Sachin Shanbhag

Stress relaxation in transient polymer networks often shows extended power-law behavior, G(t) ∼ t−β, where the exponent β frequently departs from the value 1/2 predicted by the sticky Rouse model and its variants. We introduce the fractional inhomogeneous Rouse model (FIRM), which uses a generalized Langevin equation driven by fractional Gaussian noise of exponent α, while retaining heterogeneous bead friction to represent sticky cross-links. Thus, FIRM unifies subdiffusive sticker dynamics and chain heterogeneity within a single framework. We show that the relaxation modulus G(t) can be represented as a linear combination of the squares of Mittag–Leffler functions. For homogeneous chains, it recovers two power-law regimes, t−α/2 and t−2α, on either side of the Rouse relaxation time. Fitting FIRM to stress relaxation data for an imine-based polystyrene vitrimer shows that α < 1 captures the shape of G(t) over finite observation windows. FIRM accommodates both Arrhenius and non-Arrhenius temperature dependences in the effective activation energy. We derive expressions for dynamic properties, such as mean-squared displacement and dielectric response, and outline how generalized memory kernels extend the framework to real materials. Together, these results suggest novel ways in which data from rheology, dielectric spectroscopy, scattering, and other experimental methods may be incorporated into a molecular model.

Medium amplitude oscillatory shear (MAOS) represents an elegant compromise between small amplitude oscillatory shear, which probes only linear viscoelasticity, and large amplitude oscillatory shear (LAOS), which can be compromised by experimental artifacts and challenges in interpretation. However, extracting MAOS moduli from experimental data remains challenging due to the narrowness of the accessible strain amplitude window and the need for laborious extrapolation to filter out noise and contamination by higher harmonics. We present a computational framework that bypasses this bias-variance trade-off and shifts experimental burden to numerical analysis, enabling robust extraction of MAOS moduli from conventional LAOS measurements. Our approach combines two key elements: (i) discovery of constitutive models from LAOS data using sparse polynomial approximation and (ii) Richardson extrapolation with successive refinement to isolate the asymptotic MAOS response. We demonstrate this methodology using wormlike micelles as a model system. The discovered constitutive models accurately reproduce relevant experimental data and enable systematic exploration of the frequency-dependent MAOS moduli. This framework is general and can be applied to any soft material, offering a pathway to make MAOS characterization more accessible by automating the most labor-intensive aspects of traditional experimental workflows.

Small datasets, which are common in many traditional science and engineering disciplines, present a challenge for powerful machine learning methods that are better suited for large datasets. Through the example of spontaneous colloidal sol-gel transition, we present a data-driven framework based on Gaussian Processes (GPs) for modeling and inference with limited data. GPs are well-suited for small datasets and automatically provide uncertainty quantification for model predictions. We explore GPs with two different structures (single and multi-output) to map synthesis parameters to gelation characteristics. While both GP structures quantitatively capture the relationships between the input and output variables, we favor multioutput GP, which attempts to simultaneously learn correlations between different outputs, due to its superior uncertainty calibration. To address inverse problems, we introduce a Bayesian framework that embeds a GP model into a Markov Chain Monte Carlo sampling algorithm. It enables the discovery of multiple synthesis pathways for achieving desired material properties. Therefore, this work presents a robust methodology for data-constrained modeling, with broad applicability to material design in general and the colloidal domain in particular.

Vitrimers are polymer networks connected by associative cross-links – covalent linkages that maintain network connectivity but exchange through reversible chemical reactions. Associative cross-links significantly change the dynamics of the molten polymer. This study focuses on the linear viscoelasticity of polystyrene vitrimers (PS-v) bearing imine cross-links. PS-v samples were prepared by condensation between precursor copolymers with pendant aldehydes and 1,6-hexanediamine cross-linker. The number average molecular weights of the precursors were 6 and 8 kDa, and the amine-to-aldehyde molar ratio (r) ranged between 0.8 and 2.4. The glass transition temperature exhibited a non-monotonic relationship with r. The linear viscoelasticity of PS-v was evaluated using a combination of small amplitude oscillatory shear (SAOS), stress relaxation, and creep and recovery. Time-temperature superposition analyses indicated two distinct relaxation regimes: (I) fast high frequency dynamics with a Williams-Landel-Ferry temperature dependence and (II) slow low frequency dynamics with Arrhenius behavior. The fast regime represented the segmental relaxations of the vitrimer backbone. The slow regime was described as a Slow Arrhenius Process (SAP), in which the long time dynamics have a temperature-independent rheological activation energy. For all PS-v samples in this study, the observed SAP had a much weaker temperature dependence than expected from sticky Rouse model predictions. Increasing r altered the plateau modulus and SAOS cross-over frequency but did not affect the temperature dependences of the segmental motions or SAP. To describe the origin of the SAP, three hypotheses are proposed: cross-linker diffusion, polymer matrix effects, and local elasticity fluctuations.