Advances in Condensed Matter Physics

Volume 2015, Article ID 172862, 17 pages

http://dx.doi.org/10.1155/2015/172862

## Self-Organization of Polymeric Fluids in Strong Stress Fields

Institute of Petrochemical Synthesis, Russian Academy of Sciences, 29 Leninskii Prospect, Moscow 119991, Russia

Received 15 April 2015; Revised 9 July 2015; Accepted 29 July 2015

Academic Editor: Golam M. Bhuiyan

Copyright © 2015 A. V. Semakov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Analysis of literature data and our own experimental observations have led to the conclusion that, at high deformation rates, viscoelastic liquids come to behave as rubbery materials, with strong domination by elastic deformations over flow. This can be regarded as a deformation-induced fluid-to-rubbery transition. This transition is accompanied by elastic instability, which can lead to the formation of regular structures. So, a general explanation for these effects requires the treatment of viscoelastic liquids beyond critical deformation rates as rubbery media. Behaviouristic modeling of their behaviour is based on a new concept, which considers the medium as consisting of discrete elastic elements. Such a type of modeling introduces a set of discrete rotators settled on a lattice with two modes of elastic interaction. The first of these is their transformation from spherical to ellipsoidal shapes and orientation in an external field. The second is elastic collisions between rotators. Computer calculations have demonstrated that this discrete model correctly describes the observed structural effects, eventually resulting in a “chaos-to-order” transformation. These predictions correspond to real-world experimental data obtained under different modes of deformation. We presume that the developed concept can play a central role in understanding strong nonlinear effects in the rheology of viscoelastic liquids.

#### 1. Introduction

The general task in understanding the rheology of polymeric liquids is constructing models that allow one to describe the behaviour of a material using the minimal number of physical parameters. The first fundamental step along this path was a model of an individual viscoelastic chain placed in a viscous liquid [1], known in the global literature as the Rouse model. This model predicted the existence of a relaxation spectrum (with regular distribution of relaxation times) and gave insight into the correlation of the molecular parameters of a chain and its relaxation properties. Later, numerous modifications of this model were proposed and widely discussed. The results of this line of investigations were collected and generalized in the framework of the concept of conformational statistics of polymers [2]. Further development of the statistical approach for the rheology of concentrated solutions and melts was based on the concept of entanglements, which is a physical network with some characteristic lifetimes (relaxation times) of local contacts, or nodes, formed by macromolecules [3–5]. Numerous versions and modifications of more complex entanglement models, which rather successfully explained characteristic features of polymeric fluids, were also proposed.

A radical retreat from the viscoelastic chain model and the entanglement concept was proposed by Doi and Edwards in their “tube model” and the reptation-like movement of a single chain inside a tube [6]. The physical transparency of this model was the grounds for simple scaling relationships, which proposed simple relationships between molecular parameters and relaxation properties of polymers [7]. This approach was developed in numerous publications and finally describes qualitative and in many cases semiquantitative peculiarities of the mechanical properties of liquid polymers.

The state of the art in this field based on existing concepts of the dynamics of macromolecular chains committing reptation-like motions inside a tube and scaling ideas was outlined in [8], where the universality of many physical properties of polymers has been fairly stressed. This phenomenon “arises purely from the connectivity of their molecular chains rather than from the chemistry of their monomer units.” The possibility of describing the linear viscoelastic properties of complex systems consisting of interacting elementary units with many thousands of segments up to a range exceeding 12 decimal orders by means of a rather simple model was estimated as “actually rather remarkable” [8].

Meanwhile, it was stressed that despite the general qualitative agreement between theoretical predictions and experimental data, several aspects of quantitative descriptions remain unsatisfactory [9]. There have been different attempts to modify the basic theory by introducing additional factors, such as slip-spring elements, which constrain the reptation-like movement of a chain inside a tube and so on (e.g., [10–12]).

However, the principle point is that these model concepts work marvellously only in a linear region of viscoelasticity. In the transition to nonlinear behaviour, a lot of questions and problems appear. The first complex model of nonlinear viscoelasticity, known as the well-known Kaye-Bernstein-Kearsley-Zapas equation, explored the idea of separability (or factorization) of nonlinear viscoelastic properties into a “linear” part and a modifying function. This concept was then widely used by many authors; some versions of this approach include several modifications of the so-called Wagner models. The problem of nonlinear relaxation and its quantitative description has also been discussed by numerous authors. The correspondence of this approach with the Doi-Edwards model has also been examined. This line of studies is presented in several papers [10, 13–16] which cite and summarize the numerous investigations in this field.

The method of factorization in treating nonlinear viscoelastic properties is not general [17], and failure in applications of models of this class is also known, especially when macromolecules of complicated structure are studied [18, 19]. Different approaches to fast flows have been developed based on the stochastic simulation of entangled polymeric liquids [20].

Discussions of the problem of the nonlinear behaviour of viscoelastic fluids have always been limited by the frame of the continuum mechanics. Meanwhile, the mechanism of deformations at high shear rates remains not quite clear. Several publications in the last decade based on direct measurements demonstrated the heterogeneity of flow, with jumps in shear rates. In addition, direct observation showed that self-organization actually takes place in rotational flows of polymeric viscoelastic substances.

Discussions on nonlinearity and self-organization in the deformation of polymeric fluids tacitly assume that a deformed matter reminds fluid at any rate and the issue consists only of the choice of an appropriate constitutive equation. As such, many problems of instability, bifurcation, deformation-induced structure formation, and even fracture are considered in the framework of Giesekus or FENE models (e.g., [21]) or some other continuum models.

Meanwhile, the last statement is of crucial value because it is permissible to ask: is it true at arbitrary high deformation rate (or stress)?

To answer this question, it is appropriate to return to the early conceptual results of Vinogradov et al. [22] who described the “spurt” phenomenon, the transition from shear flow to slip at high enough stresses, as well as the filament behavior at high enough deformation rates [23]. Experimental results of this kind have been treated in a rather evident manner: complex viscoelastic fluids lose the ability to flow at high deformation rates and transit into an elastic (rubbery) state, or, to formulate it more accurately, a rubber-like behavioural state occurs [24].

The dominant component of full deformation in this state is elastic (recoverable) deformations, and real flow becomes negligible. This is analogous (but not equivalent) to the flow-to-rubbery transition that takes place as a function of frequency in oscillation measurements. An obvious reflection of this analogue is the well-known Cox-Merz rule, though it has a different understanding [25, 26].

The separation of the full deformation into elastic (reversible) and flow (irreversible) components is made by measuring the reaction of a sample after cessation of the external force. The partial recovery of the initial shape is observed and this is the elastic component and the other (nonrecoverable) part of the total deformation is attributed as the flow.

The domination by elastic deformations over irreversible flow at high deformation rates is illustrated in Figure 1 for extension. The situation for shearing is quite the same for the initial stage of deformation or high deformation rates, though for low deformation rates stationary flow takes place and irreversible deformations become dominant.