Fluctuation effect

E. Brener, T. Ihle, H. Muller-Krumbhaar, Y. Saito, K. Shiraishi. Fluctuation effects on dendritic growth morphology. Physica A 200 96, 1994. 

We have seen that the cooperative region, which represents a nominal dynamical unit of liquid, is of rather modest size, resulting in observable fluctuation effects. Xia and Wolynes [45] computed the relaxation barrier distribution. The configurational entropy must fluctuate, with the variance given by the usual expression [77] 5Sc) ) = Cp barrier height for a particular region is directly related to the local density of states, and hence to... 

The mean-field SCFT neglects the fluctuation effects [131], which are considerably strong in the block copolymer melt near the order-disorder transition [132] (ODT). The fluctuation of the order parameter field can be included in the phase-diagram calculation as the one-loop corrections to the free-energy [37,128,133], or studied within the SCFT by analyzing stability of the ordered phases to anisotropic fluctuations [129]. The real space SCFT can also applied for a confined geometry systems [134], their dynamic development allows to study the phase-ordering kinetics [135]. 

Michael W. Deem, A Statistical Mechanical Approach to Combinatorial Chemistry Venkat Ganesan and Glenn H. Fredrickson, Fluctuation Effects in Microemulsion Reaction Media... 

We will demonstrate below that the Ginzburg number Gi = AT/Tc), which determines the broadness of the energy region near the critical temperature, where fluctuations essentially contribute, is Gi A(Tc/iiq)4 with A 500 in our case. To compare, for clean metals A 100, p,q — fi,., the latter is the electron chemical potential. Thus Gi 1, if Tc is rather high, Tc (f -t- )p,q, and we expect a broad region of temperatures, where fluctuation effects might be important. 

The FvdM as well as the BMVW model neglects thermal fluctuation effects both are T = 0 K theories. Pokrovsky and Talapov (PT) have studied the C-SI transition including thermal effects. They found that, for T 0 K the domain walls can meander and collide, giving rise to an entropy-mediated repulsive force of the form F where I is the distance between nearest neighbor walls. Because of this inverse square behavior, the inverse wall separation, i.e. the misfit m, in the weakly incommensurate phase should follow a power law of the form... 

The simplest data reduction algorithm for FIDs consists in averaging the magnitudes of the complex FID signal over a data window positioned within its starting portion (Fig. 26). The window can be freely positioned in a way to cut out any dead-time distortions and, at the same time, minimize field-fluctuation effects. 

Bennett, S. M., Tibbitts, T. W., Cao, W. (f99f). Diurnal temperatures fluctuation effects on potatoes grown with 12 h photoperiods. Amer. Potato J., 68, 81-86. 

Before inserting t, pBX, and bx obtained above into Eq. (44), we have to mention the fluctuation effect of the fuzzy cylinder. When a hindering fuzzy cylinder is entering the periphery of the laminar region, the hindrance that it exerts may be released by fluctuation of the segment distributions in the... 

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... 

If there is no interaction between similar reactants (traps) B, they are distributed according to the Poisson relation, Ab (r, t) = 1. Besides, since the reaction kinetics is linear in donor concentrations, the only quantity of interest is the survival probability of a single particle A migrating through traps B and therefore the correlation function XA(r,t) does not affect the kinetics under study. Hence the description of the fluctuation spectrum of a system through the joint densities A (r, ), which was so important for understanding the A4-B — 0 reaction kinetics, appears now to be incomplete. The fluctuation effects we are interested in are weaker here, thus affecting the critical exponent but not the exponential kinetics itself. It will be shown below that adequate treatment of these weak fluctuation effects requires a careful analysis of many-particle correlations. 

Therefore, the weakness of the fluctuation effects in the A + B — B reaction as compared to the case of A + B —> 0 (Section 5.1) does not permit us to restrict ourselves to the radial correlations between A s and B s subtle effects leading to equation (5.2.1) arise due to the angular correlations of reactants B in the vicinity of a single A. The demonstrated peculiarity of the A + B — B reaction requires an adequate specified mathematical approach. 

More favourable conditions of the fluctuation effect manifestation are met if one of reactants is immobile, e.g., Da = 0. In this case considerable... 

When the initial concentrations are small, fluctuation effects are not pronounced and the curve of the superposition approximation degenerates to those of the linear approximation. However, for any initial concentration no there exists distinctive time to — to(no), at which these approximations start to diverge from each other. For d = 2 parameter k slightly affects behaviour of the K = K(t) - full and broken curves 1 at long times t reveal the same slopes with the only difference in co-factors. At last, for d = 3 curves at k = 0 and k = 0.5 differ considerably and have different slopes at long t (this peculiarity of the kinetics when one of the reactants is immobile has been already stressed above). 

Therefore, the study of the stochastic Lotka and Lotka-Volterra models carried out in Chapter 8, has demonstrated that the traditional estimates of the complexity of the system necessary for its self-organisation are not correct. Incorporation of the fluctuation effects and thus introduction of a continuous number of degrees of freedom prove their ability for self-organisation and thus put them into a class of the basic models for the study of the autowave processes. 

Theory for block copolymer rheology is still in its infancy. There are no models that can predict the rheological behaviour of a block copolymer from microscopic parameters. Fredrickson and Helfand (1988) considered fluctuation effects on the low frequency linear viscoelastic properties of block copolymers in the disordered melt near the ODT. They found that long-wavelength transverse momentum fluctuations couple only to compositional order parameter fluctua-... 

Bates et al. 1990 Rosedale and Bates 1990). It was pointed out that the onset of measurable fluctuation effects depends on molecular weight in a manner that suggests that entanglements play an important role in coupling fluctuations... 

Gunkel I et al (2007) Fluctuation effects of the microphase separation of diblock-cpolymers in the presence of an electric field. Macromolecules 40(6) 2186—2191... 

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