 S Et π
(1)
where A is the projected contact area at maximum indentation (hmax), S ) dP/dh (hmax) is the stiffness of the sample at unloading (P denotes the applied indentation force), β is a geometrical factor dependent on the shape of the indenter, νs is Poisson’s ratio of the sample, and Et and νt are respectively the elastic modulus and Poisson's ratio of the indenter. For the Berkovich-like (15) VanLandingham, M. R.; Mcknight, S. H.; Palmese, G. R.; Elings, J. R.; Huang, X.; Bogetti, T. A.; Eduljee, R. F.; Gillespie, J. W., Jr. J. Adhes. 1997, 64, 31-59. (16) Du, B.; Zhang, J. Zhang, Q.; Yan, D.; He, T.; Tsui, O. K. C. Macromolecules 2000, 33, 7521-7528. (17) Drechsler, D.; Karbach, A.; Fuchs, H. Appl. Phys. A 1998, 66, S825-S829. (18) Tanaka, K.; Takahara, A.; Kajiyama, T. Macromolecules 1998, 31, 5150-5151. (19) Tsui, O. K. C.; Wang, X. P.; Ho, J. Y. L.; Ng, T. K.; Xiao, X. Macromolecules 2000, 33, 4198-4204. (20) Oliver, W. C.; Pharr, G. M. J. Mater. Res. 1992, 7, 1564-1583. (21) Pharr, G. M.; Oliver, W. C. MRS Bull. 1992, 17, 28-33. (22) VanLandingham, M. R. Microsc. Today 1997, 97-10, 12-15. (23) Sawa, T.; Akiyama, Y.; Shimamoto, A.; Tanaka, K. J. Mater. Res. 1999, 14, 2228-2232.
(2)
(3)
Here, R and R are the tip radius and the half-included angle of the three-sided pyramidal indentation tip, respectively. The tip radius, R, and the correction factor, ∆hT, were determined to be 45 ( 5 and 36 ( 4 nm, respectively, from five indentation measurements on a piece of fused silica (Valley Design Corp. Westford, MA). (Fused silica has been used as a standard sample because it is elastically isotropic and has a well-known elastic modulus of 72 GPa, which is independent of the indentation depth.20) The unloading stiffness, S ) dP/dh at h ) hmax, i.e., the initial slope of the unloading curve, can be determined from the empirically found relation for the indentation-deformation curve
P ) B(h - hf)m
(4)
with B and m being fitting parameters. Combining eqs 1-4, the Young’s modulus, Es, of the sample can be determined from the indentation-deformation curve by
Es )
(1 - νs2)S 1.053(h0 + 36) - 0.001S
(in GPa)
(5)
Results and Discussions Elastic Modulus of Thin Polymer Films. Parts a and b of Figure 1 show, respectively, the indentation curves we obtained from the PC and PS thin films at different indentation forces from 1.1 to 5.4 µN. In these figures, the deformation parameter, h, was obtained by subtracting the z scanner displacement by the bending of the cantilever, thus physically representing the amount of indentation produced in the sample by the indenter. Generally speaking, a polymer film may respond to an indentation force in one of three ways: viscoelastic deformation, viscoplastic deformation, or fracture. Nanoindentation measurements using an AFM, however, can only distinguish between recoverable deformation and unrecoverable deformation. For deformations that recover in the time scale of the measurement, they are generally taken to be elastic deformations. For those that do not recover in the time scale of the measurement, they are taken to be plastic deformations. Usually, a parameter called the “plasticity index”, ψ, is used to describe the relative plastic/elastic character of a material24,25
ψ)
A1 A1 + A2
(6)
where A1 is the area encompassed between the loading and unloading curves ()the plastic work done during indentation) and A2 is the area under the unloading curve ()the viscoelastic recovery). It follows that ψ ) 1 (i.e., A2 (24) Briscoe, B. J.; Fiori, L.; Pelillo, E. J. Phys. D: Appl. Phys. 1998, 31, 2395-2405. (25) Johnson K. L. Contact Mechanics; Cambridge University Press: Cambridge, 1985.
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Figure 2. The elastic modulus calculated by eq 5 for PC and PS thin films as a function of the indentation force, P. No systematic dependence on the indentation force is evident.
Figure 1. Indentation force curves acquired with different indentation forces (a) on a PC thin film and (b) on a PS thin film.
