Number per chain

Entanglements already exist in a system of linear chains as temporary junction points and their number per chain can be calculated from... 

Table 3. The average length of interface, bridge and loop segments (Ti, ifc and L, respectively), their average number per chain (Ni, JV), and Ni, respectively) and the average number of direct connections per chain Nd) obtained from the simulations of dense systems...

A polymer is made up of macromolecules, that is, long chains of covalently linked chemical units, called monomer units. In many synthetic macromolecules, these units are identical, and the term polymer implies their number per chain exceeds about 10, although it may reach over 10 in many instances. Branched and highly crosslinked architectures are also important in practice, but it is above all the presence of long linear monomer sequences that leads to the unique properties of polymers. For conciseness, therefore, the emphasis in the present overview of mechanical properties will be on isotropic, chemically homogeneous polymers composed of flexible linear or lightly crosslinked macromolecules. 

The statistical segment number per chain part between chemical crosslinking... 

Can branching be ruled out, since it would obviously make the number of chain ends per molecule an unknown quantity ... 

With either S or H as additives, two different kinds of chain are present A BAB. .. ABAB and A BABAB. . . ABABAA with S or B BABAB. . . ABAB with H. In these cases the total number of acid and base groups counts the total number of chain ends. There are two equivalents of total chain ends per mole of chains. Therefore, for S, [total ends] = 39.9 + 2.4 = 42.3 and... 

We need to know the number of chains per unit volume and must calculate the result on this basis ... 

This expression gives the average entropy change per chain to get the average for the sample, we multiply by the number v of subchains in the sample. The total entropy change is... 

Note that as M the absolute number of chain ends per unit volume decreases, as does the chain-end correction. 

The percolation term [p — determines the number of bonds to be broken, or disentangled such that when E chains, each with L/L entanglements per chain, interdiffuse in an interface of width X, we obtain... 

Protein M, Number of Residues per Chain Subunit Organization... 

Only one polymer molecule is produced per mole of inhibitor. The inhibitor must be at least equimolar with the number of chains formed. Concentrations must be chosen (usually very low) to give the desired molecular weight. 

The Alexander approach can also be applied to discover useful information in melts, such as the block copolymer microphases of Fig. 1D. In this situation the density of chains tethered to the interface is not arbitrary but is dictated by the equilibrium condition of the self-assembly process. In a melt, the chains must fill space at constant density within a single microphase and, in the case of block copolymers, minimize contacts between unlike monomers. A sharp interface results in this limit. The interaction energy per chain can then be related to the energy of this interface and written rather simply as Fin, = ykT(N/Lg), where ykT is the interfacial energy per unit area, q is the number density of chain segments and the term in parentheses is the reciprocal of the number of chains per unit area [49, 50]. The total energy per chain is then ... 

Fig. 38. Action of entanglements in simple shear degradation. The tension on the qth link from the center is given by fq = f0(l — 4q2/n2), where f0 qsy f(L) is the tension on center link (f(L) = function of chain length) and n = number of links per chain.

Degrees of polymerization can be calculated from quantitative 13C NMR data by considering the number of substituted (reacted) relative to unsubstituted (not yet reacted) ortho and para phenolic carbons where [5] is the sum of substituted ortho and para carbons and [5] + [f/ is the total ortho and para carbons. The fraction of reacted ortho and para sites is denoted by fs [Eq. (7.2)]. Thus, the number-average number of phenol units per chain Or) can be calculated using Eq. (7.3). This leads to a simple calculation of Mn = x x 106 — 14 ... 

A hydrogel is formed by a water-soluble polymer that has been lightly crosslinked. Hydrogels swell as they absorb water but they do not dissolve. The volume expansion is limited by the degree of crosslinking. The minimum number of crosslinks needed to form a three-dimensional matrix is approximately 1.5 crosslinks per chain, and this yields the maximum expansion possible without separation of the chains into a true solution. Thus, a hydrogel may be more than 95% water and, in that sense, has much in common with living soft tissues. 

B is again defined by Eq. (12), but with equal to the number of chain atoms per repeating unit (i.e., per pair of the alternating structural units) Mq is the mean molecular weight structural unit. It follows that... 

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