Ward aggregate model

However, at present it has not proved possible to construct a quantitative model for such complex structures, and generalised models which do not take account of fine structure at the molecular level are needed as working approximations. Of such models the Ward aggregate model and the Takayanagi two phase model have been most widely used. 

Fig. 14. Nylon 6-6. Room temperature measurements of the variation with draw ratio of e.xtensional, transverse and torsion moduli of monofilaments and comparison with Ward aggregate model assuming continuity of strain. (Adaptedfrom Hadley et al. )...

Kashiwagi et al. ° have shown that Wright s data on Perspex are consistent with the predictions of a Ward aggregate model, using orientation functions obtained by broad line nuclear magnetic resonance. 

Recent Brillouin spectroscopy measurements include those of Kumar, Renisch and Grimsditch [30] on uniaxially and biaxially stretched polypropylene films. Using the Ward aggregate model with a modified definition of molecular alignment enabled elastic anisotropy and refractive index data to be related quantitatively to molecular orientation. 

WardTordai, which implements the solution of the Ward-Tordai equation for Langmuir, Frumkin, Two-State Reorientation Quasiequilibrium, Two-State Reorientation Kinetics and Aggregation models. Using this module, the user can compare (both visually and numerically) his experimental data with the kinetic curves calculated from any of these models. 

As mentioned above, a complete set of equation involves and equation of the type of Eq. (7.35), otherwise a numerical solution of the Ward-Tordai equation is not available. The software package includes all adsorption models described in Chapter 3, i.e. the classical Langmuir and Frumkin model as well as the reorientation and 2D-aggregation models. 

Thermotropic polyesters are melt-spun from the nematic phase and orient easily in an elongational flow field (moderate drawdowns/forces are sufficient). In the fiber case, highly oriented fibers form easily with an initial modulus close to theory—typical values range from about 70 to 150 GPa. Ward [46] has shown that the tensile modulus may be described by an aggregate model, i.e., the modulus is a function of the inherent chain modulus, the molecular chain orientation, and the shear modulus (which described the stress transfer between chains). The tensile strength of LCP fibers follows the prediction of the lag-shear model [47]. Both the aggregate model and the lag shear model treat the LCP as though it... 

These earlier papers did not take into account the fact that the oriented polymers were partially oriented. It is therefore now necessary to extend the previous discussion of the aggregate model to the situation where molecular motion occurs. This has been done independently by Olf and Peterlin, McBrierty and Douglass, and Folkes and Ward. McBrierty and Douglass paper gives a formal theoretical treatment, as part of a general discussion of the influence of molecular motions on Tj, Tz and the other papers are particularly concerned with an analysis of the effects of molecular motion on the second moment anisotropy but nevertheless provide explicit results for special types of motion. The results are then applied to oriented mats of polyethylene single crystals and to... 

During the course of these and related studies, notably those concerned with the temperature dependence of the mechanical anisotropy and the identification of relaxation processes in structural terms, it became apparent that the aggregate model was successful in low density polyethylene because it described effectively the influence of the very anisotropic x-relaxation process on the mechanical behaviour. Stachurski and Ward were even able to extend the aggregate model to deal with the anisotropy of dynamic loss factor. (See Chapter 9 for further discussion.) It was, however, more in the spirit of the original conception of the aggregate model that it would deal with mechanical anisotropy in glassy polymers, where morphology was of secondary importance. 

A major part of this study was to examine the applicability of the Ward rotating element aggregate model to low density polyethylene. As discussed in Chapter 8 satisfactory agreement was obtained when account was taken of the non-affine processes which occur in the early stages of drawing." In addition, however, these experiments formed the basis of... 

The pseudo-affine deformation scheme gives a reasonable approximation to experimental data for low-density polyethylene (Figure 7.15(b)), nylon, polyethylene terephthalate and polypropylene, despite ignoring the distinction between crystalline and disordered regions. It thus provides a basis for extending Ward s aggregate model to predict the compliance constants s y and stiffiiess constants c y of the partially oriented polymer in terms of the constants sy and Cy for the anisotropic elastic unit (in practice, those of the most highly oriented sample obtainable). 

The aggregate model also has been used with success to describe the mechanical anisoti opy of several liquid crystalline polymers. Ward and co-workers [56] examined the dynamic mechanical behaviour of several thermotropic polyesters in tension and shear over a wide temperature range, and used the single-phase aggregate model to relate quantitatively the fall in tensile modulus with temperature to the corresponding fall in shear modulus. 

