Geometric mean approximation

Geometric mean approximation Dispersive and polar components of solid surface energy are found by solving yiv(l +COS0) = 2(y,Xf + 2(y Yl S An extension of GGF equation ysa predicted is significantly higher than the critical surface tension. [84]... 

Equation (8-41) is called the geometric mean approximation, it is often used in approximate theories. We apply this result to Eq.(8-40), defining AE,2 — (AE,AE2y - This gives... 

Geometric mean approximation, 29 Germane barrier of internal rotation, 391 Gibbs, free energy, 30 function, 20... 

Although it was assumed that Eq. 10 is also valid when an apolar material enters into interaction with a polar one, in practice polar surfaces interact with each other more often. Several attempts were made to generalize the correlation of Fowkes for such cases and the geometric mean approximation gained the widest acceptance. This considers only the dispersion and a polar component of the surface tension, but the latter includes all polar interactions [34]. Thus interfacial interaction can be calculated as follows ... 

The estimated values of the p s and of the original parameters are shown in Table V. Despite the wide range of initiator and monomer concentrations used, it is not possible to obtain precise estimates of this many parameters from the data. In particular, ps is very poorly defined for this system. Notice that the geometric mean approximation is equivalent to fis = 1 (see Equation 15). For a diffusion-controlled bimolecular reaction the arithmetic mean is appropriate as shown above, and this is reflected in the fact that p3 is significantly less than 1. 

Also in this case the default value of the binary parameters (fey = 0) represents the geometric mean approximation for the mixture interaction energy term. 

A work of cohesion for a liquid can be defined such that W, equals 2yLy- The work of adhesion terms in Eq. (8) are now expressed in terms of the corresponding works of cohesion of the individual liquids. In doing, so the theory diverges into two paths the harmonic mean approximation and the geometric mean approximation. In the first instance, it is assumed that can be expressed as the harmonic mean of the work of cohesion of the two liquids. 

The polar and the dispersive components of the polymer surface free energy were determined using the geometric mean approximation from the relation ... 

Data obtained from Perez-Luna el al. (Perez-I.una, 1995) The critical surface tension (p ) was estimated using Zismann s method, and the dispersive and polar components (p" and pP) were calculated using the geometric mean approximation to the w ork of adhesion. 

The authors postulate that a solid and a liquid of similar polarity will give a minimum value of y%. Kitazaki and Hata also discuss the ramifications of their postulates and experimeiital results on adhesion measurements. Overall their conclusion is that a maximum is achieved when the polarities of the solid surface and the adhesive are as similar as possible. From the Good and Girifalco theory, a maximum in would also be expected when the interparticle forces across the interface are at a maximum. Wu has also analyzed the separation of the force components of the interfacial and surface tensions. Wu, however, chooses to make an arithmetic mean approximation rather than the geometric mean approximation used by Good and by Fowkes. He bases this choice on an analogy with the form of the expressions for the interparticle forces, which are more closely mimicked by an arithmetic mean. Wu contends that his use of the arithmetic mean yields results closer to measured values. 

Model D differs from Model C in that a geometric mean approximation (eq. 30) is applied to reduce the number of penultimate termination rate coefficients from 10 to 4 ... 

In the Scatchard-Hildebrand treatment of this problem the pair potential is related to the properties of the pure components by making the geometric mean approximation ... 

For interfaces between a low and a high energy material, the geometric-mean approximation was used to give ... 

Harmonic-mean approximation Geometric-mean approximation ... 

Using a geometric mean approximation to describe the dispersion force interaction between phases 1 and 2, Fowkes also showed that for dispersion force interactions alone, the interfacial free energy is given by... 

The Lattice Fluid model (170) can be used to predict the solubilities of hydrocarbon and chlorinated hydrocarbons in nonpolar polymers. Three-dimensional solubility parameters can be used to provide an empirical correction to the geometric mean approximation (17 1). This correction predicts the solubility of polar and nonpolar solvents in polymers using only the pure component eqnation-of-state and solubility parameters. 

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