Macroscopic fluctuations transformations

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]. 

The assemblage of chains is constructed to represent the affine network model of rubber elasticity in which all network junction positions are subject to the same affine transformation that characterizes the macroscopic deformation. In the affine network model, junction fluctuations are not permitted so the model is simply equivalent to a set of chains whose end-to-end vectors are subject to the same affine transformation. All atoms are subject to nonbonded interactions in the absence of these interactions, the stress response of this model is the same as that of the ideal affine network. 

The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability a - w w) is a symmetric second-rank tensor like Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components, and If the molecule possesses one or more symmetry axes, these coincide with the main axes of the polarizability ellipsoid. Like /J is a third-rank tensor with 27 components. All coefficients of third-rank tensors vanish in centrosymmetric media effects of the molecular polarizability of second order may therefore not be observed in them. Solutions and gases are statistically isotropic and therefore not useful technically. However, local fluctuations in solutions may be used analytically to probe elements of /3 (see p. 163 for hyper-Rayleigh scattering). The number of independent and significant components of /3 is considerably reduced by spatial symmetry. The non-zero components for a few important point groups are shown in (42)-(44). 

The basic assumption behind the second hmit, the affine limit of mbberlike elasticity (sometimes also called the affine hmit of a phantom network, Hermans-Flory-Wall (Wall 1942, 1943, 1951 Hermans 1947)), is that the cross-links are firmly embedded in their surroundings and, therefore, they do not fluctuate. Their position is transformed affinely with the macroscopic strain. The elastic part of free energy is given by... 

This establishes the relationship between the density fluctuation expressed as ((A A )2), on the one hand, and the scattering intensity I(q) and the shape cr(r) of the region of volume v being assumed, on the other hand. As the volume v is increased, its Fourier transform Y,(q) becomes more sharply peaked around q = 0, and therefore we see that only the part of the intensity curve I(q) observable at very small q has a bearing on the density fluctuation for large v. We are mainly interested in the density fluctuation on a macroscopic scale, that is, in the limit of v —> oo, in which case H(q) approaches the delta function. Noting, by use of Parseval s theorem, that... 

In case a), the mean values of the chain end-to-end vectors are displaced affinely with the principal extension ratios (p = x, y, z) specifying the macroscopic strain. The fluctuations about these mean values are independent of the sample deformation. Consequently, in the free-fluctuation limit, the transformation of the actual chain vectors is not affine in the K s. The elastic free energy change for deformation results in the expression... 

Equation (22) holds for phantom networks of any functionality, irrespective of their structural imperfections. In case b), fluctuations of junctions are assumed to be suppressed fully. The junctions themselves are considered to be firmly embedded in the medium and their position is transformed affinely with the macroscopic strain. This leads to the free energy expression for an f-functional network possibly containing free chain ends... 

The so-called domain of constraints is assumed to be cubical or spherical The degree of constraint on fluctuations is affected by the degree of deformation. All centres of domains are distributed with respect to the mean positions of the junctions in the phantom network and the mean positions of the domains are assumed to transform affinely with the macroscopic strain. Further, the shapes of domains are assumed to transform non-affinely due to some relaxation of the constraints. The theory results in an elastic free energy change... 

In this paper the noise level produced by a macroscopically steady bubbly layer on a hydrofoil is predicted analytically, based on the stochastic properties of the fluctuating quantities in the layer. This analysis uses the technique of Fourier-Stieltjes transformation as used by O.M. Phillips. In this way, the sound spectrum outside the bubble layer can be correlated to the covariance of the fluctuating quantities, as gasfraction, velocity etc., assuming for instance that the bubble layer is stochastically stationairy and almost homogeneous. 

Fluctuations in thermodynamics automatically imply the existence of an underlying structure that has created them. We know that such structure is comprised of molecules, and that their large number allows statistical studies, which, in turn, allow one to relate various statistical moments to macroscopic thermodynamic quantities. One of the purposes of the statistical theory of liquids (STL) is to provide such relations for liquids (Frisch and Lebowitz 1964 Gray and Gubbins 1984 Hansen and McDonald 2006). In such theories, many macroscopic quantities appear as limits at zero wave number of the Fourier transforms of statistical correlation functions. For example, the Kirkwood-Buff theory allows one to relate integrals of the pair density correlation functions to various thermo-physical properties such as the isothermal compressibility, the partial molar volumes, and the density derivatives of the chemical potentials (Kirkwood and Buff 1951). If one wants a connection between detailed correlations and integrated moments, one may ask about the nature of the wave-number dependence of these quantities. It turns out that the statistical theory of liquids allows an answer to such a question very precisely, which leads to new types of questions. The Ornstein-Zemike equation (Hansen and McDonald 2006), which is an exact equation of the STL, introduces the concept of correlation length which relates to the spatial extension of the density and/or concentration (the latter in the case of mixtures) fluctuations. This quantity cannot be accessed from pure... 

According to the theory the mean positions of junctions transform affinely with macroscopic strain while the fluctuations are strain independent ... 

Fluctuations of such extent involve collective motion of a great number of molecules and therefore can be described by the laws of macroscopic physics, namely, thermodynamics and hydrodynamics. Thus, small parts of the system where fluctuations of the macroscopic values manifest themselves in the properties of scattered light (the Fourier transform) contain rather many molecules that enables one to speak of local values of such macroscopic terms as entropy, enthalpy, and pressure. Every point f corresponding to a small space element in liquid at an instant i can be ascribed some values of entropy density a(r,i), of molecule number density p(r,l), of energy e(f,l), of pressure P(r,i), and of the dielectric constant e(f,t). 

Fig. 5.9 Illustration of macroscopic enhancement of fluctuation from the initially microscopic one. Fluctuations in the initial and final regime can be well described by Gaussian approximation. In the transient regime fluctuation enhances macroscopi-cally, as can be calculated based on a generalised scale transformation of time, (a) Initial regime, (b) Scaling regime, (c) Final regime.

The pair correlations between atoms or sites in a molecular fluid pertain to the microscopic spontaneous fluctuations that occur in a macrosCOpically homogeneous fluid. Under certain circumstances, these fluctuations conspire collectively or in concert to form ordered phases such as crystals. The description of these transformations of phase is beyond the scope of the linear (i.e., Gaussian) theory we have outlined thus far. The incorporation of nonlinearities is the subject we turn to... 

The very existence of a unique stationary probability distribution implies that, in a statistical ensemble of systems starting with slightly different initial conditions, different members will end up in the macroscopic limit cycle with different phases. In other words the macroscopic equations are preserved by the fluctuations, but the phase variable is not asymptotically stable. While radial fluctuations transverse to the limit cycle are small as long as the system is not in the immediate vicinity of bo, phase fluctuations along the limit cycle are macroscopic, comparable to the value of the phase itself. This can be further implemented by transforming (21) into a Fokker-Planck equation, showing that while the probability of the radial variable relaxes with a finite characteristic time the probability of the phase variable is subjected to a diffusion equation [23, 30]. 

The variance of the fluctuations is controlled by the mass density associated with an LB particle. A small number of particles gives rise to large fluctuations and vice versa. For simplicity we will ignore the effects of flow on the variance of the distribution, replacing by its m = 0 value. This can be justified at the macroscopic level by the Chapman-Enskog expansion [102]. Rewriting (180) in terms of normalized variables m [see (166)], and transforming to the normalized modes [see (167) and (169)] eliminates the explicit constraints. 

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