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1. Master curves for FENE-P fluids in steady shear flow

   https://doi.org/10.1016/j.jnnfm.2022.104944  

   Yamani, Sami ; McKinley, Gareth H. ( March 2023 , Journal of Non-Newtonian Fluid Mechanics)

   Poole, Robert J (Ed.)

   The FENE-P (Finitely-Extensible Nonlinear Elastic) dumbbell constitutive equation is widely used in simulations and stability analyses of free and wall-bounded viscoelastic shear flows due to its relative simplicity and accuracy in predicting macroscopic properties of dilute polymer solutions. The model contains three independent material parameters, which expressed in dimensionless form correspond to a Weissenberg number (Wi), i.e., the ratio of the dumbbell relaxation time scale to a characteristic flow time scale, a finite extensibility parameter (L), corresponding to the ratio of the fully extended dumbbell length to the root mean square end-to-end separation of the polymer chain under equilibrium conditions, and a solvent viscosity ratio. An exact solution for the rheological predictions of the FENE-P model in steady simple shear flow is available [Sureshkumar et al., Phys Fluids (1997)], but the resulting nonlinear and nested set of equations do not readily reveal the key shear-thinning physics that dominates at high Wi as a result of the finite extensibility of the polymer chain. In this note we review a simple way of evaluating the steady material functions characterizing the nonlinear evolution of the polymeric contributions to the shear stress and first normal stress difference as the shear rate increases, provide asymptotic expansions as a function of Wi , and show that it is in fact possible to construct universal master curves for these two material functions as well as the corresponding stress ratio. Steady shear flow experiments on three highly elastic dilute polymer solutions of different finite extensibilities also follow the identified master curves. The governing dimensionless parameter for these master curves is Wi/L and it is only in strong shear flows exceeding Wi/L > 1 that the effects of finite extensibility of the polymer chains dominate the evolution of polymeric stresses in the flow field. We suggest that reporting the magnitude of Wi/L when performing stability analyses or simulating shear-dominated flows with the FENE-P model will help clarify finite extensibility effects. 

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2. Crooks Fluctuation Theorem for Single Polymer Dynamics in Time-Dependent Flows: Understanding Viscoelastic Hysteresis

   https://doi.org/10.3390/e24010027  

   Zhou, Yuecheng ; Latinwo, Folarin ; Schroeder, Charles M. ( January 2022 , Entropy)

   Nonequilibrium work relations have fundamentally advanced our understanding of molecular processes. In recent years, fluctuation theorems have been extensively applied to understand transitions between equilibrium steady-states, commonly described by simple control parameters such as molecular extension of a protein or polymer chain stretched by an external force in a quiescent fluid. Despite recent progress, far less is understood regarding the application of fluctuation theorems to processes involving nonequilibrium steady-states such as those described by polymer stretching dynamics in nonequilibrium fluid flows. In this work, we apply the Crooks fluctuation theorem to understand the nonequilibrium thermodynamics of dilute polymer solutions in flow. We directly determine the nonequilibrium free energy for single polymer molecules in flow using a combination of single molecule experiments and Brownian dynamics simulations. We further develop a time-dependent extensional flow protocol that allows for probing viscoelastic hysteresis over a wide range of flow strengths. Using this framework, we define quantities that uniquely characterize the coil-stretch transition for polymer chains in flow. Overall, generalized fluctuation theorems provide a powerful framework to understand polymer dynamics under far-from-equilibrium conditions. 

