Viscosity, Eyring

Our approach in this chapter is to alternate between experimental results and theoretical models to acquire familiarity with both the phenomena and the theories proposed to explain them. We shall consider a model for viscous flow due to Eyring which is based on the migration of vacancies or holes in the liquid. A theory developed by Debye will give a first view of the molecular weight dependence of viscosity an equation derived by Bueche will extend that view. Finally, a model for the snakelike wiggling of a polymer chain through an array of other molecules, due to deGennes, Doi, and Edwards, will be taken up. 

Of the adjustable parameters in the Eyring viscosity equation, kj is the most important. In Sec. 2.4 we discussed the desirability of having some sort of natural rate compared to which rates of shear could be described as large or small. This natural standard is provided by kj. The parameter kj entered our theory as the factor which described the frequency with which molecules passed from one equilibrium position to another in a flowing liquid. At this point we will find it more convenient to talk in terms of the period of this vibration rather than its frequency. We shall use r to symbolize this period and define it as the reciprocal of kj. In addition, we shall refer to this characteristic period as the relaxation time for the polymer. As its name implies, r measures the time over which the system relieves the applied stress by the relative slippage of the molecules past one another. In summary. 

In connection with a discussion of the Eyring theory, we remarked that Newtonian viscosity is proportional to the relaxation time [Eqs. (2.29) and (2.31)]. What is needed, therefore, is an examination of the nature of the proportionality between the two. At least the molecular weight dependence of that proportionality must be examined to reach a conclusion as to the prediction of the reptation model of the molecular weight dependence of viscosity. 

Fig. 6.4. Energy barrier between occupied and empty molecular sites u activation energy. The applied shear stress t deforms the energy barrier analogous to Eyring s theory of viscosity v activation volume...

Equation 133 is similar to the formula for the strain and temperature dependence of the yield point calculated with the thermally activated viscosity proposed by Eyring and Bauwens [37,59]. 

MSN.64. I. Prigogine, G. Nicolis, and P. Allen, Eyring s theory of viscosity of dense media and nonequilibrium statistical mechanics, in Chemical Dynamics, Papers in Honor of H. Eyring, Hirshfelder, ed., Wiley, New York, 1971. 

Eyring H. (1936) Viscosity, plasticity, and diffusion as examples of absolute reaction rates. /. Chem. Phys. 4, 283-291. 

The cooperative segmental motion in polymer molecules can be considered as a crankshaft motion of six atoms in the polymer chain. According to H. Eyring, the viscosity of a polymer melt decreases exponentially in accordance with the enthalpy of activation AHa instead of the energy of activation Ea as stated in the Arrhenius equation. 

Eyring et al. (226) examine the entanglement problem from a somewhat different point of view, the activated complex theory of liquid viscosity. In a monomeric liquid the molecules move by random jumps from one equilibrium position to another. The jump frequency is controlled by an activation barrier between neighboring sites. According to activated complex theory, a shear stress lowers the barrier in the direction of the stress and raises it in the opposite direction, producing a bias in jump frequency and a net flow of molecules in the stress direction. For low stresses, the expression for viscosity in a monomeric system is ... 

The Eyring analysis presents some problems also. Suppose one accepts their suggestion that only the movement of segments between one pair of couples, that containing the chain center, serves to advance the chain. Then, to be consistent, the correct modification of the viscosity equation for small molecules is to replace K0 by nc K0 and A by A/n, rather than the original substitutions, K0- nK0 and A- A/n(E + 1). The result is... 

Experimental viscosity-shear rate curves at high concentrations turns out to be rather similar to an expression for non-Newtonian viscosity derived from the Eyring s activated complex theory for the transport properties of liquids (341) ... 

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...

Intermolecular distances in Eyring s viscosity theory (Part 6). 

Eyring,H., Ree.T., Hirai,N. The viscosity of high polymers — Random walk of a group of connected segments. Proc. Natl. Acad. Sci. 44,1213-1217 (1959). 

The Eyring flow shows a typical dependence upon stress. With increasing stress the viscosity r decreases (structural viscosity). The calculations are in good agreement with the experimental values. Figures 21 and 22 show the influence of shearing stress on the viscosity. 

By considering that the applied stress induces molecular flow, which can be treated along the lines of the Eyring viscosity theory, the strain rate, e, and yield stress, oy, are related by the so-called Eyring equation ... 

As pointed out by Doolittle, the relationship between the viscosity of liquids and their free volume remained for a long time only an intuitive hypothesis though it described quite well numerous experimental results. A theoretical approach to the solution of the problem of the relationship between the viscosity of liquid and its free volume was generalized for the first time by Eyring [85] in terms of the absolute reaction rates theory. The formulas obtained by Eyring pointed to a qualitative relationship between viscosity and the ratio of the volume occupied by liquid molecules C to the volume occupied by holes through which molecules jump to the neighboring position ... 

The kinetic theory of gases is far more advanced than that of liquids partly because of complex interactions among the molecules of liquids. We may estimate the viscosity of a pure liquid from the following relation based on the Eyring rate theory ... 

Equation (7.3) states that the rate at which the fiber (or sheet) becomes thinner is proportional to the applied force, not the applied stress. This means that thinner and thicker regions suffer cross-sectional reduction at an equal rate. This expression also informs us that, for a given load, as the viscosity ri increases, the axial strain rate, e, decreases. This has very important implications in fiber drawing. In order to understand these implications, we need to examine the temperature dependence of viscosity. The temperature dependence of viscosity is given by the Eyring equation... 

Eyring,2 starting with the result that the energy required to form a hole of molecular size in a liquid is equal to the energy of evaporation of a molecule from the liquid, and that for a molecule to take part in liquid flow requires a hole, not necessarily the full size of the molecule, deduced that the available molar energy of activation A for viscosity will be some fraction of the molar energy of evaporation AE, A plot of the values of ... 

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