Infinite polymer

Figure 1. n band edges for neutral CnHn+2- The right hand side points correspond to an infinite polymer (CH)n. 

For low-Reynolds-number fluids the second term in the right-hand side of the Navier-Stokes equation can be neglected. Additionally, assuming that the viscous relaxation occurs more rapidly than the change of the order parameter, the acceleration term in Eq. (65) can be also omitted. Such approximations are validated in the case of polymer blends, for which they become exact in the limit of infinite polymer length, N —> oo. After these approximations, the NS equation can be easily solved in the Fourier space [160]. 

Depending on the number of carbon atoms between the two nitrogen atoms, dimers or infinite polymers have been obtained, the crystallographic structures of which have been reported [92]. 

Distorted ZnS4tetrahedra Zn-S 2.345,2.337,2.351 and 2.401 A One dithiophosphate bidentate One dithiophosphate bridges Zn atoms Infinite polymers... 

Several oxohalides, MOX2, have been recorded (Table 44) the structure of PaOCl2 consists of an infinite polymer in which the Pa atoms are seven-, eight- and nine-coordinate.143... 

Two theoretical approaches for calculating NMR chemical shift of polymers and its application to structural characterization have been described. One is that model molecules such as dimer, trimer, etc., as a local structure of polymer chains, are in the calculation by combining quantum chemistry and statistical mechanics. This approach has been applied to polymer systems in the solution, amorphous and solid states. Another approach is to employ the tight-binding molecular orbital theory to describe the NMR chemical shift and electronic structure of infinite polymer chains with periodic structure. This approach has been applied to polymer systems in the solid state. These approaches have been successfully applied to structural characterization of polymers... 

Sometimes the estimation of the electronic structures of polymer chains necessitates the inclusion of long-range interactions and intermolecular interactions in the chemical shift calculations. To do so, it is necessary to use a sophisticated theoretical method which can take account of the characteristics of polymers. In this context, the tight-binding molecular orbital(TB MO) theory from the field of solid state physics is used, in the same sense in which it is employed in the LCAO approximation in molecular quantum chemistry to describe the electronic structures of infinite polymers with a periodical structure -11,36). In a polymer chain with linearly bonded monomer units, the potential energy if an electron varies periodically along the chain. In such a system, the wave function vj/ (k) for electrons at a position r can be obtained from Bloch s theorem as follows(36,37) ... 

In this introduction, the history of dendritic molecules is limited to initial developments. It also warrants mention that theoretical considerations of infinite (polymer) networks by Flory can be traced back to 1941 [17]. 

The allyl lariat ether (Section 3.3) 3.132 forms complexes with both K+ and Ag+ (which are of similar ionic radius, Table 3.5). In the case of the K+ complex, as may be expected, the potassium ion is too large to fit snugly within the aza[15] crown-5 ring and lies somewhat above the donor atom plane. The exposed face of the metal atom is occupied by a PF6" anion. Similar coordination to the crown is exhibited by Ag+ but instead of an anion, the exposed face of the metal ion is coordinated to the allyl side chain of an adjacent molecule to give an infinite polymer in the solid state (Figure 3.91). 

An ab initio study of the geometries, polarizabilities, and rotational barriers of H(PhSiH) H oligomers has been reported.104 The trends in polarizabilities for isotactic and syndiotactic oligomers with = 4 8 were extrapolated to the infinite polymer limit. 

A more realistic situation for diffusion in a laminate is illustrated in Fig. 7-14b, which shows the solute concentration profile in the barrier layer after a short contact time t=tj. In this illustration the concentration profile of the solute just reaches the polymer/liquid interface and cL.t = 0. If we now consider a similar case with a semi-infinite polymer system with the initial solute concentration (cP>e) at the distance x < xQ = a+b/2 and cP=0 at x>x0 and t=0 (Fig.7-14c), then the possible concentration profiles for the three different times, tctj, t=t, and t>tj can be illustrated in Fig. 7-14d. If we assume a mass transfer through the interface A at x=x at t=t, in Fig. 7-14d, then mpt/A = 0.5cpepp(d-X ), which corresponds to mP, /A= cPepp(x0-a) = cPeppb/2 in Fig. 7-14c. If we combine this result with Eq. (7-54) for t=t, then we obtain the time... 

Figure 11-1 Predicted relative migration using Equations (11-2) and (11-3) (V) for many values of Dt/L2 and a. The relative migration assuming finite polymer to a finite amount of food and an infinite polymer and infinite amount of food is illustrated.

A method which would seem to have particular relevance to hydrogen-bonded systems in view of the Coulombic nature of the longer-range hydrogen-bond forces is one which evaluates long-range Coulombic interactions within the framework of the LCAO-MO method [314]. Hitherto this method has been applied to infinite polymers, where comparison with experimental structural data is not possible. 

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