Macromolecules in a Viscoelastic Liquid

The real and the imaginary components of dynamical modulus of a dilute suspension of macromolecules in a viscoelastic liquid are calculated at values of B shown at the curves and at % = 1. Adapted from the paper of Pokrovskii and Volkov (1978a). 

By studying a mixture of two polymers, one of which is present in much smaller amounts, one has a unique opportunity to obtain direct information about the dynamics of a chosen single macromolecule in a viscoelastic liquid consisting of the matrix macromolecules [59]. 

One of the first attempts to find a molecular interpretation of viscoelastic behaviour of entangled polymers was connected with investigation of the dynamics of a macromolecule in a form of generalised Rouse dynamics (Pokrovskii and Volkov 1978a Ronca 1983 Hess 1986). It formally means that, instead of assumption that the environment of the macromolecule is a viscous medium, Brownian particles of the chain are considered moving in a viscoelastic liquid with the stress tensor... 

The suspension of dilute macromolecular coils in a viscoelastic liquid is suitable for the interpretation of results on the viscoelasticity of concentrated systems with macromolecules, which are not long (M Me). This case was carefully investigated by Leonov (1994). He has confirmed the possibility of a self-consistent description for a system of very short macromolecules. 

One can notice that the dissipative terms in the dynamic equation (3.11) (taken for the case of zero velocity gradients, z/jj = 0) have the form of the resistance force (D.3) for a particle moving in a viscoelastic liquid, while the memory functions are (with approximation to the numerical factor) fading memory functions of the viscoelastic liquid. The macromolecule can be considered as moving in a viscoelastic continuum. In the case of choice of memory functions (3.15), the medium has a single relaxation time and is characterised by the dynamic modulus... 

Let us note that Eq. (41) is a generalisation of the equation for the dynamics of a macromolecule in a dilute solution (Sect. 3.1) the effective viscous liquid in which the Brownian particle moves has been replaced by an effective viscoelastic liquid in the case of a concentrated solution this introduces the concept of microviscoelasticity. Of course, if memory functions turn into -func-... 

In formulae (44) and (45), is the friction coefficient of a particle in a monomer liquid, while B and E are phenomenological parameters which will be discussed below. The correlation time r can be interpreted as relaxation time of the mechanical (viscoelastic) reaction of neighbouring macromolecules, i.e. the system as a whole. This quantity will be calculated later (see Sect 6), and the self-consistency of the theory will be demonstrated. Of course, such a choice of memory functions is eventually justified by empirical facts in later Sections, so we consider the memory functions (45) to be empirical, but to give a rather good description for the case

[Pg.165]

The final main category of non-Newtonian behaviour is viscoelasticity. As the name implies, viscoelastic fluids exhibit a combination of ordinary liquid-like (viscous) and solid-like (elastic) behaviour. The most important viscoelastic fluids are molten polymers but other materials containing macromolecules or long flexible particles, such as fibre suspensions, are viscoelastic. An everyday example of purely viscous and viscoelastic behaviour can be seen with different types of soup. When a thin , watery soup is stirred in a bowl and the stirring then stopped, the soup continues to flow round the bowl and gradually comes to rest. This is an example of purely viscous behaviour. In contrast, with certain thick soups, on cessation of stirring the soup rapidly slows down and then recoils slightly. 

In equation (6.33), the stresses in the moving viscoelastic liquid (6.31) are added to the stresses in the continuum of Brownian particles. When the equations of motion are formulated, we have to take into account the presence of the two interacting and interpenetrating continuous media formed by the viscoelastic liquid carrier and the interacting Brownian particles that model the macromolecules. However, the contribution of the carrier in the case of a concentrated solution is slight, and we shall ignore it henceforth. 

