Cylindrical distribution function

The autocorrelation function is sometimes referred to simply as the correlation function. Among those working in crystal structure analysis, the autocorrelation function is known as the Patterson function. Many of the distribution functions obtained from scattering intensity data are in the nature of the correlation function, with possible differences in the normalization constant or a constant term. Functions in this vein include the pair correlation function or the radial distribution function (and its uniaxial variant cylindrical distribution function), discussed in Chapter 4. 

Figure 4.9 Cylindrical distribution function g(R, Z) - 1 derived from the intensity data in Figure 4.7 by means of Equation (4.31). Dashed contour lines represent negative values. (From Mitchell and Windle.10)...

Wide angle X-ray scattering from an aligned sample of copolyester prepared from kO mol % polyethylene terephthalate and 60 mol % p.aceto benzoic acid was reported together with an lysis of the derived cylindrical distribution function (CDF)... 

The director determines only the direction of the preferred orientation of the molecules, and indicates nothing about the degree of orientational order in the mesophase. The order parameter, 5, which is the first moment P2 in the expansion of the cylindrical distribution function of molecules in the Legendre polynomial series, provides just such a measure of the long-range orientational order... 

For amorphous systems with cylindrical symmetry, such as aligned uniaxial nematic liquid crystals or oriented amorphous polymers, the equivalent of the radial distribution function is the cylindrical distribution function (CDF). This function can be written as [2]... 

Similarly the cylindrical distribution function can be expanded in real space as [4] 00... 

Through EQN (13a) we can obtain the (Mientationally desmeared molecular scattmng fiinctimi. The expoimental cylindrical distribution function can be sharpened to the same extent as the scattmng function. This follows from the general equivalence of rotation in real and reciprocal spaces. It is also evident from the fact that each term W2n(r) in the expansion of the CDF (EQN (10b)) is independently related to the equivalent tem l2n(Q). 

The purely intramolecular scattering pattern can be calculated from the model molecular cylindrical distribution function such as those in FIGURE 4, or directly from the molecular model, containing N atoms, through [7]... 

A microscopic description characterizes the structure of the pores. The objective of a pore-structure analysis is to provide a description that relates to the macroscopic or bulk flow properties. The major bulk properties that need to be correlated with pore description or characterization are the four basic parameters porosity, permeability, tortuosity and connectivity. In studying different samples of the same medium, it becomes apparent that the number of pore sizes, shapes, orientations and interconnections are enormous. Due to this complexity, pore-structure description is most often a statistical distribution of apparent pore sizes. This distribution is apparent because to convert measurements to pore sizes one must resort to models that provide average or model pore sizes. A common approach to defining a characteristic pore size distribution is to model the porous medium as a bundle of straight cylindrical or rectangular capillaries (refer to Figure 2). The diameters of the model capillaries are defined on the basis of a convenient distribution function. 

Perhaps the most simple flow problem is that of laminar flow along z through a cylindrical pipe of radius r0. For this so-called Poiseuille flow, the axial velocity vz depends on the radial coordinate r as vz (r) — Vmax [l (ro) ] which is a parabolic distribution with the maximum flow velocity in the center of the pipe and zero velocities at the wall. The distribution function of velocities is obtained from equating f P(r)dr = f P(vz)dvz and the result is that P(vz) is a constant between... 

In a laminar flow reactor (LFR), we assume that one-dimensional laminar flow (LF) prevails there is no mixing in the (axial) direction of flow (a characteristic of tubular flow) and also no mixing in the radial direction in a cylindrical vessel. We assume LF exists between the inlet and outlet of such a vessel, which is otherwise a closed vessel (Section 13.2.4). These and other features of LF are described in Section 2.5, and illustrated in Figure 2.5. The residence-time distribution functions E(B) and F(B) for LF are derived in Section 13.4.3, and the results are summarized in Table 13.2. 

