Chain phantom

So far, we have not introduced a specific model of the polymer network chains. This problem can be rigorously solved for cross-linked polymer networks consisting of phantom chains [13], or even in the more general case of filled networks where the chains interact, additionally, with spherical hard filler particles [15]. 

Fig. 3.14. Comparison of the torsion angle density distribution for a MC simulation of an athermal Ci0 phantom chains with Jacobian correction (the horizontal noisy line) and without Jacobian correction (the noisy curve showing two maxima at ca. 100° and 260° and a sharp minimum at 180°)...

The results of stress-strain measurements can be summarized as follows (1) the reduced stress S (A- A ) (Ais the extension ratio) is practically independent of strain so that the Mooney-Rivlin constant C2 is practically zero for dry as well as swollen samples (C2/C1=0 0.05) (2) the values of G are practically the same whether obtained on dry or swollen samples (3) assuming that Gee=0, the data are compatible with the chemical contribution and A 1 (4) the difference between the phantom network dependence with the value of A given by Eq.(4) and the experimental moduli fits well the theoretical dependence of G e in Eq.(2) or (3). The proportionality constant in G for series of networks with s equal to 0, 0.2, 0.33, and 0. Ewas practically the same -(8.2, 6.3, 8.8, and 8.5)x10-4 mol/cm with the average value 7.95x10 mol/cm. Results (1) and (2) suggest that phantom network behavior has been reached, but the result(3) is contrary to that. Either the constraints do survive also in the swollen and stressed states, or we have to consider an extra contribution due to the incrossability of "phantom" chains. The latter explanation is somewhat supported by the constancy of in Eq.(2) for a series of samples of different composition. 

According to Fixmann calculations [66], the entropy of a phantom chain S0(a) = lnZ0(a) in the region a < 1 can be written in the following form ... 

The real polymer chain may be usefully approximated for some purposes by an equivalent freely jointed chain. It is obviously possible to find a randomly jointed model which will have the same end-to-end distance as a real macromolecule with given molecular weight. In fact, there will be an infinite number of such equivalent chains. There is, however, only one equivalent random chain which will lii this requirement and the additional stipulation that the real and phantom chains also have the same contour length. 

In theta solvents, excluded volume effects vanish and flexible molecular chains behave as phantom chains (1). The characteristic ratio Cn = where (r )o is the mean square end-to-end distance of a... 

The Phantom Chain with Nearest-Neighbor Interactions The Fourier Normal Modes... 

In the next section we shall consider the equilibrium properties of some typical models of unperturbed chains with an increasing degree of complexity. They are (i) the bead-and-spring phantom chain (ii) the phantom chain with nearest-neighbor correlation and (iii) the unperturbed real chain... 

In the preceding, oc iq) (without the tilde) is the mean-square strain ratio of the q mode with respect to the unperturbed real chain with screened interactions (i.e., at r = 0), not with respect to the phantom chain accordingly, cco(q) s 1. In agreement with our assumption of a small solvent strength, will be proportional to T—, and we shall write... 

It should be commented that two distinct readjustments of the reference 0 state are implicit in Eqn. (2.2.4), both relating to the change of the lower limit of integration from k to pN. When applied to the integral of the screened interactions [last integral in Eqn. (2.2.4)], this change reflects the adoption of the unperturbed real chain instead of the phantom chain. When applied to the three-body integral [see Eqn. (2.2.7)], it implies a small shift of the 0 temperature from the phantom chain value... 

Let us first consider the phantom chain. The sharper is C(q) (see Figure 2), the larger is the number of neighboring atoms that contribute to the intramolecular force on any given atom. In particular, from Eqn. (2.1.44) we may prove that each atom exerts an elastic force on its kth neighbor with an elastic constant 12 8 cos(qk) sin (q12)/C(q), which decreases slowly with... 

The other extreme of behavior involves the "phantom chain" approximation. Here, it is assumed that the individual chains and crosslink points may pass through one another as if they had no material existence that is, they may act like phantom chains. In this approximation, the mean position of crosslink points in the deformed network is consistent with the affine transformation, but fluctuations of the crosslink points are allowed about their mean positions and these fluctuations are not affected by the state of strain in the network. Under these conditions, the distribution function characterizing the position of crosslink points in the deformed network cannot be simply related to the corresponding distribution function in the undeformed network via an affine transformation. In this approximation, the crosslink points are able to readjust, moving through one another, to attain the state of lowest free energy subject to the deformed dimensions of the network. 

The swelling factor a (JV g), defined by Eq. (5.49), is a correction factor or excess function that takes into account the interactions among non-nearest-neighbor chain sites. These are not included in the model for the corresponding ideal (or phantom) chain. The quantity g, on which the swelling factor depends, is the bare coupling parameter characteristic of the excluded-volume interaction potential v r), [cf. Eq. (5.145)]. 

For t/j = 1 (linear chains). Equation (11.9a) provides the correct value, d = 2, corresponding to a macromolecular coil at the 0-point (see Table 11.2). As noted previously, d = 4/3 for a percolation cluster, irrespective of the dimension of the Euclidean space (see Table 11.1) therefore, from Equation (11.9a), we obtain df= 4, which is consistent with the Flory-Stockmayer theory [60] for phantom chains. For three-dimensional space, d > 3 has no physical meaning because the object cannot be packed more densely than an object having a Euclidean dimension. It is evident that this discrepancy is due to the phantom nature of the polymer chains postulated by Cates [56] it is therefore, necessary to take into account self-interactions of chains due to which the dimension of a polymer fractal assumes a value that has a physical meaning. 

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