Shear modulus of the network

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 proportionality factor G is the equilibrium shear modulus of the network, given by... 

Experimental results on reactions forming tri- and tetrafunctional polyurethane and trifunctional polyester networks are discussed with particular consideration of intramolecular reaction and its effect on shear modulus of the networks formed at complete reaction. The amount of pre-gel intramolecular reaction is shown to be significant for non-linear polymerisations, even for reactions in bulk. Gel-points are delayed by an amount which depends on the dilution of a reaction system and the functionalities and chain structures of the reactants. Shear moduli are generally markedly lower than those expected for the perfect networks corresponding to the various reaction systems, and are shown empirically to be closely related to amounts of pre-gel intramolecular reaction. Deviations from Gaussian stress-strain behaviour are reported which relate to the low molar-mass of chains between junction points. 

The reduced force may be interpreted as the shear modulus of the network (13). According to equation (35), the reduced force consists of a term due to contributions from the phantom network and another from the constraints. The contribution of constraints, proportional to the term in the brackets in equation (35), decreases as the network is stretched or swollen, an important feature in comparisons of theory and experiment. 

The shear modulus of the rubber network is related to the molecular weight between cross-link points or M. The lower the molecular weight of chains between cross-links (network chains), the higher the cross-link density and the higher the modulus. This is shown in the following expression ... 

CR Shear modulus of the plastically activated entangled network... 

The shear storage modulus of the network is proportionally related to the force constant K. Thus, G is also related to the particle volume fraction via the fractal dimension D) of the network. 

The shear modulus of the phantom network is obtained from the modulus of the affine network [Eq. (7.31)] by replacing N with Nf/( f—2)b-... 

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). 

Equation (4.39) shows that for nonzero values of the parameter k the shear modulus of the constrained-junction model is larger than the phantom network shear modulus. For the affine limit, k -t- oo, the shear modulus is... 

Here X = L/Lo is the extension ratio of the sample. Note that the corresponding strain is = (L -Lo) /To = A - 1. The proportional factor G is the shear modulus of the sample. Equation (4) describes small deformation uniaxial data on polymer networks quite well. With a fit of experimental stress-strain data for low extensions it is possible to predict crosslink properties because the classical models show that the shear modulus G is proportional to both temperature and crosslink density Vc ... 

A rubber parallelepiped is deformed by forces along the x and y axes (the shear modulus of the rubber is C = 1.0 MPa). What are the nominal stresses and required to deform it to = 3/2, Ay = 2/3. Assume a Gaussian network. 

Figure 7. Dynamic shear modulus of the hybrid networks as a function of temperature 5-13 vol. % SiOi. 1 (A) ETl, 2 (O) ET2, 3 ( ) E1T2, 4 (V) crosslinked SED2000, 5 ( )DGEBA-D2000.

Here, the shear modulus of the swollen hair G = (pRT/Mc) (v2 — j)/ (1 < ) 1/3(i 2M,/M) y n is the number of segments of the network chain L x) is the anti-Langevin fimction p is the dry density of fiie sample is the number average molecular weight between crosslinks of the rubbery phase M is the primary molecular weight R is the gas constant T is absolute temperature V2 is the volume fiaction of the polymer within the gel and y is the filler effect of the HS domain that exists in the rubbery phase. Equation (3) provides y. 

The bending rigidity k of RBCs has been measured by micropipette aspiration [182] and atomic force microscopy [183] to be approximately k = 50 A T. The shear modulus of the composite membrane, which is induced by the spectrin network, has been determined by several techniques it is found to be ju = 2 x 10 N m from optical tweezers manipulation [184], while the value ju = 6 x 10 Nm is obtained from micropipette aspiration [182], Thus, the dimensionless ratio k 100, which implies that bending and stretching energies are roughly of equal importance. 

Measurement of the equilibrium properties near the LST is difficult because long relaxation times make it impossible to reach equilibrium flow conditions without disruption of the network structure. The fact that some of those properties diverge (e.g. zero-shear viscosity or equilibrium compliance) or equal zero (equilibrium modulus) complicates their determination even more. More promising are time-cure superposition techniques [15] which determine the exponents from the entire relaxation spectrum and not only from the diverging longest mode. 

Equation (40) shows that the small deformation shear modulus of an affine network increases indefinitely over the phantom network modulus as junction functionality approaches 2. 

The moduli and Tg s of the networks formed from the bulk reactions of the five systems of Figure 9 are shown in Table IV(29). The first five columns define the systems, the next two give the experimental values of G(at 298K) and Tg, and the last three give the values of pr,c, Mc, and G/G°. The last quantity is the reduction in rubbery shear modulus on the basis of that expected for the perfect network(G°). G/G° is in fact equal to M /Mc. 

The network from system 3 is distinct from the rest, being a glass at room temperature and also having a rubbery shear modulus near the value expected on the basis of G°. Possible reasons for this high value of G/G° follow those discussed previously with reference to Mc/Mc and Figure 9. The more flexible chains of the aliphatic systems give lower values of Tg, resulting in elastomers at room temperature. 

The equilibrium shear modulus of two similar polyurethane elastomers is shown to depend on both the concentration of elastically active chains, vc, and topological interactions between such chains (trapped entanglements). The elastomers were carefully prepared in different ways from the same amounts of toluene-2,4-diisocyanate, a polypropylene oxide) (PPO) triol, a dihydroxy-terminated PPO, and a monohydroxy PPO in small amount. Provided the network junctions do not fluctuate significantly, the modulus of both elastomers can be expressed as c( 1 + ve/vc)RT, the average value of vth>c being 0.61. The quantity vc equals TeG ax/RT, where TeG ax is the contribution of the topological interactions to the modulus. Both vc and Te were calculated from the sol fraction and the initial formulation. Discussed briefly is the dependence of the ultimate tensile properties on extension rate. 

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... 

Note 4 Loose ends and ring structures reduce the concentration of elastically active network chains and result in the shear modulus and Young s modulus of the rubbery networks being less than the values expected for a perfect network structure. 

---