Harmonic mean approach

It is obvious that the harmonic-mean approach has the same defects as the Owens and Wendt approach, and that internal cohesive polar interaction properties cannot determine the interfacial interaction energy between two dilferent materials. This equation was also abandoned. 

Wu proposed another equation using an harmonic mean approach to estimate the nonpolar and polar interactions across the interface [8, 109, 116]. This approach gives the same criterion for maximising adhesion. 

Finally, we should motion that in addition to the geometric mean rules there have been other combining mles recommended for the cross interactions, often known as harmonic mean approaches (see Problem 3.8). These have been considered to be variations of the Owens-Wendt approach and have found some applicability especially for mixtures of polymer melts (blends) for which there are a lot of data (see Problem 15.10), also at diverse temperatures despite often being... 

Irrespective of the method chosen, meaningful data can only be obtained if the appropriate level of signal to noise (S/N) is reached in the spectrum of each analyte. This has been achieved for Raman measurements through short data acquisition times (<1 s) and application of mathematical approaches such as If-harmonic means clustering (KHMC), factor analysis [57] and principal component analysis (PCA) [58] to the data set. Ultimately the sample response to the excitation energy determines the speed that a measurement can be made. 

This method utilizes a similar approach to Owens and Wendt s method but uses a harmonic-mean equation to sum the dispersion and polar contributions. Wu reported that the Owens and Wendt equation was giving surface tension values for polymers in error by as much as 50-100% when compared with their melt values, particularly for polar polymers. He suggested that a harmonic (or reciprocal) mean approximation might be better for polar polymers as given ... 

The empirical AN and DN indexes obtained by our approach also lend themselves easily for calculations of pair interaction numbers. Tliere is no formal theoretical guideline on how best to combine individual AN and DN numbers. Arithmetic, geometric and harmonic mean averaging may be used, with a decision as to preferred approach left to an empirical examination of results. One pair interaction number, I p, which has proven to be useful, is defined as follows ... 

Interfadal tension between two fluid phases is a definite and accurately measurable property depending on the properties of both phases. Also, the contact angle, depending now on the properties of the three phases, is an accurately measurable property. Experimental approaches are described, e.g., in Refs. 8,60, and 63 and in Ref. 62, where especially detailed discussion of the Wilhehny technique is presented. Theories such as harmonic mean theory, geometric mean theory, and acid base theory (reviewed, e.g., in Refs. 8, 20, and 64) allow calculation of the soHd surface energy (because it is difficult to directly measure) from the contact angle measurements with selected test liquids with known surface tension values. These theories require introduction of polar and dispersive components of the surface free energy. 

The enthalpy-entropy relationship has been examined using another approach. First, In X is plotted again (1/T - 1/T ,) where T, is the harmonic mean of the experimental temperatures. The AH is estimated using equation [13.21.2.18] as the slope remains the same as when 1/T is used as the independent variable, and... 

Another approach is to divide the work of adhesion and surface tensions into two terms describing the polar (p) and dispersive (d, nonpolar) interactions respectively. This gives two equations, the harmonic mean equation... 

Equations 22.3-22.14 represent the simplest formulation of filled phantom polymer networks. Clearly, specific features of the fractal filler structures of carbon black, etc., are totally neglected. However, the model uses chain variables R(i) directly. It assumes the chains are Gaussian the cross-links and filler particles are placed in position randomly and instantaneously and are thereafter permanent. Additionally, constraints arising from entanglements and packing effects can be introduced using the mean field approach of harmonic tube constraints [15]. 

At sufficiently low strain, most polymer materials exhibit a linear viscoelastic response and, once the appropriate strain amplitude has been determined through a preliminary strain sweep test, valid frequency sweep tests can be performed. Filled mbber compounds however hardly exhibit a linear viscoelastic response when submitted to harmonic strains and the current practice consists in testing such materials at the lowest permitted strain for satisfactory reproducibility an approach that obviously provides apparent material properties, at best. From a fundamental point of view, for instance in terms of material sciences, such measurements have a limited meaning because theoretical relationships that relate material structure to properties have so far been established only in the linear viscoelastic domain. Nevertheless, experience proves that apparent test results can be well reproducible and related to a number of other viscoelastic effects, including certain processing phenomena. 

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