Steady-state compliance calculated

Compliance with U.S. EPA s design performance standards can be demonstrated through one-dimensional, steady-state flow calculations, instead of field tests. For detection sensitivity, the calculation of flow rates should assume uniform top liner leakage. For detection time, factors such as drain spacing, drainage media, bottom slope, and top and bottom liners should all be considered, and the worst-case leakage scenario calculated. 

In order to calculate the concentration dependence of reduced steady-state compliance JeR, (4.5) is rewritten with the aid of eqs. (4.4) and(4.4a). One obtains ... 

Using the equations of linear viscoelasticity (Eqs. (4.30) and (4.63)), the zero-shear viscosity, 770, and the steady-state compliance, J°, can be numerically calculated from Eq. (9.19) by a computer program. If the molecular... 

As the steady-state compliance J° is sensitive to the molecular-weight distribution, the experimental results of the nearly monodisperse samples are higher than the theoretical values for ideal monodispersity. Shown in Fig. 10.11 is the comparison of the experimental data of (the experimental results shown in Fig. 10.11 are consistent with those shown in Fig. 4.12) with four theoretical curves curve 1 is calculated from Eq. (9.25) curves 2 and 3 are numerically calculated from the substitution of Eq. (9.19) into... 

Fig. 10.11 Comparison of the steady-state compliance data, x pRT, of nearly monodisperse polystyrene samples ( and from Ref. 18 A from Ref. 33) and those calculated from Eq. (9.25) (solid line 1), from the Doi-Edwards theory (the dashed line), from the Rouse theory (the dotted line), and calculated numerically from substituting Eq. (9.19) into Eq. (4.63) with K jK = 1 (line 2), and K jK = 3.3 (line 3).

Fig. 15.1 Comparison of the s (AT)/sq values (with Sq = 1,500) of polystyrene samples A(o), B(0) and C(D obtained by analyzing the J(t) line shapes A by matching the calculated and experimental steady-state compliance values) with the diffusion enhancement factors /r(AT) of OTP ( isothermal desorption A NMR) as a function of AT = T — Tg. The solid line is calculated from the modified VTF equation (Elq. (14.13)) which best fits the s (AT)/sq results of the three polystyrene samples collectively. The dashed line represents the curve calculated from the modified VTF equation best fitting the fj, AT) data of OTP.

The viscosity and the steady-state compliance are calculated from G(t),... 

The viscosity and the steady-state compliance are calculated from eqns (7.29) and (7.30). Since the contribution to the integral from the region t< T is very small, t/q and Jf are given by... 

The results of eqns (7.263) and (7.264) are in qualitative agreement with experimental results the viscosity increases steeply because of the exponential factor, and the steady state compliance is pri rtional to M. However, the quantitative agreement is not satisfactory. The observed viscosity is smaller than the calculated one, and the best fit with experiments is obtained only when the numerical coefficient in the exponential of eqn (7.263) is replaced by a smaller number (about 1/2) instead of lS/8. This suggests that relaxation mechanisms other than the contour length fluctuations are important for star polymers. Indeed it has been pointed out that in the case of star polymers the constraint release, and perhaps other tube reorganization processes, are as important as the contour length fluctuation. 

Note that the instantaneous modulus G, becomes small near the transition point due to the pretransitional effect. The viscosity i/o and the steady-state compliance are calculated as... 

Fig. 13. Steady state compliance oflinear and branched polyisoprenes in tetradecane (0.33 g/ml) at 25°. Symbols as in Fig. 8, Ref. Eteshed lines are values calculated from Rouseflam Theory...

The steady-state compliance J° can also be calculated from H by the following relation ... 

The steady-flow viscosity qo and the steady-state compliance can easily be determined from creep data in the region of linear viscoelastic behavior as shown-in Fig. 1-12, from equation 40 of Chapter 1, provided steady-state flow has been attained. However, it is easy to be misled into believing prematurely that the linear portion of the creep curve has been reached in general, it cannot be expected to become linear until the flow term t/vo is at least as large as the intercept / . It is always desirable to perform the recovery experiment shown in Fig. 1-12 to conflrm the calculation. 

For the restricted case of low molecular weights and no coupling entanglements, the viscoelastic properties of star-branched undiluted polymers can be described by a special case of the Zimm-Kilb theory o in which there is no hydrodynamic interaction. Calculations were made by Ham i by use of a method which is somewhat different from that of Rouse but yields the same results for unbranched molecules. Stars with arms of unequal length were included. For such a branched molecule, the terminal relaxation time ti, the viscosity r/o, and the steady-state compliance are always smaller than for an unbranched molecule of the same molecular weight the more branches and the more nearly equal their lengths, the... 

