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

^† 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 use of the atomic force microscope (AFM) to measure surface forces has been developed to optimize its operation as a surface imaging tool. This capability can potentially be extended to evaluate nanoscale material response to indentation. In this paper, a novel technique is used to probe local property changes in multi-component polymer systems. Changes in indentation response in interphase regions are investigated for an adhesive system involving a diffuse polymer-polymer bond and for two composite systems. To date, diamond-tipped probes with effective spring constants of 150 and 310 N/m have been used to investigate polyimide and epoxy resin matrices reinforced by carbon fibers.

Introduction

The use of polymers in multi-component systems, such as adhesives and composites, requires sensitive, nanoscale property evaluation of polymer systems. Nanoscale properties that control various aspects of material performance can be different from bulk properties. For example, the behavior of polymer composites is highly dependent on the interfacial strength between the fiber and matrix. However, interfacial strength is controlled by the matrix material adjacent to the fiber, referred to as the fiber-matrix interphase region. In several investigations, this interphase region has been shown to have significantly different properties compared to the bulk resin [1-5]. This observation is generally attributed to differences in local chemistry or microstructure. Experimental evidence of these property differences has been obtained using several different methods, each of which infer interphase properties indirectly from the test data [2-5].

The capabilities of the atomic force microscope (AFM) allow for the direct evaluation of local microstructural and property changes in complex material systems. The indentation response of a material will depend on elastic and inelastic properties and can be measured using the AFM in force mode. In this mode, 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 detection system measures the probe tip deflection. The plot of tip deflection versus piezo motion is called a force curve. 

Preliminary studies were performed to evaluate the potential of the AFM as an indentation device. In these studies, probes with spring constants less than 100 N/m, generally used for imaging, were used to indent a variety of polymer samples. For these lower stiffness probes, the applied force is insufficient to produce adequate elastic deformation in samples for which E > 0.5 GPa. Further, the results indicated that cantilever selection should be based on the expected range of sample moduli [6]. An appropriate probe spring constant can be determined using the contact stiffness, defined as the slope of the load-penetration curve:

(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. 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. From experimental observation, probes with spring constants of 20-100 N/m were generally sufficient to indent such a material while 1-5 N/m probes were not [6]. For indentation of a polymer with E = 3 GPa under similar conditions, S = 140-700 N/m, and thus much a stiffer probe would be required to produce measurable elastic deformation. In this paper, diamond-tipped probes with effective spring constants of 150 and 310 N/m are used to investigate interphases in several composite systems and in a diffuse polymer-polymer bond.

Experiment

A Digital Instruments D3000 scanning probe microscope was used to produce force curves from the interaction of two cantilever probes with a number of polymer samples. The probes were diamond-tipped stainless steel cantilevers that had widths of 80-110 mm, thicknesses of approximately 12.7 mm, and lengths of 250-300 mm. The diamond tips had heights of 100-150 mm and tip radii estimated to be 20-30 nm. The effective spring constants were estimated to be 150 ± 15 N/m (D1) and 310 ± 20 N/m (D2). These spring constants values were measured using a technique in which a digital microbalance is utilized to measure applied forces as a function of the deflection of the cantilever probe [7].

Multi-component polymer systems included two composite systems. The first system was a thermoplastic polyimide composite reinforced with a 60% volume fraction of graphite fibers (IM7/K3B) [8]. The other samples were single-fiber composite systems employing AS4 graphite fibers in an epoxy matrix (EPON 828 epoxy cured with PACM 20 amine [1, 2]). Also, an adhesive system created by allowing this same epoxy-amine system to diffuse into a fully-formed thermoplastic polymer, polysulfone, during cure at 60°C [9, 10] was investigated. The elastic moduli of neat polymer samples used in this study were determined independently using dynamic mechanical analysis (DMA) with 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 [11]) 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 nanoscale mechanical 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 Figure 1. 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 [11, 12]. 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 Figure 1. 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. To calculate the force applied to the sample, P is multiplied by cos(10°) to again account for the 10° angle of the probe.

For a sample which is infinitely stiff with respect to the probe, no indentation will occur and Dz_p = Dz_t. Therefore, the slope of the force curve, S, will have an upper limit, S*, given by

(4)

where L_c is the length of the cantilever probe and C_q is a system calibration constant [11].

For a sample that deforms due to the force applied by the probe, S will be reduced from S*. The amount of indentation, Dz_i, at different points along the force curve can thus be calculated along with the applied force, P, if k_c is known. Assuming that the unloading process is characterized by elastic recovery only, Dz_i can be related to P by the following equation generally attributed to Sneddon [13]:

(5)

where x is a constant which depends on the contact geometry, E and n are the sample elastic modulus and Poisson's ratio, respectively, and m is a power law exponent determined from a curve fit of P as a function of Dz_i. The exponent, m, is characteristic of the geometry of indentation during unloading, which is a function of the plastic hardness impression and the indenter geometry [14]. The power law parameters can then be used to calculate the contact area for a known value of E a reasonable estimate of n [7]. Without an independent assessment of the contact area, absolute modulus values cannot be determined from indentation data. However, relative stiffness values for two materials or for two different surface regions can be related by [7]

(6)

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 can be related. 

