Affine deformation of chains

With a finite value of necessarily some intramolecular hydrodynamic interaction or shielding must occur. The importance of eq. (3.53) lies at the present time, in the fact that it can be adapted for concentrated, solvent free systems like polymer melts. As Bueche (13) pointed out, in these systems every chain molecule is surrounded by chain molecules of the same sort. As all these molecules are necessarily equivalent, one cannot speak of a hydrodynamic shielding effect. This would imply that certain chains are permanently immobilized within the coils of other chains. The contrary is expected, viz. that the centre of gravity of each chain wiH independently foHow, in the average, the affine deformation of the medium as a continuum. From this reasoning Bueche deduces that the free-draining case should be applicable to polymer melts. Eq. (3.53) is then used (after omission of rj0) for the evaluation of an apparent friction factor . After introduction of this apparent friction factor into eq. (3.50), the set of relaxation times reads ... 

Figure 3.12 Affine deformation of the ends of a network chain when a shear strain y is imposed on a rubber.

A pseudo-affine model predicts the variation of 2 with the deformation of a semi-crystalline polymer. It assumes that the distribution of crystal c axes is the same as the distribution of network chain end-to-end vectors r, in a rubber that has undergone the same macroscopic strain. Figure 3.12 showed the affine deformation of an r vector with that of a rubber block. 

It has already been mentioned in Sect. 2 that the simplest assumption, affine deformation of the tubes d = dgk, yields the Mooney-Rivlin equation (1). The value V = 1/2 was obtained by a microscopic model which is briefly discussed in Sect. 2. It is suitable for the description of moderately but almost completely cross-linked networks (e.g. sulphur-, peroxide-, or radiation-crosslinked NR, PB and PDMS chains of very high degree of polymerization). 

Fig. 103. Theoretical orientation curves I rigid rodlets following affine deformation of the matrix in which they are embedded (KRATKY-theory). Other curves theory of molecular network with short chains. At each curve the number of statistical chainelements N per chain to which the curve refers is indicated.

Fig. 3. Data for the single chain from factor S,(q) in q S,(q). Representation against log qR,i, for the followingsampleofsetI. ,71(t = 10s) +,l8(t" = 00 s) A,49(9000s) and0,50(130000s), Dashed dotted line is for the isotropic Gaussian conformation and dotted lines for the completely affine deformation of that latter in parallel and perpendicular direction...

In fact, this derivation contains two main weaknesses. The first is the assumption that all chains between two crosslinks have the same end-to-md vector for the direction (parallel to the stretching axis) and modulus. This could be the case for a onedimensional material, but in an entangled or crosslinked melt the end-to-end vectors are expected to have all directions. Moreover, a distribution of the moduli is also also sensible. For the isotropic case, the end-to-end vector distribution is Gaussian, and integrating the total distribution leads back to the usual average = i — j b / 3. The simplest assumption for a deformed material is an affine deformation of this end-to-end distribution ... 

Figure 10.1 Sketch illustrating chain retraction. We see affine deformation of the matrix of constraints (represented by dots) as well as the tube, followed by retraction of the chain within the tute. Affine deformation implies that the microscopic deformation equals the macroscopic strain. After retraction, the chain deformation is non-affine, and the primitive path equals that at equilibrium (drawing from [228]).

The opposite extreme corresponds to the affine transformation of chain vectors which is reached in the limits given in equation 23). The nature of real network deformation is between affine and phantom behavior and is characterized by the set of parameters k and C, being itself linked to the network topology through the interpenetration concept (equations 112 and 113). 

A plausible assumption would be to suppose that the molecular coil starts to deform only if the fluid strain rate (s) is higher than the critical strain rate for the coil-to-stretch transition (ecs). From the strain rate distribution function (Fig. 59), it is possible to calculate the maximum strain (kmax) accumulated by the polymer coil in case of an affine deformation with the fluid element (efl = vsc/vcs v0/vcs). The values obtained at the onset of degradation at v0 35 m - s-1, actually go in a direction opposite to expectation. With the abrupt contraction configuration, kmax decreases from 19 with r0 = 0.0175 cm to 8.7 with r0 = 0.050 cm. Values of kmax are even lower with the conical nozzles (r0 = 0.025 cm), varying from 3.3 with the 14° inlet to a mere 1.6 with the 5° inlet. In any case, the values obtained are lower than the maximum stretch ratio for the 106 PS which is 40. It is then physically impossible for the chains to become fully stretched in this type of flow. 

The large deformability as shown in Figure 21.2, one of the main features of rubber, can be discussed in the category of continuum mechanics, which itself is complete theoretical framework. However, in the textbooks on rubber, we have to explain this feature with molecular theory. This would be the statistical mechanics of network structure where we encounter another serious pitfall and this is what we are concerned with in this chapter the assumption of affine deformation. The assumption is the core idea that appeared both in Gaussian network that treats infinitesimal deformation and in Mooney-Rivlin equation that treats large deformation. The microscopic deformation of a single polymer chain must be proportional to the macroscopic rubber deformation. However, the assumption is merely hypothesis and there is no experimental support. In summary, the theory of rubbery materials is built like a two-storied house of cards, without any experimental evidence on a single polymer chain entropic elasticity and affine deformation. 

In this chapter, AFM palpation was introduced to verify the entropic elasticity of a single polymer chain and affine deformation hypothesis, both of which are the fundamental subject of mbber physics. The method was also applied to CB-reinforced NR which is one of the most important product from the industrial viewpoint. The current status of arts for the method is still unsophisticated. It would be rather said that we are now in the same stage as the ancients who acquired fire. However, we believe that here is the clue for the conversion of rubber science from theory-guided science into experiment-guided science. AFM is not merely high-resolution microscopy, but a doctor in the twenty-first century who can palpate materials at nanometer scale. 

---