The Takayanagi Models

In the above, cpi = 1 and the portion of the segmental environment i with the modulus Ei contributes to the modulus E with weight 2,.  

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Lipatov et al. [116,124-127] who simulated the polymeric composite behavior with a view to estimate the effect of the interphase characteristics on composite properties preferred to break the problem up into two parts. First they considered a polymer-polymer composition. The viscoelastic properties of different polymers are different. One of the polymers was represented by a cube with side a, the second polymer (the binder) coated the cube as a homogeneous film of thickness d. The concentration of d-thick layers is proportional to the specific surface area of cubes with side a, that is, the thickness d remains constant while the length of the side may vary. The calculation is based on the Takayanagi model [128]. From geometric considerations the parameters of the Takayanagi model are related with the cube side and film thickness by the formulas ... 

The Takayanagi model parameters are related with filler concentration and interphase thickness by the following simple relationships ... 

Where a melt-crystallized polymer has been processed by drawing, rolling or other means to produce an aligned structure in which lamellae as well as polymer chains have discernible order, a pseudocrystalline unit cell is present. Provided that this unit cell contains elements of the crystals as well as the boundaries between crystals and that it is entirely typical of the material as a whole then it could be considered as a RVE within the meaning defined above. The lamella crystal itself sometimes considered as embedded in an amorphous matrix would not seem to be an acceptable RVE for reasons similar to those advanced against the Takayanagi model, namely that its modulus is dependent upon the surface tractions. The boundaries between lamella crystals in the matrix must be included in an acceptable RVE. 

In most simple applications of models all that is really being achieved is curve fitting. This applies to the Takayanagi models (which are one dimensional and assume uniformity of stress and strain within each element) as well as to simple fibre models such as the shear lag. 

Analysis of Stress—Optical Data. The slight, if indeed real, improvement of the isotropic model over the Takayanagi model would be of little consequence were it not for a more pronounced difference between the two models in their ability to describe the stress-optical data. When the parameters obtained from the dynamic data (Table IV) are substituted into Equations 8 and 9, Equation 8 produces results which are uniformly too low. Equation 9 also underestimates the magnitude of Ka but only by an average 7% (Figure 14). For most blends the discrepancy is less than 5%, and all calculated values show the characteristic elevation of the birefringence attributed to the multiphase structure. 

In this work the mechanical model proposed by Takayanagi (9) will be used. Two variants were proposed, both assuming that the two phases are connected partly in parallel and partly in series (Figures 7a and 7b). Kaplan and Tschoegl (10) have shown that the two variants of the Takayanagi model are equivalent. The series model (Figure 7b) will be used for our calculations. The modulus is given by ... 

Fig. 11.10 The series-parallel and parallel-series versions, (b) and (c), respectively, for the Takayanagi model (a), which allows for noncrystalline (shaded) material being partly in series and partly in parallel with crystalline material.

The Takayanagi model was developed to account for the viscoelastic relaxation behaviour of two phase polymers, as recorded by dynamic mechanical testing. " It was then extended to treat both isotropic and oriented semi-crystalline polymers. The model does not deal with the development of mechanical anisotropy on drawing, but attempts to account for the viscoelastic behaviour of either an isotropic or a highly oriented polymer in terms of the response of components representing the crystalline and amorphous phases. Hopefully, comparisons between the predictions of the model and experimental results may throw light on the molecular processes occurring. 

Takayanagi and co-workers transformed the spring and dashpot relaxation models (Section 1.5.6) to plastic and rubber elements in an effort to better explain the mechanical behavior of poly blends (Takayanagi et ai, 1963). Some simple combinations of the Takayanagi models are shown in Figure 2.11. The plastic phase is denoted by P and the rubber phase by R, while the quantities X and (p are functions of the volume fractions of parallel and series elements, respectively. 

In a manner similar to the application of springs and dashpots to the theory of linear viscoelasticity, we note that for units in parallel the total stress is (T = (Ti -f (T2 + (T3 -f , and that for units in series the total strain is 6 = -f 82 -f 3 -f . Finally, application of Hooke s law, cr = sE, allows the complex modulus E of the Takayanagi models in Figures 2.11a-d to be represented by the following equations, respectively ... 

One of the most important conclusions associated with the Takayanagi model is that the distribution function of the free volume fraction Fj f) is evaluated by the following equation based on the specific volume-temperature curve ... 

While the Takayanagi models have proved useful because of their simplicity, the effects of changes in mechanical behavior with composition and phase structure may also be profitably explored using several analytical relations, which include equations derived by Kerner (1956b), Hashin and Shtrikman (1963), and Halpin and Tsai (Ashton et al, 1969, Chapter 5). The most widely applied of these is the Kerner equation, which presents the... 

Figure 8.3 Schematic diagram indicating a version of the Takayanagi model.

Polymeric materials have relatively large thermal expansion. However, by incorporating fillers of low a in typical plastics, it is possible to produce a composite having a value of a only one-fifth of the unfilled plastics. Recently the thermal expansivity of a number of in situ composites of polymer liquid crystals and engineering plastics has been studied [14,16, 98, 99]. Choy et al [99] have attempted to correlate the thermal expansivity of a blend with those of its constituents using the Schapery equation for continuous fiber reinforced composites [100] as the PLC fibrils in blends studied are essentially continuous at the draw ratio of 2 = 15. Other authors [14,99] observed that the Takayanagi model [101] explains the thermal expansion. 

