Concentrated solution/melt theory

C. CONCENTRATED SOLUTION/MELT THEORIES 1. The Bird-Carreau Model... 

The status of current theories of the low shear-rate viscosities (rj0) of polymer melts (and concentrated solutions) was reviewed by Berry and Fox in 1968 (52), since when there has been little development. The viscosity of linear polymers of low MW at constant temperature (or more precisely constant free volume is proportional to Mw, but at high MW it is proportional to a higher power M, where x is empirically about 3.4-3.5 the change of slope of a ogt]0/logMw plot is fairly abrupt The high exponent 3.4—3.5 is attributed to the effects of chain... 

Molecular theories, based in large part on ideas about chain entanglements, have been constructed to explain certain of these observations. The theories must still be regarded as tentative and incomplete. They are based, first of all, on reasonable but still incompletely accepted ideas about chain organization in concentrated solutions and melts. Secondly, they deal with the response of individual chains or pairs of chains in a smoothed medium, rather than with an entire interacting ensemble. Finally, they circumvent the deep mathematical difficulties of the central problem, interaction between mutually uncrossable sequences of chain elements, by approximations which are not easy to evaluate. The purpose of this review is to summarize the present status of entanglement theories and the data upon which they are based. 

To date, there has been a very limited effort devoted to developing theory for ionic block copolymers. Gonziilez-Mozuelos and Olvera da la Cruz (1994) studied diblock copolymers with oppositely charged chains in the melt state and in concentrated solutions using the random phase approximation (RPA) (de Gennes 1970). However, this work has not been extended to dilute solutions. 

This chapter is concerned with these phases, where a substantial amount of the experimental work has been on poly(oxyethylene)-containing block copolymers in aqueous solution. From another viewpoint, the phase behaviour in concentrated block copolymer solutions has been interpreted using the dilution approximation, which considers concentrated solution phases to be simply uniformly swollen melt phases. Work on styrenic block copolymers in concentrated solution has been interpreted in this framework. There is as yet no unifying theory that treats ordered micellar phases and diluted melt phases coherently. 

Abstract The discussion of relaxation and diffusion of macromolecules in very concentrated solutions and melts of polymers showed that the basic equations of macromolecular dynamics reflect the linear behaviour of a macromolecule among the other macromolecules, so that one can proceed further. Considering the non-linear effects of viscoelasticity, one have to take into account the local anisotropy of mobility of every particle of the chains, introduced in the basic dynamic equations of a macromolecule in Chapter 3, and induced anisotropy of the surrounding, which will be introduced in this chapter. In the spirit of mesoscopic theory we assume that the anisotropy is connected with the averaged orientation of segments of macromolecules, so that the equation of dynamics of the macromolecule retains its form. Eventually, the non-linear relaxation equations for two sets of internal variables are formulated. The first set of variables describes the form of the macromolecular coil - the conformational variables, the second one describes the internal stresses connected mainly with the orientation of segments. 

The mesoscopic approach gives an amazingly consistent picture of the different relaxation phenomena in very concentrated solutions and melts of linear polymers. It is not surprising the developed theory is a sort of phenomenological (mesoscopic) description, which allows one to get a consistent interpretation of experimental data connected with dynamic behaviour of linear macromolecules in both weakly and strongly entangled polymer systems in terms of a few phenomenological (or better, mesoscopic) parameters it does not require any specific hypotheses. 

Pokrovskii VN, Pyshnograi GV (1990) Non-linear effects in the dynamics of concentrated polymer solutions and melts. Fluid Dyn 25 568-576 Pokrovskii VN, Pyshnograi GV (1991) The simple forms of constitutive equation of polymer concentrated solution and melts as consequence of molecular theory of viscoelasticity. Fluid Dyn 26 58-64... 

We will begin with a brief survey of linear viscoelasticity (section 2.1) we will define the various material functions and the mathematical theory of linear viscoelasticity will give us the mathematical bridges which relate these functions. We will then describe the main features of the linear viscoelastic behaviour of polymer melts and concentrated solutions in a purely rational and phenomenological way (section 2.2) the simple and important conclusions drawn from this analysis will give us the support for the molecular models described below (sections 3 to 6). 

The Rouse model is the earliest and simplest molecular model that predicts a nontrivial distribution of polymer relaxation times. As described below, real polymeric liquids do in fact show many relaxation modes. However, in most polymer liquids, the relaxation modes observed do not correspond very well to the mode distribution predicted by the Rouse theory. For polymer solutions that are dilute, there are hydrodynamic interactions that affect the viscoelastic properties of the solution and that are unaccounted for in the Rouse theory. These are discussed below in Section 3.6.1.2. In most concentrated solutions or melts, entanglements between long polymer molecules greatly slow polymer relaxation, and, again, this is not accounted for in the Rouse theory. Reptation theories for entangled... 

