Elastically active chains concentration

Chemical aging resulting from water absorption (i.e., hydrolysis) has not been as widely studied as physical aging. It is relatively well understood at the molecular scale (chemical mechanisms). But macromolecular (kinetics of decrease of the elastically active chain concentration) and mechanical aspects (effect of chain scissions on mechanical properties) are far from being elucidated. 

The equilibrium shear modulus of two similar polyurethane elastomers is shown to depend on both the concentration of elastically active chains, vc, and topological interactions between such chains (trapped entanglements). The elastomers were carefully prepared in different ways from the same amounts of toluene-2,4-diisocyanate, a polypropylene oxide) (PPO) triol, a dihydroxy-terminated PPO, and a monohydroxy PPO in small amount. Provided the network junctions do not fluctuate significantly, the modulus of both elastomers can be expressed as c( 1 + ve/vc)RT, the average value of vth>c being 0.61. The quantity vc equals TeG ax/RT, where TeG ax is the contribution of the topological interactions to the modulus. Both vc and Te were calculated from the sol fraction and the initial formulation. Discussed briefly is the dependence of the ultimate tensile properties on extension rate. 

Studies have been made of the elastic (time-independent) properties of single-phase polyurethane elastomers, including those prepared from a diisocyanate, a triol, and a diol, such as dihydroxy-terminated poly (propylene oxide) (1,2), and also from dihydroxy-terminated polymers and a triisocyanate (3,4,5). In this paper, equilibrium stress-strain data for three polyurethane elastomers, carefully prepared and studied some years ago (6), are presented along with their shear moduli. For two of these elastomers, primarily, consideration is given to the contributions to the modulus of elastically active chains and topological interactions between such chains. Toward this end, the concentration of active chains, vc, is calculated from the sol fraction and the initial formulation which consisted of a diisocyanate, a triol, a dihydroxy-terminated polyether, and a small amount of monohydroxy polyether. As all active junctions are trifunctional, their concentration always... 

The statistical theory of crosslinking used in the last section also gives the theoretical concentration of elastically-active chains, N, which in turn determines the rubbery modulus E = 3NRT (R is the gas constant and T is the absolute temperature). At 70% reaction one calculates E - 2 x 10 dyn/cm1 2 3 4 5 6 7 8 9 10, in agreement with the apparent level in Figure 1. 

However, in doing so one tests two theories the network formation theory and the rubber elasticity theory and there are at present deeper uncertainties in the latter than in the former. Many attempts to analyze the validity of the rubber elasticity theories were in the past based on the assumption of ideality of networks prepared usually by endllnklng. The ideal state can be approached but never reached experimentally and small deviations may have a considerable effect on the concentration of elastically active chains (EANC) and thus on the equilibrium modulus. The main issue of the rubber elasticity studies is to find which theory fits the experimental data best. This problem goes far beyond the network... 

To a first approximation, which neglects changes in average chain structure, the loss in elastically active junction point concentration may be translated directly into loss in concentration of elastically active chains and increase in the value of M, . For a perfect network in the dry state, the concentration of elastically active chains is given by the equations... 

The description of a network structure is based on such parameters as chemical crosslink density and functionality, average chain length between crosslinks and length distribution of these chains, concentration of elastically active chains and structural defects like unreacted ends and elastically inactive cycles. However, many properties of a network depend not only on the above-mentioned characteristics but also on the order of the chemical crosslink connection — the network topology. So, the complete description of a network structure should include all these parameters. It is difficult to measure many of these characteristics experimentally and we must have an appropriate theory which could describe all these structural parameters on the basis of a physical model of network formation. At present, there are only two types of theoretical approaches which can describe the growth of network structures up to late post-gel stages of cure. One is based on tree-like models as developed by Dusek7 I0-26,1 The other uses computer-simulation of network structure on a lattice this model was developed by Topolkaraev, Berlin, Oshmyan 9,3l) (a review of the theoretical models may be found in Ref.7) and in this volume by Dusek). Both approaches are statistical and correlate well with experiments 6,7 9 10 13,26,31). They differ mainly mathematically. However, each of them emphasizes some different details of a network structure. 

