Elastically effective junctions

Figure 9. Molar mass between elastically effective junction points (Mc) relative to that for the perfect network (Mc°) versus extent of intramolecular reaction at gelatin (pr,c) for polyurethane networks (29).

The experimental data to be considered are shown in Figure 1. They refer to previously published data on hexamethylene diisocyanate(HDI) reacting with polyoxypropylene(POP) triols and tetrols in bulk and in nitrobenzene(5-7,12) that is, to RA2 + RBj polymerisations. is the molar mass of chains between elastically effective junction points. A/Mj. has been determined directly from small-strain compression measurements on swollen and dry networks using the equations... 

Figure 1, Ratio of molar mass between elastically effective junctions to front factor (M(-/A) relative to molar mass between junctions of the perfect network (M ) versus extent of intramolecular reaction at gelation (pj- (.) Polyurethane networks from hexamethylene diisocyanate (HDI) reacted with polyoxpropylene (POP) triols at 80°C in bulk and in nitrobenzene solution(5-7,12). Systems 1 and 2 HDI/POP triols >i= 33, V2= 61. Systems 3-6 ...

Cardoso et al. [13] also compared the dependency of the viscoelastic properties of mature OPE/calcium gels upon the polymer and calcium concentrations to those of the LMP. They showed that, for these variables, both pectin systems exhibited a power law dependence of the G. At pH 7, for the different concentrations of non-esterified carboxyl groups available in the pectin (o-GalA ), the PPE/calcium and citrus LMP/calcium systems exhibited similar dependencies on the calcium concentration (Fig. 8a), with a power law dependence of 2.9-3.3. Still, the gelling ability of OPE/calcium systems was more dependent on the polymer concentration than the citrus pectin. For the different calcium concentrations tested, the corresponding exponents of power law dependency were approximately 3.0 and 1.9 for OPE/calcium and citrus LMP/calcium systems, respectively (Fig. 8b). These results also confirm the lower capability of the pectic olive extracts to form, under similar ionic conditions, elastically effective junctions zones. 

I - elastically effective junction points 0 - pairs of reacted groups(-AB-). 

Junctions whose path number is larger than or equal to 3 are called elastically effective junctions. Junctions with path number 1 connect dangling chains (Figure 4.10) junctions with path number 2 are not active because they merely extend the already existing paths. Both types should not be counted as effective chains. 

In Section 4.2, we introduced the Scanlan-Case (SC) criterion to find the elastically effective junctions and chains. SC proposes that a chain with both ends connected to elastically effective junctions (junctions whose path number is larger or equal to three) is elastically effective (Figure 8.4(a)) [9,10],... 

To see how these new rheological features appear, we calculate the number figff of elastically effective junctions in a unit volume from (8.31), and the number of elastically effective chains Veff from (8.32). 

Fig 2 Molar mass between elastically effective junction points(M ) relative... 

Fig 4 The occurrence of the smallest loops at complete reaction in an RA2 + RB polymerisation. Structure (a) introduces inelastic chains, and (b) contributes to the sol fraction. X denotes an elastically effective junction point. 

We next employ the criterion of Scanlan [12] and Case [13] that only subchains connected at both ends to junctions with at least three paths to the gel are elastically effective. We, thus, have i,i > 3 for an effective chain. A junction with one path (i = 1) to the gel unites a group of subchains dangling from the network matrix whose conformations are not affected by an applied stress. A junction with two paths (i = 2) to the gel merely extends the length of an effective subchain. We may call the junctions with i > 3 elastically effective junctions. An effective chain is defined as a chain connecting two effective junctions at its both ends. We, thus, find... 

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). 

Scanlan has suggested another criterion (282). An effective network junction point is a crosslink in which at least three of the four strands radiating from it lead independently to the network. A crosslink with only two strands anchored to the network simply continues an active strand a crosslink with only one anchored strand is part of a dangling end and can make no elastic contribution at equilibrium. An elastically effective strand is therefore one which joins two effective network junction points. Accordingly, the total number of active strands is simply one half the number of gel-anchored strands radiating from effective junction points ... 

