Deriving molecular motion parameters in soft materials with DF-MSE, VFT/Arrhenius, and Anderson-Weiss models

Molecular dynamics in soft matter exhibit significant variations with temperature changes, which can be described using two key models: the Vogel-Fulcher-Tammann (VFT) equation and the Arrhenius model. The VFT equation describes the viscosity of the material as a function of temperature: 
\[
\eta(T) = \eta_0 \exp\left(\frac{D T_0}{T - T_0}\right).
\]
Here, $T_0$ is the Vogel temperature, $D$ is a fragility parameter, and $\eta_0$ is a pre-exponential factor. In contrast, the Arrhenius model describes temperature dependence via:
\[
\eta(T) = \eta_0 \exp\left(\frac{E_a}{k_B T}\right),
\]
where $E_a$ is the activation energy, and $k_B$ is the Boltzmann constant.

One powerful method for probing changes in molecular dynamics is nuclear magnetic resonance (NMR), which is highly sensitive to small variations in molecular motion. Using specific pulse sequences such as Dipolar Filtered Magic-Sandwich Echo (DF-MSE), NMR can provide detailed insights into the temperature dependence of molecular dynamics in a wide range of soft materials. This pulse sequence is particularly useful in filtering out certain contributions to the NMR signal, allowing for a clearer focus on the dynamics of specific molecular interactions.

By using models like Anderson-Weiss, the NMR signal can be analyzed to extract information about the correlation times of molecular motions. Incorporating the correlation time, which can be modeled using either the VFT or Arrhenius equation, it is possible to derive important parameters such as activation energy, the correlation time at infinite temperature, and the Vogel temperature $T_0$ analytically. These parameters are critical in understanding the underlying molecular mechanisms driving the changes in molecular dynamics.

In this work, the equations that allow calculating the parameters analytically were derived, and a variety of standard samples were analyzed using the DF-MSE approach, including atactic polypropylene and a series of elastomers. These materials were chosen for their well-known behaviors, which allowed for a comparative study.