Nonlinear Polymer Molecules

Nonlinear Polymer Molecules.—The root-mean-square end-to-end distance /used above to characterize the size of a linear polymer obviously would be ambiguous if applied to nonlinear polymers, which possess a multiplicity of ends depending on the degree and functionality of branching. We resort, therefore, to the radius of 

The corresponding relation for a real linear chain may be expressed in terms of the equivalent freely jointed chain comprising n bonds each of length V as follows  

As has been pointed out on page 409, is a satisfactory alternative for as a measure of the efl ective size of a linear polymer molecule. 

A branched molecule obviously will extend over a smaller volume than would a linear one of the same molecular weight, or number of units. In other words, will be smaller for the nonlinear polymer of the same number n of equivalent elements. Hence it is convenient to write for a nonlinear polymer  

---

Number of polyfunctional units in a nonlinear polymer molecule (Chap. IX). 

Structure of each nonlinear polymer molecule can be estimated, most simply in a 0 solvent, using the structural information shown in Fig. 19. By determining the hydrodynamic size of each polymer molecule, we can also estimate the size exclusion chromatography (SEC) elution curve [302-306]. 

Nonlinear structures may arise in vinyl polymerizations through chain transfer with monomer or with previously formed polymer molecules, but such processes usually occur to an extent which is scarcely significant. A more common source of nonlinearity in the polymerization of a 1,3-diene is the incorporation in a growing chain of one of the units of a previously formed polymer molecule. The importance of both branching by chain transfer and cross-linking by addition of a polymer unit increases with the degree of conversion of monomer to polymer. 

A nonlinear molecule of N atoms with 3N degrees of freedom possesses 3N — 6 normal vibrational modes, which not all are active. The prediction of the number of (absorption or emission) bands to be observed in the IR spectrum of a molecule on the basis of its molecular structure, and hence symmetry, is the domain of group theory [82]. Polymer molecules contain a very high number of atoms, yet their IR spectra are relatively simple. This can be explained by the fact that the polymer consists of identical monomeric units (except for the end-groups). 

Nonlinear polymers in which second harmonic active molecules are incorporated into a polymer matrix offer interesting possibilities for waveguide SHG by virtue of their fabrication technology. The multi-layer films needed for waveguiding can be fabricated by simple spinning techniques already in use for decades in the electronics industry, and systematic calibration of the process al-... 

Nonlinear polymer formation in emulsion polymerization is a challenging topic. Reaction mechanisms that form long-chain branching in free-radical polymerizations include chain transfer to the polymer and terminal double bond polymerization. Polymerization reactions that involve multifunctional monomers such as vinyl/divinyl copolymerization reactions are discussed separately in Sect. 4.2.2. For simplicity, in this section we assume that both the radicals and the polymer molecules that formed are distributed homogeneously inside the polymer particle. 

If a dilute polymer solution is subjected to a imidirectional or steady flow with a velocity gradient large enough to stretch out the polymer molecule, nonlinear viscoelastic effects are observed. The simple Hookean dumbbell model, described in Section 3.4.4, can predict... 

As remarked earlier, the nonlinear viscoelastic behavior of entangled wormy micellar solutions is similar to that of entangled flexible polymer molecules. Cates and coworkers (Cates 1990 Spenley et al. 1993, 1996) derived a full constitutive equation for entangled wormy micellar solutions, based on suitably modified reptation ideas. The stress tensor obtained from this theory is (Spenley et al. 1993)... 

Cyclic or ring molecules occur in a wide range of polymers, including linear and nonlinear polymers prepared by condensation or by ring-to-chain equilibratton reactions. Cyclics also occur in many biological macromolecules, sometimes in the form of loops as in certain proteins and nucleic adds. 

Since p and E are vector quantities, a, P, 7, etc., are tensors. For example, the electric field vector in the first term will have three components in the molecular coordinate system. Each electric field component can contribute to polarization along each of the three directions in the molecular coordinate system. This triple contribution of electric field components leads to a total of nine elements to the second rank polarizability tensor. Similarly, there are 27 components to the P tensor and 81 components to 7. Molecular symmetry generally reduces these tensors to only a few independent elements. Unless the molecular coordinate system lacks an inversion center, the form of the odd-rank tensors such as P will lead to zero induced polarization in this representation of optical nonlinearities. For molecules such as benzene and polymers such as poly[bis(p-toluenesulfonate)diacetylene]... 

Doi molecular theory adds a probability density function of molecular orientation to model rigid rodlike polymer molecules. This model is capable of describing the local molecular orientation distribution and nonlinear viscoelastic phenomena. Doi theory successfully predicts director tumbling in the linear regime and two sign changes in the first normal stress difference,as will be discussed later. However, because this theory assumes a uniform spatial structure, it is unable to describe textured LCPs. 

These polymer fluids are typically called non-Newtonian or nonlinear fluids, as they show a decrease of viscosity with increasing fluid velocity (shear rate). This is also known as shear thinning. This behavior results from the fact that the polymer molecules are long and have many contact points interacting with each other, or entanglements. These molecular interactions determine the viscosity of polymers. When one moves them slowly, viscosity is still relatively high. For example, it is... 

Figure 12-3 Plots of the degree of polymerization versus the cumulative number of synthetic steps for various repetitive syntheses (a) conventional linear solid-phase synthesis (b) nonlinear straight-chain sequence synthesis (c) dendrimer synthesis (branching multiplicity of three) (d) double exponential den-drimer synthesis (branching multiplicity of three). In all cases, the degree of polymerization is defined as the total number of monomer units per polymer molecule.

Modeling Linear and Nonlinear Homo- and Copolymerizations Assuming Monofunctional Polymer Molecules and Using the PKRCM... 

Figure 12.1 Polymerization scheme for nonlinear copolymerization with crosshnking under PKRCM and monofunctional polymer molecules.

---