Relaxation of Macromolecular Coil

The results discussed in the previous section are valid in linear approximation for any concrete representations of the memory functions / (s) and p(s). To calculate relaxation times for macromolecular coil, one has to specify the memory functions and include the effect of local anisotropy. 

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The mean size and form of the macromolecular coil is characterised by the set of the tensors (pf / , a = 1,2,..., N. To describe relaxation of macromolecular coil, it is convenient to use the non-dimensional tensor variables... 

One of the prominent features of polymeric liquids is the property to recover partially the pre-deformation state. Such behaviour is analogous to a rubber band snapping back when released after stretching. This is a consequence of the relaxation of macromolecular coils in the system every deformed macro-molecular coil tends to recover its pre-deformed equilibrium form. In the considered theory, the form and dimensions of the deformed macromolecular coil are connected with the internal variables which have to be considered when the tensor of recoverable strain is to be calculated. Further on, we shall consider the simplest case, when the form and dimensions of macromolecular coils are determined by the only internal tensor. In this case, the behaviour of the polymer liquid is considered to describe by one of the constitutive equations (9.48)-(9.49) or (9.58). 

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. 

In (1.66)-(1.70), denotes solvent viscosity, c is the number of macromolecules in unit volirme, b is the length of Kuhn segments in macromolecule, N is the number of Kuhn segments in macromolecirle, r = bN is of the order of the equilibrium macromolecular coU size, 6 = 6n[irlK is the relaxation time, k the elasticity of macromolecular coils. According to the secrnid equation (1.69) macromolecules cannot be stretched beyond their fully expended length Nb. 

The theory of relaxation processes for a macromolecular coil is based, mainly, on the phenomenological approach to the Brownian motion of particles. Each bead of the chain is likened to a spherical Brownian particle, so that a set of the equation for motion of the macromolecule can be written as a set of coupled stochastic equations for coupled Brownian particles... 

This equation describes only the deformation of the macromolecular coil and therefore r11 is a relaxation time of the deformation process. It can be shown (see Appendix F) that the orientation relaxation process is characterised by the relaxation time tx. 

The rates of relaxation r7(t) in the moment t, or, in other words, the current relaxation times of the macromolecular coil can be directly calculated as... 

In relaxation processes of the macromolecular coil to equilibrium, the competing mechanisms of mobility of particles are present simultaneously. However, in the region of weakly entangled macromolecules, relaxation occurs due to isotropic mobility of particles of the chain - the diffusive mechanism -... 

So, one ought to conclude that large scale (slow) relaxation of the conformation of a macromolecular coil is realised through reptation, instead of the more slow mechanism of rearrangement of all the entangled chains, if the parameter B > 7t2/2x-... 

At constant velocity gradient Xik, the moments (pfpk) are given by relations (4.46) for the diffusive mechanism of relaxation, and by similar formula (there is a difference in definition of the relaxation times only) for the rep-tation mechanism, so that we can evaluate the expression for the tensor of gyration of the macromolecular coil, taking into account alternative mechanisms of relaxation... 

The dynamic modulus of the suspension of non-interacting macromolecular coils is determined by three sets of relaxation times... 

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 pressure p includes both the partial pressure of the gas of Brownian particles n(N +1 )T and the partial pressure of the carrier monomer liquid. We shall assume that the viscosity of the monomer liquid can be neglected. The variables xt k in equation (9.19) characterise the mean size and shape of the macromolecular coils in a deformed system. The other variables ut k are associated mainly with orientation of small rigid parts of macromolecules (Kuhn segments). As a consequence of the mesoscopic approach, the stress tensor (9.19) of a system is determined as a sum of the contributions of all the macromolecules, which in this case can be expressed by simple multiplication by the number of macromolecules n. The macroscopic internal variables x -k and u"k can be found as solutions of relaxation equations which have been established in Chapter 7. However, there are two distinctive cases, which have to be considered separately. 

Abstract Macromolecular coils are deformed in flow, while optically anisotropic parts (and segments) of the macromolecules are oriented by flow, so that polymers and their solutions become optically anisotropic. This is true for a macromolecule whether it is in a viscous liquid or is surrounded by other chains. The optical anisotropy of a system appears to be directly connected with the mean orientation of segments and, thus, it provides the most direct observation of the relaxation of the segments, both in dilute and in concentrated solutions of polymers. The results of the theory for dilute solutions provide an instrument for the investigation of the structure and properties of a macromolecule. In application to very concentrated solutions, the optical anisotropy provides the important means for the investigation of slow relaxation processes. The evidence can be decisive for understanding the mechanism of the relaxation. 

In Fig. 110 the dependence x) D is adduced for the considered den-drimers is adduced, from which file fast decay of relaxation time at macromolecule fractal dimension growth follows. Such shape of the dependence x) D can be explained by two factors influence. The macromolecular coil gyration radius is linked with dimension and polymerization degree A by the Eq. (8). As one can see, the indicated equation supposes very strong (power) dependence of on The estimation according to the Eq. (4) has shown that for the considered dendrimers Devalue varies within the limits of 1.52-2.42. Using reasonable values iV=1000 [227] and 5=0.35 [234], let us obtain variation within the limits of approx. 6-33 nm, that is, in 5.5 times, for file considered dendrimers, that corresponds well to the experimental data [227,230]. It is obvious, that reorientation of small macromolecules in solution occurs much easier than large ones and requires much less duration, which is, (x... 

Conformational relaxation of a macromolecular coil is described by the two relaxation branches (81), while one of them contains large relaxation times, the other small. Meanwhile, there could be a competing mechanism for relaxation of the macromolecular coil to equilibrium due to reptation, so the largest relaxation times, which at B > 1 can be approximated as... 

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