M. R. VanLandingham^†, S. H. McKnight^*, G. R. Palmese^‡, R. F. Eduljee^†J. W. Gillespie, Jr.^†, and R. L. McCullough ^†

^† Center for Composite Materials and Materials Science Program, University of Delaware, Newark, DE 19716-3144
^* Army Research Laboratory, Weapons and Materials Research Directorate, Aberdeen Proving Ground, MD 21005-5069
^‡ Center for Composite Materials, University of Delaware, Newark, DE 19716-3144

Abstract

The atomic force microscope (AFM) has become a popular tool for characterizing surfaces of many different types of materials. In this paper, an AFM is used to probe the mechanical properties of polymer samples through examination of force curves produced during tip-sample contact and indentation. Three types of cantilever probes with spring constants estimated to be 1-5 N/m, 20-100 N/m, and 400-500 N/m respectively, were used to study different polymer samples with known modulus values ranging from 20 MPa to 3 GPa. A technique is developed that relates the measured sample response to elastic modulus, and illuminates the importance of the relative stiffnesses of the cantilever probe and the sample to the material response.

Introduction

The characterization of polymeric materials on a sub-micron scale is necessary to evaluate their performance in a wide variety of applications. Further, nanoscale properties that control various aspects of material performance can be different from bulk properties due to differences in local chemistry or microstructure [1-4]. The imaging capabilities of the atomic force microscope (AFM) allow for the direct evaluation of local microstructural and property changes in these complex material systems [5, 6]. To date, however, the evaluation of mechanical properties with the AFM has been purely qualitative.

The operation of the AFM in force mode allows for a more quantitative characterization of polymer behavior under mechanical loads. In force mode, the AFM probe tip is first lowered into contact with the sample, then indented into the surface, and finally lifted off of the sample surface. Concurrently, a detection system measures the probe tip deflection. Our system is equipped with an optical detection system, in which a laser beam is reflected off the top of the probe and onto a photodiode. Deflection of the probe tip produces a change in the photodiode voltage, V_t, which is monitored as a function of the vertical displacement of the piezo actuator, Dz_p, to produce a force curve (see Fig. 1). As the piezo moves toward the sample surface, V_t remains constant (A to B in Fig. 1). Just before tip-sample contact is made, the probe tip can be pulled down to the surface by attractive forces, causing a small decrease in V_t (B to C in Fig. 1). Decreasing the piezo height further causes the cantilever to deflect in the opposite direction, resulting in an increased V_t signal (C to D in Fig. 1). During unloading (D to E in Fig. 1), the piezo retracts, reducing the cantilever deflection until the tip separates from the sample surface. Often, however, the tip adheres to the surface, causing a further decrease in V_t until the tip jumps out of contact (E to F in Fig. 1).

Colton and co-workers have observed that these force curves contain information regarding the nanomechanical properties of the material [7, 8]. Also, several microindentation studies [9-11] have revealed the importance of the slope of the load-penetration curve or contact stiffness,

(1)

where P is the indentation load, z is the penetration depth, r is the contact radius, and E and n are the sample elastic modulus and Poisson's ratio, respectively. Hues et al. [8] have reported that initial slopes of unloading curves for several high modulus materials (e.g., GaAs, mica, and 

silicon) indented with a diamond-tipped AFM probe did show reasonable correlation with elastic modulus. The difference in slopes for two materials, however, did not correspond directly to the difference in modulus. Further, the relationship between indentation load and penetration depth will not be linear, in general, because of continuous changes in the contact area [12]. In this paper, a more complete method for measuring nanomechanical properties with the AFM is presented and applied to a variety of polymer samples.

Experiment

A Digital Instruments D3000 scanning probe microscope was used to produce force curves from the interaction of several cantilever probes with a number of polymer samples. The effective spring constants of the probes were estimated to be 1-5 N/m (F), 20-100 N/m (T), and 400-500 N/m (S). The F and T probes were 225 mm and 125 mm, respectively, in length and were made of silicon in a single-beam configuration with widths of 30-40 mm and thicknesses of 2-5 mm. The S probes were similar to the T probes but with larger widths of 70 mm and larger thicknesses of 7 mm. The tips of these single-beam silicon probes were located near the end of the microbeams, having heights of 10-15 mm and radii of 5-10 nm. All dimensions and spring constants are as quoted by the manufacturer.

