Tetrafunctional networks

The constrained-junction model was formulated in order to explain the decrease of the elastic moduli of networks upon stretching. It was first introduced by Ronca and Allegra [39], and Flory [40]. The model assumes that the fluctuations of junctions are diminished below those of the phantom network because of the presence of entanglements and that stretching increases the range of fluctuations back to those of the phantom network. As indicated by the second part of Equation (26), the fluctuations in a phantom network are substantial. For a tetrafunctional network, the mean-square fluctuations of junctions amount to as much as half of the mean-square end-to-end vector of the network chains. The strength of the constraints on these fluctuations is measured by a parameter k, defined as... 

Figure 9. SANS measurements of R /Rt° and RL/R ° for stretched radiation cross-linked polystyrene. is determined by measurements in which the neutron is parallel (iso) and perpendicular (aniso) to the stretching direction. Mc is molecular weight between crosslinks. Theoretical curves 2 and 3 are drawn for tetrafunctional networks. Data from Ref. 21.

The broken curves in Figure 9 are the expected values of Mc/Mc for trifunctional and tetrafunctional networks, assuming that all the ring structures formed are of the smallest size (v bonds), and that only the ring structures formed pre-gel give elastically ineffective loops(29,32). They show that intramolecular... 

Figure 8. Part of a tetrafunctional network formed from an RA t and RBi polymerization corresponding to Mc°, the molar mass between junction points of the perfect network (a). Detail of the chain structure defining Mc° for HDl reacting with an OPPE, n is the number-average degree of polymerization of each arm with respect to oxypropylene units, (b). Part of the chain structure defining v, the number of bonds in the chain forming the smallest ring structure (C), for the reaction system in (b) (29). Reproduced, with permission, from Ref. 21. Copyright 1980, Stein-...

Curves 1 and 2, and 3 to 6 in Figure 1 refer, respectively, to HDI/POP triol and HDI/POP tetrol polymerisations with different values of V. Marked reductions in modulus occur even for bulk reaction systems, which give the points at the lowest values of Pj- q for the different systems. More inelastic chains are formed in trifunctional as compared with tetrafunctional networks for a given value of pp q (cf. curves 1 and 2 with 3 to 6. In addition, for a given functionality, as V decreases the proportion of inelastic loops increases. Similar results have been obtained for polyester-forming systems using POP triols and diacid chlorides(13). 

Figure 10. Significant structures for the formation of inelastic chains in trifunctional and tetrafunctional networks.

Fig. 7. Temperature dependence of Ts at a proton frequency of 88 MHz for tetrafunctional network PDMS specimens with the following numbers of dimethylsiloxane units between network points 2 (1), 3 (2), 6 (3), 9 (4), 30 (5) and for the linear PDMS (6) (reprinted from Ref.72))...

Figure 12. Intrinsic atomic stresses <7n, Vjj. <733 as determined from molecular dynamics simulation of tetrafunctional network model in uniaxial volume deformation and for corresponding melt. (After Ref. [19].)...

Similarly, from the relationship between the number of chains and number of cross-links (Problem 3.4) for a tetrafunctional network, we obtain... 

The relative posiMons of the lines for the various systems can be related to M(.o(or v), f, and the chain structures of the reactants(1.2.9-12). The slopes of the lines show that the reduction in modulus with pre-gel intramolecular reaction is larger for trifunctional compared with tetrafunctional networks (c.f. systems 1 and 2 with 4 and 5), although higher values of p- Q obtain for tetrafunctional reaction systems (c.f. Figure 2 . 

A new, valuable type of experiment are computer simulations of network properties. Computer simulations are well established both in equilibrium and non-equilibrium physics of systems of linear chains and have been used for the study of networks and melts. The effect of the topological constraints on the stress-strain behaviour of tetrafunctional networks with a regular structure has been investigated by Elyashevich and Remeev using Monte Carlo methods to generate Marko-... 

