Depth-Dependent Local Shear Modulus Profile Attained from Analyzing QCM Data with a New Transfer-Matrix Model Across a Polymer-Polymer Interface*

We present a model for analyzing quartz crystal microbalance (QCM) data that enables the determination of a local depth-dependent shear modulus G(z) in polymer films. This work builds on our group's recently published approach leveraging a continuum mechanics model to analyze QCM resonance shifts by matching boundary conditions between modeled layers. Here, we apply a transfer-matrix approach that can expand the analysis to an arbitrary number of modeled layers, allowing for a near-continuous G(z) gradient with arbitrary functional form. We demonstrate how this can be used to fit QCM data of glassy-rubbery bilayer films of polystyrene (PS) and polybutadiene (PB) to determine G(z) in that system. Despite a compositional interface of only ~5 nm between PS and PB, this new QCM analysis technique finds strong evidence for a long-ranged gradient in local modulus of ~100+ nm that corroborates previous local glass transition temperature Tg(z) measurements in this system. The ability to probe local modulus in systems such as this provides a new method to interrogate how interfaces impact the viscoelastic properties of polymer films, aiding in fundamental understanding and related industrial applications.