Analyzing QCM Data Using a New Transfer-Matrix Model: Long-Ranged Asymmetric Gradient in Shear Modulus Identified Across Immiscible Glassy–Rubbery Polymer Interface

A new approach to analyzing quartz crystal microbalance (QCM) data using an acoustic transfer-matrix model is presented that enables determining a local depth-dependent shear modulus <i>G̃</i>(<i>z</i>) profile. A strong decrease in dissipation upon annealing is observed for immiscible polymer bilayer films of rubbery polybutadiene (PB) atop glassy polystyrene (PS), reflecting large viscoelastic changes in the sample corresponding to the emergence of a broad gradient in modulus <i>G̃</i>(<i>z</i>) when the ≈5 nm compositional interface is formed. Using a new transfer-matrix form of our continuum mechanics model that matches boundary conditions of shear waves between discrete modeled layers, we computationally fit these changes in frequency Δ<i>f</i>(<i>n</i>) and dissipation ΔΓ(<i>n</i>) shifts over a range of harmonics <i>n</i> to the evolution of a modulus gradient. The <i>G̃</i>(<i>z</i>) gradient across the PS/PB bilayer, treated as a hyperbolic tangent, is observed to be broad (230 nm) and strongly asymmetric (200 nm) toward the glassy PS side, consistent with the general trends of local glass transition <i>T</i> _g(<i>z</i>) previously reported. Surprisingly, the <i>G̃</i>(<i>z</i>) gradient is found to be symmetric on a log <i>G</i> scale, with the value of <i>G</i> at the interface equivalent to the geometric mean that optimizes acoustic energy transmission.