Affine assumption

In the strained state, the chain is deformed to r/ with the chain end now at coordinates (x/, y,, z, ). To relate the microscopic strain of the chains to the macroscopic strain of the elastomer sample, we assume the deformation to be affine (assumption 4). Consider a unit cube of an isotropic rubber sample... 

The treatment of rubber elasticity presented above represents one possible extreme of behavior. The assumption that the crosslink points in the network are fixed at their mean positions and that the crosslink points deform affinely gives rise to this extreme. In real polymer networks, each crosslink point finds itself in the neighborhood of many other crosslink points. This can be verified by estimating the order of magnitude of the concentration of crosslinks and then calculating the number of crosslink points that would be found within some reasonable distance (perhaps 2 nm) of any given crosslink point. Upon deformation, the affine assumption insists that all of these crosslinks remain in the neighborhood of the particular crosslink point under consideration and, moreover, that their relative positions are fixed. 

An important consequence of the affine assumption is that the rubber model in this simple form is applicable only up to Imax = because at this draw ratio the chains parallel to the draw direction in the undeformed material are fully extended in the deformed material and cannot extend further (see equation (3.5) and the sentence following it). Further extension of the material as a whole could thus take place only non-afiinely. Equation (11.6) shows that P2 cos0)) f X /(5n) for A > 3, so that the maximum value of (P2(cosP)) to which the simple theory applies is approximately 0.2, as fig. 11.2 shows. 

The last assumption is the pseudo-affine assumption it should be seen immediately that this makes the model somewhat unphysical. It is therefore sometimes called the needles in plasticine model, in which each unit corresponds to a needle and the plasticine corresponds to the surrounding polymer. In spite of this unreality, the model often predicts well the orientation distribution of the crystallites for a drawn semicrystalline polymer. 

To quantitatively describe the change in Ny upon stretching, Kuhn and Mark [3,4] assumed (and that may be partially verified) that the affine transformation fit. This affine assumption states that the components of the displacement vectors in the bulk sample change in the same ratio as do the external dimensions of the rubber, that is. 

Due to the affineness assumption, the stress propagator reduces to an isotropic form... 

More realistic calculations need to include features which are intuitively obvious, but not easy to handle. The affine assumption can be extended to rigid junction-points, so that if there is an angular energy cos 0 where... 

The simplest molecular orbital method to use, and the one involving the most drastic approximations and assumptions, is the Huckel method. One str ength of the Huckel method is that it provides a semiquantitative theoretical treatment of ground-state energies, bond orders, electron densities, and free valences that appeals to the pictorial sense of molecular structure and reactive affinity that most chemists use in their everyday work. Although one rarely sees Huckel calculations in the resear ch literature anymore, they introduce the reader to many of the concepts and much of the nomenclature used in more rigorous molecular orbital calculations. 

Thermodynamically it would be expected that a ligand may not have identical affinity for both receptor conformations. This was an assumption in early formulations of conformational selection. For example, differential affinity for protein conformations was proposed for oxygen binding to hemoglobin [17] and for choline derivatives and nicotinic receptors [18]. Furthermore, assume that these conformations exist in an equilibrium defined by an allosteric constant L (defined as [Ra]/[R-i]) and that a ligand [A] has affinity for both conformations defined by equilibrium association constants Ka and aKa, respectively, for the inactive and active states ... 

The scheme of the interaction mechanism (Equation 88) testifies to an electro-affinity of MeFe" ions. In addition, MeFe" ions have a lower negative charge, smaller size and higher mobility compared to MeF6X(n+1> ions. The above arguments lead to the assumption that the reduction to metal form of niobium or tantalum from melts, both by electrolysis [368] and by alkali metals, most probably occurs due to interaction with MeF6 ions. The kinetics of the reduction processes are defined by flowing equilibriums between hexa-and heptacoordinated complexes. 

