Network-chain molar mass

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. 

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.

The range of semi-dilute network solutions is characterised by (1) polymer-polymer interactions which lead to a coil shrinkage (2) each blob acts as individual unit with both hydrodynamic and excluded volume effects and (3) for blobs in the same chain all interactions are screened out (the word blob denotes the portion of chain between two entanglements points). In this concentration range the flow characteristics and therefore also the relaxation time behaviour are not solely governed by the molar mass of the sample and its concentration, but also by the thermodynamic quality of the solvent. This leads to a shift factor, hm°d, is a function of the molar mass, concentration and solvent power. 

The paper first considers the factors affecting intramolecular reaction, the importance of intramolecular reaction in non-linear random polymerisations, and the effects of intramolecular reaction on the gel point. The correlation of gel points through approximate theories of gelation is discussed, and reference is made to the determination of effective functionalities from gel-point data. Results are then presented showing that a close correlation exists between the amount of pre-gel intramolecular reaction that has occurred and the shear modulus of the network formed at complete reaction. Similarly, the Tg of a network is shown to be related to amount of pre-gel intramolecular reaction. In addition, materials formed from bulk reaction systems are compared to illustrate the inherent influences of molar masses, functionalities and chain structures of reactants on network properties. Finally, the non-Gaussian behaviour of networks in compression is discussed. 

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... 

The preceding sections have shown that pre-gel intramolecular reaction always occurs in random polymerisations, and that the amount of such reaction dependes on the dilution (ce -- -), molar masses (v), chain structures (b) and functionalities (f) of the reactants. Intramolecular reaction leads to loops of finite size in the network material finally formed by a reaction mixture. Such loops may be elastically ineffective and have marked effects on the properties of the material. The present section investigates the magnitudes of such effects with regard to shear modulus and Tg. 

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-...

As is clear from the earlier discussions of pre-gel intramolecular reaction, such reaction in principle always occurs in random polymerisations, although its amount may be reduced by using reactants of higher molar mass, lower functionalities, and stiffer chain structures. Thus, the use of end-linking reactions to produce model networks (for example(35) and references quoted... 

Microgels are distinguished from linear and branched macromolecules by their fixed shape which limits the number of conformations of their network chains like in crosslinked polymers of macroscopic dimensions. The feature of microgels common with linear and branched macromolecules is their ability to form colloidal solutions. This property opens up a number of methods to analyze microgels such as viscometry and determination of molar mass which are not applicable to the characterization of other crosslinked polymers. 

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. 

The experimental data to be considered are shown in Figure 1. They refer to previously published data on hexamethylene diisocyanate(HDI) reacting with polyoxypropylene(POP) triols and tetrols in bulk and in nitrobenzene(5-7,12) that is, to RA2 + RBj polymerisations. is the molar mass of chains between elastically effective junction points. A/Mj. has been determined directly from small-strain compression measurements on swollen and dry networks using the equations... 

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... 

The positive intercepts in Figure 7 show that post-gel(inelastic) loop formation is influenced by the same factors as pre-gel intramolecular reaction but is not determined solely by them. The important conclusion is that imperfections still occur in the limit of infinite reactant molar masses or very stiff chains (vb - ). They are a demonstration of a law-of-mass-action effect. Because they are intercepts in the limit vb - >, spatial correlations between reacting groups are absent and random reaction occurs. Intramolecular reaction occurs post-gel simply because of the unlimited number of groups per molecule in the gel fraction. The present values of p , (0.06 for f=3 and 0.03 for f=4 are derived from modulus measure- ments, assuming two junction points per lost per inelastic loop in f=3 networks and one junction point lost per loop in f=4 networks. 

Polymer network comprising polymer chains having two significantly different molar-mass distributions between adjacent junction points. 

Number-average molar mass of polymer chains between two adjacent crosslinks or junction points in a polymer network. 

For simple systems, the molar mass, relative molecular mass or degree of polymerization may be included in the scheme of Rule 7 by the symbol M, M or DP, respectively, followed by the corresponding numerical values, separated by colons. Symbols are qualified if quantities refer to parts of macromolecules or assemblies, e.g., M (block), (arm), M (network chain), etc. 

These polymer networks have commercial applications. A number of experimental studies suggest that the network formation by this method proceeds in a highly non-ideal fashion. During copolymerization, a high fraction of pendant vinyls of a primary chain are consumed by intramolecular reactions, causing practically no increase in the molar mass of the system (see Chapter 7) ... 

Polyurethane Networks. Andrady and Sefcik (1983) have applied the same relationship as Rietsch et al. (1976), to the glass transition temperature of networks based on poly(propylene oxide) diols with a controlled molar mass distribution, crosslinked by aromatic triisocyanates. They obtained a Kr value of 25 K kg mol-1, about twice that for PS networks. They showed that the length distribution of elastically active chain lengths, directly related to the molar mass distribution of the starting poly(propylene oxide), has practically no effect on Tg. 

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... 

A particular polyester network may be considered as a model of degraded ideal polyester based on a prepolymer having an infinite molar mass with a number of chain scissions equal to Ave = b/2. Thus, polyester samples differing by the initial prepolymer molar mass can be used to calibrate rubber-elasticity measurements. 

M = molar mass of the whole chain (here 20 x 50 = 1000) v = a number of chain elements, separated by cross-links (here v = 8) ve = effective number of chain parts, contributing to the coherence and the elasticity of the network (here ve = 4). 

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. 

For network polymers the only meaningful molar mass is that of the polymer chains existing between junction points (i.e. network chains), since the molar mass of the network itself essentially is infinite. 

With very few exceptions, polymers consist of macromolecules (or network chains) with a range of molar masses. Since the molar mass changes in intervals of M0, the distribution of molar mass is discontinuous. However, for most polymers these intervals are extremely small in comparison to the total range of molar mass and the distribution can be assumed to be continuous, as exemplified in the figure above. 

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