Polymer samples consisted of several polyurethanes and several epoxy samples. The polyurethanes were processed from polyester polyol/TDI-based diisocyanate prepolymers with different chain flexiblities, resulting in samples with different elastic moduli [13]. Epoxy samples were processed with different ratios of epoxy to amine curing agent, again yielding samples with different elastic modulus values [1, 2]. The elastic moduli of these samples were determined independently using dynamic mechanical analysis (DMA) and ranged from 20 MPa to 3.0 GPa. DMA was performed using a three-point bend loading fixture and a frequency of 1 Hz.

To provide a calibration reference for the indentation studies, a sapphire sample (E = 470 GPa [5]) was used as an "infinitely stiff" material. After obtaining a minimum of 10 force curves for the sapphire sample, a minimum of 10 individual force curves was obtained under the same operating conditions for each polymer sample of interest. To ensure that the system did not change during testing, a second set of force curves was obtained for the sapphire sample after indentation testing was completed. Each force curve was then analyzed according to the methodology outlined in the following subsection.

AFM Indentation Technique

To extract the elastic response of the material from the contact portion of the force curve, the tip-sample interaction is modeled as two springs in series, as shown in Fig. 2. After contact is made, piezo displacement results in both probe tip deflection and sample indentation, the amounts of which depend on the relative stiffnesses of the sample and the cantilever probe [5, 14]. The relationship between the displacement of the piezoelectric actuator (Dz_p), the displacement due to tip deflection (Dz_t), and the indentation displacement (Dz_i) is simply

(2)

where the cos(10°) term corrects for the angle of the probe to the horizontal, as shown in Fig. 2. The probe force, P, can be calculated directly from the probe tip displacement by

(3)

where k_c is the spring constant of the cantilever probe. The resolution of the calculated forces is generally on the order of 1 to 50 nN, depending on cantilever stiffness, because the noise on the tip displacement detector limits depth resolution to 0.1 nm [8]. To calculate the force applied to the sample, P is multiplied by cos(10°) to again account for the 10° angle of the probe. Because of this 10° probe angle, a lateral surface force is often generated which bends the cantilever in the direction opposing the bend due to the normal force. To eliminate this effect, new software was developed [15] to provide a compensating lateral motion of the probe which counteracts the moment acting on the cantilever due to the reaction forces at the surface [13].

For a sample which is infinitely stiff with respect to the probe, no indentation will occur and Dz_p = Dz_t. From simple beam theory and considering the 10° angle of the probe, the angle change of the cantilever at the tip (Dq_t) and the tip displacement (Dz_t) are related by

(4)

where L_c is the length of the cantilever probe. This angle change is directly related to the change in photodiode voltage, DV_t, by a system calibration constant [5], C_q, such that Dq_t = DV_t / C_q. This calibration constant can be determined from the slope of the force curve, S, which has an upper limit, S*, for a sample of "infinite stiffness" given by

(5)

For a sample that deforms due to the force applied by the probe, S will be reduced from S* by a reduction factor, f (i.e., f = S / S*), such that larger values of f correspond to stiffer samples. Also, the amount of indentation, Dz_i, at different points along the force curve can be calculated along with the applied force, P, if k_c is known. Assuming that the unloading process is characterized by elastic recovery only, the indentation displacement, Dz_i, can be related to the applied force, P, by the following equation generally attributed to Sneddon [16]:

(6)

where x is a constant which depends on the geometry of contact, E and n are the sample elastic modulus and Poisson's ratio, respectively, and m is the power law exponent. The exponent, m, is characteristic of the geometry of indentation during unloading. The unloading geometry does not, in general, correspond to the indenter geometry because of plastic deformation in the vicinity of the indenter tip created during the loading process. This plastic deformation, along with the indenter geometry, determines the effective shape of indentation during unloading [17]. The power law parameters, generated from a curve fit of the unloading data, can then be used to calculate the contact area for a known value of E and a reasonable estimate of n [13]. Without an independent assessment of the contact area, absolute modulus values cannot be determined. However, relative stiffness values for two materials or for two different regions on a surface can be related by [13]

(7)

 Therefore, the elastically-recovered indentation displacements, Dz_i, for the two materials can be used to calculate the ratio of material moduli if the contact radii are similar.

