Fluctuation density

The content of this section is closely related to that of the previous section. We shall be interested in the fluctuations of the density in a given region within the system. 

Consider, for example, a system in the T, F, N ensemble. We select a region S within the vessel containing the system and inquire as to the number of particles that fall within S for a given configuration R  

Each term in the sum over i is unity whenever R is in 5, and is zero otherwise. Therefore, the sum over i counts all the particles that are within S 

The last relation holds for a homogeneous fluid, where F(5) is the volume of the region S. 

In the second step on the rhs of (3.141), we have split the double sum over i and j into two sums the first consists of all the terms with i = j and the second includes the terms with / j. We have also used the identity of the product of two Dirac delta functions  

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The scattering techniques, dynamic light scattering or photon correlation spectroscopy involve measurement of the fluctuations in light intensity due to density fluctuations in the sample, in this case from the capillary wave motion. The light scattered from thermal capillary waves contains two observables. The Doppler-shifted peak propagates at a rate such that its frequency follows Eq. IV-28 and... 

Barnes and Hunter [290] have measured the evaporation resistance across octadecanol monolayers as a function of temperature to test the appropriateness of several models. The experimental results agreed with three theories the energy barrier theory, the density fluctuation theory, and the accessible area theory. A plot of the resistance times the square root of the temperature against the area per molecule should collapse the data for all temperatures and pressures as shown in Fig. IV-25. A similar temperature study on octadecylurea monolayers showed agreement with only the accessible area model [291]. 

There are two further usefiil results related to ((N-(N)) ). First is its coimection to the isothennal compressibility = -V dP/8V) j j, and the second to the spatial correlations of density fluctuations in a grand canonical system. ... 

The correlation fiinction G(/) quantifies the density fluctuations in a fluid. Characteristically, density fluctuations scatter light (or any radiation, like neutrons, with which they can couple). Then, if a radiation of wavelength X is incident on the fluid, the intensity of radiation scattered through an angle 0 is proportional to the structure factor... 

Out of the five hydrodynamic modes, the polarized inelastic light scattering experiment can probe only the tliree modes represented by equation (A3.3.18), equation (A3.3.19) and equation (A3.3.20). The other two modes, which are in equation (A3.3.17), decouple from the density fluctuations diese are due to transverse... 

Spp(A) which is the liquid structure factor discussed earlier in section A2.2.5.2. The density fluctuation spectrum is... 

This is the result for monatomic fluids and is well approximated by a sum of tliree Lorentzians, as given by the first tliree temis on the right-hand side. The physics of these tliree Lorentzians can be understood by thinking about a local density fluctuation as made up of tliemiodynamically independent entropy and pressure fluctuations p = p s,p). The first temi is a consequence of the themial processes quantified by the entropy... 

The next two temis (Lorentzians) arise from the mechanical part of the density fluctuations, the pressure fluctuations at constant entropy. These are the adiabatic sound modes (l/y)exp[-FA t ]cos[co(A) t ] with (D(k) = ck, and lead to the two spectral lines (Lorentzians) which are shifted in frequency by -ck (Stokes line) and +ck (anti-Stokes line). These are known as the Brillouin-Mandehtarn, doublet. The half-width at... 

In this section we discuss the frequency spectrum of excitations on a liquid surface. Wliile we used linearized equations of hydrodynamics in tire last section to obtain the density fluctuation spectrum in the bulk of a homogeneous fluid, here we use linear fluctuating hydrodynamics to derive an equation of motion for the instantaneous position of the interface. We tlien use this equation to analyse the fluctuations in such an inliomogeneous system, around equilibrium and around a NESS characterized by a small temperature gradient. More details can be found in [9, 10]. 

The non-consen>ed variable (.t,0 is a broken symmetry variable, it is the instantaneous position of the Gibbs surface, and it is the translational synnnetry in z direction that is broken by the inlioinogeneity due to the liquid-vapour interface. In a more microscopic statistical mechanical approach 121, it is related to the number density fluctuation 3p(x,z,t) as... 

