Orientation Equation

The 1,3-dianions formed across the sulfone of ) -ketosulfones may be selectively dialkylated with an a, co-dihalide and thus cyclize to give 2-ketothiane dioxides . Due to its polarity, the 2-keto-substituent (or other polar group in the 2-position) adopts the axial orientation (equation 124). 

Where q0 is the length efficiency factor related to the aspect ratio of the filler and thus the stress build-up along its length it can take values from 0 to 1 and can be calculated using theoretical models, q, is an orientation factor that takes values of 1 for perfect alignment, 3/8 for alignment in the plane and 1/5 for random orientation. Equation (8.1) can be arranged as ... 

Chelation is another driving force that provides diastereoselective bromine-lithium exchange reactions to give cyclopropyl carbenoids. Thus, the exo-bromine atom in dibro-mocyclopropane 25 is exchanged exclusively due to the methoxy substituent, which encourages the lithium to occupy the cis orientation (equation 16) ° Several representative examples of cyclopropyl bromo lithium carbenoids obtained by bromine-lithium exchange reactions are given in Table 1. 

Table 17), but cyclization of -substituted carbamates is highly selective (entries 2,4 and 5), presumably by insuring reversible formation of the intermediate prior to benzyl cleavage and steric interactions between the -substituent and the alkyl substituent favoring the axial orientation (equation 61). Cycliza-tions of this type, where R = H and Y = a-phenethyl, result in little asymmetric induction. However, the diastereomeric products can be separated and used for the synthesis of chiral nonracemic amino alcohol derivatives.164... 

The next stage requires equilibration and isomerization of 69 to 68, giving a 9 1 ratio of desired to undesired product. A Ni catalyst is also used for this reaction. Finally, the third stage consists of two transformations, where 68 is first isomer-ized to 4-pentenenitrile (70) under kinetic control, fortunately without producing much of the thermodynamically more stable 2-pentenenitrile (71). Compound 70 then undergoes a second hydrocyanation with anti-Markovnikov orientation (equation 9.40). In this last Ni-catalyzed stage of the overall process, a Lewis acid, such as Ph3B, is added to ensure that linear rather than branched product (72) forms. 

In the kinetic equation (4.35), transient free energy of the elastic chain deformation controls ratio of the rate constants (4.36) while effects of molecular orientation are accounted for by the concentration factor A w 0,i). The concentration factor reduces to unity for the case of isotropic systems, assumes values above unity in the range of enhanced orientation, and below unity in the range of reduced orientation. Equation (4.35) introduces effects of molecular deformation and orientation of chain segments. 

In injection molding CAE, fiber orientation analysis is carried out combining with mold filling analysis. Using velocity distributions obtained from the mold filling analysis, the fiber orientation equation is numerically solved by a finite-difference method for predicting fiber orientation. The orientation distribution function is calculated and is applied to the estimation of mechanical properties for predicting warp age of molded parts. 

Most commonly used materials are isotropic (see section 2.5). In this case, the orientation of the load does not matter for the yield criterion. Since the principal stresses describe a stress state a completely, excepting only its orientation, equation (3.23) can be rewritten using principal stresses if the material is isotropic ... 

In going to higher Bo fields, the resolution of ENDOR signals belonging to the same type of nucleus of the same paramagnetie species only improves as a result of the inereased orientation selection (less contributing orientations equates to sharper... 

The orientation equation of the pom-pom model, Eq. 11.45, is numerically expensive to solve at every time step of a simulation, an integral over the past history of deformation must be carried out. This can be avoided by using a differential approximation [95] ... 

A third order predictor-corrector method (14) was used to Integrate the center of mass motion (equation (9)), while a second order method was used for orientational equations of motion of the molecules (equations (8) and (7)). 

The end-to-end vectors of the subchains have distributions in their length and orientation. Equation [10] clearly indicates that the deviatoric part (measurable part) of the stress tensor due to the entropy elasticity of the polymer chains, hereafter referred to as the polymeric stress, reflects the orientational anisotropy of the subchains specified by the configuration tensor S(n,t). Consequently, the polymeric stress relaxes, even though the material keeps its distorted (e.g., sheared) shape, when the orientational anisotropy induced by tbe applied strain relaxes tbrougb tbe tbermal motion of tbe cbains. (In tbis relaxed state, S(n,t) is equal to 1/3 and tbe subcbain tension is transmitted isoUopically in aU tUrecrions to balance tbe isotropic pressure.) Thus, tbe relaxation time of the polymeric stress is identical to the orientational relaxation time of the polymer chains. 

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