Predicting maximum strain hardening factor in elongational flow of branched pom-pom polymers from polymer architecture
Max G. Schußmann, Manfred Wilhelm, Valerian Hirschberg
Abstract
Abstract We present a model-driven predictive scheme for the uniaxial extensional viscosity and strain hardening of branched polymer melts, specifically for the pom-pom architecture, using the small amplitude oscillatory shear mastercurve and the polymer architecture. A pom-pom shaped polymer is the simplest architecture with at least two branching points, needed to induce strain hardening. It consists of two stars, each with $$q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> arms of the molecular weight $${M}_{w,a}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:msub> </mml:math> , connected by a backbone of $${M}_{w,b}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> </mml:math> . Despite the pom-pom constitutive model, experimental data of systematic investigations lack due to synthetic complexity. With an optimized approach, we synthesized polystyrene pom-pom model systems with systematically varied $${M}_{w,a}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:msub> </mml:math> and $${M}_{w,b}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>w</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> </mml:math> . Experimentally, we identify four characteristic strain rate dependent regimes of the extensional viscosity, which can be predicted from the rheological mastercurve. Furthermore, we find that the industrially important maximum strain hardening factor depends only on the arm number by $$[{q}^{2}/{{{{{\mathrm{ln}}}}}}({\sqrt{3}}q)]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>/</mml:mo> <mml:mi>ln</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:msqrt> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:msqrt> <mml:mi>q</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> . This framework offers a model-based design of branched polymers with predictable melt flow behavior.