The ZFS is assumed to be cylindrically symmetric (only the /q component is different from zero) and of constant magnitude. The static part of the Hzfs is obtained by averaging the Wigner rotation matrix Dq q[ pm(0] over the anisotropic distribution function, Pip pj. The principal axis of the static ZFS is, in addition, assumed to coincide with the dipole-dipole IS) axis. Eq. (48) becomes equivalent to Eq. (42), with the /q component scaled by Z)q q[ 2pm(0] The transient part of the Hzfs can be expressed in several ways, the simplest being 92) ... 

Note 5 Even for molecules with cylindrical symmetry, does not provide a complete description of the orientational order. Such a description requires the singlet orientational distribution function which can be represented as an expansion in a basis of Legendre polynomials with L an even integer. The expansion coefficients are proportional to... 

If we consider the variation of concentration along the axial direction as a distribution of concentration, we can calculate the moments for each model in terms of their respective parameters, and then compare the moments to find the relationship between parameters. Since we shall consider the concentration as a distribution function along the axis of the tube, the moments are with respect to axial distance, rather than with respect to time as used previously. Since flow in cylindrical vessels is so common, we will discuss only this case in detail. Aris (A6) gives the more general treatment in vessels of arbitrary cross section. 

Db R) Radial dispersion coefficient, general dispersion model in cylindrical coordinates Molecular diffusivity Exit age distribution function, defined in Section I... 

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

Fig. 8 Pair distribution functions of complexes of a cylindrical symmetry (57% styryl-methyl(trimethyl)ammonium, 16% methacrylic acid, 27% methyl methacrylate) and b disklike symmetry (79% styrylmethyl(trimethyl)ammonium, 13% methacrylic acid, 8% methyl methacrylate). The curves which were calculated from the scattering data are represented by triangles and squares. Solid lines represent the distribution functions of a an idealized cylinder with a diameter of 3.0 nm and of b a disk with a height of 2.2 nm. The insets depict idealized symmetries of the particles. (Adapted from Ref. [31])...

Within PB theory [2] and on the level of a cell model the cylindrical geometry can be treated exactly in the salt-free case [3, 4]. The Poisson-Boltzmann (PB) solution for the cell model is reviewed in the chapter in this volume on the osmotic coefficient. The PB approach can provide for instance new insights into the phenomenon of Manning condensation [5-7]. For example, the distance up to which counterions can be called condensed can be conveniently found via the inflection point in the log plot of the integrated radial distribution function P(r) of counterions [8, 9], defined as... 

Fig. 1 Counterion distribution function P(r) from Eq. (1) for two cylindrical cell models with R/b= 123.8,1=0.959 e0/b and the values for Bjerrum length and valence as indicated in the plots. The solid line is the result of a molecular dynamics simulation [9] while the dotted line is the prediction from Poisson-Boltzmann theory. The increased counterion condensation visible in the simulation is accurately captured by the extended Poisson-Boltzmann theory (dashed line) using the DHHC correction from Ref. [18]. An approach using the DHH correction from Ref. [16] (dash-dotted line) evidently fails to correctly describe the ion distribution...

Let N cylindrical rods be situated in volume V, their concentration being c = N/V. The polymer volume fraction in the solution is then - jrpcd V4. Let us introduce the orientational distribution function for the rods f(u) cf(u)df2 is the number of rods per unit volume, which have the orientations within the small spatial angle dQ around the unit vector u. It is dear that in the isotropic state f(u) = const = l/4a. In the liquid-crystalline state the function f(u) has two maxima along the anisotropy axis. 

It must be kept in mind, that S represents just a first-order approximation of the distribution function, and this under the additional premise of complete cylindrical symmetry only. It might be an acceptable measure when comparing cases for which a mean-field model applies. However, comparing the order parameters of liquid crystals with those of other partially ordered phases, such as stretched polymers or tribological samples can be misleading due to possibly different types of distribution functions. 

---