Geometric mean rule

Table III tests the applicability of the geometric-mean rule to the transient data. This rule states that f902 /so/gi- Table III shows that the average value of /90 (/so/oi)-172 for the transient debris is within 14% of the predicted value of unity.

The procedures outlined have a practical use. but it should be realized that the subparameter models have some empirical elements. Assumptions such as the geometric mean rule (Eq. 12-6) for estimating interaction energies between unlike molecules may have some validity for dispersion forces but are almost certainly incorrect for dipolar interactions and hydrogen bonds. Experimental uncertainties are also involved since solubility loops only indicate the limits of compatibility and always include doubtful observations. Some of the successes and limitations of various versions of the solubility parameter model are mentioned in passing in the following sections which deal brielly with several important polymer mixtures. 

A parameter, appears in the combining rule for the energy parameter. It describes the deviations of the unlike-pair interactions from the geometric-mean rule. Values less than 1 indicate that the unlike pair interactions are less favorable compared to the like-pair interactions. 

Hildebrand and Scott used the geometric mean rule to describe the interaction between molecules of two unlike species to arrive at the total solubility parameter. Strictly speaking, this was valid for nonpolar type molecules only. The fact that Equation 10.4 has produced hundreds of satisfactory correlations of solubility, swelling, permeation, surface phenomena, etc., confirms that the geometric mean rule is likewise applicable to molecules engaging in permanent dipole-permanent dipole and hydrogen bonding interactions as well. 

In effect, this takes wab to be equal to the geometric mean of waa nd wbb- The geometric mean rule, based partly on theoretical principles and partly on observation, is expected to hold, however, only in situations where dispersion forces provide the only significant interaction energy. 

Equation (72) gives some theoretical basis for the frequently applied geometric-mean rule, which is so often used in equations of state for gas mixtures and theories for liquid solutions. Berthelot was the first to use the geometric mean rule ... 

Scatchard also assumed the geometric mean rule for the cohesive energy density between the 1-2 molecules, similar to Berthelot and van Laar, in analogy with the result of London s dispersion force treatments for nonpolar molecules... 

A hard or A type system is one that is not restructured in the presence of an adsorbed film or on contact with the bulk liquid. For it, equations such as Equations 1 and 2 should apply approximately. A soft or B type system shows significant surface restructuring. Additivity and geometric mean rules are likely invalid the entropy of Interface formation will be important. Schematic illustrations of the two cases are shown in Figure 4. 

The Lorentz-Berthelot combining rules are most successful when applied to similar species. Their major failing is that the well depth can be overestimated by the geometric mean rule Some force fields calculate the collision diameter for mixed interactions as the geometric mean of the values for the two component atoms. Jorgensen s OPLS force field falls into this category [Jorgensen and Tirado-Reeves 1988]. 

The default value i/>y = 1 can be used to recover the usual first-order approximation for the characteristic interaction energy, represented by a sort of geometric mean rule. 

In some force fields, especially those using the Lennard-Jones form in eq. (2.12), the Rq parameter is defined as the geometrical mean of atomic radii, implicitly via the geometrical mean rule used for the Ci and C2 constants. 

Combination rules for a other than the arithmetic mean have also been proposed. If the geometric mean rules for C and i2(rep.) are compared to the corresponding terms in the Lennard-Jones 6—12 potential, geometric-mean rules for both 12 and cti2 are obtained. Good and Hope tested a variety of combination rules for j2 paired with both the arithmetic and geometric mean rules for geometric mean for 0 is superior to the arithmetic mean. This conclusion is open to question, however. None of the 29 systems that they examined would be expected to conform to the... 

Polar interactions where molecules having permanent dipole moments interact in solution with the dipole orientation in a symmetrical manner. It follows that the geometric mean rule is obeyed for orientation interactions and the contribution of dipole orientations to the cohesive energy and dispersion interactions. 

The absence of the volume change presumes that AU" is equal to the enthalpy or to heat of mixing which was equated to the right terms of the equation (under the equality of 8 of both components, AU = 0). The Hildebrand-Scatchard approach corresponds to the geometric mean rule. ... 

This equation satisfies the required dimensionality for and lets us treat k as a dimensionless adjustable parameter. Values of this parameter are given in Table 2.5 for some representative fluids. For simplicity, the following geometric mean rule can be adopted for k ... 

The interaction potentials between molecules have been used to determine the interactions between macroscopic bodies. In this instance, the Hamaker constants. A, derived from jS, also follow the geometric mean rule [13,18-19]. Thus,... 

Berthelofs rule is thus applied to the tensile strength between particles. We extend Berthelofs rule for an ensemble of particles assuming the geometric mean rule applies to an ensemble of adhesive contacts between particles and is weighted via the surface area fraction, thus... 

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