Subchain model

Theory for Kinetically Stiff Chains, Based on the Subchain Model 280... 

It will be obvious that the treatment of the present section can also be applied to the subchain model (77). As is well-known, this model where every junction point of subchains is assumed to interact with the surroundings, seems to provide a more realistic description of the dynamic behaviour of chain molecules than the simple model used in the proceeding paragraphs, viz. the elastic dumb-bell model where only the end-points of the chain are assumed to interact with the surrounding. One of the important assumptions of the subchain model is that every subchain should contain enough random links for a statistical treatment. From this it becomes evident that the derivations given above for a single chain, can immediately be applied to any individual subchain. In particular, those tensor components which were characterized by an asterisk, will hold for the individual subchains as well. 

Application to the Linear Subchain Model 3.2.1. Formulation of Zimm s Theory... 

A final note should be made with respect to the stress-optical coefficient. It should be clear from the derivation given in Section 2.6.1 that this coefficient is independent of the degree of hydrodynamic shielding( 3). In fact, force f is assumed to be in equilibrium with any external forces exerted by other chains and by the moving solvent. (In the case of the subchain model, these "external forces also contain the force due to the neighbouring subchain). 

At this point it seems of interest that also Fixman (107) used such a scaled Gaussian distribution. This author treated the influence of the excluded volume on the intrinsic viscosity, using the subchain model, as well. However, in this work the influence of the excluded volume is treated more correctly. The Gaussian distribution is only scaled in order to make the correction terms due to the excluded volume potential as small as possible. Fixman arrived at the following lower limit of the Flory-Fox parameter ... 

The use of. eq. (3. )) which has nothing to do with the dumb-bell model, seems justified, as eq. (5. Id) should be valid also for the subchain model. From Koyama s papers, in which use is made of the subchain model but where, unfortunately, no results suitable for a quantitative comparison with experiment are obtained, one can learn that an expression of the type of eq. (5. Id) should indeed be independent of the degree of hydrodynamic shielding of the subchain model. From these papers one can also deduce that, for the case of a non-draining polydisperse polymer, is proportional to the ratio MwjMv, where Mn is the... 

For the present purpose it should first be stated that the introduction of the subchain model does not change the character of the picture given for the elastic dumb-bell. A much more complicated situation exists, when real chain molecules are considered. It seems, however, that the statistical character of these chains, when they possess a Gaussian distribution of end-points, will suffice for an explanation of the validity of the stress-optical law. 

For our present purpose, the most interesting considerations seem those which are based on the subchain model. 

Three attempts have been made to introduce internal friction into the subchain model, in order to explain flow birefringence data. The first approach has been made by Cerf (179). who added to eq. (3.16) a third term. He obtained ... 

According to Budtov and Gotlib, this matrix should express the interaction between (equivalent) neighbour subchains only. This leads the mentioned authors to the conclusion that Cerf s second model is not compatible with the original idea of the subchain model. In fact,... 

Cerf omitted to prove that a normal coordinate transformation will, in general, be possible for a subchain model when internal friction is introduced10. 

The Gaussian subchain model and its possible generalisations are universal models, which can be applied to every macromolecule, irrespective of... 

The Gaussian subchain model and its possible generalisations allows one to calculate, in a coarse-grained approximation, the different characteristics of a macromolecule and systems of macromolecules, playing a fundamental role in the theory of equilibrium and non-equilibrium properties of polymers. The model does not describe the local structure of the macromolecule in detail, but describes correctly the properties on a large-length scale. 

The subchain model gives a more detailed description of a macromolecule and allows one to introduce, in line with the end-to-end distance (R2) = Nb2,... 

For the subchain model under consideration, an equilibrium distribution function that includes the particle interaction potential, takes the form... 

The first relaxation time is much bigger than the second one within the limits of applicability of the subchain model. So, the terms multiplied by the quantity wrm in relations (2.43) can be neglected, and expressions can be written down in the simpler form... 

The initial characteristic viscosity defined by equation (6.23) is seen to be independent of the characteristics of intramolecular friction, but this is a consequence of the simplifying assumptions. It has been shown for a dumbbell (Altukhov 1986) that, when account of the internal viscosity and the anisotropy of the hydrodynamic interaction is taken simultaneously, the characteristics of these quantities enter into the expression for a viscosity of type (6.23). This result must be revealed also by the subchain model when account is taken of the anisotropy of the hydrodynamic interaction. 

We can see that a set of constitutive equations for dilute polymer solutions contains a large number of relaxation equations. It is clear that the relaxation processes with the largest relaxation times are essential to describe the slowly changing motion of solutions. In the simplest approximation, we can use the only relaxation variable, which can be the gyration tensor (S i5J), defined by (4.48), or we can assume the macromolecule to be schematised by a subchain model with two particles. The last case, which is considered in Appendix F in more detail, is a particular case of equations (9.3) and (9.4), which is followed at N = 1, Ai = 2,... 

Now, we have to return to the subchain model of macromolecule, which was used to calculate the stresses in the polymeric system, and express the tensor of the mean orientation of the segments of the macromolecule in terms of the subchain model. 

In the simplest case, at N = 1, the considered subchain model of a macromolecule reduces to the dumbbell model consisting of two Brownian particles connected with an elastic force. It can be called relaxator as well. The re-laxator is the simplest model of a macromolecule. Moreover, the dynamics of a macromolecule in normal co-ordinates is equivalent to the dynamics of a set of independent relaxators with various coefficients of elasticity and internal viscosity. In this way, one can consider a dilute solution of polymer as a suspension of independent relaxators which can be considered here to be identical for simplicity. The latter model is especially convenient for the qualitative analysis of the effects in polymer solutions under motion. 

The expressions (F.36), calculated at the exact hydrodynamic interaction, contain the terms, which disappear, if the hydrodynamic interaction is averaged beforehand. It can, thus, be believed that, if hydrodynamic interaction is taken accurately, extra terms will also appear for the subchain model. 

When it is determined in this way, the model is called the Gaussian subchain model it can be generalised in a number of ways. When additional rigidity is taken into account, we have to add the interaction between different particles, so that matrix (8) is replaced, for example, by a five-diagonal matrix. It is also possible to take into account the finite extension of subunits by including in (9) terms of higher order in r, and so on. 

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