Vg/Vg effective concentration of elastically active chains in non-swollen sample ( 2 volume fraction of polymer in swollen sample T absolute temperature Eq modulus of the non-swollen sample. 

In general, swelling of real networks in liquids of various chemical structure not only causes the dilution of a network but also destroys physical interactions. The last is a result of interaction of fragments of polymer chains with molecules of liquid. Usually the effective concentration of elastically active chains (network density) is represented by the sum of concentrations of chains determined by the interchain chemical crosslinking and physical bonds.2 ... 

The calcnlations using Eq. [10.33] show that irrespective of polymer and plasticizer nature when m 3, the concentration of elastically active chains, (v/VX, in swollen elastomers decreases more than 30 times at the volume fraction of a solvent ( ) = (1-( )2) = 0.7. 

Hydrogen bonding makes a major contribution to the intermolecular interactions in PBU and the presence of H-bonds contributes to mechanical properties. As a result of the swelling, the concentration of elastically active chains in a labile network of elastomer is reduced. The theoretical evaluation of change in H-bond concentration in PBU on plasticization by a proton acceptor (DOS) shows that decrease in concentration of H-bonds occurs even with small amounts of plasticizer (< )i = 0.05 - 0.2). The density of a labile network determined by H-bonds decreases sharply and modulus and tensile strength of polymer decrease as well. Figures 10.71 and 10.72 show that the character of influence of a plasticizer on elastomer is similar for two very diiferent strain rates. This allows to carry out evaliration of the physical network density of elastomer under the non-equilibrium conditions of deformation using Eq. [10.33] as previously reported. ... 

Vj/V volume concentration of elastically active chains f the functionality of polymer network... 

The most important molecular parameter characteristic of a polymer network is the concentration of the elastic chains or that of the crosslinks connecting the macromolecules. An active junction is joined by at least three paths to the polymer network and an active chain is defined as one terminated by active junctions at both ends. There are several ways to express the extent of crosslinking (1) the concentration of the elastically active chains, r ei/Po, where v is the number of chains connecting two elastically active junctions and To is the volume of the dry network, (2) the molecular weight of the polymer chains between the junctions... 

Let us consider an ideal network in which every chain is connected to crosslink nodes at both extremities. Such chains are called elastically active chains (EACs). Their concentration Vq is linked to the concentration Xq of... 

The effective crosslink density, px, is defined as the concentration of elastically active chains (chains which are deformed by an applied stress) in the polymer network, and is usually reported on the basis of moles of chains per cubic centimeter of dry polymer. Network structure can also be described with a number of closely related terms [28]. For example, when linear polymers are crosslinked, it Is often desirable to express crosslinking in terms of the number average molecular weight of the polymer before crosslinking, Mp, and the number average molecular weight between crosslinks,... 

Concentration of elastically active chains in the co-polymerization of styrene with 15% p-divinylbenzene in the absence of a diluent (curve 1) and in the presence of 40% toluene (curve 2) dashed curve corresponds to the theoretical values of the overall concentration of network chains. 

For imperfect epoxy-amine or polyoxypropylene-urethane networks (Mc=103-10 ), the front factor, A, in the rubber elasticity theories was always higher than the phantom value which may be due to a contribution by trapped entanglements. The crosslinking density of the networks was controlled by excess amine or hydroxyl groups, respectively, or by addition of monoepoxide. The reduced equilibrium moduli (equal to the concentration of elastically active network chains) of epoxy networks were the same in dry and swollen states and fitted equally well the theory with chemical contribution and A 1 or the phantom network value of A and a trapped entanglement contribution due to the similar shape of both contributions. For polyurethane networks from polyoxypro-pylene triol (M=2700), A 2 if only the chemical contribution was considered which could be explained by a trapped entanglement contribution. 

To obtain accurate values of the sol, thin specimens (1 mm) in one study (13) were kept in the solvent for six weeks in another study (14), thin specimens were extracted for more than 18 days in Soxhlet extractors. When the present experimental data were obtained (6), there was little interest in knowing the sol fraction accurately. However, as discussed subsequently, to compute the extent of the curing reactions and the concentration of elastically active network chains, the sol fraction must be known accurately. 

---