Network chains molar density

The network chain molar mass can not only be calculated from the network density, but also from what is known as the cross-link index, y. The cross-link index gives the number of monomeric units which are cross-link points per primary polymer chain. A primary polymer chain is the linear macromolecule which existed before cross-linking. 

In the case of unsaturated polyesters, nondegraded samples made from a prepolymer of molar mass M and a styrene mass fraction s have a chain-ends concentration b = [2(1 — s)/M]p, where p is the density. If ve is the actual concentration of elastically active network chains, an ideal network would be obtained by welding each chain end to another one, leading to... 

The network polyurethane obtained by reacting a diisocyanate R(N=C=0)2 with pentaerythritol C(CH20H)4 contains, according to elemental analysis, 0.2% (w/w) nitrogen and has a density of 1.05 g/cm. Determine (a) polymer chain segment density in mol/cm, and (b) molar mass of chain segments between branch points. 

A cross-linked rubber of undeformed cross-section 30 mm x 2 mm is stretched at 300 K to twice its original length by subjecting it to a load of 15 N. If the density p of the rubber is 950 kg m, what is the mean molar mass M of the network chains ... 

Characterizing cross-linked networks according to molar mass is unrealistic in regard to their infinite size. Such cross-linked networks are classified according to the network chain length, kind of cross-link, and cross-link density. Here a cross-link point is defined as a group from which more than two network chains extend. A network chain corresponds to that portion of the chain which joins two cross-link points together. 

The degree of cross-linking, Xc, also known as the cross-link or network density, is the mole fraction of monomeric units which are actual cross-link points with respect to the total number of monomeric units. The number average molar mass of a network chain may be calculated from the molar mass, Mu, of monomeric units and the cross-link density ... 

The average molecular mass of the network chains between two neighboring network nodes Me and the crosslink density Vc of crosslinked polymer networks could be calculated by means of the Flory-Rehner equation [38] on the basis of swelling measurements. If 02 is the volume fraction of the polymer in the swollen system and Vi the molar volume of the solvent, the following equation holds for the crosslink density Vc in a tetra-functional network ... 

A typical network studied in this regard might have been tetrafunctionally cross-linked in the undiluted state (U2s = 1.00), and exhibit an equilibrium degree of swelhng characterized by V2m = 0.100 in a solvent having a molar volume V = SOcm mor (8.00 x 10" mm mol ) and an interaction parameter with the polymer corresponding to xi = 0-30. Calculate the network-chain density [9]. 

To determine N it is assnmed that there are no free chain ends or loops in the network. Even if these exist, they do not contribute to its elastic energy. Assuming that all network chains are fixed at two crosslinks, the density of the polymer is p = NMc/Na, where iWc is the number average molar mass of the segments between crosslinks and Na is Avogadro s number. Then Eq. (2.43) can be rewritten as... 

Telechelic polymers rank among the oldest designed precursors. The position of reactive groups at the ends of a sequence of repeating units makes it possible to incorporate various chemical structures into the network (polyether, polyester, polyamide, aliphatic, cycloaliphatic or aromatic hydrocarbon, etc.). The cross-linking density can be controlled by the length of precursor chain and functionality of the crosslinker, by molar ratio of functional groups, or by addition of a monofunctional component. Formation of elastically inactive loops is usually weak. Typical polyurethane systems composed of a macromolecular triol and a diisocyanate are statistically simple and when different theories listed above are... 

O is the stress per unit unstrained area, G the shear modulus, A the deformation ratio, p the density of the dry network. iJ>2 volume fraction of polymer present in the network, V the volume at formation. A=1 for affine behaviour (expected) and 1-2/f for phantom behaviour(1,3). is the molar mass for the perfect network, essentially the molar mass of a chain of v bonds, the number which can form the smallest loop (5-7) see Figure 2. is equal to the... 

Here, vmech is the mechanically effective chain density specified, e.g., in [168], Ac 0.67 [170] is a microstructure factor which describes the fluctuations of network junctions, Na the Avogadro number, p mass density, Ms and Zs molar mass and length of a statistic segment, respectively, kB the Boltzmann constant, and T absolute temperature. 

A model network should, at least, satisfy the following conditions [5] (i) the linear chain element of a model network should exhibit known length and, if possible, a narrow molar mass distribution ( >m)- ch elastic chain should be connected by its two ends to two different crosslink points (ii) a model network should be homogeneous crosslinking density should be constant throughout the gel and (iii) a model network should exhibit a known and constant functionality of crosslink points. 

In this equation, (l — f ) is the volume fraction of the diluent in the network phase at the moment of phase separation (or, which is the same, in the solution prior to phase separation), Uj the volume fraction of the network phase, i/ (mol/cm ) the concentration of the network active chains in unit dry volume of the gel (which is proportional to the concentration of crosshnks), Xi3 the Flory—Huggins parameter, Fj the partial molar volume of the diluent, and subscripts 1 and 3 relate to the diluent and polymer, respectively. Notably, Eq. [3.6] incorporates four independent parameters the thermodynamic quahty of the diluent Xi3> its partial molar volume Fj, the dilution of the system 1 — Vj, and the varying crosslinking density zz. ... 

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