Fitting data viscosity

Fig. 8.15. Viscosity vs shear rate in concentrated solutions of narrow distribution polystyrene The solvent in n-butyl benzene, the concentration is 0.300 gm/ml and the temperature is 30° C. The symbols are O for M = 860000 and for M = 411000 at low shear rates (155) and at high shear rates (346). The solid line for M= 860000 is the master curve for monodisperse systems from Graessley (227). The solid line for M=411000 is the master curve from Ree-Eyring (341). Either master curve fits data for both molecular weights...

Figure 1 Time profile of temperature, concentration, and viscosity during freezing of 3% sucrose. Data are calculated assuming that ice crystallization begins at -15°C and that the solution composition follows the equilibrium freezing point depression curve. Viscosities are estimated from a fit of viscosity data over a wide range of composition and temperature to a VTF-type equation (i.e., see [4]). (Data from [7,8].)...

The Adam-Gibbs equation (4-10) can be tested directly by using the calorimetrically measured entropy difference AS to compute the temperature-dependence of the relaxation time, with B then being a fitting parameter. This has been done, for example, with the data for o-terphenyl shown in Fig. 4-11, and the predicted temperature-dependence of the viscosity is found to be in qualitative, but not quantitative, agreement with the measured viscosity (see, for example. Fig 4-12). The main reason for the failure in Fig. 4-12 is that the temperature Tj at which the entropy extrapolates to zero for o-terphenyl lies below the VFTH temperature Tq required to fit the viscosity data hence the predicted viscosity does not vary as rapidly with temperature as it should. 

The excluded-volume parameter for PS was derived from viscosimetric data in good solvent collected by Nystrom and Roots for different molecular masses [113], The internal viscosity characteristic time Tq was obtained from best-fit data [12] of mechanical-dynamical results due to Massa, Schrag, Ferry, and Osaki [102,103] (see Figure 10). In analogy with what was previously found by other authors, notably Kirkwood and Riseman on the same polymer (PS) with different solvents [19], fitting the experimental data seems to require two different values of R ff = C/6to/s, one to obtain v q) through Eqn. (3.1.9) and a second one to evaluate to = I kg T, the latter... 

Equation 20 was found to fit the viscosity data of a variety of suspensions. 

Figure 13. Effect of continuous phase composition on viscosity and emulsion viscosity. Experimental data Oemulsion, o organic phase fitted data —lnrj0 = exp -K (vol%)+K2] lnrj = lnri0 + K3.

The 17, 170, A, and n are all parameters that are used to fit data, taken here as 17 = 0.05, 170 = 0.492, A = 0.1, and n = 0.4. A plot of the viscosity vs shear rate is given in Figure 10.8. For small shear rates, the viscosity is essentially constant, as it is for a Newtonian fluid. For extremely large shear rates, the same is true. For moderate shear rates, though, the viscosity changes with shear rate. In pipe flow, or channel flow, the shear rate is zero at the centerline and reaches a maximum at the wall. Thus, the viscosity varies greatly from the centerline to the wall. This complication is easily handled in FEMLAB. 

Rucker and Bike (1995) examined the rheological properties of silica-filled PMMA. They showed the existence of a yield stress and a poor fit of viscosity data to existing filler models. 

Addition of further terms in higher powers of c and [rj] to eqn. (4.15), to represent higher order interactions, will obviously improve the fit of the data. Viscosities of hyaluronan (Section 4.6.10.1.1) and xanthan (Section 4.6.10.3.1) obey the equation ... 

Finally, it is worth noting that the values of Tq or needed to fit the viscosity data are close to the temperature at which the Kauzmann temperature, Tkau is estimated from extrapolations of other properties such as those shown in Fig. 9.8, lending credence to the model. This model also provides a natural way out of the Kauzmann paradox, since not only do the relaxation times go to infinity as T approaches 7)., but also the configuration entropy vanishes since in glass at T = T only one configuration is possible. 

Table 7.9. Parameters used for curve fitting of viscosity vs. concentration data to Equations 7.9, 7.123 and 7.125 [Utracki, 1991]...

The Reynolds and Schmidt numbers can also be written in terms of the kinematic viscosity, v, which is defined as p/p. Empirical expressions for p and p were established by fitting data published in various references [26,27]. 

The first model has been explored for xanthan by Whitcomb and Macosko (1), who show that an undeformable ellipsoid of length 1.5 jm and midpoint diameter 19 A can fit data they observed for the variation of intrinsic viscosity with shear stress in distilled water. Rinaudo and Milas 2) have also adopted this model to fit their intrinsic viscosity and sedimentation data. 

