Simha equations

Here we show that it is possible to obtain information on hole morphology in polymers, as well as changes in morphology due to external stimuli, by coupling PALS and dilatometry techniques and comparing the experimental results with the theoretical evaluation of the flee volume via the S-S equation [Simha and Somcynsky,... 

Simha equation), where a/b is the length/diameter ratio of these cigarshaped particles. Doty et al.t measure the intrinsic viscosity of poly(7-benzyl glutamate) in a chloroform-formamide solution and obtained (approximately) the following results ... 

Use the Simha equation and these data to criticize or defend the following proposition These polymer molecules behave like rods whose diameter is 16 A and whose length is 1.5 A per repeat unit. The molecule apparently exists in fully extended form in this solvent rather than as random coils. 

In the last section we noted that Simha and others have derived theoretical expressions for q pl(p for rigid ellipsoids of revolution. Solving the equation of motion for this case is even more involved than for spherical particles, so we simply present the final result. Several comments are necessary to appreciate these results ... 

Using 1.32 g cm as the density of the polymer, estimate the axial ratio for these molecules, using Simha s equation ... 

The length L and diameter d of a rigid rod are related to the major axis a and minor axis b of the equivalent ellipsoid of revolution by a=L and b=(3/2)ll2d. It should be noted that, for actual molecules in solution, the quantity d can be defined only vaguely. With these relations, Simha s equation (64) for the intrinsic viscosity of an elongated ellipsoid of revolution can be rewritten for a thin rod (L/d> 1)... 

Fig. 22 Theoretical relations between Af2/[ j] and In L/d for rigid rods ellipsoid Simha equation (D-l) for equivalent ellipsoid of revolution Y-F Yamakawa-Fujii theory for straight cylinders. Dashed lines indicate asymptotes to respective solid curves...

We noted above that either solvation or ellipticity could cause the intrinsic viscosity to exceed the Einstein value. Simha and others have derived extensions of the Einstein equation for the case of ellipsoids of revolution. As we saw in Section 1.5a, such particles are characterized by their axial ratio. If the particles are too large, they will adopt a preferred orientation in the flowing liquid. However, if they are small enough to be swept through all orientations by Brownian motion, then they will increase [17] more than a spherical particle of the same mass would. Again, this is very reminiscent of the situation shown in Figure 2.4. 

Figure 4.12a shows plots of the intrinsic viscosity —in volume fraction units —as a function of axial ratio according to the Simha equation. Figure 4.12b shows some experimental results obtained for tobacco mosaic virus particles. These particles —an electron micrograph of which is shown in Figure 1.12a—can be approximated as prolate ellipsoids. Intrinsic viscosities are given by the slopes of Figure 4.12b, and the parameters on the curves are axial ratios determined by the Simha equation. Thus we see that particle asymmetry can also be quantified from intrinsic viscosity measurements for unsolvated particles. 

However, Eq. (8) leads to the paradox that overlapping does not begin until a critical volume fraction of 2.5. In order to overcome this, somewhat modified equations have been put forward by Simha (10)... 

According to the Simha-Boyer theory, AaTg=Kl. Equating Eqs. (23) and (29) through Aa, we can obtain... 

It is now generally accepted that the viscous flow of polymeric liquids is connected with chain segment rotation, i.e. with configurational entropy. From this point of view Miller concluded that the Simha-Boyer equation was not correct since the relative free-volume in SB theory equals zero at 0 K, not at T = T0. If the latter... 

Equations (65) and (66) of Simha and Weil provide the most general expression of the SB rule and free-volume ratios as a function of the thermal expansivities at and Og. The authors have shown that, since a < 8 10 4 deg-1 and Te < 5 102 K, for most polymers the exponentials may be expanded to yield ... 

Some deviations from the Simha-Boyer rule with increasing side chain length were discovered by Simha59. In so an attempt was made on the basis of a good deal of literature data to establish the range of applicability of the SB equation. 

