Crystalline polymer model

C. V Chaubal and L. G. Leal, A closure approximation for liquid crystalline polymer models based on parametric density estimation, J. Rheol. 42, 177-201 (1998) H. C. Ottinger, Stochastic Processes in Polymeric Fluids (Springer, Berlin, 1996). 

The aim of this chapter is to describe the micro-mechanical processes that occur close to an interface during adhesive or cohesive failure of polymers. Emphasis will be placed on both the nature of the processes that occur and the micromechanical models that have been proposed to describe these processes. The main concern will be processes that occur at size scales ranging from nanometres (molecular dimensions) to a few micrometres. Failure is most commonly controlled by mechanical process that occur within this size range as it is these small scale processes that apply stress on the chain and cause the chain scission or pull-out that is often the basic process of fracture. The situation for elastomeric adhesives on substrates such as skin, glassy polymers or steel is different and will not be considered here but is described in a chapter on tack . Multiphase materials, such as rubber-toughened or semi-crystalline polymers, will not be considered much here as they show a whole range of different micro-mechanical processes initiated by the modulus mismatch between the phases. 

In summary the results of our 2H NMR investigation illustrate the spacer model for liquid crystalline polymers, indicating, however, that the decoupling of the mesogenic groups from the main chain, while effective, is not complete. 

Advanced computational models are also developed to understand the formation of polymer microstructure and polymer morphology. Nonuniform compositional distribution in olefin copolymers can affect the chain solubility of highly crystalline polymers. When such compositional nonuniformity is present, hydrodynamic volume distribution measured by size exclusion chromatography does not match the exact copolymer molecular weight distribution. Therefore, it is necessary to calculate the hydrodynamic volume distribution from a copolymer kinetic model and to relate it to the copolymer molecular weight distribution. The finite molecular weight moment techniques that were developed for free radical homo- and co-polymerization processes can be used for such calculations [1,14,15]. 

Volume and mass-based expressions for the degree of crystallinity are easily derived from the experimentally measured density (p) of a semi-crystalline polymer. The method is based on an ideal crystalline and liquid-like two-phase model and assumes additivity of the volume corresponding to each phase... 

We note here that all the information presently available on high molecular weight polymer crystal structures is compatible with the bundle model. While very nearly all crystalline polymer polymorphs involve all-parallel chain arrangements, even the only known exception, namely y-iPP [104,105], where chains oriented at 80° to each other coexist, is characterized by bilayers of parallel chains with opposite orientation. This structure is thus easily compatible with crystallization mechanisms involving deposition of bundles of 5-10 antiparallel stems on the growing crystal surface. Also the preferred growth... 

If the ordered, crystalline regions are cross sections of bundles of chains and the chains go from one bundle to the next (although not necessarily in the same plane), this is the older fringe-micelle model. If the emerging chains repeatedly fold buck and reenter the same bundle in this or a different plane, this is the folded-chain model. In either case the mechanical deformation behavior of such complex structures is varied and difficult to unravel unambiguously on a molecular or microscopic scale. In many respects the behavior of crystalline polymers is like that of two-ph ise systems as predicted by the fringed-micelle- model illustrated in Figure 7, in which there is a distinct crystalline phase embedded in an amorphous phase (134). 

Figure 8 Fringe-micelle model of crystalline polymers. (PramRef. 131,)...

As previously discussed, the solids conveying rate decreases as the discharge pressure increases for all materials studied. Eor HDPE resin, the solids conveying rate decreased almost logarithmically with discharge pressure, as shown in Eig 5.12. This is a characteristic of most crystalline polymers, and has been well documented in prior literature [21] as well as predicted by all of the major models [1,14,19,20, 22, 23]. The HDPE resin demonstrated the greatest solids conveying pellet flow... 

Flory, Marie, and Abe (194) carried out a statistical mechanics analysis of vinyl polymers on the basis of a three rotational state model. Energy maps have been calculated both for m and r dyads as a function of the rotation angles around the bonds astride the methylene groups (CHR—CH2—CHR). These maps differ from those examined for crystalline polymers where rotations around... 

Here, Da is diffusion coefficient in the amorphous phase alone, oc is the volume fraction of crystalline polymer, and t is a scalar quantity that denotes the tortuosity of diffusional path of the solute. The value of Da may be estimated by the Peppas-Reinhart model if the amorphous regions of the polymer are highly swollen. This substitution yields... 

The distinct properties of liquid-crystalline polymer solutions arise mainly from extended conformations of the polymers. Thus it is reasonable to start theoretical considerations of liquid-crystalline polymers from those of straight rods. Long ago, Onsager [2] and Flory [3] worked out statistical thermodynamic theories for rodlike polymer solutions, which aimed at explaining the isotropic-liquid crystal phase behavior of liquid-crystalline polymer solutions. Dynamical properties of these systems have often been discussed by using the tube model theory for rodlike polymer solutions due originally to Doi and Edwards [4], This theory, the counterpart of Doi and Edward s tube model theory for flexible polymers, can intuitively explain the dynamic difference between rodlike and flexible polymers in concentrated systems [4]. 

Thermotropic liquid crystalline polymers, like polyesters containing mesogenic units on the main chain, may not be described by the wormlike chain model (cf. Sect. 1.2). The present article does not consider this type of polymers. 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

The dynamic behavior of liquid-crystalline polymers in concentrated solution is strongly affected by the collision of polymer chains. We treat the interchain collision effect by modelling the stiff polymer chain by what we refer to as the fuzzy cylinder [19]. This model allows the translational and rotational (self-)diffusion coefficients as well as the stress of the solution to be formulated without resort to the hypothetical tube model (Sect. 6). The results of formulation are compared with experimental data in Sects. 7-9. 

The model parameters q and ML can be estimated from experimental data for radius of gyration, intrinsic viscosity, sedimentation coefficient, diffusion coefficient and so on in dilute solutions. The typical methods are expounded in several recent articles and books [20-22], Here we refer only to the results of the application to representative liquid-crystalline polymers (See Table 1). 

We begin by formulating the free energy of liquid-crystalline polymer solutions using the wormlike hard spherocylinder model, a cylinder with hemispheres at both ends. This model allows the intermolecular excluded volume to be expressed more simply than a hard cylinder. It is characterized by the length of the cylinder part Lc( 3 L - d), the Kuhn segment number N, and the hard-core diameter d. We assume that the interaction potential between them is given by... 

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