Independent-alignment approximation

According to eqn (7.172) s and s are not equal to each other. This leads to a constitutive equation of a rather complicated form (see Section 7.9). If we disregard the difference between s and s and assume the following transformation rule 

To calculate S pis, t) we need to obtain the probability distribution function /( , s, t) that the tangent vector at the segment s is in the direction u at time t. The time evolution equation for/(n, s, t) is obtained n the same way as in Section 6.3. 

Suppose that in the time interval between t and t + At, the chain egment s moves to the position at which the chain segment f + A was ocated at time t, then u(s, t + At) is given by 

To assess the accuracy of the lA approximation, let us consider the stepwise deformation E imposed at f = 0. In this case the orientational distribution before the deformation is 

Since P(A ) becomes d(A ) for an infinitesimally small time-interval At, eqn (7.180) gives 

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Bulk material properties can be determined quite simply using this model. For example, consider the calculation of the second-moment tensor, Q = (u u ), which is required for the stress and refractive index tensors. Using the independent alignment approximation, we have... 

Retraction moves a strand from one part of the tube to another hence the strand s orientation is determined not by the orientation of the part of the tube it originally occupied, but by the orientation of the part of the tube into which it moves. To simplify the problem, however, Doi and Edwards invoked the independent alignment approximation, which assumes that after retraction each strand is oriented independently of the others, and the change in orientation produced by retraction is neglected. 

Figure 3-31 Damping function hiy) obtained by vertically shifting the time-dependent nonlinear moduli in Fig. 3-30a into superposition at long times. The data are from Fukuda et al. (1975). The solid and dashed lines are the prediction of the Doi-Edwards model, respectively, with and without the independent alignment approximation. (From Doi and Edwards 1978a, reproduced by permission of The Royal Society of Chemistry.)...

The Doi— Edwards equation predicts that the ratio 2 is —2/7 = —0.29 at low shear rates. This changes to 4 2/ l i = —1/7 = —0.14 when the independent alignment approximation is dropped (Osaki et al. 1981). With or without the independent alignment approximation, the ratio —is predicted to decrease towards zero as the shear rate increases. The prediction of fof entangled solutions contrasts with that predicted... 

Doi and Edwards were able to develop an equation for the stress relaxation modulus G(t) of monodisperse entangled linear chain liquids in the terminal region without resorting to the independent alignment approximation From G(t), expressions can be obtained for the plateau modulus, the steady-state viscosity and steady-state recoverable compliance. The following dependences on chain length are obtained ... 

Curtiss and Bird introduce reptation in a maimer which does not involve the tube concept, at least not in an explicit way. Their model leads to a constitutive equation in which the stress is the sum of two contributions. On contribution is exactly 1/3 the expression obtained by Doi and Edwards when those workers invoke the independent alignment approximation, i.e., that contribution is a special case of the BKZ relative strain equation. The othCT contribution depends on strain rate and is proportional to a link tension coefficent c (0 < e < 1) which diaracterizes the forces along the chain arising from the continued displacements of chain relative to surroundings. 

Part I summarizes the main ideas of de Gennes, Doi and Edwards about tube models and reptation in entangled polymer systems. Attention has been limited to properties for which predictions can be made without invoking the independent alignment approximation macromolecular diffusion, linear viscoelasticity in the plateau and terminal regions, stress relaxation following a step strain from rest of arbitrary magnitude, and equilibrium elasticity in networks. 

Q (E) represents the result of the independent alignment approximation. A comparison of Q(E) with Q (E) as well as with experimental results through the damping function h ) for step shear will be presented in the next section. 

Eq. (18.17) is simply the definition of the Doi-Edwards damping function with the independent-alignment approximation (Chapters 8 and 12). The damping functions of Doi and Edwards with and without the independent-alignment approximation both explain well the experimental results of a well-entangled nearly monodisperse polymer system (Fig. 9.2). Note that here the unit vector u° represents the orientation of a Fraenkel segment as opposed to the orientation associated with an entanglement strand or primitive step in the Doi-Edwards theory. 

Fig. 18.13 Comparison of the simulation values ( ) of —N2 t,X)/Ni(t,X) in the slowmode region obtained from the present study and the experimental valnes (o) in the terminal region of the entangled system studied by Osaki et al. with the curve (solid line) numerically calculated from the Doi—Edwards expression (Elq. (12.17)) with the independent-alignment approximation.

Fig. 7.16. h(Y) determined from the procedure explained in Fig. 7.15. Filled circles represent polystyrene of molecular weight 8.42 xlO and the unfilled circles of 4.48 x 10 . Directions of pips indicate concentrations which range from 0.02 g cm to 0.08 g cm. The solid curve rqnesents the theoretical value (eqn (7.131)), and the dashed curve the result of the independent alignment approximation (eqn 7.187). Reproduced from ref. 69.

Fig. 7.17. Quantity -N2/N1 is plotted against magnitude of shear y. Sample polystyrene solution in chlorinated biphenyl (Af = 6.7 x 10, p = 0.40gcm" ). The number of entanglements Z corresponds to about 14. The solid line is the theoretical value, - Qyyiy) - Gzz(y))/(C (y) Qyyiv))- The dashed line is the result of the independent alignment approximation. Reproduced from ref. 67.

Step-Up. Figure 49 shows the response for a two-step history in which the second step is approximately the same magnitude as the first step, ie, K2 = 2y i. The results are from Osaki s work (138) on polystyrene solutions and illustrate that both the shear stress response and the normal stress response are well represented with the DE independent alignment approximation (ie, the K-BKZ equations). This result is similar to what was found previously for the K-BKZ model (Fig. 34) (138). 

Figure 11.2 Damping function h y> obtained from step-shear experiments on an entangled 20% solution of polystyrene of molecular weight 1.8 10 in chlorinated diphenyl (symbols) (data of Fukuda etal. [19]) compared to the predictions of the Doi-Edwards theory with (solid line) without (dashed line) the independent alignment approximation. From Doi and Edwards [131.

Because s and s are not equal to each other, the general constitutive equation becomes very complicated and requires numerical solution. The independent alignment approximation allows us to ignore this difference. Then, with s = s the transformation equation 99 becomes... 

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