In general, enzymes are proteins and cany charges the perfect assumption for enzyme reactions would be multiple active sites for binding substrates with a strong affinity to hold on to substrate. In an enzyme mechanism, the second substrate molecule can bind to the enzyme as well, which is based on the free sites available in the dimensional structure of the enzyme. Sometimes large amounts of substrate cause the enzyme-catalysed reaction to diminish such a phenomenon is known as inhibition. It is good to concentrate on reaction mechanisms and define how the enzyme reaction may proceed in the presence of two different substrates. The reaction mechanisms with rate constants are defined as ... 

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. 

In this equation, r) the absolute hardness, is one-half the difference between /, the ionization potential, and A, the electron affinity. The softness, a, is the reciprocal of T]. Values of t) for some molecules and ions are given in Table 8.4. Note that the proton, which is involved in all Brdnsted acid-base reactions, is the hardest acid listed, with t — c (it has no ionization potential). The above equation cannot be applied to anions, because electron affinities cannot be measured for them. Instead, the assumption is made that t) for an anion X is the same as that for the radical Other methods are also needed to apply the treatment to polyatomic... 

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 section, we show the morphological changes of stretched NR without filler by AFM. Two-dimensional mappings of topography and elasticity for elongated NR will be given to confirm the breakdown of the long-beheved assumption of affine deformation. 

This equation illustrates the components of a competitive protein binding assay system. That is, the reaction system contains both radioactive and non-radioactive free ligand (P and P) and both radioactive and non-radioactive protein bound ligand (P Q and PQ). This type of assay assumes that binding protein will have the same affinity for the labeled or non-labeled material that is being measured. Although this assumption is not always completely valid, it usually causes no problems of consequence with most radioassays or radioimmunoassays. 

The interpretation of much of the binding data given so far is based upon the assumption that the high affinity binding sites represent a population of independent sites. In the unphosphorylated II" " these sites would open up either to the periplas-mic or cytoplasmic side of the membrane independently of each other. The assumption ignores the evidence that the enzyme is, in fact, multimeric and that the data... 

Stokes law is rigorously applicable only for the ideal situation in which uniform and perfectly spherical particles in a very dilute suspension settle without turbulence, interparticle collisions, and without che-mical/physical attraction or affinity for the dispersion medium [79]. Obviously, the equation does not apply precisely to common pharmaceutical suspensions in which the above-mentioned assumptions are most often not completely fulfilled. However, the basic concept of the equation does provide a valid indication of the many important factors controlling the rate of particle sedimentation and, therefore, a guideline for possible adjustments that can be made to a suspension formulation. 

Equations (2.10) and (2.12) are identical except for the substitution of the equilibrium dissociation constant Ks in Equation (2.10) by the kinetic constant Ku in Equation (2.12). This substitution is necessary because in the steady state treatment, rapid equilibrium assumptions no longer holds. A detailed description of the meaning of Ku, in terms of specific rate constants can be found in the texts by Copeland (2000) and Fersht (1999) and elsewhere. For our purposes it suffices to say that while Ku is not a true equilibrium constant, it can nevertheless be viewed as a measure of the relative affinity of the ES encounter complex under steady state conditions. Thus in all of the equations presented in this chapter we must substitute Ku for Ks when dealing with steady state measurements of enzyme reactions. 

Successful lead optimization can drive the affinity of inhibitors for their target enzymes so high that the equilibrium assumptions used to derive the equations for calculating enzyme-inhibitor K, values no longer hold. 

In this chapter we consider the situation where this assumption is no longer valid, because the affinity of the inhibitor for its target enzyme is so great that the value of K w approaches the total concentration of enzyme ( / T) in the assay system. This situation is referred to as tight binding inhibition, and it presents some unique challenges for quantitative assessment of inhibitor potency and for correct assessment of inhibitor SAR. 

We may, I think, without too much rashness, assume that there is some substance or substances in the nerve endings or [salivary] gland cells with which both atropine and pilocarpine are capable of forming compounds. On this assumption, then, the atropine or pilocarpine compounds are formed according to some law of which their relative mass and chemical affinity for the substance are factors. 

---