Gg in different ways. By fixing the weighting ratio according to Eq. 7, we also avoid the need to retain it as a fitting parameter. By contrast, the Takayanagi model (28) demands the unique assignment of this parameter to each microstructure, but such numerical values may have little relevance to the macroscopic bulk sample. 

More quantitative information depends on the use of models. The Takayanagi models were already mentioned in connection with Figure 6.29. More analytical models have been evolved by Kerner/ Hashin and Shtrikman/ and Davies.Briefly, the first two theories assume spherical particles dispersed in an isotropic matrix. From the modulus of each phase, the composite modulus is calculated. An upper or lower bound modulus is arrived at by assuming the higher or lower modulus phase to be the matrix, respectively. The theory is reviewed elsewhere. 

The second method to determine interpenetration involves a mechanical property such as tensile modulus. The load is shared between both polymers, approaching the theoretical upper limit of the Takayanagi model for co-continuous versus parallel structures. 

Figure 8.8 Schematic representations of change in modulus E with temperature on the Takayanagi model for (a) the and (h) the L situations corresponding to Eo and 90, respectively. Calculations assume amorphous relaxation at temperature r(aa) and crystalline relaxation at temperature r( c) and (c) shows combined results. C, crystalline phase A, amorphous phase. (Reproduced with permission from Takayanagi, Imada and Kajiyama, J. Polym. Sci. C, 15, 263 (1966).

The Takayanagi models were remarkably successful in providing a simple interpretation of the d3mamic mechanical behaviour of crystalline polymers and polymer blends. The theoretical basis is contained in Equations (8.1) to (8.6) of Section 8.2, and is deficient in two respects. First, only tensile deformations are considered and shear deformations are ignored. Secondly, as emphasized in Chapter 7, Voigt and Reuss schemes (i.e. parallel and series) only provide bounds to the true behaviour. 

In contrast with the Takayanagi model, which considers only extensional strains, a major deformation process involves shear in the amorphous regions. Rigid lamellae move relative to each other by a shear process in a deformable matrix. The process is activated by the resolved shear stress a sin y cosy on the lamellar surfaces, where y is the angle between the applied tensile stress o and the lamellar plane normals, which reaches a maximum value for y = 45° (see Chapter 11 for discussion of resolved shear stress in plastic deformation processes). 

Other applications of the Takayanagi model to oriented polymers have included linear polyethylene that was cross-linked and then crystallized by slow cooling from the melt under a high tensile strain [22], and sheets of nylon with orthorhombic elastic symmetry [23]. A fuller discussion is given in the more advanced text by Ward [24]. 

The Takayanagi Models Most polymer blends, blocks, grafts, and interpenetrating polymer networks are phase-separated. Frequently one phase is elastomeric, and the other is plastic. The mechanical behavior of such a system can be represented by the Takayanagi models (6). Instead of the arrays of springs and dashpots, arrays of rubbery (R) and plastic (P) phases are indicated (see Figure 10.6) (7). The quantities A and

indicated multiplications indicate volume fractions of the materials. 

As with springs and dashpots, the Takayanagi models may also be expressed analytically. For parallel model Figure 10.6a, the horizontal bars connecting the two elements must remain parallel and horizontal, yielding an isostrain condition (e = e ). Then... 

Figure 10.6 The Takayanagi models tor two-phase systems (a) an isostrain model (b) isostress model (c, d) combinations. The area of each diagram is proportional to a volume traction ot the phase.

In multicomponent polymeric systems such as polymer blends or blocks, each phase stress relaxes independently (39-41). Thus each phase will show a glass-rubber transition relaxation. While each phase follows the simple superposition rules illustrated above, combining them in a single equation must take into account the continuity of each phase in space. Attempts to do so have been made using the Takayanagi models (41), but the results are not simple. 

Derive equations to express Young s modulus as a function of rubber and plastic composition using the Takayanagi models (c) and (d) in Figure 10.6. 

Qualitatively, the modulus of polymer blends and composites is expected to be intermediate between the modulus of the materials involved. Quantitatively, the picture is more complex, depending on phase morphology and continuity. The Takayanagi models (Section 10.1.2.3) explore the basic methods of calculating not only the modulus, but many other viscoelastic quantities. While the original models were developed with glassy and rubbery polymers in mind, they are quite general and useful for composite systems as well. 

First, assume the longitudinal direction. In response to a unidirectional stress in the longitudinal direction, both the fibers and the matrix are continuous. The Takayanagi model shown in Figure 10.6a will be assumed, with A equal to 0.62. TTie basic relation is given by equation (10.6). Then... 

Thus, to a first approximation, the modulus is given by the modulus of the fiber times its volume fraction. However, in the transverse direction the fibers are discontinuous, while the matrix retains its continuity. The Takayanagi model Figure 10.6b and equation (10.8) hold. Then... 

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