) 0) for a fully plastic deformation, ψ ) 0 (i.e., A1 ) 0) for a fully elastic case, and 0 < ψ < 1 for one with intermediate behaviors. From parts a and b of Figure 1, the plasticity indices of PS and PC thin films are estimated to be about 0.47 ( 0.07 and 0.41 ( 0.04 by averaging values obtained from different loading-unloading curves with different indentation forces, respectively, using eq 6. Hence both values fall in the range 0 < ψ < 1, demonstrating the typical viscoelastic-plastic character of polymers. The closeness of the two values of ψ found suggests that PC and PS have similar plasticity. It has been reported that when the indentation depth exceeds 10-25% of the thickness of the test film, effects of the supporting substrate underneath should be considered.22 Since the thickness of the thin film samples used in this experiment is about 10 µm, the indentation depth should thus be less than 1 µm according to this criterion. From Figure 1, it can be seen that the indentation depth that took place in the PC and PS thin films under the maximum indentation force used (i.e., 5.4 µN) is 65.6 and 61.5 nm, respectively, which are far less than 1 µm. Therefore, effects of the substrate on the indentation measurements can be ignored in this work. On the other hand, the ratio hf/hmax determines how accurately one can determine the elastic modulus of a sample using the Oliver-Pharr method 26 (where hf is the permanent deformation of the sample after complete unloading and hmax the maximum deformation at the peak indentation force, Pmax). It was found that when hf/hmax is 0.7, the Oliver-Pharr method will lead to large errors.26 From Figure 1, the hf/hmax ratios were about 0.4-0.6. Therefore, the Oliver-Pharr method is appropriate for use in the present study to evaluate the elastic modulus, Es, of PC and PS thin films. The values we obtained for Es of the samples based on six measurements as a function of the indentation force, P, are shown in Figure 2. (Possion’s ratio of PC and PS has been assumed to be 0.31 and 0.28, respectively.27) As seen from the data, PS is slightly stiffer than PC. Values of Young’s modulus found for the PC and PS films are 2.2 ( 0.1 and 2.6 ( 0.1 GPa, respectively, which agree well with data reported in the literature acquired by macroscopic measurements (see Table 1). It has been suggested that the physical properties of thin polymer films may differ from those of the bulk. For instance, the glass transition temperature of polymer thin films may be lower or higher than that of the bulk, which depends on the thickness of the film, influence of the polymer-air interface, and the specific interactions between the polymer and the supporting substrate.28-32 At the polymer-air interface, it was proposed that a liquidlike layer, extending a few segmental lengths into the film,28,29 exists where the local density is smaller than that in the bulk. This may lead to a systematic reduction in the measured value of Young’s modulus with reduced indentation force as the indentation depth becomes increasingly comparable to the thickness of the liquidlike layer (∼5 nm33). At the smallest indentation force used (Figure 1), the correction may be up to ∼20%, which should be discernible within the error bar of this experiment. However, these deviations are not notable in our data (see Table 1 and Figure 2). During the measurements, the AFM indentation produces a local high pressure (∼hundreds atmospheres) at the surface region of the polymer that is in contact with the tip. This local pressure will decrease the free volume of the polymer surface and (27) Whitney, W.; Andrews, R. D. J. Polym. Sci., Part C: Polym. Symp. 1965, 16, 2981-2990. (28) Keddie, J. L.; Jones, R. A. L.; Cory, R. A. Faraday Discuss. 1994, 98, 219-230. (29) Keddie, J. L.; Jones, R. A. L.; Cory, R. A. Europhysics Lett. 1994, 27, 59-64. (30) van Zanten, J. H.; Wallace, W. E.; Wu, W. Phys. Rev. E 1996, 53, R2053-R2056. (31) Forrest, J. A.; Dalnoki-Veress, K.; Stevens, J. R.; Dutcher, J. R. Phys. Rev. Lett. 1996, 77, 2002-2005. (32) Porter, C. E.; Blum, F. D. Macromolecules 2000, 33, 7016-7020. (33) Kajiyama, T.; Tanaka, K.; Takahara, A. Macromolecules 1997, 30, 280-285.
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Figure 3. (a) Illustration for how the plastic zone size, c, for a symmetric conical or spherical indent was determined. (b) Illustration showing how the plastic zone size, c*, of an asymmetric indent is similarly obtained.
