Volume distribution function

Unlike surface area calculations, the volume distribution function and all subsequently discussed functions are based on the model of cylindrical pore geometry. 

Another useful function sometimes used in place of the volume distribution function is (Inr) which can be expressed as... 

The volume distribution function (r) represents the volumetric uptake in a unit interval of pore radii, irrespective of the variation in the number or the length of the pores. When (r) is divided by nr, the mean cross-... 

The set of formulas derived in the preceding section is self-sufficient for numerical evaluation of the linear and cubic magnetic responses as soon as the material parameters are known. Here we assume that the system is dilute enough and interparticle interaction may be neglected. In this framework, to construct the dependencies to be compared to the experimental data, one has, first, to calculate the responses for a subsystem of identical grains with the volume Vm, and then perform the averaging with an appropriate volume-distribution function f(Vm). 

The theory was tested with the aid of an ample data array on low-frequency magnetic spectra of solid Co-Cu nanoparticle systems. In doing so, we combined it with the two most popular volume distribution functions. When the linear and cubic dynamic susceptibilities are taken into account simultaneously, the fitting procedure yields a unique set of magnetic and statistical parameters and enables us to conclude the best appropriate form of the model distribution function (histogram). For the case under study it is the lognormal distribution. 

Table I Parameters of the pore volume distribution function...

Thus the volume distribution function is constant over the particle size range where the power law exponent p = —4. Can the constant volume distribution for p = —4 be compatible with the bimodal volume distribution that covers much the same particle size range The power law and bimodal volume distributions are equivalent only as a very rough approximation. Most of the aerosol volume is present in the accumulation... 

Calculate the volume distribution function rty for the power-law distribution. 

For layered or laminar mixtures, one can also define mixedness by the striation thickness, s (defined as 1/2 of the layer thickness). It can be shown that in these systems there is a simple relation Ay = 1/s. Since in real blends there is a variety of striation thicknesses, s should be expressed by a volume distribution function. 

Figure 4.1. Volume distribution functions evaluated by GNOM for the PDADMACI-SDS gel containing Pt nanoparticles derived from H2PtCl6 (1), (NH4)2PtCl4 (2), and Na2PtCl6 (3). (Reprinted with permission from reference 15. Copyright 2000, American Chemical Society.)...

Fractal dimensions Fd were used for the calculation of the pore volume distribution functions in relation to their radii from equation ... 

Substituting the Gaussian distribution of the volume distribution function (eq. 6.9-4) into the above equation, we get ... 

Frere, M. Jadot, R., and Bougard, J., Determination of the micropore volume distribution function of activated carbons by gas adsorption. Adsorption, 3(1), 55-66 (1997). 

Figure 1.16. Micropore volume distribution function of activated carbon ACTW 52 and certain of its modifications calculated from CO2 adsorption data by the DR-method [1.3, 1.59].

Dimensionless Volume Distribution Function vs. Dimensionless Particle Volume... 

Hie pore volume distribution function Fy of a mat, ddetmined by liquid extnision (or by memuiy intiusion), is expiessed as follows (Jena and Gupta 1999) ... 

Khater et al. calculated the time dependence of TRM for an assembly of noninteracting uniaxial particles with a Poisson volume distribution function [P(V) dV= (4V/VQ)exp(-2VIVg) dV]. Transforming the volume distribution in a distribution over relaxation times t, described... 

The model has some limits, mainly as far as the calculation of in the blocked state ( t j = M, I2>K for uniaxial symmetry) is concerned. The calculation does not account for vibrations of the magnetic moment m in the potential well, that is, for transverse relaxation. This leads to an abrupt variation of x close to Tg) (for a single particle the variation should be steplike). Such a very rapid variation is never observed experimentally, neither for samples with a narrow volume distribution. In our opinion it is important to take into account the effect of transverse relaxation, which should smooth the x variation below Tg. Moreover, the volume distribution function assumed in the Gittleman calculation is not realistic. Finally, interactions between particles, almost always present in real systems, are neglected. 

It is clear that and its relationship with Tg), depends on the form of the volume distribution. Gittleman et al. " gave some simple relationships between T and Tg) for different types of volume distribution functions, for example, - AK V) lk n o)TQ) = ATg, where the constant A depends on the form of the size distribution and is equal to 2 and 1.8 for a rectangular and for a Poisson distribution, respectively. 

Figure 8-6. Time-dependent volume distribution functions D(R) ofZnO-based colloidal particles in liquid suspension maintained inside a sealed cell during SAXS measurements. The time of growth increases from 10 up to 120 minutes. The volume functions were derived, using the GNOM package, from the corresponding set of experimental SAXS curves. (Reprinted with permission from Tokumoto (1999). Copyright 1999 by Elsevier).

The only remaining step in the design of our weight distribution step is a normalization procedure that extends the usability of the method to general unstructured meshes as well, which is achieved by means of multiplying the previous weight distribution w with the element volume distribution function which one can obtain by means of an L2 projection from the Qo (piecewise constant) space into the Qi (trilinear) space. 

In (7) and (8) the independent variables have been decomposed into the irreducible trace ( 0 and v ), deviator (denoted by ) and skew (denoted by square brackets on subscripts) components. In the application below, each of the phenomenological coefficients is assumed to be proportional to the appropriate volume distribution function (t> . Thus = (l) etc., where are constants. This has the appealing property that the theory automatically reduces to micropolar or Navier Stokes when the concentration of the solid constituents goes to zero. As will be seen shortly, it has the nasty property of making even the simplest flow problem inherently nonlinear. 

Equations (11) and (12) are the linear momentum balances in the flow direction. Here the total pressure head has been partitioned into partial pressure components through the use of the solid volume distribution function . Equation (13) is the balance of linear momentum for the solid phase (the fluid phase is the same but of opposite sign) in the y direction. In this equation, we have allowed for an interphase pressure effect (incorporated in the parameter E) to ensure that the mixture remains saturated at all times [8]. Equations (14) and (15) account for the angular momentum balance in the binormal direction for the two constituents. Finally, the primes imply differentiation with respect to 3 ... 

---