Table 4. Values p for universal Kun and experimental Kex constants of the Simha-Boyer equation...

These equations are not valid at high concentrations of dispersed phase. A number of expressions have been discussed in detail by Sherman (27), Goodwin(28) and Frisch and Simha(29). One such equation, for a solid dispersed phase, is due to Mooney... 

Simha et al. (1973) showed that the Tait relation is also valid for polymers in the glassy state. In this case the value of bx is about the same as for polymer melts, but b2 is smaller ( 2 3 x 10 3). Even nowadays frequently use is made of the Tait equation. 

It is not surprising that attempts have been made to derive equations of state along purely theoretical lines. This was done by Flory, Orwoll and Vrij (1964) using a lattice model, Simha and Somcynsky (1969) (hole model) and Sanchez and Lacombe (1976) (Ising fluid lattice model). These theories have a statistical-mechanical nature they all express the state parameters in a reduced dimensionless form. The reducing parameters contain the molecular characteristics of the system, but these have to be partly adapted in order to be in agreement with the experimental data. The final equations of state are accurate, but their usefulness is limited because of their mathematical complexity. 

A very interesting semi-empirical equation of state was derived by Hartmann and Haque (1985), who combined the zero-pressure isobar of Simha and Somcynsky (1969) with the theoretically derived dependence of the thermal pressure (Pastime and Warfield, 1981). This led to an equation of state of a very simple form ... 

The majority of theories describing the concentration dependence of viscosity of diluted and moderately concentrated disperse systems is based on the hydrodynamic approach developed by Einstein [1]. Those theories were fairly thoroughly analyzed in the reviews written by Frish and Simha [28] and by Happel and Brenner [29], In a fairly large number of works describing the dependence of viscosity on concentration the final formulas are given in the form of a power series of the volume concentration of disperse phase particles — [Pg.111]

Figure 1 shows a comparison, published by Mori and Ototake [13], of the experimental dependences of viscosity on concentration of dispersions of solid particles based on the data of Vand [34], Robinson [12], Orr and Blocker [5], Dalla Valle and Orr [17] with the theoretical equations based on the hydrodynamic approach used by Einstein (1), Simha (30), Vand (31), Roscoe (44) and the phenomenological equation of Mori and Ototake (14). A more complicated form of the theoretical dependence, naturally makes it possible to describe experimental results over a wider range, but for concentrated dispersions most of theoretical equations remain inapplicable. 

The analysis by gel permeation chromatography, which provides a convenient way of determining parameters a and K of Eq, 25 has indicated35 tentatively an a value of 1.7 for polymers Pt-D1 and Pt-D2. This suggests a rod-like structure which has been supported by the theoretical treatment of the hydrodynamic property of the polymer based on a stretched ellipsoid of a revolution model (Simha s equation)47. The molecular dimension of Pt-D1 polymer, e.g. the minor axis (18 A) obtained from the calculation is in good agreement with that determined by the X-ray structural analysis... 

Nanda, V. S. Simha, R., "Equation of State of Polymer Liquids and Glasses at Elevated Pressures," J. Chem. Phys., 41, 3870 (1964). 

FIGURE 12.4 Alpha, a, from the equation tj/tJs = 1 + a. versus axial ratio for ellipsoids of revolution according to Simha theory [12],... 

Variants of 12.4.33] have also been obtained by others, including Everett who subsumed the difference between z° and in K and, for rod-like molecules parallel to the surface, by Prigogine and Marechal ). Frisch, Simha and Eirich ) derived (2.4.33] for 1 in an embryonic polymer adsorption theory. An equation resembling (2.4.33). 

Simha (1940) employing Eq. (10) solved the equation for the viscosities of solutions of ellipsoids of revolution for the limiting case a —> 0. Under this condition the distribution of the particles can be regarded as almost completely random. Simha also found that for large axial ratios the viscosity increment at a0 for very dilute solutions or suspensions (i.e., 4> —> 0) can be approximately represented by... 

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