arXiv subjects

Sachin Shanbhag

Publications and source records attributed to Sachin Shanbhag.

10 recordsLinked to original sources

arXiv2026

When Spectroscopies Speak the Same Language: Unifying Rheology, Electrochemical Impedance, and Dielectrics

Spectroscopic techniques measure the dynamical response of physical systems subjected to oscillatory perturbations. For small perturbations around the equilibrium state, these spectroscopic methods are unified by the common mathematical framework of linear response theory. This work presents a unified perspective on rheological or mechanical spectroscopy, electrochemical impedance spectroscopy, and broadband dielectric spectroscopy through the lens of linear response theory. Subtle conceptual similarities and differences among these techniques are highlighted by analyzing their mapping to the linear response theory, basic building blocks, elementary models, and time and frequency domain response functions. Data validation and analysis approaches, including Kramers-Kronig relations, equivalent circuits, and multimode models are discussed. The shared fundamentals of different spectroscopies enable seamless exchange of ideas across domains.

cond-mat.soft↗

arXiv2026

Fractional Viscoelasticity in Transient Unentangled Polymer Networks

Stress relaxation in transient polymer networks often shows extended power-law behavior, $G(t) \sim t^{-β}$, 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 Mittag-Leffler functions. For homogeneous chains, it recovers two power-law regimes, $t^{-α/2}$ and $t^{-2α}$, on either side of the terminal relaxation time. Fitting FIRM to stress relaxation data for an imine-based polystyrene vitrimer shows that $α< 1$ is required to capture the shape of the terminal relaxation. It also accommodates both Arrhenius and non-Arrhenius temperature dependence in the rheological 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 chemistry-specific molecular model.

cond-mat.soft↗

arXiv2024

Sparse Regression for Discovery of Constitutive Models from Oscillatory Shear Measurements

We propose sparse regression as an alternative to neural networks for the discovery of parsimonious constitutive models (CMs) from oscillatory shear experiments. Symmetry and frame-invariance are strictly imposed by using tensor basis functions to isolate and describe unknown nonlinear terms in the CMs. We generate synthetic experimental data using the Giesekus and Phan-Thien Tanner CMs, and consider two different scenarios. In the complete information scenario, we assume that the shear stress, along with the first and second normal stress differences, is measured. This leads to a sparse linear regression problem that can be solved efficiently using $l_1$ regularization. In the partial information scenario, we assume that only shear stress data is available. This leads to a more challenging sparse nonlinear regression problem, for which we propose a greedy two-stage algorithm. In both scenarios, the proposed methods fit and interpolate the training data remarkably well. Predictions of the inferred CMs extrapolate satisfactorily beyond the range of training data for oscillatory shear. They also extrapolate reasonably well to flow conditions like startup of steady and uniaxial extension that are not used in the identification of CMs. We discuss ramifications for experimental design, potential algorithmic improvements, and implications of the non-uniqueness of CMs inferred from partial information.

cond-mat.soft↗

arXiv2024

Harmonic Balance for Differential Constitutive Models under Oscillatory Shear

Harmonic balance (HB) is a popular Fourier-Galerkin method used in the analysis of nonlinear vibration problems where dynamical systems are subjected to periodic forcing. We adapt HB to find the periodic steady-state response of nonlinear differential constitutive models subjected to large amplitude oscillatory shear flow. By incorporating the alternating-frequency-time scheme into HB, we develop a computer program called FLASH (acronym for Fast Large Amplitude Simulation using Harmonic balance), which makes it convenient to apply HB to any differential constitutive model. We validate FLASH by considering two representative constitutive models, viz., the exponential Phan-Thien Tanner model and a nonlinear temporary network model. In terms of accuracy and speed, FLASH outperforms the conventional approach of solving initial value problems by numerical integration via time-stepping methods often by several orders of magnitude. We discuss how FLASH can be conveniently extended for other nonlinear constitutive models, which opens up potential applications in model calibration and selection, and stability analysis.

cond-mat.soft↗

arXiv2023

The Method of Harmonic Balance for the Giesekus Model under Oscillatory Shear

The method of harmonic balance (HB) is a spectrally accurate method used to obtain periodic steady state solutions to dynamical systems subjected to periodic perturbations. We adapt HB to solve for the stress response of the Giesekus model under large amplitude oscillatory shear (LAOS) deformation. HB transforms the system of differential equations to a set of nonlinear algebraic equations in the Fourier coefficients. Convergence studies find that the difference between the HB and true solutions decays exponentially with the number of harmonics ($H$) included in the ansatz as $e^{-m H}$. The decay coefficient $m$ decreases with increasing strain amplitude, and exhibits a "U" shaped dependence on applied frequency. The computational cost of HB increases slightly faster than linearly with $H$. The net result of rapid convergence and modest increase in computational cost with increasing $H$ implies that HB outperforms the conventional method of using numerical integration to solve differential constitutive equations under oscillatory shear. Numerical experiments find that HB is simultaneously about three orders of magnitude cheaper, and several orders of magnitude more accurate than numerical integration. Thus, it offers a compelling value proposition for parameter estimation or model selection.

