Harmonic mean equation, surface tension

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 ... 

Harmonic means Geometric means Critical surface tension Equation of state... 

The subscripts d and p refer to dispersive and polar components of the surface tension coefficients. Equation 4.38 is called the harmonic-mean equation. The idea can also be written in the form of a geometric mean equation ... 

The harmonic mean equation is generally considered to be applicable to low surface tension materials such as organic polymers and liquids. If y and yP are known for two liquids, and the contact angles of those liquids on the solid of interest are measured, equation (36) produces two simultaneous equations that can be solved to find the surface tension and polarity of a solid polymer surface. Numerous assumptions have been made in developing the theory of fractional polarity. For example, it ignores the possibility of induced polarity at the interface between polar and nonpolar materials (80). These assumptions limit the application of equation (36) to systems where at least one and preferably both of the... 

The initial interfacial tension can be estimated from the surface tensions of the matrix (y ) and dispersed phase (y ) according to the harmonic mean equation ... 

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... 

The interfacial tension is related to the surface tension and the polarity of the two contiguous phases by the harmonic-mean equation (1),... 

In the CAB model the breakup condition is determined by means of the drop deformation dynamics of the standard Taylor analogy breakup model [5] (cf. TAB model above). In this approach, the drop distortion is described by a forced, damped, harmonic oscillator in which the forcing term is given by the aerodynamic droplet-gas interaction, the damping is due to the liquid viscosity and the restoring force is supplied by the surface tension. More specifically, the drop distortion is described by the deformation parameter, y = Ixjr, where x denotes the largest radial distortion from the spherical equilibrium surface, and r is the drop radius. The deformation equation in terms of the normalized distortion parameter, y, as provided in Eq. 9.29 is... 

The theory of fractional polarity combined with Yoimg s equation provides another means of estimating the surface tension of the sohd (3). Apphcation of the harmonic mean approximation leads to the relation... 

In addition to the methods discussed above, there are a few other solid surface tension determination methods, such as the Wu method [29, 30] and the Schultz methods [31, 32], which also fall into the category of partitioning surface tensions into independent components. Wu used the harmonic means to describe the interfacial surface tension instead of the geometric mean, based on a few slightly different assumptions to derive the equations for Wu s model. The Schultz methods can be considered as a special case of the extended Fowkes method. The contact angle of a polar liquid (usually water) on the solid is conducted in another nonpolar liquid medium (e.g., pure hydrocarbon compounds), or the contact angle of a nonpolar liquid on the solid is measured in another polar liquid medium. 

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