While the VFT equation provides a good fit to viscosity data over a wide temperature range, it should be used with caution for temperatures at the lower end of the transformation region, where AH becomes constant. The VFT equation always overestimates the viscosity in this temperature regime. 

Originally, the various constants had no particular physical meaning, though one does note that the viscosity becomes infinite when T = Tj,. The VFT equation runs into some difficulties in fitting data near the glass transition temperature but the success of the empirical approach has often led subsequent theoreticians to interpret their models in terms of the VFT equation. 

Table 1. Coefficients C, a and K obtained by Fitting Experimental Viscosity Data of Aitgelt and Harle ...

M-Base Engineering + Software GmbH in Aachen, Germany (www.m-base.de) makes available a program, MCBase, that allows the user to search, compare, and perform queries of all CAMPUS data that the user has loaded into the databank. MCBase has several features not available in CAMPUS such as WEE and power-law curve fitting of viscosity data, substitute grade search, exclude function, and enhanced text search options. 

Figure 12.15 Critical temperature To determined by fitting the viscosity data in Figure 12.14 from [8, 15] with the Vogel-Tammann-Fulcher equation. Glass transition temperatures taken from [42].

Fig. 5 Experimental data and fitted shear viscosity function of a carbon black filled polybutadiene compound at 100 °C, as obtained in the author s laboratory...

The parameter is a time constant or characteristic time, and tJo and tJqo represent the viscosity, rj, when(Ac-y) 1 and (Ac-y) 1, respectively. For mid-range values of (Acy), the equation generates a power-law region with a log-log slope of n— 1. This model has been quite successful in fitting data for polymer melts and solutions over at least three or four decades of shear rate. 

Equations (4.8)-(4.10) have been solved in simple steady state shear flow using Mathematica software (Leonov and Chen 2010). The stress components are expressed as function of shear rate y with the values of constitutive parameters 00, a,p, r, r2,Xe, and t o. Here Oq and t]o represent relaxation time and zero shear viscosity respectively. The other parameters XgandXv represent the tumbling for elasticity and viscosity. Rest of the characteristic parameters a,p,ri,r2 represent anisotropic properties of liquid crystal polymers. Among the eight parameters, only relaxation time and zero shear viscosity are determined from experimental data. The other six parameters can be obtained from cinve fitting data using the Mathematica software. 

Wu [49] proposed estimating G° as the storage modulus at the frequency where tan 5= minimum, but often there is no minimum in the data. Using this method, Fuchs et al. [38] reported a modest effect of the tacticity of PMMA on the plateau modulus. They also found that Eq. 5.4 fitted their viscosity data with a = 3.4 but that the constant K depended on tacticity. Plazek et al. [50] studied the effect of tacticity on the creep behavior of PMMA and noted that absorbed water is always a problem in making measurements on polar polymers. Wu [51] later proposed an empirical equation between the ratio (G Gq) of the plateau modulus to the crossover modulus, G, (where G = G") and the polydispersity index. 

Montfort et al. [138] used Cole-Cole complex viscosity plots to analyze data for binary polystyrene blends. They superposed data at several temperatures by plotting rff versus rf and obtained curves showing contributions from the two components of the blend. Figure 5.30 shows plots of data for three blends. Blend (a) contains 5% polystyrene with M = 4 10 in a matrix of a polystyrene with M= 3.5 10. Labaig etal. [141] used Eq. 5.65 to fit data for branched polyethylenes. 

We note the appearance in these models of a material constant, A, with units of time. As mentioned above, such a constant is an essential feature of a rational model for the shear rate dependency of viscosity. Elberli and Shaw [68] compared a number of empirical viscosity equations. They observed that time constant values obtained by fitting data to two-parameter viscosity models were less sensitive to experimental error than those based on more complex models. The data at low shear rates and in the neighborhood of the reciprocal of the time constant are most critical in obtaining meaningful values of the parameters, while the high shear rate data are not as important. 

Several methods of estimating LCB levels using linear viscoelastic data are described in Section 5.12. A technique based on the shape of the viscosity curve has been proposed for single-site polyethylenes with low levels of LCB. Lai etal [80] found that for strictly linear polyethylenes prepared using single-site catalysts, the Cross equation (Eq. 10.55) gives a reasonably good fit to viscosity data. They further showed that the characteristic time A of the cross equation is proportional to the zero shear viscosity for these materials ... 

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