Adams and Olmsted Reply to comment on article "A non-monotonic constitutive model is not necessary to obtain shear banding phenomena in entangled polymer solution" [Phys. Rev. Lett. 102, 067801 (2009), arXiv:0805.0679]

TL;DR

This paper is a reply clarifying misconceptions about shear banding in entangled polymer solutions, emphasizing that non-monotonic models are not necessary to explain shear banding phenomena, and addressing experimental and parameter disagreements.

Contribution

The authors defend their original claim that shear banding can occur without non-monotonic constitutive models, countering critiques about stress gradients and experimental interpretations.

Findings

Shear banding can occur without non-monotonic flow curves.

Experimental observations of shear banding do not always require stress gradients.

Parameter disagreements do not invalidate the original model.

Abstract

Wang [Phys. Rev. Lett. 103, 219801 (2009)] makes the following points about our Letter [Phys. Rev. Lett. 102, 067801 (2009), arXiv:0805.0679]: (1) He infers that, "contrary to its title, shear banding emerged from monotonic curves only if there was a stress gradient", and he points out that nonquiescent relaxation was found (experimentally) after step strain in geometries without a stress gradient. (2) He disagrees with the values of the parameters we used. (3) In some recent experiments the flow was homogeneous after cessation of step strain, and only subsequently developed nonquiescent macroscopic motion. We only showed step strains that developed an inhomogeneity before cessation of flow. In this reply we address these points.

Figures

Full text

Adams and Olmsted Reply: Wang Wang (2009) makes the following points about our Letter Adams and Olmsted (2009): (1) He infers that, “contrary to its title, shear banding [in Adams and Olmsted (2009)] emerged from monotonic curves only if there was a stress gradient”, and he points out that nonquiescent relaxation was found (experimentally) after step strain in geometries without a stress gradient Boukany et al. (2009). (2) He disagrees with the values of the parameters we used. (3) In some recent experiments the flow was homogeneous after cessation of step strain, and only subsequently developed nonquiescent macroscopic motion Boukany et al. (2009). We only showed step strains that developed an inhomogeneity before cessation of flow, as in Wang et al. (2006).

(1) As our title stated Adams and Olmsted (2009), we showed that a fluid with a monotonic constitutive curve based on Doi-Edwards (DE) theory can have signatures similar to shear banding. These signatures arise from a stress gradient (e.g. the bowed steady state velocity profile obtained in the stress gradient of a cone and plate rheometer Tapadia and Wang (2006) or transient banding-like profiles during startup). Flat geometries can have transient banding-like signatures: e.g. two clearly defined bands of shear rates during large amplitude oscillatory shear (LAOS) Adams and Olmsted (2009); Hu et al. (2007); *tapadia:196001; *zhou:189, or inhomogeneous banding-like transients during startup flows in presence of inhomogeneous spatial fluctuations (noise) (Fig. 1) Adams and Olmsted (2009).

(2) Our parameters were matched to experiment, for a nonlinear model in which the parameters $\tau_{d}$ and $\tau_{R}$ roughly correspond to their rigorously defined counterparts in linear rheology. Because we use (the best available) crude nonlinear theory, the parameters do not correspond precisely. We used $\epsilon=\eta/(G\tau_{d})\simeq 10^{-5}$ based a plateau modulus $G\simeq 3\textrm{kPa}$, reptation time $\tau_{d}\simeq 20\,\textrm{s}$, and solvent viscosity $\eta\simeq 1\textrm{Pa s}$ Ravindranath and Wang (2008). Although $\tau_{d}/\tau_{R}\sim 10^{3}$ implies too many entanglements, it fits the experimental nonlinear rheology well Tapadia and Wang (2006). This inconsistency is an unsatisfactory feature of current theory.

(3) The step strain results in Adams and Olmsted (2009) should be compared with Wang et al. (2006)-(Fig. 5), where the velocity profile became inhomogeneous before cessation. Fig. 2 shows a calculation in which inhomogeneities develop only after cessation of flow, during a strong recoil. This is for startup in a flat geometry, with noisy initial conditions, and resembles Wang et al. (2006)-(Fig. 3) if there were no experimental wall slip.

Wang’s newest experiments show dramatic rupture and internal fracture, despite a homogeneous velocity before cessation Boukany et al. (2009) (similar fracture planes could be interpreted in Ravindranath and Wang (2007)-(Fig. 3f), but in a cone-and-plate geometry; moreover, those data are also consistent with wall slip and simple recoil). Our calculations (Fig. 2) go some way towards modelling this phenomenon, but do not capture this rupture, and have not yet been adequately modified to incorporate slip. It remains a strong challenge to distinguish which experimental features are captured by tube models, and which (e.g. rupture) require new physical insight. One suggestion is the “elastic yielding” in Wang (2009) which may be similar to modifying the DE model to incorporate the instability of the spatial distribution of entanglements de Gennes (2007). In fact, the instability in the DE model occurs when the shear rate greatly exceeds the reptation time, which is one criterion for elastic yielding postulated in Wang (2009).

J. M. Adams1 and P. D. Olmsted2

1Department of Physics, University of Surrey,

Guildford, GU2 7XH, United Kingdom

2School of Physics & Astronomy, University of Leeds

Leeds, LS2 9JT, United Kingdom

Recieved 9 October 2009

DOI: 10.1103/PhysRevLett.103.219802

PACS numbers: 61.25.he, 83.50.Ax