Extended volume

It is clear that the introduction of a critical volume fraction is a step toward dealing with percolation on a continuum. To this end Zallen and Scher (1971) considered the motion of a classical particle in a random potential, V r), and introduced a function, which detines the fraction of space accessible to particles of energy E. The connection with percolation is in the fact that, for energies such that 4>(E) > c(.E), there are infinitely extended volumes of allowed (V < T) space. The critical value c is identified with 0.15 for d = 3, and delocalized states appear above c ... 

The key to controlling two-phase mixture properties throughout an extended volume is modularity of design. A two-phase mixture can be controlled on a small scale far more readily than on a large scale. In this facility, the mixture is controlled in a number of small modules. Several modules can be combined... 

Extend volume calculation to selected nonideal flow models from the ideal combinations listed in Table 16.3 this is optional... 

Fig. 6.4 High-temperature nematic GLCs with benzene and extended volume-excluding cores accompanied by DSC data from second heating scans...

For the case where abundant nuclei are present in the initial stages of the transformation, which is termed site saturation, the extended volume of one sphere of product phase is modeled by changing the sign for the rate constant in the shrinking particle model (Chapter 6) to make it a growing particle model. The linear growth rate of each sphere of the product phase is G (m/sec) and its volume is V (m ). 

An inorganic, low cost, inert material added to polymers to improve properties, and to extend volume to reduce costs. Fillers are usually solid, particulate materials. 

A review article (59) on the prediction of Tg by extending volume-temperature curves generated by molecular dynamics simulations to low temperatures. 

To treat the regime beyond the early transformation the extended volume concept is adopted. In this case, the nucleation and growth rates are separated from geometrical considerations such as impingement. The extended volume (14) is the volume that would have been formed if the entire volume had participated in nucleation and growth, even that portion transformed (VP). In this case. 

Or the Johnson-Mehl equation [72]. In these equations, the concept of extended volume ftaction is adopted. By using this concept, the hard impingement can be taken into consideration indirectly. The extended volume ftaction is the sum of the volume ftaction of all new phases without direct consideration of the hard impingement between new particles and is related to the actual volume fraction by... 

In his calculation, firstly, the area at the distance of y from the nucleation site B is calculated. The summation of this area for all nuclei gives the total extended area. From this value, the actual area can be calculated. The extended volume can be obtained by integrating the area for all distances. Finally, the actual volume fraction can be derived. 

Multiplying by the area of nucleation site, the extended volume fraction is obtained as ... 

A single Bragg peak as shown In Fig. 13 Is too narrow for irradiation of extended tumor volumes, which typically have extensions in the order of a few centimeters. Therefore, special beam delivery techniques had to be developed in order to achieve a homogenous irradiation of extended volumes. In depth this is achieved by superposition of several Bragg peaks with different peak positions, as illustrated in Fig. 30. The position of the Individual Bragg peaks is defined by the energy of the primary particles the required energy variation can be achieved by passive as well as by active methods [148,149]. 

Fig. 30. Irradiation of extended volumes with homogenous dose deposition by superposition of Bragg peaks at different depths, corresponding to different beam energies. (Courtesy U. Weber)...

Light scattering experiments are well known in the study of critical phenomena, where an increasing density-density correlation lenght and a slowing down of the decay rate of fluctuations are observed. In difference to critical phenomena the freezing transition can be studied neither at equilibrium conditions nor in an extended volume. 

In order to calculate X one needs to calculate the extended volume of the polymer... 

If it can be assumed that the time taken by a crystallite to grow to its full size is much shorter than that taken by a chain scission, then the extended volume is simply the... 

Figure 7.2 Scheme explaining the eoneept of extended volume. Extended volume V x of four domains is equal to the sum of their volumes, Vi + V2 + "3 + V4, and exeeeds the transformed volume. 

Equation (7.18) can be transformed into Equation (7.7a) by integration by parts, followed by the substitution r= t-r) G. The same reasoning applied to the 2D case gives a result equivalent to Equation (7.7b). As in the extended volume approach, instantaneous nucle-ation can be considered either by neglecting all nucle-ation events except those occurring exactly at the onset of crystallization or by substituting D5(t) for F(t). 

One notes that V(xd) is equal to the volume of unimpinged spherulite expressed by Equation (7.2a) and Equation (7.2b). In derivations based on probability calculus it is also assumed, as in the extended volume approach, that a growing sphere that passes through an arbitrary point as the first one represents a real spherulite. It appears that the concept of extended volume and probability calculus yield the same result if applied to crystallization in infinite volume with the nucleation and growth rate independent of spatial coordinates. 

To predict the conversion of melt into spherulites during isothermal crystallization of fiber-reinforced composites, the concept of extended volume was used... 

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