M. R. VanLandingham¹, S. H. McKnight², G. R. Palmese¹, R. F. Eduljee¹, J. W. Gillespie, Jr.¹, and R. L. McCullough¹

¹Center For Composite Materials, University of Delaware, Newark, DE 19716
²Army Research Laboratory -- Materials Directorate, Wilmington, DE 19898

Atomic force microscopy (AFM) has become an extremely useful tool in materials science applications. In addition to imaging surfaces with nanometer resolution, the AFM can also be used to determine surface roughness [1], probe local changes in friction [2], measure surface forces [1, 3], and assess changes in local elasticity over a sample surface. Recently much attention has been focused on the interpretation of AFM force curves for characterizing tip-sample interactions [1, 3-7]. Through appropriate analysis of force curves, a wealth of information can be obtained regarding the mechanical, chemical, and adhesive properties of the surface. To date, most studies using force mode AFM [3-6] have concentrated on the attractive and transition regions of the force curve. Very little effort has been dedicated to the examination of the contact portion (or so-called repulsive region) of the force curve. This region contains valuable information regarding the nanoscale mechanical response of the sample [7]. In this paper, a technique is developed that (1) relates the elastic modulus of the sample to the sample response measured using AFM force curves; and (2) illuminates the importance of the relative stiffnesses of the cantilever probe and the sample to the determination of elastic modulus.

During the force mode of AFM, the 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 laser beam is reflected off the top of the probe and onto a segmented photodiode, thus producing a measurement of the probe tip deflection. A plot of this tip deflection signal as a function of the vertical displacement of the piezo actuator is produced. Such a plot, termed a "force curve," is shown in Fig. 1. As the piezo displacement in the z-direction increases, the tip deflection voltage (V_td) remains constant until the probe tip makes contact with the surface (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_td (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_td 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_td until the tip jumps out of contact (E to F in Fig. 1).

To extract the nanoscale mechanical response of the material from the contact portion of the resulting force curve, the tip-sample interaction is modeled as two springs in series, as shown in Fig. 2. After contact is made between the probe tip and the sample surface, 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. 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

(1)

The tip displacement can be directly related to the applied force, P, by

(2)

where k_c is the spring constant of the cantilever probe. The accuracy of the calculated forces is generally on the order of 1 to 10 nN, depending on cantilever stiffness, because the noise on the tip displacement detector limits depth resolution to 0.1 nm [1].

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, the angle change (Dq_t) and the tip displacement (Dz_t) are related by

(3)

where L_c is the length of the cantilever probe. This angle change is directly related to the change in V_td by system conversion and amplification factors which can be lumped into a constant, C_q, such that

(4)

Combining Equations 3 and 4, the slope of the force curve, S, is shown to have an upper limit, S^*, 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 using Equation 1. The applied force, P, can also be calculated using Equation 2 if k_c is known. Using contact mechanics for the indentation of an elastic half space [8, 9], the indentation displacement, Dz_i, can be related to the applied force, P, through the relation

(6)

where x is a constant which depends on probe tip geometry, E and n are the sample elastic modulus and Poisson's ratio, respectively, and m is a power law exponent characteristic of the indentation behavior. For behavior characteristic of a flat cylindrical punch, m = 1, while for behavior characteristic of spheres and paraboloids of revolution, m = 1.5. Normally, m is determined from a curve fit of P as a function of Dz_i for a material of known E and n.

For the experimental investigations, a Digital Instruments D3000 scanning probe microscope was used to produce force curves from the interaction of two separate silicon cantilever probes and a number of polymer samples. The spring constants of the two probes were estimated to be 1 and 60 N/m respectively. The polymer samples consisted of polyurethanes, visible-light-cured acrylated epoxies, thermally-cured epoxies, and polyetheretherketone (PEEK). The elastic moduli of these polymer samples were determined independently using dynamic mechanical analysis and range from 0.02-3.0 GPa. In order to provide a calibration reference, titanium (E = 110 GPa [10]) was used as an "infinitely stiff" material for the determination of C_q from Equation 4. For each sample, a minimum of 10 individual force curves was obtained. Each force curve was then analyzed according to the methodology outlined previously.

The results obtained show promising correlation between the generated force curve data and the sample moduli. The slope reduction factor, f, was observed to increase as the stiffness of the sample increased. For each cantilever probe, however, the f values leveled off above a certain modulus value (approximately 0.5 GPa for the 60 N/m probe and 0.1 GPa for the 1 N/m probe). The response of polymers with moduli greater than 0.5 and 0.1 GPa, respectively for the 60 N/m and 1 N/m probes, approaches that of the "infinitely stiff" titanium sample. Therefore, above these modulus values, samples of differing moduli become indistinguishable.

The use of the reduction factor (f), however, does not take into account the actual indentation process. Using Equations 1 through 4, the loads applied to the sample surface and the corresponding indentation displacements can be determined. The ratio of the maximum indentation displacement relative to a maximum probe tip displacement of 100 nm versus sample modulus is plotted in Fig. 3. This ratio is shown to be quite sensitive to changes in modulus up to 0.5 and 0.1 GPa for the 60 N/m and 1 N/m probes, respectively. When the amount of probe tip displacement exceeds the indentation displacement, Dz_i / Dz_t becomes insensitive to modulus changes. In other words, the force applied by the probe to the sample surface is insufficient to produce adequate deformation of the stiffer samples. Because the 60 N/m probe applies higher forces to the sample than does the 1 N/m probe for a given tip displacement, it is sensitive up to higher modulus values and thus can be used to evaluate higher modulus samples. Also shown in Fig. 3 are curves generated using the spring model shown schematically in Fig. 2 and outlined in Equations 1, 2, and 6. Input values typical of experimentally measured responses (m = 1.2 and x/(1-n²) = 80) were used to calculate values of Dz_i corresponding to Dz_t = 100 nm for probe spring constants of 1, 60, and 500 N/m. The model results correspond well to the experimental data and indicate that extremely stiff probes (k_c = 500 N/m) will be required to probe modulus changes in polymers commonly used in engineering applications. Both the experimental results and the results of the spring model indicated that cantilever selection should be based on the expected range of moduli.

Future work will address the use of stiffer cantilevers to probe higher modulus (E > 1.0 GPa) polymeric materials. While this technique shows promise, several factors must be addressed to extend the capabilities of the AFM as a probe of local property variations. One such area to be studied will be the stochastic aspects of the indentation process. Material heterogeneity and other stochastic effects can be significant when probing nanoscale properties and must be addressed. A second concern is the effect of time-dependent behavior on the measured response. However, changing several independent force mode parameters should allow for a more detailed characterization of the relative amounts of viscoelastic and plastic behavior involved. Finally, techniques currently under development should allow for more accurate measurements of the spring constants of cantilever microbeams so that the applied forces can be determined more precisely. With a firm understanding of these effects, a more quantitative evaluation of the indentation mechanics can be made. Calibration curves can then be generated which relate the nanoscale mechanical response of the sample to values of modulus for various probe-polymer combinations. These calibration curves will allow for the study of modulus changes in heterogeneous regions typically found in polymer blends, adhesive systems, coatings, and in the vicinity of fibers in composite materials. 

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