Results

Polymer-Polymer Adhesive System

Indentation was performed as part of a study in which the properties of a bond between epoxy and polysulfone were evaluated. The extent of the diffusion of the epoxy into the polysulfone was estimated using several methods to be between 2 and 3 mm [9, 10]. Indentations were made over a total distance of approximately 5 mm, stepping laterally with increments of 200 to 300 nm. The D1 probe was used to apply a maximum indentation force of 4500 nN. Also, the surface was imaged after indentation to evaluate changes in the sizes of the plastic impressions. 

A variety of responses was measured across the polymer-polymer bond. The elastic response of the bulk materials, characterized by power law front factors (a = xE/(1-n²)) for various power law exponents (m), is shown in Fig. 2. A curve fit of a as a function of m, while perhaps not physically meaningful, does show two different trends for the two bulk materials. The bulk epoxy, in general, exhibited slightly larger amounts of elastic recovery and less inelastic deformation than the bulk polysulfone. This observation is most likely due differences in mechanical behavior. Although the modulus values are similar (2.3 GPa for the epoxy versus 2.5 GPa for the polysulfone), thermosetting polymers, such as the epoxy, are generally much more brittle, exhibiting much lower levels of plasticity than thermoplastic polymers due to the nature of crosslinking versus chain entanglement. Thus, the different trends in the a versus m curves stem from differences in both modulus and contact geometry. The responses from the interphase region tend to lie between these two curves.

(a)

(b)

Figure 2 – Plot of the power law front factor, a, as a function of the power law exponent, m, for the epoxy-polysulfone system; (a) two different dependencies exist for the bulk material; and (b) the response of the interphase material lies between the responses of the bulk polymers.

The overall responses measured in the interphase region were similar to the bulk epoxy in some cases and the bulk polysulfone in other cases, but in most cases were different from either of the bulk materials. In particular, the sizes of the plastic impressions were often larger in the interphase region than in either of the bulk materials, and thus were used to estimate the width of the interphase to be approximately 3 mm. This estimation agrees with estimations of the interphase size from energy-dispersive x-ray spectroscopy, electron microscopy, and AFM imaging [10].

Polymer Composite Systems

Indents were made near a graphite fiber in a 60% volume fraction thermoplastic polyimide composite (IM7/K3B) [8] using the D1 probe. Maximum applied loads were approximately 3000 nN, and indents were made radially outward from the fiber using step sizes of less than 100 nm. For indents nearest the fiber, deformation was significantly restricted by the presence of the fiber because the fiber modulus is much greater than the matrix modulus. Using the indentation mechanics presented previously, an apparent stiffening of the matrix material results. This effect was modeled using finite element analysis. Results from a two-dimensional axisymmetric model are shown in Fig. 3, along with the experimental results, using constitutive properties representative of the IM7/K3B system [8]. In the case where displacement continuity is maintained across the fiber-matrix interface, a significant increase in the apparent matrix modulus is expected as the fiber surface is approached [5]. For the model, normalized stiffness is defined as the force divided by the indentation displacement normalized to that in the bulk polymer. For the experimental data, a normalized stiffness (similar to contact stiffness) is calculated using Equation 7, where material 2 is the bulk polymer and material 1 is the material indented near the fiber. The normalized distance is then just the distance from the fiber divided by the fiber diameter, which is 5 mm for IM7. The results of the model and the experimentally measured responses are in good agreement.

A single-fiber specimen, consisting of an AS4 carbon fiber in an epoxy matrix, was sectioned, mounted in epoxy potting compound, and polished to a 0.05 mm finish. The D2 probe was used to indent the sample, starting from the edge of the fiber and moving radially outward from using step sizes of less than 100 nm. This system was chosen because the interphase formed near the fiber has been shown to have a lower modulus and a lower glass transition temperature than those of the bulk material [1-3]. The stiffening effect of the fiber, however, overwhelmed the modulus changes in the interphase. Thus, differences in modulus near the fiber were not observed.

Figure 3 – Plot showing the stiffening effect of the fiber to local deformation of the matrix. Experimental data is shown along with the predictions of a finite element model. Stiffness is normalized with respect to the bulk behavior, and normalized distance is the distance from the fiber divided by the fiber diameter.

Conclusions

A technique that utilizes the atomic force microscope (AFM) as a nanoindentation device has been presented. This technique was utilized to study the local indentation response in multi-component polymer systems. Probes with spring constants of 150 and 310 N/m were used to indent across interphase regions. Indentation across a polymer-polymer adhesion system produced a variety of responses. The response of the interphase material was different than the responses of either the bulk epoxy or the bulk polysulfone. These differences were used to estimate the width of the interphase at approximately 3 mm. Indents were also made near graphite fibers in two composite systems. Deformation was significantly restricted for indents nearest the fibers. This effect leads to an apparent stiffening of the matrix material which was measured experimentally and modeled using finite element analysis. The step size across interphase regions is limited by the size of the plastic impressions and the geometry of the interphase. Step sizes much less than 100 nm were achieved with a lateral resolution of less than 1 nm.

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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