Tetrafunctional cross-links

Figure 4. Frequency dependence of the storage modulus G at 303 K. Key PDMS-B11, (comblike crosslinks) , PDMS-C1, (tetrafunctional cross-links, randomly introduced) PDMS-A2 (tetrafunctional cross-links, end-linked network).

Figure 3.1 Covalent cross-linking reaction, (a) Cross-linking point between two macromoiecules. (b) Several tetrafunctional cross-linking points between macromo-lecular chains, which form a polymer network.

A 2.830 g sample of a poly(methyl methacrylate) (PMMA) network immersed in ethyl acetate (EtAc) reaches swelling equilibrium after 24 h, the final weight being 7.336 g. The cross-linking reaction was carried out with a tetrafunctional cross-linking agent in the undiluted state (v2s = ) Calculate M,.. 

These contributions are valid for the most common case of tetrafunctional cross-links. In the general case of /-functional cross-links the Vi in the parentheses in Equations 2.53 and 2.55 should be replaced by the ratio 2//. 

The polyacrylamide gels were prepared by free-radical polymerization [3]. Acrylamide, the linear constituent N,N -methylenebisacrylamide, the tetrafunctional cross-linking constituent and ammonium persulfate and N,N,N, N -tetramethylethylenediamine (TEMED), the initiators, were dissolved in water. Micropipettes with a well-defined diameter (1.4 mm) were immersed in this solution. Within 5 min, the solution gelled. After an hour, the gels were removed from the micropipettes and immersed in water to wash away residual monomers. The gels then underwent hydrolysis in a 1.2% solution of TEMED (pH 12) for more than a month. Approximately 20% of the acrylamide groups were converted to acrylic acid groups, some of which were ionized in water. 

At least one tetrafunctional cross-linking site per primary molecule is necessary for all the primary chains to be bonded together. If the expectation e is less than 1, then cross-linking cannot include all the primary molecules, and the result is that branched molecules, not cross-linked networks, are formed. Cross-linked networks are always produced at e > 1. The limiting condition for the occurrence of a cross-linked network is therefore that cr,t = 1. Equation (23-44) then becomes... 

A sample of a certain rubbery polymer has tetrafunctional cross-links (that is, four chains meet at each cross-link). The rubber is found to have shear modulus G = 800 kPa at 40°C. Estimate the number of cross-links per m. ... 

Typical configurations of four chains emanating from a tetrafunctional cross link in a polymer network. In the left sketch, the network was prepared in the undiluted state, and in the right sketch, it was prepared in solution and then dried. When a network is cross linked in solution and the solvent then removed, the chains collapse in such a way that there is reduced overlap in their configurational domains. 

If the tetrafunctional cross-links to which are connected four clmins leading to the network are suppressed, the network (first-stage) is formed only with the crosslinks having three chains connected to the mesh. Each chain end is connected to this type of cross-link, whose number density is therefore... 

Figure 2.30 Idealized network structure of a crosslinked polymer. indicates a crosslink (junction) and —> signi es continuation of the network structure. Wavy lines between crosslinks are active network chain segments. (Note that for a tetrafunctional cross-link, as shown here, the number of crosslinks is one-half the number of active network chain segments.)...

The quantity n in the above represents the number of active network chains per unit volume, sometimes called the network or cross-link density. The number of cross-links per unit volume is also of interest. For a tetrafunctional cross-link (see Figure 9.3), the number of cross-Unks is one-half the number of chain segments (see Section 9.9.2). Equation (9.34) holds for both extension and compression. 

The interpretation of the constants 2Ci and 2C2 has absorbed much time the results are inconclusive (42). JtJs tempting but generally considered incorrect to equate 2Ci and nRT rflrl). The original derivation of Mooney (46) shows that 2C2 has to be finite, but it does not indicate its value relative to 2Ci. According to Flory (41), the ratio 2C2/2C1 is related to the looseness with which the cross-links are embedded within the structure. Trifunctional cross-links have larger values of ICJlCi than tetrafunctional cross-links, for example (58). 

Assuming a tetrafunctional cross-linking mode (four chain segments emanating from the locus of the hydrogen bond) ... 

Table A9.1.2 also demonstrates that at each gelatin concentration, the number of bonds per gelatin molecule is relatively constant. This number, of course, is the number of bonds taking part in three-dimensional network formation. Not all the gelatin chains are bound in a true tetrafunctionally cross-linked network. Many dangling chain ends exist at these low concentrations, and the network must be very imperfect.

Fig. 1.14. Typical configurations of four chains emanating from a tetrafunctional cross-link in a polymer network prepared in the undiluted state [67].

Since dangling chains constitute imperfections in a network structure, one would expect their presence to have a detrimental effect on the ultimate properties (//A )r and Qfr of an elastomer. This expectation is confirmed by an extensive series of results obtained on PDMS networks that had been tetrafunctionally cross-linked using a variety of techniques [130]. The largest values of the ultimate strength... 

Suppose that a network having tetrafunctional cross-links (f) = 4,A = ) and a density of 0.900gcm has [/ ] (a = oo) = O.lOONmm" (10 Nm (Pa) = 10 MNm (MPa) = 1.02kgcm ) at 298.2K. Calculate the network-chain density, the cross-link density, and the average molecular weight between cross-links [9]. 

A typical network studied in this regard might have been tetrafunctionally cross-linked in the undiluted state (U2s = 1.00), and exhibit an equilibrium degree of swelhng characterized by V2m = 0.100 in a solvent having a molar volume V = SOcm mor (8.00 x 10" mm mol ) and an interaction parameter with the polymer corresponding to xi = 0-30. Calculate the network-chain density [9]. 

The early treatments of trapped entanglements were based on the criterion that any entanglement on a strand running between two chemical cross-links is trapped, as in Fig. 14-4-1 this led to the relation (for tetrafunctional cross-linking)... 

In gels with permanent chemical tetrafunctional cross-links, the equilibrium shear elasticity would be expected to be given by equation 33 of Chapter 10,... 

If a cross-linked network is produced from originally linear rubber molecules by the addition of n" moles of tetrafunctional cross-linking agent per cm of rubber, then each cross-link originates four network chains, and each chain belongs to two cross-links. Consequently, there are 2Mc moles of network chains per cm in the cross-linked network, that is, 2nc = n= d/Mc-... 

In a tetrafunctional cross-linking process, formation of an elastically effective junction will create two effective chains. Generalization of this to a ( -functional process leads to equation (54), a relationship between and p which is similar to that for perfect networks (equation 44). Combination of equations (53) and (54) then yields equation (55). 

The additional term on the right-hand side of the above equation compared to Equation 2.17 is negligible for small values of V2 or for cross-links of high functionality. Equations 2.17 and 2.18 can be represented in the form of nomographs for ease of determination of an unknown parameter given that the other parameters in the equation are known. Figure 2.4 represents such a nomograph based on Equation 2.18 for tetrafunctionally cross-linked networks (f= 4). 

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