Model parallel voids

The suspension model of Einstein and Guth is extended into 4-th order function of v, and the parallel voids model is extended into the 3-rd order function of v. The fraction of reinforcing material in the filler space Cr is considered as a measure of the efficiency of the reinforcing material for a given two-component polymer system, indicating also the state of adhesion at the phase boundary. 

We have therefore attempted to compare the obtained results with the parallel voids model (cf. Table 1), for which the reinforcement function e can be written as... 

According to the parallel voids model (Table 1) we have... 

Thus, assumptions of anisotropic hole expansion yield a different relationship between hole volume and the characteristic dimension(s) s, which are allowed to increase. For example, by framing the holes as cylinders with a fixed height a, their volumes will increase according to the square of the radius s Vh oc af. As a further, naive example, if the cavities are modeled as voids between two parallel planes (e.g., the interlayer gaps in clays [Consolati et al., 2002]), the increase in their volume... 

The zeolites discussed so far have all relied on exotic organocations to function as SDAs. However, sometimes the inorganic cations can have a greater influence as demonstrated by the 12-MR zincosilicate, VPI-8 (65-68). The synthesis of VPI-8 still requires an organic additive but its role as template may be as a void filler. A recently determined model for the structure of VPI-8 viewed along the 001 direction is shown in Fig. 10 (69). Much like SSZ-31, the structure of VPI-8 involves ID channels running in parallel that are defined by odd-shaped 12-MR structures (6.2 x 5.97 x 5.88 A). An unusual feature of this structure is the pinwheel building unit that is composed of four 5-MR stmctures surrounded by another four 5-MR structures. 

Desig- nation Nominal Size Surface Area (m /g) Total Void Fraction Dt X 10s (cm2/sec) Average Tortuosity Factor t, Parallel-Path Pore Model r = 2K./5. (A) r Based on Average Pore Radius... 

With the aid of the parallel-pore model, it is possible to calculate the product of Papp and Sm, which allows one to express heterogeneous kinetic rate laws in pseudo-volumetric form. The parallel-pore model is discussed in greater depth later in this chapter. For the present discussion, it is only necessary to visualize straight cylindrical pores of length L and radius (raverage)- If E represents the total number of pores, then the void volume is... 

The parallel-pore model provides an in-depth description of the void volume fraction and tortuosity factor Tor based on averages over the distribution in size and orientation, respectively, of catalytic pores that are modeled as straight cylinders. These catalyst-dependent strncture factors provide the final tools that are required to calculate the effective intrapellet diffusion coefficients for reactants and prodncts, as well as intrapellet Damkohler numbers. The following conditions are invoked ... 

Larger voids can be modeled as delaminations. Their effect on stiffness can be accounted for by combining the sublaminates into which they divide the original laminate to determine the corresponding stiffness properties (combination of springs in parallel and/or in series). For example, with reference to Figure 6.9, the laminate stiffness in the x direction accounting for individual voids can be approximated by ... 

The parallel array of fibrils and voids may be the more appropriate model for the fiber after orientational drawing, for example. In any event, fibers that tend to have relatively small values of R, whether it be Rr, Rc, or Rs, and large values of can be described as having a fine microvoid structure, and vice versa large R values and small Ns values are characteristic of a coarse microvoid structure. 

Damkohler (1935) is perhaps one of the pioneers to develop diffusion models to understand adsorption process within a particle. In his development he assumed a parallel transport of molecules occurring in both the void space as well as the adsorbed phase (Figure 9.2-1). 

First works modelled the small plasticizer molecules as being partly attached to the macromolecules with a pending portion acting as lubricants. Thereby, plasticizer molecules should have a specific chemical structure in order to create points of attraction with the polymer chains and leave an unattached portion. Then, this lubrication model was extended with the notion of voids filled by plasticizer between the macromolecules, establishing a structure of parallel alternate layers of polymers lubricated by strata made of plasticizers, which break intermolecular bonds between polymer chains. Furthermore, for Houwink the quantify of broken links between macromolecules was dependent on the swelling rate induced by the dissolving ability of the plasticizer, which is higher when the respective polarities of polymer and plasticizer are similar. 

Gm) and micropores (Gp), as well as separate mean pore radii for macropores (am) and micropores (a ). In the microregion, df, is calculated in the same manner as in the parallel-pore model using Sg and Vg, while in the macroregion related to the void spaces between primary particles, Am is obtained from pore-volume distribution. When Gm = 0, a monodisperse model similar to the parallel-pore model is obtained [9, 17, 34]. 

One important, but often not clearly defined variable in the characterisation of porous membranes, is the shape of the pore or its geometry. In order to relate pore radii to physical equations, several assumptions have to be made about the geometry of the pore. For example, in the Poiseuille equation (see eq. IV 4) the pores are considered to be parallel cylinders, whereas in the Kozeny-Carman equation (eq. IV - 5) the pores are die voids between the close-packed spheres of equal diameter. These models and their corresponding pore geometries are extreme examples in most cases, because such pores do not exist in practice. However, in order to interpret the characterisation results it is often essential to make assumptions about the pore geometry. In addition, it is not the pore size which is the rate-determining factor, but the smallest constriction. Indeed some characterisation techniques determine the dimensions of the pore entrance rather than the pore size. Such techniques often provide better information about permeation related characteristics. 

Fiber-reinforced systems have been modeled with use of an MC method to place parallel fibers into a polymer matrix, with a finite element algorithm (FEA) then being used to compute elastic properties (274). A generic meshing algorithm for use in FEA studies of nanoparticle reinforcement of polymers has been developed (275) and applied to the calculation of mechanical properties of whisker and platelet filled systems. The method should be applicable to void-containing low dielectric materials of such great utility in the semiconductor industry. 

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