Dumb-bell model

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. 

This latter result has been obtained with the aid of the dumb-bell model by Hermans (61) as early as in 1943. Introducing the well-known molecular weight averages ... 

C0Pi6 (137) was the first to give an expression for the contribution of the form birefringence to the Maxwell constant. His theory is based on the elastic dumb-bell model, which has been used in early theories on flow birefringence and viscosity and which is identical with the model used in Sections 2.6.1 and 2.6.2. The ratio of Maxwell constant to intrinsic viscosity is probably unaffected by this simplification, when also the viscosity is calculated with the same model, as Copi6 did. For the absence of the form effect, this has strictly been shown in the mentioned Sections. In fact, in the case of small shear rates the situation is rather simple To a first approximation with respect to shear rate, the chain molecules are only oriented, their intramolecular distances which are needed for the calculation of form birefringence, being unaffected. 

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... 

In any case, it can be demonstrated with the aid of the dumb-bell model that eq. (5.10) is a much better approximation for statistical coil molecules than eq. (5.11 a) for rigid rods. Two cases are considered for the purpose A rigid dumb-bell of fixed length hr and an elastic dumb-bell, according to the usual definition, possessing a root mean square length (Kyi. ... 

Thus, the observations made at the beginning of this section [see eqs. (5.10) and (5.11a)] with respect to coil molecules and rigid rods, are confirmed for the behaviour of the dumb-bell models. In particular, a comparison of eqs. (5.18) and (5.22) shows that, other than for the intrinsic viscosity, the Maxwell constant can only be calculated when, besides (hi) also (hi) is known. This remark will be of importance for the next section, where theories for short chain molecules will be discussed. 

For the calculation of the Maxwell-constant an assembly of frozen random conformations is considered. Brownian motion is taken into account only so far as rotary diffusion of the rigid conformations is concerned. In this way a first order approximation of the distribution function with respect to shear rate is obtained. This distribution function is used for the calculation of the Maxwell-constant, [cf. the calculation of the Maxwell-constant of an assembly of frozen dumb-bell models, as sketched in Section 5.I.3., eq. (5.22)]. Intrinsic viscosity is calculated for the same free-draining model, using average dimensions [cf. also Peter-lin (101)]. As for the initial deviation of the extinction angle curve from 45° a second order approximation of the distribution function is required, no extinction angles are given. 

We consider now the case of a system which is subject to internal forces only. The above considerations are then applicable to the axis of the resultant angular momentum, where, in place of , the angle denoted above by ifi appears and the quantum condition (8) applies. The polarisation of the light cannot be observed, however, since the atoms or molecules of a gas have all possible orientations. The case mentioned above, where all the particles of the system move in planes perpendicular to the axis, is of frequent occurrence, e.g. in the case of the two-body problem (atom with one electron) and in that of the rigid rotator (dumb-bell model of the molecule) the transition j->j is then impossible. 

Fig. 7.9 Illustration of the dumb-bell model for the deformation of polymer coils...

Apparently, the extensional flow causes the deformation of polymer coils. The simplest dumb-bell model can be used to describe this deformatimi. As illustrated in Fig. 7.9, two beads with a distance R, and the entropic elastic recoveiy force uprni deformation of polymer coils is... 

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. 

One remark remains to be made in this connection In the qualitative consideration just made as well as in the used model considerations (dumb-bell, subchain) one point is disregarded, viz. the influence of the... 

Stractural models of the Pd(l 1 l)-c(2V3x3)-iect-C6H6 adlattice are proposed in Figs. 9(A) and 9(B). In Fig. 9(A), each benzene molecule is spread out over four metal-surface atoms but centered on two-fold bridging sites this helps rationalize the dumb-bell shape of the aromatic molecules. To account for the pseudo-triangular shape of the molecular image, the model in Fig. 9(B) is... 

Exercises like the ones discussed have also been carried out for 2D molecules of other shapes (ellipses, dumb bells, needles,. ..) for some of these, 2D equations of state have also been derived. However, for our present purpose they are of no consequence. Such 2D models of monolayers are still far from reality. 

In 2-17 compounds, the atoms located in the dumb-bell sites make a strong contribution to the anisotropy. These sites are preferentially occupied by the substituents when Co is partially replaced by other TM atoms, and this can even change the sign of 7C1, inducing EA behavior (Deportes et al. 1976). This effect has also been described in terms of a single-ion, local crystal-field model but certain results seem to require that band-structure changes also be considered (Perkins and Strassler 1977) i.e., that 3d-electrons be treated as collectivized. 

This phenomenon has been discovered in the liquid crystal phases consisting of so-called banana (or bent-core) shape molecules [17, 27]. A mechanical model in Fig. 4.39a illustrates the idea. Each of the two dumb-bells has symmetry Do h with infinite number of mirror planes containing the longitudinal rotation axis and one mirror plane perpendicular to that axis. Imagine now that one of the dumb-bells is lying on the table and we try to put another one on the top of the first one parallel to... 

Further models for polymer dynamics include the incorporation of stiffness parameters for both local and collective modes, and the approach of Bird and co-workers using the finitely extensible non-linear elastic (FENE) dumb-bell. The latter has been used to reproduce the non-Newtonian viscosity observed with polymer solutions even at the 6-temperature at high shear rates (frequencies), but not given by the simple (infinitely extensible) bead spring. 

Tensile strength, modulus and ultimate elongation were measured on dumb-bell specimens (cut from the sheets using type-A die) and tested on a Zwick/Roell ZOlO model at a crosshead speed of 500mm/min according to ASTM D412. 

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