Volume fraction of spheres

Thus, the perpendicular conductivity is always less than the parallel conductivity. If the second component is a number of spheres embedded in a matrix of the first component, then the composite conductivity is given by the Maxwell equation when the volume fraction of spheres is very small ... 

In these Equations, G is the modulus of the syntactic foam, G0 is the modulus of the polymer matrix, v0 is Poisson s ratio of the polymer matrix, and 9 is the maximum packing fraction of the filler phase. For uniform spheres, 9 0.64 (see Sect. 3.6). The volume fraction of spheres in the syntactic foam is 9sph. The slope of the G/G0 vs. 9sph curve depends strongly upon whether or not G/G0 is greater or less than 1.0. The slope is negative if the apparent modulus of the hollow spheres is less than the modulus of the polymer matrix. 

Since the volume fraction of spheres is true absorption, and disappears for any material with a purely real dielectric function, since "=0. Equations (11) and (12) validate Beer s law for dilute colloids - the absorbance is proportional to however for higher values of ... 

As a simple example of the additive version of the RG technique we consider a suspension of hard spheres dispersed in a continuous, Newtonian solvent. The effective viscosity of this composite system is a function of the solvent viscosity, rj, the number of hard spheres, N, and a couphng parameter, g = na l() V, defined as the ratio of the volume of a hard sphere of diameter a to the total volume of the suspension. This suspension can be treated as an effectively continuous fluid with a viscosity g = rj N g,rj ) that approaches the value rjo as the volume fraction of spheres, (j) = Ng, tends to zero. 

A similar analysis predicts the stiffening effect of spherical inclusions with a much higher modulus than the polymer (Fig. 4.9b) the modulus increases by about 50% at a 0.2 volume fraction of spheres. However, Section 4.2.3 shows that it is more efficient to use continuous, aligned fibres to stiffen thermoplastics. 

The fraction of delocalized or free charge is approximately a constant with a logarithmic increase as the volume fraction of spheres is decreased. This occurs because the entropy of the counterions becomes more important for small , thus stabilizing the free charge. The fraction of charge that is free or unbound is much larger than that of the one-dimensional case where the entropy is much more restricted and either another variational calculation... 

Einstein equation n. An equation relating the viscosity rjf of a sphere-filled, Newtonian liquid to that of the unfilled liquid r o, for volume fractions / of spheres up to about 10%. It is... 

Volume fraction of sphere or filler d> Maximum filler fraction... 

Volume fraction of spheres in suspension Angular velocity (1 /T)... 

The Dancoff factors calculated using the Monte Qirlo method and those calculated using Eq. (3) are shown in Fig. 1. Values obtained from the two methods compare very well. Equation (3), therefore, is a good representation of the Dancoff factor even for large volume fractions of spheres. 

The volume fraction of spheres (polymer coils containing entrained solvent) can be found from... 

Substitution of Equation 10.9 for the volume fraction of spheres into the Einstein-Batchelor equation along with the use of the root mean squared end-to-end distance for the sphere diameter gives rise to Equation 10.10. 

Figure 17 Calculated state diagram for hard globules with diameter a dispersed In nematic rods with axial ratio L/D = too. 0s is the volume fraction of spheres, and 0r that of the rods. L/a = 10. The solid line marks the spinodal instability to the lamellar phase, and the dash-dotted line that to macroscopically demixed phases. The dotted line separates the region where self-assembled chains are of (a) the open type from that where they are of (b) the dense type. (From P. van der Schoot. J. Chem. Phys. 117 3537, 2002. With permission.)... 

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