Analysis of startup shear flow data of linear polystyrene blends: is shear stress undershoot caused by transient shear banding?

Abstract Startup shear stress data of a well-defined set of binary polystyrene blends consisting of monodisperse blend components were reported recently by Parisi et al. (J Non-Newtonian Fluid Mech 315:105028, 2023). They presented convincing evidence that in the fast flow of melts with narrow molar mass distribution, shear stress undershoot is observed after the overshoot and before approaching the steady state. For blends with broad relaxation time spectra, no undershoot was found. We analyze this data set by comparison with predictions of the rotation zero stretch (RZS) model (Wagner et al. Rheol Acta 63:573–584, 2024), which is generalized here to the multi-mode MRZS model for polymer blends. We confirm that the steady-state shear viscosity of the monodisperse blend components as well as of the binary blends agrees with the viscosity predicted by the Doi-Edwards independent alignment model. As long as there is no undershoot, the RZS model (monodisperse melts) and the MRZS model (binary blends) result in a quantitative description of the full startup curve of the shear stress growth $${\sigma }_{12}^{+}(\dot{\gamma },t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>γ</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> including overshoot and steady state, based solely on the linear viscoelastic characterization. The shear stress undershoot observed at higher shear rates in melts with narrow molar mass distribution is not described by the RZS or MRZS model. However, the analysis of the experimental data shows clear evidence that undershoot occurs only if after the overshoot, the decreasing shear stress at a higher shear rate undercuts the shear stress at lower rates, i.e., only if $$\partial {\sigma }_{12}^{+}(\dot{\gamma },t)/\partial \dot{\gamma }&lt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>γ</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mi>∂</mml:mi> <mml:mover> <mml:mi>γ</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . For blends with broad relaxation time spectra, $$\partial {\sigma }_{12}^{+}(\dot{\gamma },t)/\partial \dot{\gamma }\cong 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>γ</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mi>∂</mml:mi> <mml:mover> <mml:mi>γ</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>≅</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and no undershoot is observed. The hypothesis is made that undershoot is due to transient shear banding, which is initiated in shear stress regimes characterized by $$\partial {\sigma }_{12}^{+}(\dot{\gamma },t)/\partial \dot{\gamma }&lt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>γ</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mi>∂</mml:mi> <mml:mover> <mml:mi>γ</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>&lt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and which disappears at larger strains when the shear stress growth $${\sigma }_{12}^{+}(\dot{\gamma },t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mn>12</mml:mn> </mml:mrow> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>γ</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> approaches the steady state $${\sigma }_{12}(\dot{\gamma })$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>σ</mml:mi> <mml:mn>12</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>γ</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> with $$\partial {\sigma }_{12}(\dot{\gamma })/\partial \dot{\gamma }&gt;0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:msub> <mml:mi>σ</mml:mi> <mml:mn>12</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>γ</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:mi>∂</mml:mi> <mml:mover> <mml:mi>γ</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . Graphical Abstract