increase the local density.34,35 which suppresses the effect of the liquidlike layer at the polymer-air interface, leading to the observed constancy in the measured elastic moduli (Figure 2). On the other hand, high surface energy chain ends36 and a small proportion of low-Mn components in the polymer37 may also moderate the effect of the surface liquidlike layer. More studies will be needed in order to distinguish which of these three is the dominant factor. It should be remarked that since the indentation depth is very low compared to the film thickness as mentioned above, influence of the specific interactions between the polymer and the substrate on the measurement should be negligible. Plastic Zone Size and Yield Strength of Thin Polymer Films. One of the physical responses of a glassy polymer to an externally applied stress is yield strength. Above this physical limit, the plastic deformation occurs. More recently, indentation methods have been developed to include this common mechanical state of polymers in the analysis, wherein the plastic zone size, c, surrounding an indent is correlated with the yield strength of thin films by using Johnson’s conical spherical cavity model for elastic-plastic materials.38,39 According to the model
c)
( ) 3P 2πσys
1/2
(7)
Here, P is the indentation force, like before, and σys is the uniaxial yield strength of the sample. Figure 3a illustrates how the plastic zone size, c, can be determined from the section profile of a conical or spherical indent. The plastic zone boundary was taken to be where the pile-up around the indent ceases and the profile returns to parallelism (34) Gracias, D. H.; Somorjai, G. A. Macromolecules 1998, 31, 12691276. (35) Mears, D.; Pae, K.; Sauer, J. A. J. Appl. Phys. 1969, 40, 42294237. (36) Tanaka, K.; Jiang, X.; Nakamura, K.; Takahara, A.; Kajiyama, T. Macromolecules 1998, 31, 5148-5149. (37) Tanaka, K.; Takahara, A.; Kajiyama, T. Macromolecules 1997, 30, 6626-6632. (38) Johnson, K. L. J. Mech. Phys. Solids 1970, 18, 115. (39) Kramer, D.; Huang, H.; Kriese, M.; Robach, J.; Nelson, J.; Wricht, A.; Bahr, D.; Gerberich, W. W. Acta Mater. 1999, 47, 333-343.
Figure 4. (a) AFM topographical image of a 3 × 3 array of AFM indents created in a PS thin film at various indentation forces: Indents along the same column are produced by the same indentation force. Along any one row, in going from right to left, the indentation forces applied are 2.2, 2.7, and 3.2 µN. (b) Section profile of the three indents along the line drawn in the topographical image shown in Figure 4a.
with the sample surface. Similar AFM section profiles have been taken for all indents made in order to determine the extent of the plastic zone for the two polymers under different experimental conditions. Figure 4a shows a set of nine indents produced in a PS thin film. Figure 4b shows the section profile of the three indents along the line drawn in Figure 4a, acquired under indentation forces of 2.2, 2.7, and 3.2 µN (from right to left). There are a few points that are noteworthy. First, all three indents are asymmetric with most of the excavated material accumulating only on one side of the indent. This is attributable to the asymmetric shape of the tip. For asymmetric indents, we simply assume the plastic zone size to be c* as exemplified in Figure 3b for use in eq 7 to calculate the yield strength of polymer thin films. Second, depths of the indents measured by section analysis are consistently less than those revealed by the corresponding indentation-force curves. Two possible reasons may explain this: (1) There was a certain amount of viscoelastic recovery that took place during the 5-10 s elapse time between the act of indentation and the subsequent imaging of the indent. (2) The fact that the imaging tip has a radius of curvature comparable to that of the indent (N.B. The same tip was used for making the indent and for imaging.) and that it is scanning over the surface at a finite speed (∼3.0 µm/s) may easily cause the tip to miss the vertex of the indent, resulting in underestimation of the indentation depth. Between these two possible causes, the former should be relatively unimportant for this study. All indentation measurements had been carried out at room temperature, which would condition the polymers well within the glassy state. Most of the recovery should therefore be elastic within the 5-10 s elapse time. On the other hand, since the apex angle of the tip used is 60°, the amount of underestimation of the indentation depth can be anywhere
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Figure 5. Plots of the measured plastic zone size, c*, as a function of the indentation force, P, for the PC (open squares) and PS (solid circles) thin films. The solid and dotted lines are best fits of eq 7 to the data of PC and PS, respectively. Table 1. Elastic Moduli and Yield Strengths of the PC and PS Thin Films Measured in This Work by AFM Indentation Poisson’s ratio, ν PC PS
0.31 0.28
elastic modulus, Es (GPa) ref 27 exptl 2.0 2.8
2.2 ( 0.1 2.6 ( 0.1
yield strength, σys (MPa) ref 27 exptl 72.5 88.5
141.2 178.7
between 0 and ∼45/tan 60° (nm). Assuming the medium value for the amount of underestimation, the error in the measurement by section analysis can be comparable to the actual value of the indentation depth. Therefore, indentation depths are determined directly from the indentation-force curves in this work. Finally, we note that fracturing may also occur in the polymer during indentation. However, the process is uncontrollable and so will be beyond the scope of the present analysis. Figure 5 shows the plastic zone size, c*, as a function of the indentation force, P, for the PC and PS thin films. Fitting these data to eq 7, we obtained estimates for the yield strength of PC and PS to be 141.2 and 178.7 MPa, respectively. The results are summarized in Table 1. It is seen that both the yield strengths of the PC and PS thin films are about twice the literature values, which are 72.5 and 88.5 MPa, respectively.27 In the following, we will provide an explanation for the discrepancy. Johnson38 had found that when the ratio (Eeff tan β)/σys is greater than 40, the conical spherical cavity model is not valid for the analysis of elastic-plastic indentations. Here, β denotes the angle between the face of the conical indenter and the indented surface (β ) 90° - R). The effective elastic modulus, Eeff, takes into account the elastic deformation occurring in both the sample and the indentation tip and is related to Young’s modulus of the sample, Es, and that of the tip, Et, according to
1 - νs 1 - νt 1 ) + Eeff Es Et
(8)
By use of the literature values quoted above for Es of the two polymers, the ratio (Eeff tan β)/σys comes out to be 77 and 86 (definitely >40), respectively, for PC and PS. On the other hand, Kramer et al.39 had been able to apply eq 7 to cases where the ratio was ca. 330, but the material investigated was a metal. It is unclear whether similar extension of the analysis can be applied to polymer samples, for which the plastic zone will be far more
Figure 6. Plots of the rescaled plastic zone size, c′ (see text) vs the measured plastic zone size, c* for the PC (open squares) and PS (solid circles) thin films. Straight lines are linear fits to the data.
