Network chain parameters

Once P(F ° ) and P(Fg° ) have been calculated, it is possible to calculate a number of network structure parameters including the weight fraction of sol, wg, and the "effective" crosslink density. A given polymer or crosslinker will be part of the sol only if all of its groups are attached to finite chains. Thus, the weight of the sol is given by... 

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... 

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 40. A simple example Cellular metabolism is modeled as a linear chain of reactions, with long range interactions mimicking the cellular environment and interactions within the metabolic network. The parameters are the number of metabolites m, the number of regulatory interactions, the probability p of positive versus negative interaction, as well as the maximal displacement ymax from equilibrium for each reaction. Each reaction is modeled as a reversible Michaelis Menten equation according to the methodology described in Section VIII.

Let us first consider a network immersed in a melt of polymer chains with degree of polymerization p. In the athermal case, the network should be swollen. As polymer-network interaction parameter Xnp increases, the volume of the network decreases until a practically complete segregation of the gel from polymer melt occurs. It has been found [34, 35] that two qualitatively different regimes can be realized either a smooth contraction of the network (Fig. 8, curve 1) or a jumpwise transition (Fig. 8, curve 2). The discrete first order phase transition takes place only for the networks prepared in the presence of some diluent and when p is larger than a critical value pcr m1/2. The jump of the... 

Fig. 8. Dependence of the volume traction of network chains within the network, On, on the network-polymer interaction parameter, xnp, for p < p (curve /) and p > pcr (curve 2). Reproduced with permission from Ref. [35]...

The concentrations of elastically active network chains related to the dry state vd series A-F were [26] vd = 3.6, 5.7, 6.3, 7.1, 10.9 and 15.2 x 10 5 molcm 3 (structure formation at high dilution in the system. Using vd values together with other molecular parameters, the dependences of y vs ip 2 were calculated and both the extent of the collapse, A, and the critical value,... 

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. 

For networks in which the average crosslink functionality exceeds the above-mentioned values, the ratios of the slopes lead to functionalities which are obviously too high. The assumption according to which the connectivity factor x2 is equal to unity cannot hold any more. This means that a fraction of the network chains connect nodules which are not first neighbors. The probability for permanent entanglements to occur is increased. No quantitative treatment of the experimental data is possible in that case, because of the number of parameters to consider, none of them being accessible by independent experiments. 

The synthesis and characterization of these ABA block copolymers of styrene and dienes have been described elsewhere (JO, 11). Since the polystyrene end blocks aggregate into glassy domains which act as network junctions, the elastic center blocks must virtually represent the network chains/ The polystyrene domains should also act as a finely divided filler. Hence it might be expected that the mechanical properties of these materials could depend on the two basic parameters polystyrene content and length of center block ( molecular weight between crosslinks ). 

From a fit of Equation (10) to spatially resolved relaxation curves, images of the parameters A, B, T2, q M2 have been obtained [3- - 32]. Here A/(A + B) can be interpreted as the concentration of cross-links and B/(A + B) as the concentration of dangling chains. In addition to A/(A + B) also q M2 is related to the cross-link density in this model. In practice also T2 has been found to depend on cross-link density and subsequently strain, an effect which has been exploited in calibration of the image in Figure 7.6. Interestingly, carbon-black as an active filler has little effect on the relaxation times, but silicate filler has. Consequently the chemical cross-link density of carbon-black filled elastomers can be determined by NMR. The apparent insensitivity of NMR to the interaction of the network chains with carbon black filler particles is explained with paramagnetic impurities of carbon black, which lead to rapid relaxation of the NMR signal in the vicinity of the filler particles. 

Figure 15.7 Variation of the splitting A in an elongated PDMS model network containing a small amount (9%) of free PDMS chains, as a function of (A2-A 1), with X the elongation ratio. A pair of identical network/free chains systems are used A deuterated network chains (Mn = 10500 g.mol 1) o deuterated free chains (M = 10500 g.mol"1). S is the chain segment order parameter...

H and 2H NMR have been used in styrene-butadiene rubber (SBR) with and without carbon-black fillers to estimate the values of some network parameters, namely the average network chain length N. The values obtained from both approaches were checked to make sure that they were consistent with each other and with the results of other methods [71, 72, 73]. To this purpose, a series of samples with various filler contents and/or crosslink densities were swollen with deuterated benzene. The slopes P=A/ X2-X 1) obtained on deuterated benzene in uniaxially stretched samples were measured. The slopes increase significantly with the filler content, which suggests that filler particles act as effective junction points [72, 73]. 

It is demonstrated that the quasi-static stress-strain cycles of carbon black as well as silica filled rubbers can be well described in the scope of the theoretic model of stress softening and filler-induced hysteresis up to large strain. The obtained microscopic material parameter appear reasonable, providing information on the mean size and distribution width of filler clusters, the tensile strength of filler-filler bonds, and the polymer network chain density. In particular it is shown that the model fulfils a plausibility criterion important for FE applications. Accordingly, any deformation mode can be predicted based solely on uniaxial stress-strain measurements, which can be carried out relatively easily. 

Epoxy networks may be expected to differ from typical elastomer networks as a consequence of their much higher crosslink density. However, the same microstructural features which influence the properties of elastomers also exist in epoxy networks. These include the number average molecular weight and distribution of network chains, the extent of chain branching, the concentration of trapped entanglements, and the soluble fraction (i.e., molecular species not attached to the network). These parameters are typically difficult to isolate and control in epoxy systems. Recently, however, the development of accurate network formation theories, and the use of unique systems, have resulted in the synthesis of epoxies with specifically controlled microstructures Structure-property studies on these materials are just starting to provide meaningful quantitative information, and some of these will be discussed in this chapter. 

The stress optical coefficient C merits special attention, because it leads directly to the parameter F2 that characterizes the optical anisotropy of the network chain under strain. F2 is defined by (13)... 

There are two parameters used as a measure of cross-Unk density the number of network chains, v, usually expressed as v/ V, where V is the volume of the unstrained network and the number of cross-links (p) per unit volume, p/F. The relationship between p and v is established by knowing the number of chains starting from a particular cross-linking point, (functionality). The two most important types of network are the tetrafunctional (c ) =z 4) and the trifunctional ( = 3). Another characteristic parameter of a network is the number-average molecular weight between cross-links,... 

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. 

The parameter rl r, sometimes referred to as the front factor, can be regarded as the average deviation of the network chains from the dimensions they would assume if they were isolated and free from all constraints. For an ideal elastomer network, the front factor is unity. 

The entropy of a network-solvent system will increase because of the tendency of the solvent molecules to disperse in the network. This is in analogy to thermodynamics of the dissolution process of macromolecules in a solvent. In reality, it is necessary to take into consideration the additional effect of interaction between polymer segments and solvent molecules, e.g., by introducing an interaction parameter. The dilation gives rise to an elastic response from the network chains which will oppose the tendency for dilation. 

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). 

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