Results

AFM Probe Selection

Preliminary studies were performed to evaluate the potential of the AFM as an indentation device. In these studies, F and T probes were used to indent a variety of polymer samples. The slope reduction factor, f, was observed to increase as the stiffness of the sample increased. However, this observation was only true up to a certain value of modulus for a given probe spring constant. Similarly for a given probe force, the maximum indentation displacement approached the measurement limits above a certain modulus value [18]. For lower stiffness probes, the applied force is insufficient to produce adequate elastic deformation in samples for which E > 0.5 GPa. An appropriate probe spring constant for indentation can be determined using the contact stiffness, defined in Equation 1. For example, a polymer with E = 0.3 GPa and n = 0.4 which is indented with a probe producing contact radii ranging from 20-100 nm will have a contact stiffness during indentation of S = 14-70 N/m. T probes (k_c = 20-100 N/m) are sufficiently stiff to indent this material. For indentation of a polymer with E = 3 GPa under similar conditions, S = 140-700 N/m, and thus a much stiffer probe would be required to produce measurable elastic deformation.

Polyurethanes

From the results of DMA, the room-temperature modulus values of the five polyurethane samples were 0.02, 0.05, 0.11, 0.26, and 0.56 GPa. The indentation responses of these systems were similar. Large amounts of elastic deformation, as measured from the unloading response, were observed, and the amount of inelastic deformation, as measured by the hysteresis between the load and unload curves, depended on the magnitude of the maximum load. Typical load-indentation curves for the 0.26 GPa and 0.56 GPa samples, measured using a T probe and a maximum load of 5000 nN, are shown in Fig. 3. The calculated m values are quite similar, indicating similar unloading geometries, but the total amount of indentation, as well as the amounts

of elastic deformation and load-unload hysteresis, is larger for the lower modulus material. This observation indicates that for these materials, as the penetration depth increases, the deformation zone increases such that both elastic and inelastic deformation increase.

Epoxies

Three epoxies samples were processed with 20, 28, and 50 parts amine, respectively, to 100 parts of epoxy; the 100/28 epoxy-amine mixture is the stoichiometric ratio [1, 2]. These samples had a modulus values of 3.0, 2.3, and 2.0 GPa, respectively. Indentation was performed with an S probe at several load levels. For each sample, the elastic unloading response, Dz_i, increased as the maximum indentation load was increased, as shown in Fig. 4. Further, at intermediate load levels, correlation between Dz_i and sample modulus, using Equation 7, was achieved. At the lowest load level, the sample responses were indistinguishable due to scatter in the data. At the highest load level, the elastic response from the off-stoichiometry samples leveled out, while that of the 100/28 sample continued to increase at a steady rate. Also at the higher load levels, the load-unload hysteresis, or inelastic response, increased significantly for the 100/20 and 100/50 samples, such that the contact radius values for the three samples were no longer similar.

Conclusions

The development of a technique that utilizes the atomic force microscope (AFM) as a nanoindentation device has been presented. This technique allows for the indentation response to be extracted from AFM force curve data. Unloading curves were characterized by a power law relation between load and indentation displacement that was developed from elasticity theory. The contact stiffness was shown to be a useful parameter for estimating the resistance of a polymer to indentation by a probe with a known spring constant.

Indentation tests were performed on several homogeneous polymeric materials. A set of polyurethanes with modulus values ranging from 0.02 to 0.56 GPa responded in a similar manner to indentation loads. Both elastic and inelastic deformation increased with penetration depth of the indenter. Three epoxy samples with slightly different modulus values were indented with a 450 N/m probe at several load levels. At the intermediate load levels, good correlation between modulus and elastic indentation response was observed. At low load levels, measurement error rendered the sample responses indistinguishable, while at higher levels, inelastic deformation differences convoluted the results.

Acknowledgments

The authors gratefully acknowledge the support of the U. S. Army Research Laboratory (ARL) under the Composite Materials Research Collaborative Program (CMRCP), ARL agreement number DAAL01-96-2-0048. Also, the authors at the University of Delaware appreciate the close interactions with scientists and engineers at Digital Instruments which continue to play a large roll in the development of this technique. 

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