The relaxation dynamics of single flow-stretched polymers in semidilute to concentrated solutions

Abstract

Recent experiments on the return to equilibrium of solutions of entangled polymers stretched by extensional flows [Zhou and Schroeder, Phys. Rev. Lett. 120, 267801 (2018)] have highlighted the possible role of the tube model's two-step mechanism in the process of chain relaxation. In this paper, motivated by these findings, we use a generalized Langevin equation (GLE) to study the time evolution, under linear mixed flow, of the linear dimensions of a single finitely extensible Rouse polymer in a solution of other polymers. Approximating the memory function of the GLE, which contains the details of the interactions of the Rouse polymer with its surroundings, by a power law defined by two parameters, we show that the decay of the chain's fractional extension in the steady state can be expressed in terms of a linear combination of Mittag-Leffler and generalized Mittag-Leffler functions. For the special cases of elongational flow and steady shear flow, and after adjustment of the parameters in the memory function, our calculated decay curves provide satisfactory fits to the experimental decay curves from the work of Zhou and Schroeder and earlier work of Teixeira et al. [Macromolecules 40, 2461(2007]. The non-exponential character of the Mittag-Leffler functions and the consequent absence of characteristic decay constants suggest that melt relaxation may proceed by a sequence of steps with an essentially continuous, rather than discrete, spectrum of timescales.

Statistical Dynamics of Flow-Driven Globular Polymers

“…Expressions for the X n ( t ) variables themselves are found from eqs – by applying Laplace transforms to the equations, solving the resulting algebraic equations by matrix inversion, and then Laplace inverting the result. In this way, if one defines the Laplace transform of a function f ( t ) by the relation , it can be shown that where and a ̂ n ( s ) = s ζ + b λ n +k , with denoting the inverse Laplace transform operation.…”

“…Within this approximation, it is readily shown that Here, τ̅ is a dimensionless time defined as τ̅ = τ/τ 1 , with τ 1 given by τ 1 = ζ N 2 l 2 /(3π 2 k B T ). In the special limit α = 1 corresponding to elongational flow, eq reduces to while in the limit α = 0 corresponding to shear flow, it reduces to Furthermore, in the limit κ → 0, both eqs and recover results derived earlier for the case of non-self-interacting chains. , …”

“…The b and Wi values that correspond to these κ and x values [and that must be substituted in eqs and to construct C (τ)] are obtained from the data used to calculate the dependence of x on Wi . (In our earlier calculations on chain relaxation in entangled solutions of flow-stretched polymers, the average of the projection of R (τ) onto the steady-state end-to-end distance R of the chain at τ = 0, viz., ⟨ R · R (τ)⟩, was identified as ⟨ R 2 (τ)⟩ in comparisons of theory and experiment; though the two quantities are not the same, ⟨ R · R (τ)⟩ was found to accurately track the experimental trends in the decay of ⟨ R 2 (τ)⟩).…”

“…To calculate ⟨R(t − τ)•R(t)⟩, one proceeds as before from eq 9a by first performing the indicated inverse Laplace transforms (by means of partial fractions and the convolution theorem), but without setting the variable τ to 0, and next passing to the limit t → ∞ (so that the system can be assumed to have been prepared in a steady state 35 ). In this limit, contributions to the time correlation function from the initial (t = 0) chain configuration again drop out.…”

“…Analytical models of globule–flow interactions could be especially helpful in realizing these outcomes, as they may be able to link specific model parameters to various empirical observations and, in the process, establish a well-defined molecular basis for interpreting experimental and numerical data. Few, if any, such models appear to have yet been formulated, and the present paper is a preliminary effort at developing a framework for doing so, motivated in part by the discovery that an early exactly solvable model of collapsed polymers works nicely within an approach introduced in this lab to calculate flow-driven chain properties, − including the dependence of chain stretching on the strength of an external flow field, and the relaxation to equilibrium of chains stretched by and then released from the action of the flow. The calculations are to a large extent exact, but they include an approximation (that can be treated analytically) to account for the finite extensibility of the chains.…”