Effective network chains

Networks obtained by anionic end-linking processes are not necessarily free of defects 106). There are always some dangling chains — which do not contribute to the elasticity of the network — and the formation of loops and of double connections cannot be excluded either. The probability of occurrence, of such defects decreases as the concentration of the reaction medium increases. Conversely, when the concentration is very high the network may contain entrapped entanglements which act as additional crosslinks. It remains that, upon reaction, the linear precursor chains (which are characterized independently) become elastically effective network chains, even though their number may be slightly lower than expected because of the defects. 

Swelling data indicate that crosslink density in the continuous phase of the 70 30 and 60 A0 networks is high. Crosslink densities were estimated from the data in Table III by the method of Hill and Kozlowski ( ). Results were for 80 20, "Vg = 10" moles of elastically effective network chains/cm for 70 30, Vg = 2.5 x lO" chains/cm for 60 AO, Vg = A.3 x 10" chains/cm. These estimates suggest that the crosslink densities are within the range reported for conventional, highly crosslinked acrylic HMMM and polyester HMMM enamels (19,20). 

Load Sharing of Filler Particles. Comparison of ultimate strength of a propellant and its unfilled binder matrix almost always shows that the propellant has up to several times the tensile strength of the matrix. This filler reinforcement is presently thought to stem from additional crosslinks formed between filler particles and the network chains of the binder matrix (5, 8, 9, 34). Effective network chains are defined as the chain segments between crosslinks. From the classical theory of elasticity, the strength and/or modulus of an elastomer is proportional to the number of effective network chains per unit volume, N, or... 

Model networks are tridimensional crosslinked polymers whose elastically effective network chains are of known length and of narrow molecular weight distribution. The techniques used to synthesize such networks are derived from those developed for the synthesis of star shaped macromolecules, whereby the initiator used must be bifunctional instead of monofunctional. ... 

Structures of Type (b) can neither be counted as a single elastically effective network chain, nor, if the chain lengths differ, as two chains, because affine deformation for the chain end-to-end vectors does not apply. 

In this Chapter we will consider the elasticity and swelling of networks which deviate from our definition of ideal networks only because of a certain number of network defects (see Chapter II, Section 2). We will designate by v the number of elastically effective network chains, which in an ideal network equals the number of chains because of the absence of defects. 

To predict macrosyneresis in the case of dissolved polymer chains being crosslinked it is necessary to know the change in the number of elastically effective network chains as a result of changing the number of chemical crosslinks and the effect of v on (r2),. and possibly on y. Moreover, if the molecular weight of primary chains is finite and the sol fraction is important, a complete description of the phase separation requires treating the system as a multicomponent one, containing a network phase and a multicomponent diluent with branched polymer species. If additional crosslinking is carried on in an extracted network the latter complication can be eliminated. 

The elastically effective network chains should obey Gaussian statistics. They should therefore be long enough, and their average degree of polymerization should be known. In addition the distribution of chain-lengths is expected to be rather narrow. 

The functionality of the crosslinks should be known, and constant throughout the geL The functionality is the number of elastically effective network chains which are tied to one given crosslink. 

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 ... 

The modulus G defined above is proportional to the number of elastically effective network chains v, and to the 2/3 power of the term ft. If the networks investigated are sufficiently close to ideality, the number v of network chains is equal to the number of precursor drains and v may be replaced by (M V02) l, yielding ... 

This review was devoted to tridimensional polymeric networks of well-defined structure, called for that reason model networks. These networks are synthesized by endlinking processes, whereby well-characterized linear precursor chains become elastically effective network chains. The model network can be considered close to... 

Theory The swelling behavior of polymer networks is described by several network parameters Xg> a polymer network-solvent interaction parameter u, the concentration of elastically effective network chains and qQ, a reference degree of swelling which is related to the unperturbed end-to-end distance of the polymer chains during network formation. 

Summing the individual [TQnl s from m = 3 to fk then gives the total crosslink density [X]. At p = 1, P(F ) becomes zero, so that in the limit of complete reaction, one can write, theoretically, [X j = [A/ ]q. The crosslink density is an important parameter as it can be related to the concentration of effective network chains and hence to shear modulus of the crosslinked polymer (Miller and Macosko, 1976 Langley, 1968 Langley and PoUmanteer, 1974). 

