Perfect network

The next step in the development of a model is to postulate a perfect network. By definition, a perfect network has no free chain ends. An actual network will contain dangling ends, but it is easier to begin with the perfect case and subsequently correct it to a more realistic picture. We define v as the number of subchains contained in this perfect network, a subchain being the portion of chain between the crosslink points. The molecular weight and degree of polymerization of the chain between crosslinks are defined to be Mj, and n, respectively. Note that these same symbols were used in the last chapter with different definitions. 

Comparing this result with Eq. (3.1) shows that the quantity in brackets equals Young s modulus for an ideal elastomer in a perfect network. Since the number of subchains per unit volume, i /V, is also equal to pN /Mj, where M, is the molecular weight of the subchain, the modulus may be written as... 

The shear modulus for an ideal elastomer in a perfect network is not difficult to derive ... 

A constant force is applied to an ideal elastomer, assumed to be a perfect network. At an initial temperature Tj the length of the sample is Ij. The temperature is raised to Tf and the final length is If. Which is larger Ij or If (remember F is a constant and Tf > Tj) Suppose a wheel were constructed with spokes of this same elastomer. From the viewpoint of an observer, the spokes are heated near the 3 o clock position-say, by exposure to sunlight-while other spokes are shaded. Assuming the torque produced can overcome any friction at the axle, would the observer see the wheel turn clockwise or counterclockwise How would this experiment contrast, in magnitude and direction, with an experiment using metal spokes ... 

Molecular weight per cross-linked unit. Hence also the molecular weight per chain of a perfect network. 

Number (or number of moles) of cross-linked or branched units (Chaps. IX and XI). Hence, also the number of chains in a perfect network structure (Chap. XI). Effective number (or number of moles) of chains in a real network (Chaps. XI and XIII). 

The structure of a perfect network may be defined by two variables, the cycle rank and the average junction functionality (f>. Cycle rank is defined as the number of chains that must be cut to reduce the network to a tree. The three other parameters used often in defining a network are (i) the number of network chains (chains between junctions) v, (ii) the number of junctions p, and (iii) the molecular weight Mc of chains between two junctions. They may be obtained from and using the relations... 

It is always easy to calculate idealized scattering curves for perfect networks. The experimental systems vary from the ideal to a greater or lesser degree. Accordingly, any estimate of the correctness of a theoretical analysis which is based on an interpretation of experiment must be put forth with caution since defects in the network may play a role in the physical properties being measured. This caveat applies to the SANS measurement of chain dimensions as well as to the more common determinations of stress-strain and swelling behavior. 

The properties of a polymer network depend not only on the molar masses, functionalities, chain structures, and proportions of reactants used to prepare the network but also on the conditions (concentration and temperature) of preparation. In the Gaussian sense, the perfect network can never be obtained in practice, but, through random or condensation polymerisations(T) of polyfunctional monomers and prepolymers, networks with imperfections which are to some extent quantifiable can be prepared, and the importance of such imperfections on network properties can be ascertained. In this context, the use of well-characterised random polymerisations for network preparation may be contrasted with the more traditional method of cross-linking polymer chains. With the latter, uncertainties can exist with regard to the... 

Figure 8 illustrates(29) the close relationship between the chain defining the molar mass between junction points of the perfect network (M ) and the chain of v bonds of the preceding sections. The illustrations in (b) and (c) are for the RA2 +... 

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-...

Figure 9. Molar mass between elastically effective junction points (Mc) relative to that for the perfect network (Mc°) versus extent of intramolecular reaction at gelatin (pr,c) for polyurethane networks (29).

Even reactions in bulk, as indicated by the points at the lowest values of pr,c for the various systems, yield networks with moduli less than those predicted for the perfect networks for given reactants. 

The moduli and Tg s of the networks formed from the bulk reactions of the five systems of Figure 9 are shown in Table IV(29). The first five columns define the systems, the next two give the experimental values of G(at 298K) and Tg, and the last three give the values of pr,c, Mc, and G/G°. The last quantity is the reduction in rubbery shear modulus on the basis of that expected for the perfect network(G°). G/G° is in fact equal to M /Mc. 

The factors which influence pre-gel intramolecular reaction in random polymerisations are shown to influence strongly the moduli of the networks formed at complete reaction. For the polyurethane and polyester networks studied, the moduli are always lower than those expected for no pre-gel intramolecular reaction, indicating the importance of such reaction in determining the number of elastically ineffective loops in the networks. In the limit of the ideal gel point, perfect networks are predicted to be formed. Perfect networks are not realised with bulk reaction systems. At a given extent of pre-gel intramolecular... 

The reactants used to form the networks studied are generally of lower molar mass than those used by other workers to form perfect networks (e.g. (35)). However, the present results do indicate that very little pre-gel intramolecular reaction can have a marked effect on modulus. For example, for pr,c = 0.05, or ac = 0.58, with a trifunctional polyurethane-forming system of Me = 635g mol l, the modulus is reduced by a factor of five below that calculated on the basis of the small-strain(affine) behaviour of the perfect network. As a result, it is recommended that the effective absence of pre-gel intramolecular reaction in endlinking reactions to form perfect networks be confirmed by experiment rather than be assumed. 

The Gge contribution of the form given Eq.(2) represents the simplest form of permanent interchain interactions. The value of Gee at Teg=1 and w =1, i.e. the Gge contribution of a perfect network, has been assumed equal to the plateau modulus of the corresponding linear polymer (10,15,23). This assumption has not always been confirmed and, therefore, for the purpose of this work we prefer to consider g of g" as proportionality constants. 

