Bulk zero-shear viscosity

Figure 14.7 Dependence of the zero-shear viscosity, uo, on molecular weight, M, for different dendrimer systems. (1) Dendrimers of different chemical composition but in the same state (i.e. PAMAM, PPI and PBzE dendrimers in bulk D, C and E, respectively). (2) Compositionally identical dendrimers (i.e. PAMAMs) in solutions and in the bulk state (A, B and D, respectively). (3) Compositionally identical dendrimers and linear polymers of comparable molecular weights (i.e. PAMAMs in the bulk state D and F, respectively)...

The motion of polymers in concentrated solution and bulk is of major theoretical and practical concern. For example, the strong dependence of zero-shear viscosity on molecular weight (approximately the 3.4 power) and the marked decrease of viscosity 1) with shear rate y not only bespeak some of the unusual properties of long-chain molecules but also are of essential importance in virtually every processing operation. Yet the reasons for these unusual behaviors have become clear only recently. The reptation con-... 

For more concentrated suspensions, other parameters should be taken into consideration, such as the bulk (elastic) modulus. Clearly, the stress exerted by the particles depends not only on the particle size but on the density difference between the partide and the medium. Many suspension concentrates have particles with radii up to 10 pm and a density difference of more than 1 g cm . However, the stress exerted by such partides will seldom exceed 10 Pa and most polymer solutions will reach their limiting viscosity value at higher stresses than this. Thus, in most cases the correlation between setfling velocity and zero shear viscosity is justified, at least for relatively dilute systems. For more concentrated suspensions, an elastic network is produced in the system which encompasses the suspension particles as well as the polymer chains. Here, settling of individual partides may be prevented. However, in this case the elastic network may collapse under its own weight and some liquid be squeezed out from between the partides. This is manifested in a dear liquid layer at the top of the suspension, a phenomenon usually... 

Rheological measurements are used to investigate the bulk properties of suspension concentrates (see Chapter 7 for details). Three types of measurements can be applied (1) Steady-state shear stress-shear rate measurements that allow one to obtain the viscosity of the suspensions and its yield value. (2) Constant stress or creep measurements, which allow one to determine the residual or zero shear viscosity (which can predict sedimentation) and the critical stress above which the structure starts to break-down (the true yield stress). (3) Dynamic or oscillatory measurements that allow one to obtain the complex modulus, the storage modulus (the elastic component) and the loss modulus (the viscous component) as a function of applied strain amplitude and frequency. From a knowledge of the storage modulus and the critical strain above which the structure starts to break-down , one can obtain the cohesive energy density of the structure. 

Bruce Bersted [1, 2] in 1975 proposed a method for calculating the viscosity of a linear polyethylene from its MWD. The basic idea was that each molecule makes a contribution to the bulk viscosity equal to its zero-shear viscosity but that as the shear rate increases from zero, the maximum length of molecule that makes such a contribution decreases. In other words, as the shear rate increases, the effects of progressively shorter molecules on the viscosity are eliminated. He said that this is equivalent to cutting off the relaxation spectrum at progressively shorter times as the shear rate increases. 

The concept of chain entanglements influencing the line-widths, or T2 s, can be examined more directly by studying the influence of molecular weight. It is well established that the zero shear bulk viscosity of all amorphous polymers is directly proportional to the molecular weight below a critical low molecular weight, M., and above this molecular weight increases as the 3.5 power of M. ( ) ( ) (50) M, represents approximately... 

Fig. 5.4. Ratio of Maxwell-constant to intrinsic viscosity as a function of molecular weight for linear oligomers of poly oxy-propylene glycol in cyclo-hexanol at 20° C (full circles). Ratio of birefringence to shear stress in the limit of zero shear stress for the same oligomers in bulk (temperatures 20—60° C) (open squares). The experimental point at the righthand side is obtained on a high molecular weight sample, Tsvetkov, Garmonova and Stankevich (166)...

The second viscosity coefficient X is related to the shear viscosity l (first viscosity coefficient) by A, = 2/3 t if the bulk viscosity coefficient defined by K = A, + 2/3 p is zero. Otherwise, X is given by... 

Figure 8.3 Flow profile, v(z), above a solid surface with slip. The viscosity T][z) close to the surface is lowerthan in the bulk. The shear gradient, dv/dz, is increased correspondingly. Solid line The viscosity is reduced inside a hypothetical discrete layer of thickness df. The slip length, bs, is the distance from the surface to the extrapolated plane of zero shear (dotted). Dashed line The viscosity gradually decreases at the interface.

When a fluid flows past a solid surface, the velocity of the fluid in contact with the wall is zero, as must be the case if the fluid is to be treated as a continuum. If the velocity at the solid boundary were not zero, the velocity gradient there would be infinite and by Newton s law of viscosity, equation 1.44, the shear stress would have to be infinite. If a turbulent stream of fluid flows past an isolated surface, such as an aircraft wing in a large wind tunnel, the velocity of the fluid is zero at the surface but rises with increasing distance from the surface and eventually approaches the velocity of the bulk of the stream. It is found that almost all the change in velocity occurs in a very thin layer of fluid adjacent to the solid surface ... 

An extrapolation of the form r] k) = /(0)/(l + a k ) where a is a state-dependent constant enabled the zero-wave vector Newtonian viscosity to be calculated. The fit gave 9.71 mP which compares well with the experimental value (8.9 mP). Guo and Zhang used equilibrium molecular dynamics to calculate the shear and bulk viscosities of liquid water. Using the SPC/E model they fotmd that these were 6.5 and 15.3 mP, respectively, as opposed to the experimental values 8.9 and 21.3 mP. 

Assuming constant viscosity and permittivity, and negligible pressure gradients as a result of the fluid flow, integration of equation (19.1) from the hydrodynamic plane of shear where the fluid velocity is zero (v = 0), to a point in the bulk where the potential is zero and fluid velocity is constant (Ueo) gives the following ... 

In Figure (5), it becomes obvious that the minima of the viscosity are directly related to the zero values of the horizontal velocity and not to the vertical velocity of the bulk solid in vertical direction, although it is 20 times higher than the horizontal velocity. This results demonstrate that the viscosity depends mainly on the shear deformations D. ... 

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