Intrinsic viscosity hard sphere

The intrinsic viscosity of microgels described in [9] decreased with increasing fractions of the crosslinking monomer to about 8 ml/g which was still above the theoretical value for hard spheres of about 2.36 ml/g according to the Einstein equation and assuming a density of 1.1 g/ml. Obviously, due to the relatively low fraction of the crosslinking monomer, these microgels did not behave like hard spheres and were still swellable to some extent. 

The solution properties of dendrigraft polybutadienes are, as in the previous cases discussed, consistent with a hard sphere morphology. The intrinsic viscosity of arborescent-poly(butadienes) levels off for the G1 and G2 polymers. Additionally, the ratio of the radius of gyration in solution (Rg) to the hydrodynamic radius (Rb) of the molecules decreases from RJRb = 1.4 to 0.8 from G1 to G2. For linear polymer chains with a coiled conformation in solution, a ratio RJRb = 1.48-1.50 is expected. For rigid spheres, in comparison, a limiting value RJRb = 0.775 is predicted. 

Similar to what was done with the intrinsic viscosity we may compare Eq. (24) with the corresponding equation for hard spheres which is given by [3,74,75]... 

The physical properties of these polymeric dendrimers have been studied to some extent. Intrinsic viscosity measurements combined with MW afford values of according to Eq. (5). Alternatively, the translational diffusion coefficient leads to Rh according to Eq. (6). These equations may well be applicable, since it is observed that Rn and Rh scale with the 1/3 power of MW in support of the equal density hard-sphere assumption [88]. 

Fig. 37. The ratio of the equivalent hard sphere volume fraction based on the measured intrinsic viscosity as a function of for polyfmethyl methacrylate) spheres with grafted poly( 12-hydroxy stearic add) layers such that a/L = 4.7 (Mewis et ai, 1989). Open and closed circles correspond to the low and high shear limits of suspension viscosity.

The intrinsic viscosity can be related to the overlap concentration, c, by assuming that each coil in the dilute solution contributes to the zero-shear viscosity as would a hard sphere of radius equal to the radius of gyration of the coil. This rough approximation is reasonable as a scaling law because of the effects of hydrodynamic interactions which suppress the flow of the solvent through the coil, as we shall see in Section 3.6.1.2. The Einstein formula for the contribution of suspended spheres to the viscosity is... 

Figure 7.7. Relative viscosity of hard-sphere suspension in Newtonian fluid as a function of the volume fraction. Thomas curve represents the generalized behavior of suspensions as measured in 19 laboratories. The remaining curves were computed from Simha s, Mooney s and Krieger-Dougherty s relations assuming Einstein value for intrinsic viscosity of hard spheres, [T ] = 2.5, but different values for the maximum packing volume fraction, ([) = 0.78, 0.91, and 0.62 respectively.

The relationships between 17 and ( ) have been derived for suspensions of monodispersed hard spheres in Newtonian liquids. However, most real systems are polydispersed in size, and do not necessarily consist of spherical particles. It has been found that here also Simha s Eq 7.24, Mooney s Eq 7.28, or Krieger-Dougherty s Eq 7.8 are useful, provided that the intrinsic viscosity and the maximum packing volume fraction are defined as functions of particle shape and size polydispersity. For example, by allowing ( ) to vary with composition, it was possible to describe the vs. ( ) variation for bimodal suspensions [Chang and Powell, 1994]. Similarly, after values... 

In agreement with the decrease of of microgels on increasing the amoimt of EUP in the monomer mixture (Fig. 26), their mean particle diameter likewise decreases (Fig. 30). With the molar mass of microgels also their diameter increases (Fig. 31). Howeve a 20-fold increase of the M, corresponds to only less than a 3-fold increase of d. These results illustrate results that microgels from EUP are rather compact globular particles with intrinsic viscosities closely approaching that of hard spheres. 

FIGURE 16.1 Intrinsic viscosity for monodispersed rods, disks, and hard spheres. The empirical dependence for polydispersed disks is also shown. 

Viscosity measurements on this O/W microemulsion showed that the extrapolated intrinsic viscosity [rj] was somewhat higher than that expected for hard spheres but was still in agreement with spherical droplets if one assumed some hydration of the ethylene oxide headgroups, an effect that is to be expected because the water-soluble poly (ethylene oxide) is very hydrophilic [50]. For a given alkane/surfactant ratio, almost perfect hard-sphere behavior is obtained under this assumption close to the solubilization boundary... 

Similar results, which showed behavior of the viscosity according to a hard-sphere model, have been observed for a variety of O/W microemulsion systems such as Brij 96-butanol-hexadecane-water [52], Tween 60-Span 80-glycerol-paraffin-water [53], and Brij 96-pentanol-hexadecane-water [54]. Again the extrapolated intrinsic viscosity is found to be about 60% greater than expected according to Einstein s equation and the Huggins coefficient kn [cf Eq. (13)] to be about 1.8 [52]. This shows that such behavior will quite generally be observed for droplet-type microemulsions irrespective of the inter-... 

In conclusion, chain molecules are isolated in the limit of high dilution. They behave as hard spheres. Hence, intrinsic viscosity reflects in that limit the effective hydrodynamic volume of isolated chains. In other words, intrinsic viscosity characterizes the increase in viscosity compared to solvent due to isolated chain molecules of solute. 

These calculations still assume that the polymer coil in solution is a hard sphere with an even density throughout the sphere and with a fixed boundary to the solvent. For a more realistic discussion of the dimensions of a real polymer coil in solution, the reader is referred to Chap. 8. In particular, the correlation of diameter dy molar mass M and intrinsic viscosity [rj] in Eq. (7.4) is discussed in detail in Chap. 8 in the form of the Fox-Flory equation that correlates the intrinsic viscosity with the radius of gyration Rq of a polymer coil and with the molar mass ... 

Therefore the critical concentration is proportional to the reciprocal intrinsic viscosity. The factor of 2.5 assumes that the polymer coils behave like hard spheres in solution. Viscosimetric measurements for the determination of the intrinsic viscosity have to be performed in dilute solutions at concentrations clearly below c for an exact linear extrapolation according to the Huggins equation (Eq. 4.9). This condition is fulfilled for example in Fig. 4.2, where it is shown that the data points for the viscosimetric determination are below the critical concentration calculated from Eq. (7.7). 

The assumption in Determination of the polymer coil dimensions from the intrinsic viscosity in Chap. 7 that the polymer coils in solution behave like hard spheres with a constant density inside the coil and a fixed boundary to the solvent is only a simple approximation. In reality, a polymer chain shows a dynamic behavior with fast and statistically changing conformations. 

PJ. Fox and T.G. Flory developed a theoretical correlation of the intrinsic viscosity [ri]y the radius of gyration Rq and the molar mass M [75]. For the case of hard spheres, the intrinsic viscosity is related to Rq and M if the volume of a polymer coil in Eq. (7.3) is replaced with the volume of an equivalent sphere with a radius of gyration according to Eq. (8.28) ... 

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