We address three problems that limit the use of the atomic force microscope when measuring elastic moduli of soft materials at microscopic scales. The first concerns the use of sharp cantilever tips, which typically induce local strains that far exceed the linear material regime. We show that this problem can be alleviated by using microspheres as probes, and we establish the criteria for their use. The second relates to the common use of the Hertz contact mechanics model, which leads to significant errors when applied to thin samples. We develop novel, simple to use corrections to apply for such cases. Samples that are either bonded or not bonded to a rigid substrate are considered. The third problem concerns the difficulty in establishing when contact occurs on a soft material. We obtain error estimates for the elastic modulus resulting from such uncertainty and discuss the sensitivity of the estimation methods to error in contact point. The theoretical and experimental results are compared to macroscopic measurements on poly(vinyl-alcohol) gels.

The swelling behavior of fully neutralized sodium polyacrylate gels was investigated in aqueous solutions of alkali metal (LiCl, NaCl, KCl, CsCl) and alkaline earth metal salts (CaCl2, SrCl2, BaCl2). The total salt concentration and the ratio of monovalent to divalent cations were varied in the biologically significant range. It is found that the concentrations of both monovalent and divalent cations vary continuously and smoothly in the gel despite the abrupt change in the gel volume. The individual elastic, mixing, and ionic contributions to the free energy of the gel were separately determined as a function of the degree of network swelling to elucidate the thermodynamics of swelling. Shear modulus measurements performed at different Ca2+ concentrations suggest that Ca2+ does not form stable cross-links between the polymer chains. At low and moderate swelling ratios the concentration dependence of the shear modulus follows a power law behavior, G variation of phi n, with n = 0.34 +/- 0.03. At high swelling degrees, however, the shear modulus increases with increasing swelling. The value of the Flory-Huggins interaction parameter, chi, determined from osmotic swelling pressure and shear modulus measurements, strongly depends on the ionic composition of the equilibrium solution and increases with increasing Ca2+ concentration.

The atomic force microscope (AFM) has found wide applicability as a nanoindentation tool to measure local elastic properties of soft materials. An automated approach to the processing of AFM indentation data, namely, the extraction of Young's modulus, is essential to realizing the high-throughput potential of the instrument as an elasticity probe for typical soft materials that exhibit inhomogeneity at microscopic scales. This paper focuses on Hertzian analysis techniques, which are applicable to linear elastic indentation. We compiled a series of synergistic strategies into an algorithm that overcomes many of the complications that have previously impeded efforts to automate the fitting of contact mechanics models to indentation data. AFM raster data sets containing up to 1024 individual force-displacement curves and macroscopic compression data were obtained from testing polyvinyl alcohol gels of known composition. Local elastic properties of tissue-engineered cartilage were also measured by the AFM. All AFM data sets were processed using customized software based on the algorithm, and the extracted values of Young's modulus were compared to those obtained by macroscopic testing. Accuracy of the technique was verified by the good agreement between values of Young's modulus obtained by AFM and by direct compression of the synthetic gels. Validation of robustness was achieved by successfully fitting the vastly different types of force curves generated from the indentation of tissue-engineered cartilage. For AFM indentation data that are amenable to Hertzian analysis, the method presented here minimizes subjectivity in preprocessing and allows for improved consistency and minimized user intervention. Automated, large-scale analysis of indentation data holds tremendous potential in bioengineering applications, such as high-resolution elasticity mapping of natural and artificial tissues.

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Facile synthesis and detailed characterization of photopolymerizable and biocompatible poly(ethylene glycol) dimethacrylates (PEGDM) and poly(ethylene glycol) urethane-dimethacrylates (PEGUDM) are described. Poly(ethylene glycol)s of various molecular masses (M(n) = 1000 to 8000 g/mol) were reacted with methacrylic anhydride or with 2-isocyanatoethyl methacrylate to form PEGDMs and PEGUDMs, respectively. PEGDMs were also prepared by a microwave-assisted route to achieve fast reaction conversions under solvent free conditions. Combined analyses of (1)H NMR and MALDI-TOF MS confirmed the formation of prepolymers of high purity and narrow mass distribution (PD < 1.02). Aqueous solutions of the PEGDMs and PEGUDMs (10% and 20% by mass fraction) were photopolymerized to yield hydrogels. Bovine chondrocytes, seeded in the hydrogels, were used to assess the biocompatibility. Preliminary rheology and uniaxial compression measurements showed varied mechanical response, and biocompatibility studies showed that cells are completely viable in both types of hydrogels after two weeks.

The volume transition induced by monovalent-divalent cation exchange of fully neutralized polyacrylate hydrogels was investigated in aqueous NaCl solutions. The variation of the osmotic swelling pressure, shear modulus, and mixing pressure was measured when Na(+) ions were substituted by divalent or trivalent cations. Alkali metal salts move freely throughout the entirely network, and alkaline earth metal salts (CaCl(2), SrCl(2)) promote aggregation of polyacrylate chains, but these aggregates are relatively weak. Transition metal salts (CoCl(2), NiCl(2)) form stronger interchain associates. Rare earth cations (La(3+) and Ce(3+)) bind practically irreversibly to the polymer. Experimental data indicate that transition metal cations modify both the elastic and mixing components of the free energy, while alkaline earth metal cations affect primarily the mixing term. The behavior of freely swollen gels was compared with similar gels subjected to uniaxial compression. In uniaxially compressed gels, volume transition occurs at lower cation concentrations than in the corresponding undeformed gels. The shift of the transition point increases with the deformation ratio and is larger for Co(2+) than for Ca(2+).

The lack of practicable nonlinear elastic contact models frequently compels the inappropriate use of Hertzian models in analyzing indentation data and likely contributes to inconsistencies associated with the results of biological atomic force microscopy measurements. We derived and validated with the aid of the finite element method force-indentation relations based on a number of hyperelastic strain energy functions. The models were applied to existing data from indentation, using microspheres as indenters, of synthetic rubber-like gels, native mouse cartilage tissue, and engineered cartilage. For the biological tissues, the Fung and single-term Ogden models achieved the best fits of the data while all tested hyperelastic models produced good fits for the synthetic gels. The Hertz model proved to be acceptable for the synthetic gels at small deformations (strain < 0.05 for the samples tested), but not for the biological tissues. Although this finding supports the generally accepted view that many soft materials can be assumed to be linear elastic at small deformations, the nonlinear models facilitate analysis of intrinsically nonlinear tissues and large-strain indentation behavior.

An investigation is described of a set of chemically cross-linked polyacrylamide hydrogeis prepared by copolymerization of acrylamide in presence of different amounts of bisacrylamide. These networks were studied by using osmotic and mechanical measurements, dynamic light scattering, and small-angle X-ray scattering (SAXS) techniques. The form of the concentration dependence of the swelling pressure was found to vary with the cross-linking density, water being a good solvent for the loosely cross-linked gels, while at higher bisacrylamide content the solvent power diminishes. The ratio of the longitudinal osmotic modulus obtained from mechanical and osmotic observations to that from dynamic light scattering w a~ found to increase with increasing cross-linking, in agreement with SAXS observations of the amplitude of the concentration fluctuations associated with the static heterogeneities in the sample. Furthermore, for all the gels studied, the values of the correlation lengths determined by three independent methods-SAXS, dynamic light scattering intensity, and collective diffusion coefficient measurements-were consistent with each other.