Submolecule model

Further development and application of the submolecule model of exchange interaction. 

Validity of Heitler-London Wave-Functions and the submolecule model. 

Cohen-Addan JP, Dupeyre R. Strongly entangled polymer chains in a melt. Description of njnx properties associated with a submolecule model. Polymer 1983 24 400. 

Here t, is still a ratio of a viscosity to a modulus, as in the spring-dashpot model of Figure 1, but each sprint has the same (shear) modulus, pRTfM and the steady-flow viscosity T] of equation (16) is the sum of the viscosities of the individual submolecules. Molecular theories are discussed more fully in Section X. 

N is the number of statistical segments in a chain, 1 is their length and m the number of submolecules. This model includes the front factor only in C, but the parameter K0 is also temperature dependent it depends on the thermal expansion. Thus, the Priss tube model predicts different temperature dependences of C, and C.. It means that the entropy and energy contribution have to be dependent on X and include a considerable intermolecular part. 

In section 3.1.3. we proposed a simple model to calculate the anisotropic form factor of the chains in a uniaxially deformed polymer melt. The only parameters are the deformation ratio X of the entanglement network (which was assumed to be identical to the macroscopic recoverable strain) and the number n, of entanglements per chain. For a chain with dangling end submolecules the mean square dimension in a principal direction of orientation is then given by Eq. 19. As seen in section 3.1.3. for low stress levels n can be estimated from the plateau modulus and the molecular weight of the chain (n 5 por polymer SI). 

In Fig. 3-5a, the polymer coil is modeled as a series of beads equally spaced along the polymer backbone and connected to each other by springs. The beads account for the viscous forces and the springs the elastic forces in the molecule the portion of the chain represented by a single spring is called a submolecule. The bead-spring model is... 

Hermans and Van Beek626 have recently used the new model of polymer molecules suggested by Rouse.46 At high frequencies the whole molecule cannot follow the field so it is divided into a number of submolecules small enough to follow the field and yet sufficiently large to have a Gaussian distribution. Dielectric relaxation for the case of dipoles parallel to the chain has been calculated by Founder sum transforms. The distribution of relaxation modes appears though the multiplicity of the mathematical solution for the diffusion equation. 

Both authors find that rm is equal to the relaxation time of a submolecule and should be quite independent of molecular weight. From this point of view the agreement between theory and practice is quite good, particularly for 1000. Unfortunately the calculations have only been carried out for the case of dipoles parallel to the chain, and Rouse s model does not apply when the dipole is in the side-groups, as is the case for all polymers that have been studied in dilute solutions. 

Having proceeded this far, we are now in a position to write the equation of motion for our linear bead-and-spring model. To do this we divide the polymer molecule into z submolecules so that there are z springs and z + 1 beads. We now introduce a unidirectional deformation. The restoring force on each of the beads is given by ... 

The most studied relaxation processes from the point of view of molecular theories are those governing relaxation function, G,(t), in equation [7.2.4]. According to the Rouse theory, a macromolecule is modeled by a bead-spring chain. The beads are the centers of hydrodynamic interaction of a molecule with a solvent while the springs model elastic linkage between the beads. The polymer macromolecule is subdivided into a number of equal segments (submolecules or subchains) within which the equilibrium is supposed to be achieved thus the model does not permit to describe small-scale motions that are smaller in size than the statistical segment. Maximal relaxation time in a spectrum is expressed in terms of macroscopic parameters of the system, which can be easily measured ... 

For the Rouse model the submolecule is the shortest length of chain that can undergo relaxation, and the motion of segments within a submolecule is ignored. This limitation implies that the theory is applicable only when m 1, and means that the equation for xp reduces to... 

Consider a system of rods having an orientation with respect to a preferred axis measured by an angle of inclination 0. The rods are broken up into a sequence of y consecutive submolecules. The diagram of the model is as follows ... 

In 1953 Rouse published a paper to describe theoretically the flow of polymers in dilute solutions. The polymer molecule is assumed to exist as a statistical coil and is subdivided into N submolecules. Each submolecule is thought of a solid bead. The beads behave as Gaussian chains and their entropy-elastic recovery can be described by a spring with a spring constant hkT/cP-, where a is the average end-to-end distance of a submolecule and k is the Boltzmann constant. The model is shown in Figure 8.9. 

Zimm s model (1956) is also a chain of beads connected by ideal springs. The chain consists of N identical segments joining + 1 identical beads with complete flexibility at each bead. Each segment, which is similar to a submolecule, is supposed to have a Gaussian probability function. The major difference between the two models lies in the interaction between the individual beads. In the Rouse model, such interaction is ignored in Zimm s model, such interaction is taken into consideration. 

The bead-spring model with dominant hydrodynamic interaction was extended to star-shaped branched molecules by Zimm and Kilb. For/branches of equal length, each with Nb submolecules, equations 15,16,36, and 38 are replaced by... 

---