Affinity fluctuation

The final assumption is that each affinity fluctuation is correlated only with fluctuation of its associated external parameters and that cross correlations of the type... 

The polarizable fluctuating charge model in CHARMM results from the work of Patel, Brooks and co-workers [92, 214], The water model is based on the TIP4P-FQ model of Rick, Stuart and Berne [17], In the development of the force field the electronegativities and hardnesses were treated as empirical parameters and do not have any association with experimental or QM values, for example, from ionization energies and electron affinities of single atoms. 

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

According to the phantom network model, the fluctuations Ar are independent of deformation and the mean f deform affinely with macroscopic strain. Squaring both sides of Equation (23) and averaging overall chains gives... 

IR dichroism has also been particularly helpful in this regard. Of predominant interest is the orientation factor S=( 1/2)(3—1) (see Chapter 8), which can be obtained experimentally from the ratio of absorbances of a chosen peak parallel and perpendicular to the direction in which an elastomer is stretched [5,249]. One representation of such results is the effect of network chain length on the reduced orientation factor [S]=S/(72—2 1), where X is the elongation. A comparison is made among typical theoretical results in which the affine model assumes the chain dimensions to change linearly with the imposed macroscopic strain, and the phantom model allows for junction fluctuations that make the relationship nonlinear. The experimental results were found to be close to the phantom relationship. Combined techniques, such as Fourier-transform infrared (FTIR) spectroscopy combined with rheometry (see Chapter 8), are also of increasing interest [250]. 

B. The mean vector connecting the chain ends is deformed affinely. The crosslink junctions fluctuate according to the theory of Brownian motion (2, 5, 7) ... 

C. Fluctuations are partially damped by chain interferences. This leads to a result intermediate between A. and B. To be specific, a model proposed by Flory (16) is adopted here. In this model, the inhibitory influence on junction fluctuations is assumed to be affine in the strain. One finds... 

In this review, we have given our attention to Gaussian network theories by which chain deformation and elastic forces can be related to macroscopic deformation directly. The results depend on crosslink junction fluctuations. In these models, chain deformation is greatest when crosslinks do not move and least in the phantom network model where junction fluctuations are largest. Much of the experimental data is consistent with these theories, but in some cases, (19,20) chain deformation is less than any of the above predictions. The recognition that a rearrangement of network junctions can take place in which chain extension is less than calculated from an affine model provides an explanation for some of these experiments, but leaves many questions unanswered. 

Our results tend to approach the affine theory with increasing z. That means, that the fluctuationsof crosslinks are more and more restricted, the reason for this being the change in microstructure. (This is quite different from the strain dependent restriction of fluctuations as predicted by Flory s recent theory.)... 

Therefore, Flory s theory concludes that as the functionality of a network increases, the constraint contribution, fc, should decrease and eventually vanish. Furthermore, in the extreme limit in which junction fluctuations are totally suppressed, the Flory theory reduces to the affine network model ... 

The phantom network behaviour corresponding to volumeless chains which can freely interpenetrate one through the other and thus to unrestricted fluctuations of crosslinks should be approached in swollen systems or at high strains (proportionality to the Mooney-Rivlin constant C-j). For suppressed fluctuations of crosslinks, which then are displaced affinely with the strain, A for the small-strain modulus (equal to C1+C2) approaches unity. This situation should be characteristic of bulk systems. The constraints arising from interchain interactions important at low strains should be reflected in an increase of Aabove the phantom value and no extra Gee contribution to the modulus is necessary. The upper limit of the predicted equilibrium modulus corresponds therefore, A = 1. 

When the network junctions are entirely immobilized by the surrounding chains, h equals zero. Then the junctions in a deformed specimen are displaced in proportion to the macroscopic strain, i.e., the deformation is affine. Alternatively, h equals unity when junction fluctuations are not impeded, the defining characteristic of a phantom network (16, 17). The parameter h was introduced (13) to allow empirically for different degrees of fluctuations. For undiluted networks at small deformations, h should usually be small, though not necessarily zero. 

Ronca and Allegra (12) and Flory ( 1, 2) assume explicitly in their new rubber elasticity theory that trapped entanglements make no contribution to the equilibrium elastic modulus. It is proposed that chain entangling merely serves to suppress junction fluctuations at small deformations, thereby making the network deform affinely at small deformations. This means that the limiting value of the front factor is one for complete suppression of junction fluctuations. 

James and Guth showed rigorously that the mean chain vectors in a Gaussian phantom network are affine in the strain. They showed also that the fluctuations about the mean vectors in such a network would be independent of the strain. Hence, the instantaneous distribution of chain vectors, being the convolution of the distribution of mean vectors and their fluctuations, is not affine in the strain. Nearly twenty years elapsed before his fact and its significance came to be recognized (Flory, 1976,... 

The relationship between fluctuation and dissipation is reminiscent of the reciprocal Onsager relations that link affinity to flux. The two relationships become identical under Onsager s regression hypothesis which states that the decay of a spontaneous fluctuation in an equilibrium system is indistinguishable from the approach of an undisturbed non-equilibrium system to equilibrium. The conclusion important for statistics, is that the relaxation of macroscopic non-equilibrium disturbances is governed by the same (linear) laws as the regression of spontaneous microscopic fluctuations of an equilibrium system. In the specific example discussed above, the energy fluctuations of a system in contact with a heat bath at temperature T,... 

Two-dimensional heteronuclear ( H- N) nuclear magnetic relaxation studies indicate that the dihydrofolate reductase-folate complex exhibits a diverse range of backbone fluctuations on the time-scale of picoseconds to nanoseconds To assess whether these dynamical features influence Michaelis complex formation, Miller et al used mutagenesis and kinetic measurements to assess the role of a strictly conserved residue, namely Gly-121, which displays large-amplitude backbone motions on the nanosecond time scale. Deletion of Gly-121 dramatically reduces the hydride transfer rate by 550 times there is also a 20-times decrease in NADPH cofactor binding affinity and a 7-fold decrease for NADP+ relative to wild-type. Insertion mutations significantly decreased both... 

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