Concentrated solutions and melts

The Zimm model predicts correctly the experimental scaling exponent xx ss M3/2 determined in dilute solutions under 0-conditions. In concentrated solution and melts, the hydrodynamic interaction between the polymer segments of the same chain is screened by the host molecules (Eq. 28) and a flexible polymer coil behaves much like a free-draining chain with a Rouse spectrum in the relaxation times. 

A comparison of the solution behaviour of PS in both solvents, toluene and frans-decalin, reveals that the limiting power of the molar mass dependence of r 0 (3.35 and 3.28, respectively) is very close to the value of 3.4 observed in highly concentrated solutions and melts. The concentration dependence of r 0, however, is clearly different in each of the solvents ... 

Conformation and Deformation of Linear Macromolecules in Concentrated Solutions and Melts in the Self-Avoiding Random Walks Statistics... 

Polymeric chains in the concentrated solutions and melts at molar-volumetric concentration c of the chains more than critical one c = (NaR/) ] are intertwined. As a result, from the author s point of view [3] the chains are squeezed decreasing their conformational volume. Accordingly to the Flory theorem [4] polymeric chains in the melts behave as the single ones with the size R = aN112, which is the root-main quadratic radius in the random walks (RW) Gaussian statistics. 

Medvedevskikh Yu. G. Conformation and deformation of linear macromolecules in concentrated solutions and melts in the self-avoiding random walks statistics (see paper in presented book)... 

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. 

Bueche et al. (33) determined chain dimensions indirectly, through measurements of the diffusion coefficient of C1 Magged polymers in concentrated solutions and melts.The self-diffusion coefficient is related to the molar frictional coefficient JVa 0 through the Einstein equation ... 

Experiments due to neutron scattering by the labelled macromolecules allow one to estimate the effective size of macromolecular coils in very concentrated solutions and melts of polymers (Graessley 1974 Maconachie and Richards 1978 Higgins and Benoit 1994) and confirm that the dimensions of macromolecular coils in the very concentrated system are the same as the dimensions of ideal coils. It means, indeed, that the effective interaction between particles of the chain in very concentrated solutions and melts of polymers appears changes due to the presence of other chains in correspondence with the excluded-volume-interaction screening effect. The recent discussion of the problem was given by Wittmer et al. (2007). 

Now one can return to dynamic equation (3.4) of a macromolecule in very concentrated solutions and melts of polymers, which can be rewritten in the form... 

As was demonstrated by Pyshnograi (1994), the last term in (6.7) can be written in symmetric form, if the continuum of Brownian particles is considered incompressible. In equation (6.7), the sum is evaluated over the particles in a given macromolecule. The monomolecular approximation ensures that the stress tensor of the system is the sum of the contributions of all the macromolecules. In this form, the expression for the stresses is valid for any dynamics of the chain. One can consider the system to be a dilute polymer solution or a concentrated solution and melt of polymers. In any case the system is considered as a suspension of interacting Brownian particles. 

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. 

We should pay special attention to the last relation in (8.20), which is a relaxation equation for the variable One can find examples of relaxation equations in Section 2.7 for dilute solutions of polymers and in Chapter 7 for concentrated solutions and melts of polymers. The presence of internal variables and equations for their change are specific features of the liquids we consider in this monograph. 

The set of internal variables is usually determined when considering a particular system in more detail. For concentrated solutions and melts of polymers, for example, a set of relaxation equation for internal variables were determined in the previous chapter. One can see that all the internal variables for the entangled systems are tensors of the second rank, while, to describe viscoelasticity of weakly entangled systems, one needs in a set of conformational variables xfk which characterise the deviations of the form and size of macromolecular coils from the equilibrium values. To describe behaviour of strongly entangled systems, one needs both in the set of conformational variables and in the other set of orientational variables w fc which are connected with the mean orientation of the segments of the macromolecules. 

The internal variables for this case are governed by relaxation equations (7.29) and (7.40) which are valid for the small mode numbers a2 -C i/ /x- This is a case of very concentrated solutions and melts of polymers. Keeping only the zero-order terms with respect to the ratio B/E, the set of relaxation equations for the internal variables can be written in the simpler form... 

Of course, these relations are trivial consequences of the stress-optical law (equation (10.12)). However, it is important that these relations would be tested to confirm whether or not there is any deviations in the low-frequency region for a polymer system with different lengths of macromolecules and to estimate the dependence of the largest relaxation time on the length of the macromolecule. In fact, this is the most important thing to understand the details of the slow relaxation behaviour of macromolecules in concentrated solutions and melts. 

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... 

The generalization of the microscopic approaches for description of real many-chain polymer systems such as networks, concentrated solutions and melts... 

The first phenomenological model for description of polymer dynamics in concentrated solutions and melts was proposed in 1971 by P.de Gennes [50]. In this classical work, it was assumed that due to entanglements, the chain motions in the direction normal to the chain contour are blocked up and only tangential ones are possible. This kind of chain motion in the effective tube was called reptation. In the absence of external fields, the chain can escape from the tube by either of the free tube ends. 

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