Journal of Applied Mathematics and Computing (JAMC) is a broad-based journal covering all branches of computational or applied mathematics with special encouragement to researchers in theoretical computer science and mathematical computing. It covers all major areas, such as numerical analysis, discrete optimization, linear and nonlinear programming, theory of algorithms, computational logic, applied combinatorics, coding theory, cryptograhics, fuzzy theory with applications, differential equations with applications.
Engineering Mathematics Das Pal Vol 1 Pdf Free 44
Free volume is a significant boundary that portrays polymer properties. The free volume idea has been effectively applied in polymer material science to portray different properties of polymers like versatility, consistency, and so on. The quantitative measurement of free-volume size and number has grown in popularity and significance. The free-volume size determination can be performed using the positron annihilation lifetime (PAL) technique. It is a non-destructive method with a minimum free-volume size determination of around 1 Å. The positron annihilation lifetime fundamentals will be discussed in the second section. Li et al. [18] concentrated on the free-volume varieties of Nylon-12/PVA melt-blends treated in supercritical carbon dioxide (ScCO2) using PAL spectroscopy. They found that the free volume opening size and its items declined with the increment in PVA content. The free-volume holes in Nylon-12/PVA melt-blends expand to varying degrees after 1 h in ScCO2 at 50 C and 20 MPa, depending on the weight proportion of PVA. After delivering CO2, the free-volume size diminishes as a component of the passed time because of subatomic unwinding, and the unwinding time steadily relies upon the substance of PVA in the melt-blends.
The component associated with the annihilation of the o-Ps state is generated by pick-off with the encompassing electrons. The o-Ps lifetime is supposed to rely upon the free-volume hole sizes in polymer materials because the annihilation rate λ3 (λ3 = 1/τ3) of o-Ps is relative to the crossover of the positron and pick-off electron wave capabilities. A connection between the free volume in subatomic frameworks and the noticed o-Ps lifetimes has been figured out from the positronium hole theory [41]. This gives quantitative data about the free-volume size from the PAL spectrum. The average of the o-Ps hole volume radius (R), which was tested using the o-Ps lifetime (τ3), was determined from the accompanying connection [42,43]:
In a polymer blend, the free volume does not simply comprise the addition of each individual free volume; instead, the relative fractional of the free-volume hole of the blend can be expressed as [81]:
All these results reflect that the compatibilizer had a strong effect on the free-volume properties in the blend. However, the information about the free volume that we obtained from the finite term analysis of the positron lifetime spectra is limited because it gives only a few averaged lifetimes. In other words, what we obtained from this analysis is the average free-volume size. However, most of the time, the free-volume size has a distribution, and obtaining this information will be more helpful for us to analyse the detailed microstructural information of polymers and polymer blends. In the present work, the LT10 program [38] was used to analyse the measured spectra. The lifetime distribution data are shown in Figure 9. As shown in Figure 9A, in the pure and blended polymers, the o-Ps lifetime has one clear peak located in the range from 0.5 to 3.0 ns. The shorter lifetime peak is close to the o-Ps lifetime in PVA, and the longer lifetime peak is related to the o-Ps lifetime in Nylon-6,6. In addition, the o-Ps lifetime distributions for the blended samples were located between these peaks for the pure polymers. By increasing the Nylon-6,6 content in the blended PVA, the o-Ps distribution became narrower. The behaviours for the o-Ps hole radius [Figure 9B] and o-Ps hole volume size [Figure 9C] are similar to those of the o-Ps lifetime distribution. The one clear o-Ps lifetime peak (free-volume size peak) for the blended samples shows strong evidence of no phase separation. Because this blend is miscible, the dispersed Nylon-6,6 tends to have as small a domain size as possible in the interaction with PVA polymeric chains. Therefore, o-Ps will form and annihilate in one phase. 2ff7e9595c
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