For a one-component fluid, the vapour-liquid transition is characterized by density fluctuations here the order parameter, mass density p, is also conserved. The equilibrium structure factor S(k) of a one component fluid is... 

If we consider the scattering from a general two-phase system (figure B 1.9.10) distinguished by indices 1 and 2) containing constant electron density in each phase, we can define an average electron density and a mean square density fluctuation as ... 

Wilding N B and Bruce A D 1992 Density fluctuations and field mixing in the critical fluid J. Phys. Condens. Matter 4 3087-108... 

Term N J, Fairclough P A, Towns-Andrews E, Komanshek B U, Young R J and Ryan A J 1998 Density fluctuations the nucleation event in isotactic polypropylene crystallization Polymer 29 2381- 5... 

Molecules are in continuous random motion, and as a result of this, small volume elements within the liquid continuously experience compression or rarefaction such that the local density deviates from the macroscopic average value. If we represent by 6p the difference in density between one such domain and the average, then it is apparent that, averaged over all such fluctuations, 6p = 0 Equal contributions of positive and negative 6 s occur. However, if we consider the average value of 6p, this quantity has a nonzero value. Of these domains of density fluctuation, the following statements can be made ... 

In the next section we shall pursue the scattering by fluctuations in density. In the case of solutions of small molecules, it is the fluctuations in the solute concentration that plays the equivalent role, so we shall eventually replace 6p by 6c2. First, however, we must describe the polarizability of a density fluctuation and evaluate 6p itself. 

The quantity G - Gq = 6G is the change in G associated with the fluctuation, and the term (9G/dp)o 6p = 0 because of the cancellation of positive and negative density fluctuations. Therefore we obtain... 

Lateral density fluctuations are mostly confined to the adsorbed water layer. The lateral density distributions are conveniently characterized by scatter plots of oxygen coordinates in the surface plane. Fig. 6 shows such scatter plots of water molecules in the first (left) and second layer (right) near the Hg(l 11) surface. Here, a dot is plotted at the oxygen atom position at intervals of 0.1 ps. In the first layer, the oxygen distribution clearly shows the structure of the substrate lattice. In the second layer, the distribution is almost isotropic. In the first layer, the oxygen motion is predominantly oscillatory rather than diffusive. The self-diffusion coefficient in the adsorbate layer is strongly reduced compared to the second or third layer [127]. The data in Fig. 6 are qualitatively similar to those obtained in the group of Berkowitz and coworkers [62,128-130]. These authors compared the structure near Pt(lOO) and Pt(lll) in detail and also noted that the motion of water in the first layer is oscillatory about equilibrium positions and thus characteristic of a solid phase, while the motion in the second layer has more... 

The Pink model is found to exhibit a gel-fluid transition for lipids with sufficiently long chains, which is weakly first order. The transition disappears in bilayers of shorter lipids, but it leaves a signature in that one observes strong lateral density fluctuations in a narrow temperature region [200,201]. In later studies, the model has been extended in many ways in order to explore various aspects of gel-fluid transitions [202]. For example, Mouritsen et al. [203] have investigated the interplay between chain melting and chain crystallization by coupling a two-state Doniach model or a ten-state Pink model to a Potts model. (The use of Potts models as models for... 

Note that large density fluctuations are suppressed by construction in a random lattice model. In order to include them, one could simply simulate a mixture of hard disks with internal conformational degrees of freedom. Very simple models of this kind, where the conformational degrees of freedom affect only the size or the shape of the disks, have been studied by Fraser et al. [206]. They are found to exhibit a broad spectrum of possible phase transitions. 

Incompressible Limit In order to obtain the more familiar form of the Navier-Stokes equations (9.16), we take the low-velocity (i,e. low Mach number M = u I /cs) limit of equation 9,104, We also take a cue from the continuous case, where, if the incompressible Navier-Stokes equations are derived via a Mach-number expansion of the full compressible equations, density variations become negligible everywhere except in the pressure term [frisch87]. Thus setting p = peq + p and allowing density fluctuations only in the pressure term, the low-velocity limit of equation 9,104 becomes... 

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