Particle number fluctuations

Fig. 6.12. A Typical CARS signal trajectory revealing the particle number fluctuations of 110-nm polystyrene spheres undergoing free Brownian diffusion in water. The epi-detected CARS contrast arises from the breathing vibration of the benzene rings at 1003cm 1. B Measured CARS intensity autocorrelation function for an aqueous suspension of 200-nm polystyrene spheres at a Raman shift of 3050 cm-1 where aromatic C-H stretch vibrations reside. The corresponding translational diffusion time, td, of 20 ms is indicated. (Panel B courtesy of Andreas Zumbusch, adapted from [162])...

A— wavelength of radiation and —scattering angle. Particle number fluctuations, Equation (5) are given by / = / ( 0). 

The Fluctuation Theory of Solutions—also known as Fluctuation Solution Theory, Kirkwood-Buff Theory, or simply Fluctuation Theory— provides an elegant approach relating solution thermodynamics to the underlying molecular distributions or particle number fluctuations. Here, we provide the background material required to develop the basic theory. More details can be found in standard texts on thermodynamics and statistical mechanics (Hill 1956 Munster 1970). Indeed, the experienced reader may skip this chapter completely, or jump to Section 1.2. A list of standard symbols is also provided in the Prolegomenon to aid the reader, and we have attempted to use the same set of symbols and notations in all subsequent chapters. Throughout this work we refer to a collection of species (1, 2, 3,...) in a systan of interest. We consider this to represent a primary solvent (1), a solute of interest (2), and a series of additional cosolutes or cosolvents (3,4,...) which may also be present in the solution. However, other notations such as A/B or u/v is also used in the various chapters. All summations appearing here refer to the set of thermodynamically independent components (n in the mixture unless stated otherwise. Derivatives of the chemical potentials with respect to composition form a central component of the theory. The primary derivative of interest here is defined as... 

The KBIs quantify the average deviation, from a random distribution, in the distribution of j molecules surrounding a central i molecule summed over all space. In this respect they are more informative than the particle number fluctuations as they can then be decomposed and interpreted in terms of spatial contributions— using computer simulation data, for example. They clearly resemble the integrals... 

Finally, we note that particle number fluctuations increase, and many properties diverge, as one approaches a critical point. O Connell and cowoikers have shown that one can still apply FST under these circumstances by using integrals over direct... 

Hence, the derivative of the solute chemical potential (or activity) with respect to solute concentration can be expressed in terms of a combination of number densities and particle number fluctuations or KBIs. The ability to express thermodynamic properties in terms of KBIs is the major strength of FST. This has been achieved without approximation and the relationship holds for any stable binary solution at any composition involving any type of components. Derivatives of other chemical potentials can be obtained by application of the GD equation, or by a simple interchange of indices. The same approach can be applied to the second expression in Equation 1.48, with a subsequent application of Equation 1.27, to provide chemical potential derivatives with respect to other concentration scales. 

The KB/FST inversion procedure is the process of obtaining expressions for the particle number fluctuations or KBIs in terms of experimentally available (isothermal-isobaric) data. Again, there are multiple approaches to the inversion procedure (Ben-Naim 1977 O Connell 1994 Smith 2008). Arguably, the simplest approach involves the pseudo chemical potential and partial molar volumes (Ben-Naim 2006). First, we note that combining Equations 1.46 and 1.47 provides... 

Computer simulations represent one of the most common approaches to determining the local fluctuations. However, there are some technical difficulties, which can arise during the analysis of a typical simulation (see Chapter 6 for a full discussion). Most evaluations of the KBls have used the integration approach, in contrast to the actual particle number fluctuations. Furthermore, as the vast majority of simulations are performed for closed periodic systems, one is naturally limited to performing the integration out to some cutoff distance from the particle of interest. This seems reasonable given the similarities between the RDFs in open and closed systems (Weerasinghe and Pettitt 1994). Hence, one can define distance-dependent KBIs (and even distance-dependent thermodynamic functions) such that... 

In addition to the particle size and the particle number fluctuation in the measurement volume, boundary layer, overlapping, and other effects, as mentioned earlier, have an impact on the fluctuating transmission signals as well. Therefore, the determination of the particle sizes from the transmission signals requires knowledge of quantitative correlation between these factors and the transmission signal. 

For most problems that involve the effects of molecular fluctuations on reactive dynamics, it is not necessary to revert to a full molecular dynamics description of the system. We are interested in particle number fluctuations of reactive chemical species that arise from reaction and diffusion processes and occur in, small fluid volume elements. The most appropriate scale for the consideration of fluctuations is the mesoscopic scale, the regime that lies... 

For a single-component system, we can use these results and the thermodynamic relation N(dix/dN)vj — V dP/dN)yj to cormect the isothermal compressibility icj = — /F)(9 F/8P)m,t to particle number fluctuations and microscopic interactions ... 

The other approach uses kinetic theory to calculate the transport coefficients in a stationary non-equilibrium situation such as shear flow. The first application of this approach to SRD was presented in [21], where the collisional contribution to the shear viscosity for large M, where particle number fluctuations can be ignored, was calculated. This scheme was later extended by Kikuchi et al. [26] to include fluctuations in the number of particles per cell, and then used to obtain expressions for the kinetic contributions to shear viscosity and thermal conductivity [35]. This non-equilibrium approach is described in Sect. 5. 

Similar to the NVT ensemble, fluctuations in the energy vanish at the thermodynamic limit, as do particle number fluctuations. Indeed,... 

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