Statistics volume fraction

This model then leads us through a thicket of statistical and algebraic detail to the satisfying conclusion that going from small solute molecules to polymeric solutes only requires the replacement of mole fractions with volume fractions within the logarithms. Note that the mole fraction weighting factors are unaffected. 

Thus for an ideal two-phase system the total calibrated intensity that is scattered into the reciprocal space is the product of the square of the contrast between the phases and the product of the volume fractions of the phases, Vi (1 — Vi) = V1V2. V1V2 is the composition parameter66 of a two-phase system which is accessible in SAXS experiments. The total intensity of the photons scattered into space is thus independent from the arrangement and the shapes of the particles in the material (i.e., the topology). Moreover, Eq. (8.54) shows that in the raw data the intensity is as well proportional to the irradiated volume. From this fact a technical procedure to adjust the intensity that falls on the detector is readily established. If, for example, we do not receive a number of counts that is sufficient for good counting statistics, we may open the slits or increase the thickness of a thin sample. 

In the complete Eulerian description of multiphase flows, the dispersed phase may well be conceived as a second continuous phase that interpenetrates the real continuous phase, the carrier phase this approach is often referred to as two-fluid formulation. The resulting simultaneous presence of two continua is taken into account by their respective volume fractions. All other variables such as velocities need to be averaged, in some way, in proportion to their presence various techniques have been proposed to that purpose leading, however, to different formulations of the continuum equations. The method of ensemble averaging (based on a statistical average of individual realizations) is now generally accepted as most appropriate. 

We have introduced a statistical mechanical approach, illustrating how the material properties and rheology play a role at the microscopic level. Our main reason for doing this is to determine the microstructure and calculate the macroscopic rheological properties. We can now evaluate the coordination number z from Equation (5.30) for our colloid pair potential in Figure 5.9. The variation of z with volume fraction is shown in Figure 5.10. 

Due to their high aspect ratio, nanocarbons dispersed in a polymer matrix can form a percolating conductive network at very low volume fractions (< 0.1 %). The conductivity of a composite above the transition from an insulator can be described by the statistical percolation using an excluded volume model [22,23] to yield the following expression ... 

Validation of the database. This is the final part in producing an assessed database and must be undertaken systematically. There are certain critical features such as melting points which are well documented for complex industrial alloys. In steels, volume fractions of austenite and ferrite in duplex stainless steels are also well documented, as are 7 solvus temperatures (7 ) in Ni-based superalloys. These must be well matched and preferably some form of statistics for the accuracy of calculated results should be given. 

Coalescence Growth Mechanism. Following the very early step of the growth represented by Eq. (1), many nuclei exist in the growth zone. Hence Eq. (2) would be a major step for the crystal growth. Since there are many nuclei and embryos with various sizes in the zone, Uy in Eq. (2) can be assumed to be a random variable. Due to mathematical statistics, the fraction of volume approaches a Gaussian after many coalescence steps (3). A lognormal distribution function is defined by... 

The Peclet number compares the effect of imposed shear (known as the convective effect) with the effect of diffusion of the particles. The imposed shear has the effect of altering the local distribution of the particles, whereas the diffusion (or Brownian motion) of the particles tries to restore the equilibrium structure. In a quiescent colloidal dispersion the particles move continuously in a random manner due to Brownian motion. The thermal motion establishes an equilibrium statistical distribution that depends on the volume fraction and interparticle potentials. Using the Einstein-Smoluchowski relation for the time scale of the motion, with the Stokes-Einstein equation for the diffusion coefficient, one can write the time taken for a particle to diffuse a distance equal to its radius R, as... 

Fig. 3. The dependence of % on the volume fraction of the dry polymer tp2 for i - 0.012 and c = 0. Numbers at curves denote the number of monomers in the statistical segment, s. From Ilavsky [34]...

The radial distance distribution in simple atomic and molecular fluids is determined essentially by the exclusion volume of the particles. Zemike and Prins [12] have used this fact to construct a one-dimensional fluid model and calculated its radial distance correlation function and its scattering function. The only interaction between the particles is given by their exclusion volume (which is, of course, an exclusion length in the one-dimensional case) making the particles impenetrable. The statistical properties of these one-dimensional fluids are completely determined by their free volume fraction which facilitates the configurational fluctuations. 

The conclusion that the free-volume fraction at Tg is not a universal parameter for linear polymers of differing molecular structure can be qualitatively confirmed by the following arguments71. Assume that at temperatures far below Tg polymeric chains are in a state of minimum energy of intramolecular interaction, Le. the fraction of higher-energy ( flexed ) bonds is zeroS4. On the other hand, let the equilibrium fraction of flexed bonds at T> Tg obey the Boltzmann statistics and be a function of Boltzmann s factor e/kT. Thus, the fraction of flexed bonds at Tg can be estimated from the familiar expression ... 

The Theory of Porous Media is the Mixture Theory, restricted by the concept of the Volume Fractions. Hereby, we have a look at a continuum which consists of several constituents. In this investigation we deal with a solid phase a = S), a Liquid phase (a = L) and a Gas phase (a = G). The components of the real structure will be statistically distributed over the control space, so that we gain to a smeared model of the real structure. 

We can now ask the question What is the magnitude of the coil density How far is the statistical coil, which we have considered so far, diluted How many monomer units are present in a unit volume It is possible to calculate how the density, p, depends on the distance to the centre of gravity and on the number of links n. It appears that the volume fraction in the centre equals /o = Ihfn, i.e. ... 

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