All of these considerations refer to ideal networks, that is, networks without free chain ends. The proportion of free chain ends is inversely proportional to the molar mass of the primary polymer chain molecule. For < Mn)cXMc n, the effective network chain concentration, [MJeff, can be calculated from the total chain concentration, [Me], via... 

An old point of controversy in rubber elasticity theory deals with the value of the so-called front factor g = Ap which was introduced first in the phantom chain models to connect the number of elastically effective network chains per unit volume and the shear modulus by G = Ar kTv. We use the notation of Rehage who clearly distinguishes between A andp. The factor A is often called the microstructure factor. One obtains A = 1 in the case of affine networks and A = 1 — 2/f (f = functionality) in the opposite case of free-fluctuation networks. The quantity is called the memory factor and is equal to the ratio of the mean square end-to-end distance of chains in the undeformed network to the same quantity for the system with junction points removed. The concept of the memory factor permits proper allowance for changes of the modulus caused by changes of experimental conditions (e.g. temperature, solvent) and the reduction of the modulus to a reference state However, in a number of cases a clear distinction between the two contributions to the front factor is not unambiguous. Contradictory results were obtained even in the classical studies. 

Effective network chains concentration in moles/cm from compression measurements. Ci and C2in Pa. 

All the quantities so far defined relate to ideal networks, i.e., continuous branched structures without free chain ends. In reality, the number of such free chain ends increases with decreasing primary chain molecular weight. The molar concentration MX niol/g of effective network chains can, according to P. J. Flory, be calculated from the molar concentration [MJ of all the chains present for M )q > (M ) ,... 

Ve is XLD expressed as the number of moles of elastically effective network chains per cubic centimeter of film. Since E" is low at temperatures well above Tg, Ef E. XLD can also be calculated from the extent of swelling of a film by solvent. While cross-linked films do not dissolve in solvent, solvent dissolves in a cross-linked film. As cross-links get closer together the extent of swelling decreases. Equation (5) can be used to predict the storage modulus above Tg from the XLD. 

Domain Size Theory. Assuming spherical domains, Yeo and co-workers (38) derived equations for the domain size in sequential IPNs. The domain diameter of polymer II, Dn, was related to the interfacial tension y, the absolute temperature T times the gas constant R, and the concentration of effective network chains, ci and cn, occupying volume fractions v and Pn, respectively ... 

Figure 10.7 (8) illustrates the stress relaxation of a poly(dimethyl siloxane) network, silicone rubber, in the presence of dry nitrogen. The reduced stress, o(t)/(T(0),is plotted,so that under the initial conditions its value is always unity. Since the theory of rubber elasticity holds (Chapter 9), what is really measured is the fractional decrease in effective network chain segments. The bond interchange reaction of equation (10.2) provides the chemical basis of the process. While the rate of the relaxation increases with temperature, the lines remain straight, suggesting that equation (10.2) can be treated as the sole reaction of importance. 

Here v represents the total number of elastically effective network chains per unit volume, k is the Boltzmann constant and T the absolute temperature. 

For cross-links of types (a), (b), (c) and (I) depicted in Fig. 8.1 each cross-link is attached to four network chain ends. The effect of each cross-link is therefore to introduce two extra network chains and since if there were no cross-links there would be no effective chains, in the sense used in this chapter and Chapter 3, the density of cross-links v will be half the effective network chain density. Hence ... 

It is however important to appreciate that the analysis demands a knowledge of the functionality of the cross-link or alternatively requires that an arbitrary assumption be made. This is because information is required on the cross-link density. Linkages of types (a), (b), (c) and (1) of Fig. 8.1 are linked to four network chains, each link generating two effective network chains. In this case therefore the cross-link density will be half the network chain density and this will be related to by a combination of eqns (8.3) and (8.4). In turn Me may be determined from stress-strain measurements. These measurements of course yield Mc,phys and since when studying chemical cross-links we are interested in Mc,chem it is necessary to invoke the Moore-Watson-Mullins calibration (8.8) or obtain Me directly from eqn (8.12). In linkages of types (d) and (e) of Fig. 8.1 the second cross-link does not further reduce the Me and in this case the cross-link density equals the network chain density. 

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