Constitutive Models

PUA nanocomposites exhibit microphase separation into a soft-phase matrix and a stiffer, percolated hard phase due to strong hydrogen bonding within the hard phase. Consistent with this morphology, we use two models, each with two parallel fractional branches: FMG–FMG model has an FMG branch for each of the soft-phase matrix and percolated hard phase; FMM–FMG model has an FMM branch for the soft-phase matrix and an FMG branch for the percolated hard phase. At the relatively low nanofiller loadings considered here, no separate nanoparticle branch is introduced, and fillers are assumed to be uniformly dispersed within the two phases. The schematic of the models are provided as follows:

FMM–FMG and FMG–FMG models
Figure 1: FMM–FMG and FMG–FMG models

The resulting models use six to seven parameters: characteristic moduli \((E_{c_1}, E_{c_2})\), characteristic relaxation times \((\tau_{c_1}, \tau_{c_2})\), power exponents \((\alpha_1, \alpha_2)\), and an additional exponent \(\beta_1\) associated with the second spring-pot element in the FMM branch. The equivalent storage and loss moduli are given by

\[ E'_{\mathrm{model}}(x) = \sum_{k=1}^{2} E_{c_k} \frac{ (x\tau_{c_k})^{\alpha_k}\cos\!\left(\frac{\pi\alpha_k}{2}\right) + (x\tau_{c_k})^{2\alpha_k-\beta_k}\cos\!\left(\frac{\pi\beta_k}{2}\right) }{ 1 + (x\tau_{c_k})^{\alpha_k-\beta_k}\cos\!\left(\frac{\pi(\alpha_k-\beta_k)}{2}\right) + (x\tau_{c_k})^{2(\alpha_k-\beta_k)} }, \]
\[ E''_{\mathrm{model}}(x) = \sum_{k=1}^{2} E_{c_k} \frac{ (x\tau_{c_k})^{\alpha_k}\sin\!\left(\frac{\pi\alpha_k}{2}\right) + (x\tau_{c_k})^{2\alpha_k-\beta_k}\sin\!\left(\frac{\pi\beta_k}{2}\right) }{ 1 + (x\tau_{c_k})^{\alpha_k-\beta_k}\cos\!\left(\frac{\pi(\alpha_k-\beta_k)}{2}\right) + (x\tau_{c_k})^{2(\alpha_k-\beta_k)} }. \]

Here, \(x = a_T \omega\) is the shifted frequency. The shift factor \(a_T\) is modeled using the two-state, two-time-scale (TS2) function, which captures the time–temperature superposition behavior of neat and nanocomposite PUAs over a broad temperature range. In TS2, the glass transition is represented as a smooth transition between low- and high-temperature Arrhenius regimes:

\[ \ln(a_T) \equiv \ln\!\left(\frac{\tau[T]}{\tau[T_0]}\right) = \frac{E_1}{RT} + \frac{E_2 - E_1}{RT} \left( \frac{1}{1 + \exp\!\left\{\frac{\Delta S}{R}\left(1 - \frac{T^*}{T}\right)\right\}} \right) - \frac{E_1}{RT_0} - \frac{E_2 - E_1}{RT_0} \left( \frac{1}{1 + \exp\!\left\{\frac{\Delta S}{R}\left(1 - \frac{T^*}{T_0}\right)\right\}} \right). \]

Here, \(E_1\) and \(E_2\) are activation energies, \(\Delta S / R\) is the dimensionless transition entropy, \(T^*\) is the transition temperature, and \(T_0\) is the reference temperature, taken as \(T_g\).