Einstein value

The intrinsic viscosity is the Einstein value [rj] = 2.5 and the packing fraction cpm(0) is that in the low shear limit. As the volume fraction approaches the maximum packing fraction, the viscosity rapidly... 

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. 

The variations in the intrinsic viscosity predicted by the primary electroviscous effect are often small, and it is difficult to attribute variations in the experimentally observed [17] from the Einstein value of 2.5 to the above effect since such variations can be caused easily by small amounts of aggregation. Nevertheless, Booth s equation has been found to be adequate in most cases. Further discussions of this and related issues are available in advanced books (Hunter 1981). 

Riseman (139) or Kuhn and Kuhn (153 , 153"). For the sake of comparison, Fig. 12 also shows the theta-solvent intrinsic viscosities of polystyrene in cyclohexane [experimental valuesofKRiGBAUMandFuoRY (149), small black points theoretical values, broken line] and the theoretical intrinsic viscosities of rigid ellipsoids with axial ratios p = M/500 (chain curve). As a matter of course, the chain curve reduces to the Einstein value of [rf in the range of M below500 [see, for example, Petehlin (16) ]z. 

The Einstein value of [jj] is 0.025q where g, is the density of the solute molecule, equal to 1.05 g./ml. for polystyrene. The value of 500 was rather arbitrarily chosen for the purposes of illustration, but it happens to be quite close to the molar weight of the so-called preferred statistical segment of polystyrene [see Kurata and Yamakawa (157)). 

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.

Importantly, Yoshizaki and Yamakawa [25] found that, in contrast to /, [77] of a wormlike cylinder undergoes significant end surface effects until the axial ratio p reaches about 50, on the basis of numerical solutions to the Navier-Stokes equation with the no-slip boundary condition for spheroid cylinders, spheres, and prolate and oblate ellipsoids of rotation. They constructed an empirical interpolation formula for [ y] of a spheroid cylinder which reduces to eq 2.37 for p > 1 and to the Einstein value at p = 1. Then, with its aid, Yamakawa and Yoshizaki [4] formulated a modified theory of [77] for wormlike cylinders which agrees with the Yamakawa-Fujii theory [3] for Lj lq > 2.278 and with the Einstein value at Ljd = 1, regardless of dj2q smaller than 0.1. However, no formulation has as yet been made for L/2q < 2.278 and d/2q > 0.1, i.e., for short flexible cylinders. In what follows, the Yamakawa-Yoshizaki modification is referred to as the Yamakawa-Fujii-Yoshizaki theoiy. 

Calculations have been carried out by a statistical method [23,a] and also by a random-walk method [23,d] (Section 2.5.1.2). They indicate that, for values of ab less than 0.1, the ratio (k/ko)/0A0B would be in the region of 3-10. According to the latter treatment, for example, the factor (for equal-sized molecules) is [1 - - (2r g/3)( ) g/Z)AB)] which with Equations (2.4) and (2.26) becomes [1 - - (8/3)(n/m)]. With the Stokes-Einstein values n = 6 and m = 8, the factor is therefore 3 with the more realistic values n = 3 — 4 and m = 3 — 4, it is in the region 4 1. The effect is appreciable it imphes, for instance, that reactions with 0a0b down to 0.1 would be difficult to differentiate experimentally from reactions with 0a0B 1 ... 

Figure 3.6 Deviations of diffusion coefficients from the Stokes-Einstein value (1). Variation of numerical factor n of modified Stokes equation with van der Waals radius r of molecules diffusing in carbon tetrachloride. Solid circles compounds 1-23 of Table 3.1 half-open circles compounds 24-26 open circles compounds 27-36. Theoretical curve according to Wirtz (Ref. [12,a]) labelled SW. See text. From Ref. [1 l,a].

Figure 3.7 Deviations of diffusion coefficients from the Stokes-Einstein value (2). Comparison of the diffusion coefficients of symmetrical solutes in five protic and aprotic solvents with the prediction of the Stokes-Einstein equation. The diffusion coefficient — viscosity product Drj is plotted against the reciprocal of the solute radius the predictions of Stokes law for perfect stick (kT/6n) and perfect slip kT/An) are shown by dotted lines. See text. From Ref. [13,b].

Figure 3.8 Deviations of diffusion coefficients from the Stokes-Einstein value (3). Plot of log D/T against log rj for spherical and symmetrical solutes in various solvents. The diffusion coefficients of rare gases, CH4, CCI4, Et4C, and the tetraalkyltins are plotted as a function of solvent viscosity in alcohols, hydrocarbons, acetone, and acetonitrile. (1) n-Hexane, 25 °C (2) acetone, 25 °C (3) acetonitrile, 25 °C (4) acetone, 10 °C (5) methanol, 40 °C (6) methanol, 25 °C (7) methanol, 10 °C (8) n-decane, 25 °C (9) ethanol, 25 °C (10) n-tetradecane, 40 °C (11) 2-propanol, 25 °C (12) n-tetradecane, 25 °C (13) 1-butanol, 25 °C (14) 2-propanol, 10 °C (15) 1-butanol, 10 °C (16) 1-octanol, 40 °C (17) 1-octanol, 25 °C (18) 1-octanol, 10 °C. See text. From Ref. [13,a].

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