For a tetrafunctional network where v= No/2, the change in Snet on stretching is ... 

Real networks always contain molecular imperfections, such as pendant chains bound to the network at one end only, intramolecular loops formed by linking of two units of the same chain, and intermolecular entanglements. For an imperfect tetrafunctional network Hory [4,65] proposed a simple formula for correction for pendant chains... 

A tetrafunctional network containing cyclics (heavy lines). Cyclics a and b were trapped by linear chains that passed through them prior to end linking into a network structure. 

Sharaf, M. A. Mark, J. E. Hosani, Z. Y. A., Regular Bimodal Polydimethylsiloxane Networks. Elastomeric Properties of the Tetrafunctional Networks. 

In a perfect tetrafunctional network, Pa/Vo = 2q/M and thus equation 33 leads to Mo = MJ2, which means that chain ends have been connected two by two to obtain this perfect network. 

Fig. 5. Determination of the parameters of the Flory theory for perfect tetrafunctional networks Mark (O) M = 4000, 4700, 18500gmor ...

Fig. 11. Dependence of the structure factor at zero deformation on M, for imperfect tetrafunctional networks...

As a numerical example, the affine shear modulus of a perfect poly (dimethylsiloxane) tetrafunctional network of density p — 0.97 g/cm and Me = 11,300 g/mol is Gaff = 0.212 x 10 N/m. However, imperfections such as chain ends exist in typical networks, as illustrated in Figure 6. The following correction can be made to account for this circumstance (16). Before the cross-linking reactio it is assumed that Vm chains of length are present in the melt, with... 

The fluctuation range (Ar = rg/2 for a tetrafunctional network) is generally quite large and of considerable importance. The instantaneous distribution of chain vectors r is not affine in the strain because it is the convolution of the distribution of the affine mean vector r with the distribution of the fluctuations Ar, which are independent of the strain. The elastic free energy of such a network is... 

For example. Gaff is twice Gph for a perfect tetrafunctional network. 

We propose the use of a cluster approach (28). The simplest cluster will contain four chains for a tetrafunctional network. Higher order clusters will contain larger numbers of chains with different contributions of short and long chains. Preliminary results based on such a cluster approach have shown that r decreases as one considers clusters with increasing number of chains. Such an analysis clearly indicates the necessity of taking into account the detailed network structure in describing some properties of a network. 

Thus, only half of q is subject to alteration by deformation of the network. The other half, <( ) >, representing the fluctuations, is invariant with deformation. In consequence of this circumstance, the elastic free energy for a perfect tetrafunctional network is half of the first term of eq. 6 i.e.,... 

Stress-strain relations derived from eq. 10 are of the same form as eqs. 7 and 8 derived from eq. 6. They differ only by a numerical factor arising from replacement of v by 5. For a tetrafunctional network this factor is one-half. Hence, the relationship of the retractive force to the extension for simple elongation of a tetrafunctional phantom network is given by eq. 7 or eq. 8 modified by this 16 20... 

In the ladder model treatment of Blizard, an alternative termination of the line of springs (Fig. 10-3) was considered in which each end, rather than being fixed, is attached to three other such lines, each of these to three more, and so on indefinitely, thus reproducing the connectivity of a tetrafunctional network. This.proyision increases the equilibrium compliance by a factor of 2 (corresponding-fo the factor of (/ — 2)//mentioned in Section I above), and it modifies the frequency dependence, which is now expressed by a rather complicated combination of hyperbolic functions. This frequency dependence of J" is also shown in Fig. 10-7 the maximum is slightly broader than for fixed cross-links (i.e., cross-links with affine deformation). 

FIG. 10-8. Chompff-Duiser network model. (I) Unit of tetrafunctional network (II) decoupled mathematical equivalent. ... 

Sharaf MA, Mark JE, Hosani ZYA. Regular bimodal polydimethylsiloxane networks. Elastomeric properties of the tetrafunctional networks. Eur Polym J 1993 29 809-17. 

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