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3. Dimethylformamide-Mediated Polymer Microstructure Dictates the Extensional Rheology of Aqueous PNIPAM Solutions

   https://doi.org/10.1021/acs.macromol.4c01658  

   Zhang, Diana Y ; Schwendinger, Alec J ; Calabrese, Michelle A ( October 2024 , Macromolecules)

   Polymer solution processability in extensional-flow dominated operations is strongly influenced by polymer conformation and solution phase behavior. Cosolvent addition can be used to tailor polymer conformation and solution phase behavior to yield formulations that are amenable to processes such as spraying and atomization, coating, and fiber spinning. The addition of N,N-dimethylformamide (DMF) to aqueous poly(N-isopropylacrylamide) (PNIPAM) solutions induces unique phase behavior and microstructure formation, yet the effects on solution processability have remained unexplored. In this work, the effect of DMF cosolvent content on the rheology (both shear and extensional) and microstructure of PNIPAM solutions is investigated. While all examined PNIPAM solutions exhibit nearly Newtonian steady shear behavior regardless of DMF content, the same solutions exhibit varying degrees of extensibility. Surprisingly, the extensional relaxation time increases by more than twenty-fold with increasing DMF content in the water-rich regime. In the DMF-rich regime, however, solution extensibility dramatically decreases. Interestingly, this unique variation in extensional flow behavior does not scale as expected based on changes in the measured intrinsic viscosity and radius of gyration. Instead, a mechanism is proposed that relates the extensional flow behavior to the solution microstructure, which is found to vary with DMF content in light scattering measurements. In the water-rich regime, DMF molecules are proposed to bridge PNIPAM chains via hydrogen bonding and hydrophobic interactions, resulting in physically crosslinked aggregates. In extensional flows, these aggregates behave like a polymer with higher apparent molecular weight, increasing the extensional relaxation time. In the DMF-rich regime, non-bridging DMF molecules increasingly solvate individual PNIPAM chains; consequently, more individual chains are stretched in extensional flows, leading to a reduction in the extensional relaxation time. These findings demonstrate that interactions between components in these ternary systems have unexpected but significant implications in solution extensional flow behavior. Additionally, in the case of PNIPAM/DMF/water, the processability of polymer-containing formulations can be modulated for spraying or for fiber spinning applications just by varying cosolvent (DMF) content. 

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4. Non-local continuum modelling of steady, dense granular heap flows

   https://doi.org/10.1017/jfm.2017.554  

   Liu, Daren ; Henann, David L. ( November 2017 , Journal of Fluid Mechanics)

   Dense granular heap flows are common in nature, such as during avalanches and landslides, as well as in industrial flows. In granular heap flows, rapid flow is localized near the free surface with the thickness of the rapidly flowing layer dependent on the overall flow rate. In the region deep beneath the surface, exponentially decaying creeping flow dominates with characteristic decay length depending only on the geometry and not the overall flow rate. Existing continuum models for dense granular flow based upon local constitutive equations are not able to simultaneously predict both of these experimentally observed features – failing to even predict the existence of creeping flow beneath the surface. In this work, we apply a scale-dependent continuum approach – the non-local granular fluidity model – to steady, dense granular flows on a heap between two smooth, frictional side walls. We show that the model captures the salient features of both the flow-rate-dependent, rapidly flowing surface layer and the flow-rate-independent, slowly creeping bulk under steady flow conditions. 

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5. Physically motivated models of polymer networks with dynamic cross-links: comparative study and future outlook

   https://doi.org/10.1098/rspa.2021.0608  

   Hui, Chung-Yuen ; Cui, Fan ; Zehnder, Alan ; Vernerey, Franck J. ( November 2021 , Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Polymer networks consisting of a mixture of chemical and physical cross-links are known to exhibit complex time-dependent behaviour due to the kinetics of bond association and dissociation. In this article, we highlight and compare two recent physically based constitutive models that describe the nonlinear viscoelastic behaviour of such transient networks. These two models are developed independently by two groups of researchers using different mathematical formulations. Here, we show that this difference can be attributed to different viewpoints: Lagrangian versus Eulerian. We establish the equivalence of the two models under the special situation where chains obey Gaussian statistics and steady-state bond dynamics. We provide experimental data demonstrating that both models can accurately predict the time-dependent uniaxial behaviour of a poly(vinylalcohol) dual cross-link hydrogel. We review the advantages and disadvantages of both approaches in applications and close by discussing a list of open challenges and questions regarding the mathematical modelling of soft, viscoelastic networks. 

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