Number average crosslinking

If Mw and are known, then a can be calculated with the aid of Eq. (13 thereupon from the equilibrium shear modulus, Ge, the sol fraction, w, can be calculated with the aid of Eqs. (14)-(17) provide the crosslink functionality, f, is known. Subsequently the we t average crosslinking index in the system as a whole, and the number average crosslinking index in the networic fraction,... 

If we assume a random distribution of crosslinking events, the average number of crosslinks along the length of a chain is proportional to its length. Thus, we would expect a chain with... 

For a given initial molecular weight distribution, the weight fraction of chains incorporated into a network increases as the number of crosslinking events increases. Similarly, the number of crosslinking events required to create a network decreases as the average molecular weight increases. These relationships are illustrated schematically in Fig. 5.16. 

Figure 5.16 Schematic representation of the effect of number of crosslinks and initial average molecular weight on network formation... 

Figure 8. SANS measurements of R /Fg° and R /Rg° versus A of tetrafunctional polydimethylsiloxane networks. Mw and Mn are weight and number average molecular weights before crosslinking, cp is the polymer fraction in solution before cross-linking. Data from Ref. 20.

It is possible to uniquely determine both the scission efficiency Gis) and the crosslinking efficiency G(x) via Equations 9 and 10 providing both the number average molecular weight (M ) and the weight average molecular weight (Af ) can be measured. 

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

BB] is the concentration of BB, and [B] = 2[BB], The number of crosslinks is simply the number of BB monomer molecules in which both B double bonds are reacted, that is, //[BB. The number of polymer chains is the total number of A and B double bonds reacted divided by the degree of polymerization, ([A] + [B])ppCw (the weight-average degree of polymerization is employed for reasons previously described in Sec. 2-10). The critical extent of reaction at the gel point pc occurs when the number of crosslinks per chain is and thus is given by... 

Mc number average molecular weight between crosslinks... 

In both equations, Dw is the solute diffusion coefficient in pure water, rs is the molecular radius of the solute, ls is its characteristic size, Vw is the water free volume, Mc is the molecular weight between crosslinks in the amorphous phase, Mn is the number average molecular weight of the polymer before crosslinking, M is the minimum value of Mc below which the solute cannot diffuse, 4>(V) is the free volume function mentioned earlier, and k3 is a constant. 

The number of crosslinked units per primary molecule denoted by crosslinking index y is equal to qP (jP is the number average). At... 

Taken from Ref.19 b taken from Ref.69 c taken from Ref. 8 d taken from Ref.67 taken from Ref, 731 r taken from Ref.,0) phr of sulfur h % of dicumyl peroxide number of crosslinking units per weight average PMA chain j average number of repeat units between crosslinks k concentration of chains between network joints, (mole/cm3)x 104 1 methyl group rotation m segmental motion motions within the crystalline part of the polymer... 

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. 

Klc = critical stress intensity factor in mode I, MPam1/2 Km = stress concentration factor Me = average molar mass between crosslinks, kg mol 1 Mn = number-average molar mass, kg mol 1 p = hydrostatic pressure, Pa Tg = glass transition temperature, K... 

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