Spatial fluctuations

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... 

Figure 5.4. Scheme of (a) a spatial fluctuation and (b) a local depletion zone due to monolayer density fluctuations in thin liq-nid tihns. 

In the vicinity of bifurcation points, fluctuations play an important role. A stochastic analysis 1 3 reveals that as the instability is approached from below, the system develops long-range spatial correlations and a concomitant divergence of the variance of fluctuations. The calculation of the divergence exponents indicates classical values in the presence of fluctuations varying in the hydrodynamic range.4 For spatial fluctuations of shorter range the problem remains open. 

In the amorphous pattern we observe a spatial fluctuation of density as well as different repulsive and attractive forces. Describing the amorphous structure with (n — m)/n = x and using the 6—12-potential of Eq. (5) we obtain an internal shearing stress... 

Described in Section 2.1.1 the formal kinetic approach neglects the spatial fluctuations in reactant densities. However, in recent years, it was shown that even formal kinetic equations derived for the spatially extended systems could still be employed for the qualitative treatment of reactant density fluctuation effects under study in homogeneous media. The corresponding equations for fluctuational diffusion-controlled chemical reactions could be derived in the following way. As any macroscopic theory, the formal kinetics theory operates with physical quantities which are averaged over some physically infinitesimal volumes vq = Aq, neglecting their dispersion due to the atomistic structure of solids. Let us define the local particle concentrations... 

As it was first noted by Zeldovich [33] it is not easy to distinguish experimentally between exponents 1 and 3/4 (equations (2.1.8) and (2.1.77)). The approach just presented cannot be applied to charged reactants since their electrostatic attraction cuts off spatial fluctuation spectrum at the Debye radius. 

As it was noted in [77], reduction of the reaction rate with time observed for non-interacting particles at high concentrations/long reaction times, Section 6.2, is unlikely to occur for charged particles since spatial fluctuations in particle densities are now governed not by (i) but the screening radius in other words, the Coulomb repulsion of similar particles prevents their aggregation. 

Equations (7.3.23) and (7.3.24) actually imply that one- and two-dimensional cases actually exhibit already macroscopic separation of the system into regions consisting of only A particles and only B particles. This is also confirmed by the fact that the integral over the spectrum of spatial fluctuations diverges in the cases at small k. On the other hand, to find the aggregation of particles in numerical experiments in the f/iree-dimensional case we must treat the deviations from the Poisson distribution in large volumes. More detailed field-theoretical formalism has confirmed this conclusion [15]. 

Inhomogeneous open networks characterized by spatial fluctuations of the crosslink density (nodular/globular morphologies, microgels, see Chapter 7). 

One possible way of characterizing the network inhomogeneity linked to spatial fluctuations of the crosslink density (Chapter 7) could be to study the width of the a dissipation band in the DMTA or DS spectra. The comparison between tetraglycidyl methylene dianiline (TGMDA) DDS... 

The existence of temporal and spatial fluctuations of analytes in the investigated media... 

Let us first consider pattern formation in systems with base geometry. Furthermore, let us assume that the CE is placed in a plane parallel to the WE and has about the same geometry as the WE. Then, there exist states with uniform potential and current distribution, i.e. homogeneous states, and we can investigate the response of a system to local perturbations or spatial fluctuations. 

In the case of the single phase alloy Smo.5Ybo.5Cu2Ge2 the NMR signal due to copper was not detected and the resonance line broadening beyond the point of detection due to spatial fluctuation of EFG at the copper nuclei explains the reason why rare earths in EuCu2Si2 and YbCu2Si2 appear identical to the copper nuclei. 

In the concentrated and marginal regimes of fig 5,3 a mean-field description, which neglects any spatial fluctuations, is appropriate. In these regimes, the solution is homogeneous and there is no chain-length dependence. Neither does the persistence p of the chains play a role since the Floiy-Huggins expressions do not contain the chain flexibility. This is so because the flexibility is assumed to be the same in the solution and in the reference state, so that p cancels in the entropy difference between the two states. 

In atomic and simple molecular liquids, the Kerr effect, electric saturation, etc., are caused not only by non-linearities directly induced in the atoms and molecules by the external electric field (the Voigt effect), but are primarily due to Yvon-Kirkwood fluctuational-statistical processes. One should keep in mind, however, that the intrinsic electric polarizability of non-dipolar molecules is subject to modification by at least three factors (i) the effect of translational fluctuations, (//) intermolecular attractive or repulsive forces, and (in) non-linearities induced by the tenq>oraIly and spatially fluctuating electric fields of neighbouring molecules. Thus, the Yvonr-Kirkwood process discussed in Section 4 leads, in dipolar approximation, to the following variation of the electric polarizability tensor of a molecule immersed in a statistically noit-uniform medium ... 

Visscher P. T. and Van den Ende F. (1994) Diel and spatial fluctuations of sulfur transformations. In Microbial Mats. Structure, Development, and Environmental Significance (eds. L. J. Stal and P. Caumette). Springer, Berlin, pp. 353-360. 

If entanglements acted like ordinary crosslinks (vN/2) per unit volume) the stored energy function would be given by the usual expression for tetrafunctional phantom networks with the spatial fluctuations of junctions suppressed ... 

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