Figure 6. Expected change in the equilibrium modulus, Gg, with respect to its ideal value for a perfect network, Gg produced by a 3% change in conversion, AC, functionality, Af the molar ratio [OH]/[NCO], Ar, and cyclization, AC in dependence on conversion. The data refer to a system composed of. trifunctional telechelic polymer and difunctional coupling agent. (Reproduced with permission from Ref. 42. Copyright 1987 CRC Press.)...

It is shown that model, end-linked networks cannot be perfect networks. Simply from the mechanism of formation, post-gel intramolecular reaction must occur and some of this leads to the formation of inelastic loops. Data on the small-strain, shear moduli of trifunctional and tetrafunctional polyurethane networks from polyols of various molar masses, and the extents of reaction at gelation occurring during their formation are considered in more detail than hitherto. The networks, prepared in bulk and at various dilutions in solvent, show extents of reaction at gelation which indicate pre-gel intramolecular reaction and small-strain moduli which are lower than those expected for perfect network structures. From the systematic variations of moduli and gel points with dilution of preparation, it is deduced that the networks follow affine behaviour at small strains and that even in the limit of no pre-gel intramolecular reaction, the occurrence of post-gel intramolecular reaction means that network defects still occur. In addition, from the variation of defects with polyol molar mass it is demonstrated that defects will still persist in the limit of infinite molar mass. In this limit, theoretical arguments are used to define the minimal significant structures which must be considered for the definition of the properties and structures of real networks. 

Networks formed from stoichiometric, end-linking polymerisation are often assumed to be perfect networks 1-4). However, such an assumption means that intramolecular reaction within gel molecules, which must occur for a network to be formed, never leads to inelastic chains. The assumption is unlikely to be true. The smallest loops which can occur must be elastically ineffective(5-9) and from chain... 

The moduli of model polyurethane networks clearly show reductions below the values expected for perfect networks, with the reductions increasing with pre-gel intramolecular reactlon(5-7). The reductions can be shown to be too large to come solely from pre-gel loop forma-tion( ), some must occur post-gel. In addition, extrapolation to conditions of zero pre-gel intramolecular reaction, by increasing reactant concentrations, molar masses of reactants or chain stiffness, still leaves a residual proportion of inelastic chains due to gel-gel intramolecular reaction. It is basically a law-of-mass-action effect( ). The numbers of reactive groups on gel molecules are unlimited. Intramolecular reaction occurs, and some of this gives Inelastic chains. Only a small amount of such reaction has a marked effect on the modulus. 

In the present paper, theoretical arguments and modulus measurements are used to deduce the significant gel structures which lead to inelastic loop formation and to quantify the network defects and reductions in modulus which may be expected, even in the limit of no pre-gel intramolecular reaction. In this limit all the existing theories and computer simulations of polymerisations including intramolecular reactlon(8,10,ll) predict that perfect networks are formed. 

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... 

Figure 1, Ratio of molar mass between elastically effective junctions to front factor (M(-/A) relative to molar mass between junctions of the perfect network (M ) versus extent of intramolecular reaction at gelation (pj- (.) Polyurethane networks from hexamethylene diisocyanate (HDI) reacted with polyoxpropylene (POP) triols at 80°C in bulk and in nitrobenzene solution(5-7,12). Systems 1 and 2 HDI/POP triols >i= 33, V2= 61. Systems 3-6 ...

Figure 2. Illustrating the equivalence between the chain forming the smallest loop of v bonds, and the chain between junction points in the perfect network (of molar mass M°). (a) RA + RB and (b) RAj + RB polymerisations and networks.

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... 

Table I. Values of parameters characterising pre-gel intramolecular reaction (v,b,(f-2)/(vb ) ) (5-7,12) and the extents of post-gel intramolecular reaction which, in the limit of ideal gelling systems, lead to inelastic loop formation at complete reaction (p g). The values of pj g define the indicated values of Mg/M° and the reductions in shear moduli of the dry networks relative to those of the perfect networks (G/G° = Mc/Mc). The values of Pr g in the limit of reactants of infinite molar mass (v = < ) are denoted p°>°° in the text...

The last two columns in Table I give the intercept values of Mg/Mg in Figure 6 and the corresponding values of shear moduli of the dry networks relative to those of the perfect networks. The values of G/G° for V = > are estimates of the maximum values obtainable for... 

IntrEunolecular reaction always occurs in non-linear polymerisations and because of this end-linking polymerisations never lead to perfect networks. The amount of pre-gel intramolecular may be small... 

The case of RAf polymerisations has been evaluated and the experimentally deduced values of Pg g interpreted in terms of one-membered loop formation within structures of defined size. The knowledge of such structures is important for theories of non-linear polymerisations and network formation. Theories have to simplify in some way the infinite numbers of structures which actually occur and it is important that those significant for intramolecular reaction are retained. In this context, it should be noted that present theories assume that ideal gelling systems lead to perfect networks. The approximation needs to be removed as it is inconsistent with the structure of the growing gel, a molecule with unlimited pairs of groups for intramolecular reaction. 

Note 2 A model network is not necessarily a perfect network. If a non-linear polymerization is used to prepare the network, non-stoichiometric amounts of reactants or incomplete reaction can lead to network containing loose ends. If the crosslinking of existing polymer chains is used to prepare the network, then two loose ends per existing polymer chain result. In the absence of chain entanglements, loose ends can never be elastically active network chains. 

Note 4 Loose ends and ring structures reduce the concentration of elastically active network chains and result in the shear modulus and Young s modulus of the rubbery networks being less than the values expected for a perfect network structure. 

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