math.NA↗

arXiv2022

Large Amplitude Oscillatory Shear Study of a Colloidal Gel at the Critical State

We investigate the nonlinear viscoelastic behavior of a colloidal dispersion at the critical gel state using large amplitude oscillatory shear (LAOS) rheology. The colloidal gel at the critical point is subjected to oscillatory shear flow with increasing strain amplitude at different frequencies. We observe that the first harmonic of the elastic and viscous moduli exhibits a monotonic decrease as the material undergoes a linear to nonlinear transition. We analyze the stress waveform across this transition and obtain the nonlinear moduli and viscosity as a function of frequency and strain amplitude. The analysis of the nonlinear moduli and viscosities suggests intracycle strain stiffening and intracycle shear thinning in the colloidal dispersion. Based on the insights obtained from the nonlinear analysis, we propose a potential scenario of the microstructural changes occurring in the nonlinear region. We also develop an integral model using the time-strain separable K-BKZ constitutive equation with a power-law relaxation modulus and damping function obtained from experiments. At low strain amplitudes, this model compares well with experimental data at all frequencies. However, a stronger damping function, which can be efficiently inferred using a spectral method, is required to obtain quantitative fits across the entire range of strain amplitudes and the explored frequencies.

cond-mat.soft↗

arXiv2022

Kramers-Kronig Relations for Nonlinear Rheology: 1. General Expression and Implications

The principle of causality leads to linear Kramers-Kronig relations (KKR) that relate the real and imaginary parts of the complex modulus $G^{*}$ through integral transforms. Using the multiple integral generalization of the Boltzmann superposition principle for nonlinear rheology, and the principle of causality, we derived nonlinear KKR, which relate the real and imaginary parts of the $n^\text{th}$ order complex modulus $G_{n}^{*}$. For $n$=3, we obtained nonlinear KKR for medium amplitude parallel superposition (MAPS) rheology. A special case of MAPS is medium amplitude oscillatory shear (MAOS); we obtained MAOS KKR for the third-harmonic MAOS modulus $G_{33}^{*}$; however, no such KKR exists for the first harmonic MAOS modulus $G_{31}^{*}$. We verified MAPS and MAOS KKR for the single mode Giesekus model. We also probed the sensitivity of MAOS KKR when the domain of integration is truncated to a finite frequency window. We found that that (i) inferring $G_{33}^{\prime\prime}$ from $G_{33}^{\prime}$ is more reliable than vice-versa, (ii) predictions over a particular frequency range require approximately an excess of one decade of data beyond the frequency range of prediction, and (iii) $G_{33}^{\prime}$ is particularly susceptible to errors at large frequencies.

cond-mat.soft↗

arXiv2022

Kramers-Kronig Relations for Nonlinear Rheology: 2. Validation of Medium Amplitude Oscillatory Shear (MAOS) Measurements

The frequency dependence of third-harmonic medium amplitude oscillatory shear (MAOS) modulus $G_{33}^{*}(ω)$ provides insight into material behavior and microstructure in the asymptotically nonlinear regime. Motivated by the difficulty in the measurement of MAOS moduli, we propose a test for data validation based on nonlinear Kramers-Kronig relations. We extend the approach used to assess the consistency of linear viscoelastic data by expressing the real and imaginary parts of $G_{33}^{*}(ω)$ as a linear combination of Maxwell elements: the functional form for the MAOS kernels is inspired by time-strain separability (TSS). We propose a statistical fitting technique called the SMEL test, which works well on a broad range of materials and models including those that do not obey TSS. It successfully copes with experimental data that are noisy, or confined to a limited frequency range. When Maxwell modes obtained from the SMEL test are used to predict the first-harmonic MAOS modulus $G_{31}^{*}$, it is possible to identify the range of timescales over which a material exhibits TSS.

cond-mat.soft↗

arXiv2018

Stable and contact-free time stepping for dense rigid particle suspensions

We consider suspensions of rigid bodies in a two-dimensional viscous fluid. Even with high-fidelity numerical methods, unphysical contact between particles occurs because of spatial and temporal discretization errors. We apply the method of Lu et al. [Journal of Computational Physics, 347:160-182, 2017] where overlap is avoided by imposing a minimum separation distance. In its original form, the method discretizes interactions between different particles explicitly. Therefore, to avoid stiffness, a large minimum separation distance is used. In this paper, we extend the method of Lu et al. by treating all interactions implicitly. This new time stepping method is able to simulate dense suspensions with large time step sizes and a small minimum separation distance. The method is tested on various unbounded and bounded flows, and rheological properties of the resulting suspensions are computed.

math.NA↗

arXiv2008

Self-diffusion in binary blends of cyclic and linear polymers

A lattice model is used to estimate the self-diffusivity of entangled cyclic and linear polymers in blends of varying compositions. To interpret simulation results, we suggest a minimal model based on the physical idea that constraints imposed on a cyclic polymer by infiltrating linear chains have to be released, before it can diffuse beyond a radius of gyration. Both, the simulation, and recently reported experimental data on entangled DNA solutions support the simple model over a wide range of blend compositions, concentrations, and molecular weights.

cond-mat.soft↗