Elastic chain

Elastic chains 163 Elastomers 27, 32, 33, 63, 65 Electron affinity, monomers 150 Electrophiles 155-157, 162 —, plurifunctional 162 Electrophilicity 149... 

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

The equality of the values for f=3 implies that random gel-gel reaction in fact leads to an average of two elastically active junctions or three elastic chains lost per pair of gel-gel groups reacted. For f=4, the experimentally derived value 0.03 was on the basis of one elastically active junction lost per pair of gel-gel groups reacted. Hence, the ratio of the experimentally derived and the calculated values, 0.03/0.13 = 0.23, is the average number of elastic-... 

An alternative pathway for entanglement loss is chain scission (Fig. 3.2, process B), in which a covalent bond along the polymer main chain is broken and a stress-bearing, otherwise elastic, chain is lost. Chain scission reactions, for example, homolytic carbon-carbon cleavage, have obviously high activation energies. The stress-free rates of these reactions are therefore typically extremely low. 

We did not differentiate between the various modes of vibration (longitudinal, transversal, acoustical, optical) for the sake of simplicity. The vibrational states in a crystal are called phonons. Figure 5-2 illustrates the collective, correlated transversal vibrational motion of a linear elastic chain of particles. 

They should consist of elastically effective chains only. An elastically effective chain should connect two different crosslinks, and two such crosslinks should be tied by only one elastic chain. This means that the gel should contain no defects such as pendant chains (one end of which only is connected with a crosslink), loops (chains linked at both ends to the same crosslink), or double connections. Physical crosslinks (permanent entanglements) should be prohibited, too. 

This method was applied to synthesize various networks, with elastic chains of different nature polystyrene, polymethacrylates, polyvinylpyridine, and more recently polydienes. In some cases ethylene dimethacrylate is used to achieve cross-linking9, because of its higher electrophilicity. 

Adequate mixing has also to be provided for. Upon reaction, the polymer solution gels, and it can be expected that the precursor chains are converted into elastic chains, whereas the crosslinks should exhibit the functionality of the electrophilic reagent15,16. Suitable electrophilic reagents are tris(allyloxy)s-triazine(trifunctional) and a similar tetrafunctional compound. Plurifunctional isocyanates have also been used successfully for such syntheses17. ... 

It has been found that for a homologous series of model networks the value of d increases as the average length of the elastic chains increases. For a given network d is found to increase as the degree of swelling becomes larger. 

If the elastic chains solely connect first neighbor nodules, x2 = 1. If the proportion of chains linking crosslinks which are not first neighbors increases, the value of x2 also increases. The parameter x2 therefore characterizes the degree of interpenetration of the network chains, and it should be related to the proportion of entanglements present in the network. In practice, it can be expected that for networks in which the functionality of the crosslinks is low, entanglements are quite unlikely, and the value of x2 should stay close to unity. 

It follows that the nascent b networks are best fitted for defining a reference state, because at the swelling ratio Qe the elastic chains are supposed to have undergone the smallest conformational changes with respect to the free chains they were before crosslinking. 

Another possible choice for the reference state is the equilibrium swelling degree of the network. In that case the thermodynamic characteristics of the actual polymer-swelling solvent system would be taken into account. However this choice does not consider the conditions of network formation. Moreover when the gel is swollen to equilibrium, the elastic chains are extended with respect to the corresponding free chains, and the extension ratio is not easy to evaluate13. ... 

Attention should be drawn to the fact that if the state with the swelling ratio Qc( u"1) of the nascent network is chosen as the reference state, the memory term h2 3 should be independent of the molecular weight of the elastic chains. This point is still somewhat controversial, and though experimental data support this statement in several cases35,36 there are other cases in which the opposite was observed14,22. ... 

Another approach to this problem arises from the fact that the number of crosslinks in a network can be evaluated directly. The overall concentration of polymer in the swollen network being v (the reciprocal of the volume swelling degree Q), the number of elastic chains per unit volume of swollen network is given by ... 

This expression can be used to evaluate the intercrosslink distance dc in the nascent gel, whereby Qc = t "1 is the swelling degree of the gel upon its formation. However the intercrosslink distance d can only be considered identical with (r2)1/2 - the root mean square end-to-end distance of the elastic chains - if the overwhelming majority of the elastic chains connect first neighbor crosslinks, i.e., when the elastic chains hardly interpenetrate. 

If the network — obtained by endlinking — is ideal, all precursor chains have been converted into elastically effective network chains, and v can be expressed in terms of the molecular weight M of the elastic chains by ... 

From Eq. (11), and taking into account the expression v = (M F02)-1, valid for ideal networks, it can be seen that the analytical relationship between the equilibrium swelling degree Q and the molecular weight M of the elastic chains is rather complicated. Several additional points have to be considered ... 

The experimentally observed proportionality between Q and M3,s requires that the difference (A h2 3 — B Q 2 3) varies only very little with the molecular weight of the elastic chains or the swelling degree, respectively. If the segment concentration of the nascent network is taken as the swollen reference state, h can be considered as a constant. This implies that either B is zero, or B Q 2 3 is very small compared with A h2 3, provided x is a constant. If h is not a constant — this will be discussed in the next section — one has to assume that the molecular weight dependence of A h2/3 and of B Q 2 3 are compensating, in order for the difference to remain independent of the molecular weight. 

It is thus possible to account for the type of variation of Q with the molecular weight of the elastic chains, but a quantitative check is doubtful in most cases, because of the number of parameters which have to be introduced. 

Here b and bQ are the average volumes occupied by an elastic chain in the dry undeformed state and in the swollen reference state, respectively. The ratio 60 should therefore be identical with the term h, previously discussed. 

Comparison is thus rendered possible between networks with elastic chains of the same average length, but exhibiting different crosslink functionalities. From the ratio... 

Table 2. Comparison of PS networks prepared with various amounts of DVB with three-functional PS networks containing elastic chains of comparable length. /ca c. has been determined from the F(f) curve in Fig. 9...

This means that the connectivity factor x2 — which was not taken into account — should be close to unity the vast majority of the elastic chains connect first neighbor crosslinks, as long as the average functionality does not exceed S or 6. For PDMS networks the fit is only good up to / = 4. For polystyrene networks the limiting functionality is attained for a proportion of DVB per living end of the order of 5. (Table 3). 

Homologous series of networks, prepared under the same experimental conditions, and differing only by the length of their elastic chains have been made available, and were used to investigate their structure, their swelling behavior, and their mechanical deformation properties. It was assumed that within a homologous series of networks the crosslink functionality remains constant, an assumption which is supported by results on star-polymers11. ... 

A model network can thus be schematized by an ensemble of spring-suspended beads the elastic chains act as entropy springs, and they connect beads of constant — as yet unknown — functionality. 

If the functionality of the crosslinks is low, the probability for elastic chains to connect first neighbor crosslinks is close to unity. Consequently there is little interpenetration of the elastic chains and few permanent entanglements are expected. However this does not mean that all first neighbor crosslinks in the gel are connected by elastic chains. 

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