Elastic particle model for coil-stretch transition of dilute polymers in elongational flows

The phenomenon of the "coil-stretch'" transition, wherein a long-chain polymer initially in a coiled state undergoes a sudden configuration change to become fully stretched under steady elongational flows, has been widely recognized. This transition can display intricate hysteresis behaviors under specific critical conditions, giving rise to unique rheological characteristics in dilute polymer solutions. Historically, microscopic stochastic models and Brownian dynamics simulations have shed light on the underlying mechanisms of the transition by uncovering bistable configurations of polymer chains. Following the initial work by Cerf(1952), we introduce a continuum model in this study to investigate the coil-stretch transition in a constant uniaxial elongational flow. Our approach involves approximating the unfolding process of the polymer chain as an axisymmetric deformation of an elastic particle. We make the assumption that the particle possesses uniform material properties, which can be represented by a nonlinear, strain-hardening constitutive equation to replicate the finite extensibility of the polymer chain. Subsequently, we analytically solve for the steady-state deformation using a polarization method. By employing this reduced model, we effectively capture the coil-stretch transition and establish its specific correlations with material and geometric properties. The hysteresis phenomena can be comprehended through a force-balance analysis, which involves comparing the externally applied viscous forces with the intrinsic elastic responsive forces. While simple, we demonstrate that our model provides a comprehensive understanding of the elastohydrodynamic nature of the coil-stretch transition, which is independent of any stochastic properties of polymers considered in microscopic models.