One Equation Now Predicts a Plastic's Stiffness From a Femtosecond to 15 Minutes. It Starts With 9,920 Atoms and Spans 20 Orders of Magnitude in Time.

Ask how stiff a piece of plexiglass is and the honest answer is: it depends how fast you push. Poly(methyl methacrylate), the polymer sold as Plexiglas and Perspex, resists a sudden shove almost like a rigid crystal, because at those speeds its atoms and bonds have no time to do anything but stretch. Push slowly and its molecules find time to shuffle and rearrange, and the material softens. Push slower still and whole segments of the polymer chains relax. There is no single stiffness, only a stiffness curve that runs across an enormous range of timescales.

Nobody has been able to measure that whole curve with one instrument, and nobody has been able to compute it from first principles either. Molecular dynamics simulations, which track every atom, can only reach the fastest motions, because their time steps are femtoseconds and a laboratory experiment lasts seconds or minutes. Dynamic mechanical analysis, the standard engineering test, works at the slow end. Brillouin light scattering covers gigahertz frequencies, ultrasound and high-strain-rate rigs fill other slices. Each community gets its own snapshot of the same material, and the snapshots have never been joined by a single theory.

A paper in The Journal of Chemical Physics by Alessio Zaccone of the University of Cambridge and colleagues, including Tim Sirk of the U.S. Army Research Laboratory, reports that they have now done it. Starting from an atomistic model of PMMA containing 9,920 atoms, the team calculated the shear modulus, the resistance to sliding deformation, from above a terahertz down to the millihertz range. That is more than 20 orders of magnitude in frequency, from the vibration of a single carbon-hydrogen bond to a deformation lasting about a quarter of an hour, and the single red curve their theory produces passes through every published measurement along the way.

The framework is called non-affine lattice dynamics, or NALD, which Zaccone and Sirk developed over the past decade. In a perfect crystal, every atom follows the imposed deformation uniformly; that is an "affine" response. In a disordered glass like PMMA, atoms sit in lopsided local environments and, when the material is squeezed, make additional small displacements of their own. Those non-affine motions collectively soften the material, and NALD computes that softening by connecting the vibrational modes of the atomic structure to the forces a deformation generates. Rather than simulating ever-longer deformations, which becomes impossible, the method extracts the vibrational spectrum once and propagates it analytically.

The new ingredient that reaches the slow end is memory. In a simple model of friction, the drag on an atom depends only on what is happening at that instant. Glassy polymers do not behave that way; their response carries information about what happened before. The team built that in with a power-law memory kernel, making the atomic equations of motion non-Markovian, so that the theory represents the broad spread of relaxation processes that define glasses. That is what lets it capture the β relaxation, a secondary softening near one hertz that corresponds to a rearrangement taking about a second, roughly 15 orders of magnitude longer than a molecular dynamics time step.

Along the way the calculation shows what is physically happening at each scale. At the highest frequencies PMMA behaves as an almost instantaneous elastic network of bonds, with terahertz resonances of carbon-oxygen and carbon-hydrogen vibrations visible in the curve. Moving slower, non-affine rearrangements progressively lower the modulus. Around a hertz the β relaxation appears, and the theory then meets the dynamic mechanical analysis data that engineers actually use to design parts.

"Atomistic simulation is often thought of as a tool for understanding very short times and very small length scales, while engineering tests describe materials at macroscopic scales," Zaccone wrote in a summary of the work. "Our work suggests that the divide between the two does not have to be fundamental." The authors say the framework still needs to be pushed closer to the glass transition, where the dynamics become even messier, and extended to more complex polymer architectures. But for a material that has been made by the ton since the 1930s, it is the first time anyone has drawn its mechanical behavior in one unbroken line from its atoms to the bench.