The relaxation time that we have determined may be referred to as the terminal viscoelastic relaxation time it is equal to the relaxation time which was introduced to characterise the medium surrounding the chosen macromolecule. Thus, for >—>00, the theory is self-consistent and this confirms the statement of Section 3.1.1 that chains of Brownian particles are moving independently in a liquid made of interacting Kuhn segments. 

It is natural that estimates (9.30)-(9.32) practically coincide with estimates (6.39) and (6.40), at x C 1, for corresponding quantities for a system of macromolecules in viscoelastic liquid. 

We can see that Eqs. (2 101) (2-104) are sufficient to calculate the continuum-level stress a given the strain-rate and vorticity tensors E and SI. As such, this is a complete constitutive model for the dilute solution/suspension. The rheological properties predicted for steady and time-dependent linear flows of the type (2-99), with T = I t), have been studied quite thoroughly (see, e g., Larson34). Of course, we should note that the contribution of the particles/macromolecules to the stress is actually quite small. Because the solution/suspension is assumed to be dilute, the volume fraction is very small, (p 1. Nevertheless, the qualitative nature of the particle contribution to the stress is found to be quite similar to that measured (at larger concentrations) for many polymeric liquids and other complex fluids. For example, the apparent viscosity in a simple shear flow is found to shear thin (i.e., to decrease with increase of shear rate). These qualitative similarities are indicative of the generic nature of viscoelasticity in a variety of complex fluids. So far as we are aware, however, the full model has not been used for flow predictions in a fluid mechanics context. This is because the model is too complex, even for this simplest of viscoelastic fluids. The primary problem is that calculation of the stress requires solution of the full two-dimensional (2D) convection-diffusion equation, (2-102), at each point in the flow domain where we want to know the stress. 

Chemical reaction in TGDDM-diamine systems leads to the formation of a tridimensional network, which is characteristic of thermosets. The appearance of a tridimensional macromolecule of infinite size is defined as the gelation of the system. At this moment, the viscosity suddenly increases to high values. Gelation in TGDDM-diamine systems can be detected by viscoelasticity, and the gel point corresponds to the time when tan 8 is independent of frequency. This criterion describes the passage from a liquid to a viscoelastic material and is characteristic of many other systems (13-16). 

Graebling, D., Muller, R., and Palierne, J.F. (1993) Linear viscoelastic behavior of some incompatible polymer blends in the melt interpretation of data with a model of emulsion of viscoelastic liquids. Macromolecules. 26 (2), 320-329. 

This volume represents the culmination of four decades study of the dynamics of macromolecules in nondilute solution. I am profoundly grateful to the staff of the Library, Worcester Polytechnic Institute, for their assistance with my more esoteric search inquiries. I am very grateful to my few graduate students and modestly more numerous undergraduates for their research on exemplary polymer solutions, as described and cited below in the appropriate chapters. The work here benefited from interactions over four decades with many colleagues. The treatment of small-molecule and ion diffusion in extremely viscous liquids grew largely from conversations many years ago with Dr. Bret Berner. Preliminary studies on aspects of viscoelasticity represent a collaboration with Dr. P. Peczak(lO). However, the analysis in this volume is almost entirely my own work. 

Polymers differ from other substances by the size of their molecules which, appropriately enough, are referred to as macromolecules, since they consist of thousands or tens of thousands of atoms (molecular weight up to 106 or more) and have a macroscopic rectilinear length (up to 10 4 cm). The atoms of a macromolecule are firmly held together by valence bonds, forming a single entity. In polymeric substances, the weaker van der Waals forces have an effect on the components of the macromolecules which form the system. The structure of polymeric systems is more complicated than that of low-molecular solids or liquids, but there are some common features the atoms within a given macromolecule are ordered, but the centres of mass of the individual macromolecules and parts of them are distributed randomly. Remarkably, the mechanical response of polymeric systems combines the elasticity of a solid with the fluidity of a liquid. Indeed, their behaviour is described as viscoelastic, which is closely connected with slow (relaxation time to 1 sec or more) relaxation processes in systems. 

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