measuring G

Complex Shear Modulus of Commercial Gelatin by Instrumented Indentation Application Note Introduction All gels are comprised of a three-dimensional cross-linked polymer network and a liquid fi ller. Because of the structure provided by the polymer network, gels can behave like solids even though they are substantially liquid by composition. Gels are classifi ed according to their liquid fi llers: hydrogels incorporate water, organogels incorporate oil, and aerogels incorporate air. Many practical applications derive from the mechanical similarity between gel and biological tissue. For example, gels are commonly used as tissue substitutes for evaluating both ballistics and armor [1–4]. When gel is employed as a tissue substitute, mechanical characterization of both tissue and gel is essential. Ideally, one would measure the mechanical properties of the tissue that is to be mimicked and then develop a gel which behaves similarly. It is reasonable to expect that developing a tissue substitute might require testing many different gels. Small-scale mechanical testing by dynamic instrumented indentation presents a number of practical advantages for characterizing both biological tissue and gel. First, the necessary volume of material is small. This is especially important if the application of interest constrains the material to a small volume, such as a thin fi lm. Also, minimal sample preparation is required; only a fl at surface must be presented to the indenter. Finally, instrumented indentation holds the possibility of mapping out the spatial variation of properties in the test material; this ability is especially relevant for characterizing biological tissue. This note presents the theory behind measuring complex shear modulus by dynamic instrumented indentation and applies that theory to the characterization of commercial gelatin using a punch which is only 100 µm in diameter. What makes these measurements so challenging is the combination of the compliance of the test material and the small contact size. Big contacts on compliant materials are not very diffi cult; neither are small contacts on stiff materials. Small contacts on compliant materials are extremely challenging, because the contact stiffness is small relative to the stiffness of the instrument. Thus, great care must be taken in characterizing the instrument. One gains a defi nite advantage by operating the instrument where it is most compliant, i.e. at its resonant frequency. But even with the instrument stiffness minimized, the instrument still dominates the measurement, so the instrument must be accurately characterized, and this characterization must be immediately relevant. That is, it should be at the same position, frequency, and temperature as the actual test. Thus, a new test method, “G-Series DCM CSM Flat Punch Complex Modulus, Gel”, is used in this work to seamlessly integrate instrument characterization and testing. Jennifer Hay Theory The complex shear modulus ( G*) has real and imaginary components which manifest the intrinsic elastic and viscous natures of the material: G* = G ’ + iG ”. Eq. 1 When a material is indented by a fl at- ended cylindrical punch, the relationship between the shear modulus ( G’), Poisson’s ratio ( /H9263), elastic contact stiffness ( S), and punch diameter ( D) is [5] G’ = S(1–/H9263)/(2D). Eq. 2 Many have demonstrated the validity of an analogous defi nition for G” that depends on contact damping ( C /H9275) [6–8]: G’ = C/H9275(1–/H9263)/(2D). Eq. 3 Proper dynamic analysis of the Agilent G200 NanoIndenter reveals that the contact stiffness ( S) must be obtained by subtracting the instrument stiffness (K i) from the total measured stiffness (Ks): S = K s – Ki. Eq. 4 Similarly, the contact damping ( C/H9275) must be obtained by subtracting the instrument damping ( C i/H9275) from the total measured damping ( Cs/H9275): C/H9275 = C s/H9275 – C i/H9275. Eq. 5 Logistically, stiffness and damping are obtained by oscillating the indenter. This is accomplished electromagnetically. The amplitude of the force oscillation (F o) is set, and the amplitude ( zo) and phase shift ( /H9278) of the resulting displacement oscillation are measured. The values for instrument stiffness and damping are obtained by oscillating the indenter alone—that is, not in contact with any test material. Thus the instrument stiffness and damping are given by: K i = [(Fo/zo).cos/H9278]free-hanging , Eq. 6 and Ci/H9275 = [(Fo/zo).sin/H9278]free-hanging . Eq. 7 The test method “G-Series DCM CSM Flat Punch Complex Modulus, Gel” includes a “self-calibration” phase in which K i and Ci/H9275 are automatically evaluated according to Eqs. 6 and 7. 2Figure 2. Gel sample, ready for testing in inverted NanoVision puck with extended epoxy lip.The system stiffness and damping are obtained by oscillating the indenter while in full contact with the test material: K s = [(Fo/zo).cos/H9278]in-contact , Eq. 8 and C s/H9275 = [(Fo/zo).sin/H9278]in-contact . Eq. 9 Ks and Cs/H9275 are evaluated according to Eqs. 8 and 9 during the “testing” phase of the method “G-Series DCM CSM Flat Punch Complex Modulus, Gel”. Substituting the expressions for K i (Eq. 6) and Ks (Eq. 8) into Eq. 4, and using the resulting expression for S in Eq. 2 gives a practical expression for measuring the elastic shear modulus by instrumented indentation: G’ = ([(F o/zo).cos/H9278]in-contact – [(Fo/zo).cos/H9278]free-hanging ) (1 –/H9263)/(2D) Eq. 10 Likewise, the expression for the shear loss modulus is given by: G” = ([(Fo/zo).sin/H9278]in-contact – [(Fo/zo).sin/H9278]free-hanging ) (1 –/H9263)/(2D) Eq. 11 Finally, the loss factor, tan /H9254, expresses the ratio of the loss modulus to the storage modulus: tan /H9254 = G ”/G’. For a perfectly elastic material, the loss factor would be zero. The loss factor increases with the damping capacity of the material; a loss factor greater than 1 means that the material damps more energy than it stores. In an instrumented indentation test, the loss factor is particularly useful in that it is independent of contact area and its determination. Experimental Method In order to contain the gels for testing, pucks that are normally used with the NanoVision (scanning) option were modifi ed for this application. First, they were used as “cups” rather than “pucks”. Second, the rim of the cup was extended using 5-minute epoxy in order to provide an adequate surface for the cleaning material (i.e. tape). To make the epoxy rim, the cup was placed bottom-side up on a piece of Saran Wrap. Then 5-minute epoxy was mixed and a toothpick was used to dab the epoxy around the rim. When the epoxy was fully cured, the Saran Wrap was removed. Two versions of Knox ® gelatin (Figure 1) were tested in this work. The fi rst gel (“1X Gel”) was made following package directions: one package of Knox gelatin was dissolved in 8 fl uid ounces (240ml) of near-boiling water. Once dissolved, the gel was poured into a modifi ed NanoVision cup, fi lling it to the rim. A second gel (“2X Gel”) was made by dissolving one package of Knox gelatin in only four fl uid ounces (120ml) of near-boiling water to create a “double concentration” version of the same gel. The second gel was poured to the rim of a second modifi ed NanoVision cup, and then both cups were set on a plate. A little gel was poured on the plate around the two cups; then they were covered with a plastic container. The extra gel and covering provided a sealed, humid environment in which the gels could set without drying. The gels were poured about six hours prior to testing. Just before testing, a piece of double-sided tape was adhered to the epoxy lip. Figure 2 shows a gel sample ready for testing. Figure 1. Standard and double-concentration versions of this gel were tested. Double-sided tape, mounted on extended epoxy lip. 3An Agilent G200 NanoIndenter was used for all testing. The system was confi gured with a DCM II actuator, fl at-ended cylindrical punch (D = 101.1µm), and CSM option. The CSM option allowed the superposition of an oscillating force. A fl at-ended cylindrical punch was employed in order to generate a contact area which was known and independent of penetration depth. The NanoSuite test method “G-Series DCM CSM Flat Punch Complex Modulus, Gel” was used for all testing, because it seamlessly integrates dynamic self-calibration, testing, and tip cleaning. Fifteen different sites were tested on each gel. Test sites were separated by 500µm. The testing frequency was 110Hz, because that is the resonant frequency of the DCM II actuator. (Equipment is most compliant at its resonant frequency.) Table 1 summarizes the details of testing. Results and Discussion Table 2 summarizes the results for this testing. Not surprisingly, the standard-concentration gel (1X) had a modulus that was about half that of the double-concentration gel (2X). Interestingly, increasing the gel concentration had the effect of reducing the loss factor by about half. The loss factor for the 1X gel was 0.245±0.030; the loss factor for the 2X gel was only 0.132±0.016. These results are consistent with sensory perception; that is; the 2X gel felt stiffer and more “bouncy” than the 1X gel. Figure 3 shows the results of preliminary testing on the gel. These tests are obviously spaced too closely together (200µm). Although these tests did not provided useful quantitative results, they did provided qualitative feedback which was valuable for performing and interpreting later tests. Fortuitously, the scale bar in Figure 3 is about the same length as the diameter of the indenter face. The visible deformation occurred when the indenter was withdrawn from the gel. Because the gel adhered to the tip as it was withdrawn from the sample, each test left behind a protrusion of gel having about the same diameter as the tip. (The fact that it was a protrusion, not an impression, was discerned by moving the optical microscope up and down to change the focal plane.) The tensile stress induced at the surface left behind circumferential “wrinkles” when the gel fi nally broke away from the indenter. For these tests, the indenter was cleaned between each test. The apparent adhesion verifi ed the necessity of such cleaning. The fact that subsequent tests all left similar traces confi rmed that in fact, the tip was being successfully cleaned. (If bits of gel remained on the tip from one test to another, later tests would show less adhesion than earlier tests.) As a result of these preliminary tests, the test-to-test spacing was increased to 500µm so that there would be no interference between tests. Conclusions The Agilent G200 NanoIndenter was used to measure the complex shear modulus of edible gelatin. The combination of the compliance of the test material (on the order of 1kPa) and the scale of the test (100µm) makes these results novel in the fi eld of mechanical testing. These extraordinary measurements required (1) a dynamically compliant actuator/transducer (the DCM II head, operating at its resonant frequency) and (2) a test method which integrated self-calibration, testing, and tip cleaning. The same equipment and test method may be used to characterize other kinds of gels and, most interestingly, biological tissue. Figure 3. Residual traces from preliminary testing; scale bar is the same length as the diameter of the indenter. Tensile stresses induced by pull-off leave circumferential “wrinkles”.Table 1. Summary of required inputs. Table 2. Summary of results. Uncertainty range represents ±1 /H9268. Sample Temp. G’ G” tan /H9254 C kPa kPa — 1X Gel 23.9 1.822±0.023 0.446±0.055 0.245±0.030 2X Gel 24.0 3.811±0.058 0.504±0.055 0.132±0.016 Input Value UnitsClean Tip Between Tests? (yes=1; no=0) 1 Oscillation Amplitude (in material) 500 nmPhase Change For Contact 2 degreesPoisson’s Ratio 0.5 Pre-test Compression 5 µmPre-test Compression Retracted? (yes=1; no=0) 0 Punch Diameter 101.1 µmSurface Approach Excitation 20 µNSurface Approach Frequency 110 HzTesting Frequency 110 HzX Move to Cleaning Material 1.5 cmY Move to Cleaning Material 0 cm Nanomeasurement Systems from Agilent Technologies Agilent Technologies, the premier measurement company, offers high precision instruments for nanoscience research in academia and industry. Exceptional worldwide support is provided by experienced application scientists and technical service personnel. 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