A mathematical expression relating forces and deformation motions in a material is known as a constitutive equation. However, the establishment of constitutive equations can be a rather difficult task in most cases. For example, the dependence of both the viscosity and the memory effects of polymer melts and concentrated solutions on the shear rate renders it difficult to establish constitute equations, even in the cases of simple geometries. A rigorous treatment of the flow of these materials requires the use of fluid mechanics theories related to the nonlinear behavior of complex materials. However, in this chapter we aim only to emphasize important qualitative aspects of the flow of polymer melts and solutions that, conventionally interpreted, may explain the nonlinear behavior of polymers for some types of flows. Numerous books are available in which the reader will find rigorous approaches, and the corresponding references, to the subject matter discussed here (1-16). 

A major goal in the physics of polymer melts and concentrated solutions is to relate measurable viscoelastic constants, such as the zero shear viscosity, to molecular parameters, such as the dimensions of the polymer coil and the intermolecular friction constant. The results of investigations to this end on the viscosity were reviewed in 1955 (5). This review wiU be principaUy concerned with advances made since in both empirical correlation (Section 2) and theory of melt flow (Section 3). We shall avoid data confined to shear rates so high that the zero shear viscosity cannot be reliably obtained. The shear dq endent behavior would require an extensive review in itself. 

In the case of concentrated solutions or melts of liquid crystalline polymers, the Onsager theory and hence the above arguments are not applicable. It is also not applicable to the rather flexible liquid crystalline polymers. 

The negative first normal stress difference under a medium shear rate, characterized by liquid crystalline polymers, makes the material avoid the Barus effect—a typical property of conventional polymer melt or concentrated solution, i.e., when a polymer spins out from a hole, or capillary, or slit, their diameter or thickness will be greater than the mold size. The liquid crystalline polymers with the spin expansion effect have an advantage in material processing. This phenomenon is verified by the Ericksen-Leslie theory. On the contrary, the first normal stress difference for the normal polymers is always positive. 

In an excellent review article, Tirrell [2] summarized and discussed most theoretical and experimental contributions made up to 1984 to polymer self-diffusion in concentrated solutions and melts. Although his conclusion seemed to lean toward the reptation theory, the data then available were apparently not sufficient to support it with sheer certainty. Over the past few years further data on self-diffusion and tracer diffusion coefficients (see Section 1.3 for the latter) have become available and various ideas for interpreting them have been set out. Nonetheless, there is yet no established agreement as to the long timescale Brownian motion of polymer chains in concentrated systems. Some prefer reptation and others advocate essentially isotropic motion. Unfortunately, we are unable to see the chain motion directly. In what follows, we review current challenges to this controversial problem by referring to the experimental data which the author believes are of basic importance. 

At present, no reported data on ring self-diffusion in polymer concentrates are available other than those of Mills et al. and no theory of this subject exists other than Klein s. Thus we see a virgin field of research open before us. What seems most needed is experimental data for self-diffusion in the melt and concentrated solutions of rings. Diffusion of linear chains in ring chain matrices should also be instructive, as pointed out by Mills et al. The reptation idea now dominating the study of polymer self-diffusion will face crucial tests when accurate and systematic diffusion data on these systems become available. 

Doi and Edwards developed the most extensive theory of dynamics of rod-like macromolecules in concentrated solutions [2,12], As discussed in the previous section, the majority of commercial TLCPs are not perfect rigid rods but are semi-rigid rods having some degree of flexibility. Although several experimental results support the validity of the rigid rod approximation for both lyotropic [13-15] and thermotropic liquid crystalline polymers [16-18], there is no complete theory on the dynamics of TLCPs. To better understand the dynamics of TLCPs in the melt, the Doi-Edwards theory on the dynamics of rod-like polymers in solution is summarized here. Readers may find further details of the theory in the original reference [2,12]. 

The zero shear viscosity scales with Nf" to contrast Af dependence for isotropic polymers [20] So far, we have examined the dynamics of rod-Uke macromolecules in isotropic semi-dilute solution. For anisotropic LCP solutions in which the rods are oriented in a certain direction, the diffusion constant increases, and the viscosity decreases, but their scaling behavior with the molecular weight is expected to be unchanged [2,17], Little experimental work has been reported on this subject. The dynamics of thermotropic liquid crystalline polymer melts may be considered as a special case of the concentrated solution with no solvent. Many experimental results [16-18] showed the strong molecular weight dependence of the melt viscosity as predicted by the Doi-Edwards theory. However, the complex rheological behaviors of TLCPs have not been well theorized. 

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