The equilibrium positions of the four junction points define the spatial relationship between the strands of each pair. For simplicity, the mean relative positions (internal coordinates) of the junction points of each pair are taken to be the same. The junction points of each strand are separately anchored to the network by at least two of their remaining strands, so each is an elastically effective strand according to Scanlan s criterion. The network itself in effect completes the loop for each strand, making the A, B, and C pairs as structurally distinct as catenane molecules (301). 

The presence of states 4 and 6 in a network at complete reaction is presented schematically in Figure 7. It can be seen that over the complete reaction system the number of junction points lost is Ng.Pe, where Na is the number of monomer units initially and Pe is the fraction of units in state 6. Hence at complete reaction, the number of elastically effective function points in Na(P - Ps), where Pi, + P = 1. Thus,... 

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. 

With the advancement of a curing reaction, the of the resin will increase, but the goal is to quantitatively predict the of a resin as a function of cure conversion. Several models have been proposed to correlate the with the conversion or extent of curing (a). With the increase in conversion, the concentration of reactive functionalities decreases, and crosslinks or junction points are formed, leading to the departure from Gaussian behaviour. Steric hindrance affects chain conformation at high crosslink densities. The models are based on the statistical description of network formation and calculation of the concentration of jrmction points of different functionalities as a function of conversion. However, one issue that complicates the calculation and which is not fully resolved is whether to consider all the junction points or only those which are elastically effective. 

Flory clarified the activity of subchains by using the words elastically effective chain, or active chain [ 1 ]. An elastically effective chain is a chain that connects two neighboring cross-link junctions in the network. 

The Scanlan-Case (SC) criterion states that a subchain is elastically effective if both its ends are connected to the elastically active junctions, i.e., whose path numbers / and i are larger than or equal to 3 (Figure 4.11). The criterion leads to... 

The number of elastically effective chains = v(l — 2/(p) in phantom network theory is smaller than its affine value v. In an affine network, all junctions are assumed to displace under the strict constraint of the strain, while in a phantom network they are assumed to move freely around the mean positions. In real networks of rubbers, the displacement of the junctions lies somewhere between these two extremes. To examine the microscopic chain deformation and displacement of the junctions, let us consider deformation of rubbers accompanied by the sweiiing processes in the solvent (Figure4.14) [1,5,14,25]. 

This chapter studies the local and global structures of polymer networks. For the local structure, we focus on the internal structure of cross-Unk junctions, and study how they affect the sol-gel transition. For the global structure, we focus on the topological connectivity of the network, such as cycle ranks, elastically effective chains, etc., and study how they affect the elastic properties of the networks. We then move to the self-similarity of the structures near the gel point, and derive some important scaling laws on the basis of percolation theory. Finally, we refer to the percolation in continuum media, focusing on the coexistence of gelation and phase separation in spherical coUoid particles interacting with the adhesive square well potential. 

In the present poblem of telechelic polymers, the A-state corresponds to the bridge chain connecting the micellar junctions, while the B-state is the dangling chain. In affine network theory, va = ( X ) r as in (9.3), and vb = 0. But vb may also be affine if the B-state is another type of the elastically effective state, such as helical conformation or globular conformation of the same chain. We can study the stress relaxation in rubber networks in which chains change their conformation by deformation [30]. 

On the other hand, at high frequencies o) > po, the storage modulus (7.5a) gives the equilibrium number of polymer chains in the network. This number, however, must be regarded as the number of elastically effective chains that connect two junctions in the network since only these chains can support stress. Another important feature of the network is the number and structure of elastically inactive chains that are dangling from the network. We call them dangling ends. We next study how such topological structures of the network depend on the temperature and concentration. 

In a randomly cross-linked network, the number density of chain ends is related to the number-average molecular weight A of the N primary linear chains and to the polymer density p by equation (49). The vertices in the network consist of the junctions and the 2N chain ends. This case is described by equation (50), and combination of equations (43), (46), (49) and (50) yields equation (51). The number of junctons p in the spanning tree is obtained by setting equal to zero in equation (51), resulting in equation (52). The number of junctions which are elastically effective is the difference between p and p and is related to by equation (53). 

---