Affine deformation method

Affine deformation method, description, 109 Agitated glass ampule method apparatus, 510,511/312 limitations, 515... 

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. 

The simplest explanation is that there is a rubber-like network present and that this has a maximum extensibility due to the degree of entanglement, which is constant for a given grade of polymer and depends on its molar mass and method of polymerisation. This limiting extensibility is not to be confused with the limit of applicability of the affine rubber model for predicting orientation distributions discussed in section 11.2.1 because the limiting extension can involve non-affine deformation. 

Using these v-dependent weight functions, the weighting scheme described by eq. (97) ensures that each nuclear position v(i) is transformed to its counterpart nuclear position t(j), while the entire electron density is deformed continuously. This method of weighted affine transformations has no origin or coordinate dependence. 

Empirical Relationship - Empirical relationships correlating glass transition temperature of an amorphous viscoelastic material with measurement temperature and frequency, such as the William Landel Ferry equation (17) and the form of Arrhenius equation as discussed, assume an affine relationship between stress and strain, at least for small deformations. These relationships cover finite but small strains but do not include zero strain, as is the case for the static methods such as differential scanning calorimetry. However, an infinitely small strain can be assumed in order to extend these relationships to cover the glass transition temperature determined by the static methods (DSC, DTA, dilatometry). Such a correlation which uses a form of the Arrhenius equation was suggested by W. Sichina of DuPont (18). 

The relationship between chromatographic retention and n/Tj was far from perfect. The poor correlation (r=0.198) indicated that that the log k values measured on an immobilized-HSA column could not be correlated with the binding affinity values measured with a modified Hummel-Drq er method using purified HSA. Lidocaine, quinine, and scoporamin were outliers. Possible reasons are the immobilization deformed the binding sites or... 

The strain in the crystal that results from the deformation can be calculated by a method [36] similar to that of Section 5, by minimizing the quantity, (d,- Dj), where are vectors between a particle and its 12 nearest neighbors, D, are these vectors in the perfect fee lattice, and a is the optimal affine transformation tensor. The local strain tensor s is the symmetric part of a. This tensor can be transformed... 

Chains in Networks. One of the first studies of chains in networks is by Gao and Weiner (235) where they performed extended simulations of short chains with fixed (affinely moving) end-to-end vectors. The first extensive molecular dynamics simulations of realistic networks were performed by Kremer and collaborators (236). These calculations were based on a molecular dynamics method that has been applied to study entanglement effects in polymer melts (237). The networks obtained by cross-linking the melts were then used to study the effect of entanglements on the motion of the cross-links and the moduU of the networks. The moduli calculated without any adjustable parameters were close to the phantom network model for short chains, and supported the Edwards tube model for long ones. Similar molecular dynamics analyses were used to understand the role of entanglements in deformed networks in subsequent studies (238-240). 

Medalia [16] demonstrated a further usefulness of this approach by using graphitised carbon blacks, which are known to have a weaker surface-affinity towards rubber. Thus, the method provides a means of examining the filler-rubber interaction. However, by the conditions set for deriving Equation 8.10, its applicability is limited to a small deformation. Also, only bi-particle interactions are considered in the equation and multi-particle interactions are assumed to be negligible. For the filled-rubber with the normal loading of 40-50 phr, the carbon black particles are crowded and the multi-particle interaction is important. 

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