Nobile and Cocchini [33] used the double reptation model to calculate the relaxation modulus, the zero-shear viscosity and the steady-state compliance for a given MWD. They compared three forms of the relaxation function for monodisperse systems the step function, the single integral, and the BSW. In the BSW model, they set the parameter j8 equal to 0.5, which gives /s° G equal to 1.8. The molecular weight data were fitted to a Gex function to facilitate the calculations (see Section 2.2.4 for a description of distribution functions). For the step function form of the relaxation function is given by Eq. 8.37. 

A good diagnostic for creep and stress relaxation tests is to plot them on the same scales as a function of either compliance (J) or modulus (G), respectively. If the curves superimpose, then all the data collected is in the linear region. As the sample is overtaxed, the curves will no longer superimpose and some flow is said to have occurred. These data can still be useful as a part of equilibrium flow. The viscosity data from the steady-state part of the response are calculated and used to build the complete flow curve (see equilibrium flow test in unit hi.2). 

With these equations the steady state viscority and recoverable compliance for sufh-dently long diains (so that relaxaticm jmK es for t te make negligible contributions to the integrals) can be calculated ... 

Basically, a constant stress cr is applied on the system and the compliance J(Pa ) is plotted as a function of time (see Chapter 20). These experiments are repeated several times, increasing the stress in small increments from the smallest possible value that can be applied by the instrament). A set of creep curves is produced at various applied stresses, and from the slope of the linear portion of the creep curve (when the system has reached steady state) the viscosity at each applied stress, //, can be calculated. A plot of versus cr allows the limiting (or zero shear) viscosity /(o) and the critical stress cr (which may be identified with the true yield stress of the system) to be obtained (see also Chapter 4). The values of //(o) and <7 may be used to assess the flocculation of the dispersion on storage. 

The steady-state recoverable compliance, which is a measure of the elastic strain, can be calculated using... 

The renormalized horizontal and vertical factors used to build up the master curve follows the longest relaxation time, Xz, and the steady-state creep compliance Je, respectively (d, 19, 4S). Therefore, the scaling relationships expressed by eqs.4 and 5 can be used to calculate the static exponents s and t. Figure 10 shows the log-log plot of ah and ay versus log e, in case of sample G1 (Table III), for which t=1.95 and s=0.69. The values of s, t and A are listed in Table III not for all the samples, but only when enough experimental data were available in the sol-gel transition so as to construct accurately the master curves. 

Fig. 2.11. Creep-compliance measurements at several temperatures (indicated in the figure) on a polystyrene sample with molecular weight 46 900, reduced to 100 °C with shift factors calculated from the steady-state viscosity. Subscript p denotes multiplication by Tp/ TqPq). From Plazek [53], by permission.

For a Newtonian low molar mass liquid, knowledge of the viscosity is fully sufficient for the calculation of flow patterns. Is this also true for polymeric liquids The answer is no under all possible circumstances. Simple situations are encountered for example in dynamical tests within the limit of low frequencies or for slow steady state shears and even in these cases, one has to include one more material parameter in the description. This is the recoverable shear compliance , usually denoted and it specifies the amount of recoil observed in a creep recovery experiment subsequent to the unloading. Jg relates to the elastic and anelastic parts in the deformation and has to be accounted for in all calculations. Experiments show that, at first, for M < Me, Jg increases linearly with the molecular weight and then reaches a constant value which essentially agrees with the plateau value of the shear compliance. 

Figure 6.10 Creep compliance (/) as a function time (t). Calculation of zero-shear-rate viscosity and steady-state recoverable shear compliance.

Masao Doi and Sam F. Edwards (1986) developed a theory on the basis of de Genne s reptation concept relating the mechanical properties of the concentrated polymer liquids and molar mass. They assumed that reptation was also the predominant mechanism for motion of entangled polymer chains in the absence of a permanent network. Using rubber elasticity theory, Doi and Edwards calculated the stress carried by individual chains in an ensemble of monodisperse entangled linear polymer chains after the application of a step strain. The subsequent relaxation of stress was then calculated under the assumption that reptation was the only mechanism for stress release. This led to an equation for the shear relaxation modulus, G t), in the terminal region. From G(t), the following expressions for the plateau modulus, the zero-shear-rate viscosity and the steady-state recoverable compliance are obtained ... 

M being the molecular weights of components A and B, respectively. The steady-state shear compliance J° can be calculated by... 

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