Time-temperature-composition equivalence of the glass transition in digitally mixed 3D-printed polymers

Abstract

For more than seventy years, time-temperature equivalence (TTE) has enabled a few measurements to describe the linear viscoelasticity of polymers over many decades of frequency. Multimaterial 3D printing now combines elastomeric and glassy resins voxel by voxel into digitally mixed polymers. Composition shifts the glass transition along the frequency axis, as temperature does, but also broadens it and increases the rubbery plateau modulus. A translation, the only operation of TTE, cannot capture these changes. Here we establish a time-temperature-composition equivalence (TTCE) from dynamic mechanical measurements of five compositions of a printed family. A shift of the transition along the reduced-frequency axis, a stretch of its breadth and a normalization between its plateaus collapse the storage modulus onto one master curve. The collapse holds at every measured temperature, frequency and composition, through a single reduced variable. The position and breadth of the transition change linearly with the volume fraction of the glassy resin. The phase angle and the relaxation spectrum superpose with the same parameters, so the family shares one distribution of relaxation times. The resulting closed form gives the storage and loss moduli continuously in composition, temperature and frequency, and quantitatively predicts two further mixtures withheld from the calibration. The same shift, stretch and normalization also collapse published temperature sweeps of a patterned composite family of different resins, pointing to broader applicability. Composition thus joins temperature and frequency as a third variable of one equivalence, and the closed form can guide the design of graded and architected printed structures.

arXiv comment15 pages, 6 figures; Supporting Information appended (7 pages)