extended. Equally unclear is whether eq 7 will be suitable for the analysis of asymmetric indenters, with which the excavated material accumulates mostly on one side of the resultant indent. Notwithstanding the aforementioned legitimate criticisms of Johnson’s model (eq 7), we note that it fits the data of both polymer thin films very well, except for an apparent underestimate of the data of the plastic zone size, c* (which result in the ultimate overestimate of σys found). It suggests that we may correct the effects due to asymmetry of the indent phenomenologically by replacing the measured plastic zone size, c* by an apparent one, c′, that will give the correct value of σys by eq 7. Figure 6 shows plots of c′ vs c* for the PC (open squares) and PS (solid circles) thin films assuming the yield strengths to be 72.5 and 88.5 MPa, respectively. Subjecting the data to linear fits, one obtains c′ ) 0.97 + 1.38c* (in nm) for PC and c′ ) 2.4 + 1.39c* (in nm) for PS. For all measurements of this study, c* is always bigger than 50 nm. Therefore, the constant term in either expression of c′ is negligible. We incorporate a correction factor, , into the model, as shown below, to account for the phenomenologically found correction of c′
c′ )
( ) 3P 2πσys
1/2
(9)
Clearly, should be 1/1.38 - 0.72 for PC and 1/1.39 - 0.72 for PS. While the reason why this simple correction works for asymmetric (Berkovich-like) indenters and the value for the correction factors found (∼0.72) is not exactly clear, the present result demonstrates that Johnson’s conical spherical cavity model can be modified simply (as in eq 9) for the calculation of yield stress of PC and PS thin films. A better understanding of how the correction factor, , worked here will certainly be beneficial to extending AFM indentation methods to the determination of yield stress for all other materials. Conclusions In this work, we have investigated the use of atomic force microscopy (AFM) indentation for the determination of elastic moduli and yield strengths of polymer thin films. Polycarbonate and polystyrene thin films that are ∼10 µm thick have been studied. By application of the theory of Oliver and Pharr to parameters derived from the AFM indentation-force curves, Young’s moduli of PC and PS thin films were found to be 2.2 ( 0.1 and 2.6 ( 0.1 GPa,
Elastic Moduli and Yield Strengths of Thin Films
respectively, which are in good agreement with the literature values. However, a straightforward application of Johnson’s conical spherical cavity model to the measured data of the plastic zone size (which were obtainable from the sectional profile of the indents) leads to yield strengths of the polymer films that are about twice the values reported in the literature. The deviations are attributable to the ratio (Eeff tan β)/σys being >40 and the indent being not symmetric, which had been assumed in the model. By introduction of an empirical correction factor, ) ∼0.72 (which is essentially used to scale c* down to c′ ) c*), to Johnson’s conical spherical cavity model, good agreement between theory and experiment was achieved. This may suggest that a simple modification to Johnson’s model may be sufficient to extend its validity to polymerssfor which the ratio (Eeff tanβ)_/σys tends to be smaller than that of what the theory had originally assumed, namely hard materialssand the asymmetric Berkovich-like in-
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denters that are popularly used in AFM indentation studies. Acknowledgment. We thank Professor C. M. Chan and Ms. Pauline S. T. Leung of the Chemical Engineering Department, Hong Kong University of Science and Technology (HKUST), for technical assistance. This work was supported by the National Science Foundation of China. O.K.C.T. acknowledges support by the Polymer Physics Laboratory, Changchun Institute of Applied Chemistry, under Project No. R120001W, and Hong Kong University of Science and Technology (HKUST) through the Research Grant Council of Hong Kong under Project No. DAG98/99.SC24. The sapphire sample for calibrating the detection sensitivity of SPM was kindly supplied by Drs. S. Mogonov and L. Huang of Digital Instruments (Veeco Instruments Inc.). LA001434A
