Modeling soft, permeable matter with the proper generalized decomposition (PGD) approach, and verification by means of nanoindentation

Abstract

Understanding sliding and load-bearing mechanisms of biphasic soft matter is crucial for designing synthetic replacements of cartilage, contact-lens materials or coatings for medical instruments. Interstitial fluid pressurization is believed to be the intrinsic load-carrying phenomenon governing the frictional properties. In this study, we have characterized permeability and identified the fluid contribution to the support of load during Atomic Force Microscopy (AFM) nanoindentation of soft polymer brushes in aqueous environments, by means of the Proper Generalized Decomposition (PGD) approach. First, rate-dependent AFM nanoindentation was performed on a poly(acrylamide) (PAAm) brush in an aqueous environment, to probe the purely elastic as well as poro-viscoelastic properties. Second, a biphasic model decoupling the fluid and solid load contributions was proposed, using Darcy's equation for liquid flow in porous media. Using realistic time-dependent simulations requires many direct solutions of the 3D partial differential equation, making modeling very time-consuming. To efficiently alleviate the time-consumption of multi-dimensional modeling, we used PGD to solve a Darcy model defined in a 7D domain, considering all the unknowns and material properties as extra coordinates of the problem. The obtained 7D simulation results were compared to the experimental results by using a direct Newton algorithm, since all sensitivities with respect to the model parameters are readily available. Thus, a simulation-based solution for depth- and rate-dependent permeability can be obtained. From the PGD-based model permeability is calculated, and the velocity- and pressure-fields in the material can be obtained in real-time in 3D by adjusting the parameters to the experimental values. The result is a step forward in understanding the fluid flow, permeability and fluid contributions to the load support of biphasic soft matter.