Universality of Time–Temperature Scaling Observed by Neutron Spectroscopy on Bottlebrush Polymers

The understanding of materials requires access to the dynamics over many orders of magnitude in time; however, single analytical techniques are restricted in their respective time ranges. Assuming a functional relationship between time and temperature is one viable tool to overcome these limits. Despite its frequent usage, a breakdown of this assertion at the glass-transition temperature is common. Here, we take advantage of time- and length-scale information in neutron spectroscopy to show that the separation of different processes is the minimum requirement toward a more universal picture at, and even below, the glass transition for our systems. This is illustrated by constructing the full proton mean-square displacement for three bottlebrush polymers from femto- to nanoseconds, with simultaneous information on the partial contributions from segmental relaxation, methyl group rotation, and vibrations. The information can be used for a better analysis of results from numerous techniques and samples, improving the overall understanding of materials properties.


Samples
The samples used for quasi-elastic neutron scattering (QENS) experiments, were PDMS-g-PDMS bottlebrush polymer with varying side chain length, ℎ = 298, 1800, 11500 g/mol, and similar backbone length, = 16500, 13500 g/mol, synthesized based on anionic ring opening polymerization and characterized by Gel Permeation Chromatography (GPC), and Nuclear Magnetic Resonance (NMR). A more detailed description of the synthesis can be found in Jakobi et al. S1

Neutron Scattering Experiments
Dynamical studies, based on the intermediate scattering function, ( , ), on these samples have been performed by QENS and have been published in Bichler et al. S5 Hereby, a combination of three different spectrometers has been used to capture a time range of three orders of magnitude, i.e., = 1 ps to = 1 ns. The short times were covered by the time-of-flight spectrometer Pelican S2 . This allows a combination of the data of the three instruments and subsequently enables the combined analysis of ( , ) based on the different relaxation processes. S5 The time dependence of ( , ) for all temperatures measured and for all three samples can be found in Bichler et al. S5 S3 Supporting Analysis Steps for the example PDMS-g-PDMS with = g/mol

Mean-Square Displacement Analysis
The mean-square displacement (

Adjustment for Fast Vibrations
In order to adjust the mean-square displacement of the methyl group and the segmental dynamics for fast vibrations, the atomistic mean-square displacement has been used. It is included in the Debye-Waller Factor and obtained from the incoherent intermediate scattering function analysis. These values have been subtracted from the respective temperatures of the original meansquare displacement data (Fig. 2a) to gain the single process mean-square displacement. The resulting temperature dependence of the atomistic mean-square displacement is shown in Fig. S2 including a dashed line, which serves as a guide for the eye.

Shift Parameter
To create the mean-square displacement of the single processes, time, , and displacement, , scaling has been applied. Hereby, for the methyl group mean-square displacement shift factors for both directions have been used, while for the segmental mean-square displacement only the time axis has been scaled. In case of the methyl group, time shift factor, , and relaxation time, ℓ , show a similar temperature scaling, which points to a time-temperature superposition principle (purple squares in Fig. S3a). The temperature dependence can be described with an Arrhenius law, with the activation energy, , being the same as obtained for the temperature dependence of the relaxation times, ℓ .
The shift factors, , show a linear relationship with temperature as illustrated in Fig. S3b, which is described by The associated fit parameter, describing the temperature dependence for both shift factors are summarized in Table S1.
The mean-square displacement for the segmental dynamics needs only to be shifted along the time axis. The shift parameters used follow the Vogel-Fulcher-Tammann (VFT) behavior known from dielectric spectroscopy experiments on this sample. S1  To simplify the summation of the pure methyl group mean-square displacement and the pure segmental mean-square displacement, the latter one was interpolated by the mathematical expression ⟨ 2 ( )⟩ = + 1 1 + 2 2 (S3) with = 502, = 0.6, 1 = −800, 1 = 0.7, 2 = 324, and 2 = 0.74 as illustrated in Fig. S4.
The aim was to find a parametrization of the segmental mean-square displacement for further data treatment, allowing to create the master curve of the segmental mean-square displacement in the same time range with exactly the same points as obtained for the methyl group mean-square displacement master curve. This allows easier pointwise addition of the partial meansquare displacements, resulting in the overall mean-square displacement (Fig. 4b).