In a second version of the aggregate model Ward and co-workers assumed, on the basis of observation of the X-ray diffraction pattern, that the aggregate unit averages the deformation over a length of 8 10 monomer units. The chain modulus in this case can be determined experimentally by measuring the change in the X-ray dififaction pattern under stress, and a temperature-dependent was observed. By rearrangement of Equation (7.11) a further plot of... 

The fibres in short fibre composites are not usually fiilly aligned. An early attempt to deal with such systems was made by Brody and Ward [12], who applied the aggregate model of Section 7.5 above assuming that the elastic constants of a representative unit of structure could be predicted by Equations (8.7a)-(8.7e). It was found that this simple theory fitted the results for composites reinforced with short fibres of carbon or glass reasonably well, with the moduli lying close to the lower Reuss bounds. Recent studies by Ward and co-workers [13] have shown that this approach is viable provided that the elastic constants of the representative unit are calculated more exactly. 

Quantitative studies of the elastic anisotropy of short fibre composites have been undertaken by combining quantitative measurements of fibre orientation (using optical image analysis methods) with aggregate models. Ward and co-workers have been very active in this area [12, 14], as have Tucker and co-workers [15]. 

Demonstrated tensile modulus of injection-molded bars increases with increasing HBA content (based on three compositions), attributable to orientation differences. Tracked orientation through the cross section using x-ray analysis, elucidating up to nine-layer structure. Modeled mechanical properties based on orientation and layer thickness using Ward s aggregate model. 

The relationship among tensile, chain, and shear modulus with orientation may be explained by an aggregate model. Ward s [64] model considers the polymer to consist of an aggregate of highly oriented units that differ in their... 

The technique of Brillouin spectroscopy (Section 6.3.3) has been applied to determine the elastic constants of oriented polymer fibres. Early studies of this nature were undertaken by Kruger et al. [46,47] on oriented polycarbonate films, also determining the third-order constants, which define the elastic non-linear behaviour. Wang, Liu and Li [48,49] have described measurements on oriented polyvinylidene fluoride and polychlorotrifluoroethy-lene films. In the latter case the results were interpreted using an aggregate model differing in detail from that of Ward discussed in Section 8.6.2. 

For amorphous polymers, Ward et al. [96] and Kausch [88] and later Rawson and Rider [95] are in agreement that the mechanical anisotropy can be discussed very satisfactorily by the aggregate model. Moreover, the development of anisotropy with draw ratio can often be described by the pseudo-affine deformation scheme [94]. 

Figure 830 A plot of I/E3 against 1/C showing the validity of the aggregate model analysis for a range of R006-60 extrudates previously pressure annealed at 238°C for 15 min at 450 MPa pressure (0)10 1 (A) 7 1 (o) 5 1. (Reproduced from Powell, A.K., Craggs, C., Ward, I.M. (1990) The structure and properties of oriented chain-extended polyethylene ). Mater. Sci., 25, 3990. Copyright (1990) Springer Science and Business Media.)...

Davies, G.R. and Ward, I.M. (1988) Structure and properties of oriented thermotropic liquid crystal polymers in the solid state, in High Modulus Polymers (eds A.E. Zachariades and R.S. Porter), Marcel Dekker, New York, Chap. 2 Troughton, M.J., Davies, G.R. and Ward, I.M. (1989) Dynamic mechanical properties of random copolyesters of 4-hydroxybenzoic acid and 2-hydroxy 6-naphthoic acid. Polymer, 30, 58 Green, D.I., Unwin, A.P., Davies, G.R. and Ward, I.M. (1990) An aggregate model for random liquid crystalline copolyesters. Polymer, 31, 579. 

The theoretical estimation of the elastic constants of a partially oriented short fibre reinforced composite is most simply undertaken by considering that it consists of an aggregate of units of structure, each of which consists of a perfectly aligned short fibre composite. This approach is analogous to the aggregate model for oriented polymers described in Section 8.6.1. It was first proposed by Brody and Ward [15] and subsequently by Advani and Tucker [16], all these workers setting out the tensor equations for the elastic constants, as described in Section 8.6.3. 

To derive an adsorption kinetics model the Ward and Tordai equation (4.1) is again the main relationship between the dynamic adsorption r(t) and the subsurface concentration c(0,t). As it was described in detail in paragraph 4.1.2, an adsorption isotherm as additional function r(c) is needed for a kinetic model. The isotherm equations (2.110) - (2.112) given in Chapter 2 represent such type of function, which accounts for a 2D-aggregation in the adsorption layer [48]. The set of equation is too complex to find an analytical solution. Only for the short time range and for low adsorption layer coverage, the following approximation is valid [65]... 

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