Fluctuation Theory Debye

Polarizability a is usually measured in terms of refiractive index n or dielectric constant s and the light-scattering apparatus can be designed in relation to the measurement of refractive index. (Note according to Maxwell s theory.) For that reason, we utilize the relation 

The fluctuation of concentrations is always accompanied by a change in free energy, AG. We now expand AG in terms of Ac around the equilibrium concentration (c) using Taylor s series  

The first term is zero for a closed term at constant temperature, whereas the higher terms, including (Ac), may be neglected since the fluctuations are rather small. Thus, only the second term is physically meaningful  

Using the Boltzmann expression, we obtain a distribution function of concentration  

The two integrals in the equation of (Ac ) can be evaluated by using the two formulas and we obtain 

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FLUCTUATION THEORY (DEBYE) 325 We may change 6 into d, since there is only one independent variable involved now ... 

Peter Debye in 1944 further extended the work of Rayleigh and the fluctuation theory of Smoluchowski and Einstein to include the measurement of the scattering of light by macromolecular solutions for determining molecular size. 

To obtain a correct form of Eq. (22) allowing for thermodynamic non-ideality of the solution, fluctuation theory originally developed by Einstein, Zernicke, Smoluchowski and Debye has been adapted to polymer solutions. 

Tn the critical region of mixtures of two or more components some physical properties such as light scattering, ultrasonic absorption, heat capacity, and viscosity show anomalous behavior. At the critical concentration of a binary system the sound absorption (13, 26), dissymmetry ratio of scattered light (2, 4-7, II, 12, 23), temperature coefficient of the viscosity (8,14,15,18), and the heat capacity (15) show a maximum at the critical temperature, whereas the diffusion coefficient (27, 28) tends to a minimum. Starting from the fluctuation theory and the basic considerations of Omstein and Zemike (25), Debye (3) made the assumption that near the critical point, the work which is necessary to establish a composition fluctuation depends not only on the average square of the amplitude but also on the average square of the local... 

The 13C nuclear resonance studies in my report provide some informations on lipid membrane fluctuations in binary mixtures. Totally unsolved problems include an appropriate two-dimensional Debye-Huckel theory for membranes, and theoretical treatments of boundary free energies (between proteins and lipids, and between solid and fluid phase lipids). 

Compared with the momentum of impinging atoms or ions, we may safely neglect the momentum transferred by the absorbed photons and thus we can neglect direct knock-on effects in photochemistry. The strong interaction between photons and the electronic system of the crystal leads to an excitation of the electrons by photon absorption as the primary effect. This excitation causes either the formation of a localized exciton or an (e +h ) defect pair. Non-localized electron defects can be described by planar waves which may be scattered, trapped, etc. Their behavior has been explained with the electron theory of solids [A.H. Wilson (1953)]. Electrons which are trapped by their interaction with impurities or which are self-trapped by interaction with phonons may be localized for a long time (in terms of the reciprocal Debye frequency) before they leave their potential minimum in a hopping type of process activated by thermal fluctuations. 

In the years 1910-1917 Gouy2 and Chapman3 went a step further. They took into account a thermal motion of the ions. Thermal fluctuations tend to drive the counterions away form the surface. They lead to the formation of a diffuse layer, which is more extended than a molecular layer. For the simple case of a planar, negatively charged plane this is illustrated in Fig. 4.1. Gouy and Chapman applied their theory on the electric double layer to planar surfaces [54-56], Later, Debye and Hiickel calculated the potential and ion distribution around spherical surfaces [57],... 

The suspended particles are small compared with distances Z over which the potential

in potential from mean value, the ion concentrations deviate as in linearized Debye-Htickel theory ... 

Hill, 1986). We have emphasized that fluctuation contributions, e.g. Eq. (4.71) p. 90, have a definite sign. This Debye-Hiickel theory treats correlations between ionic species, and here we observe again that treatment of correlations lowers this free energy. 

Lee, B.P., and Fisher, M.E. Density fluctuations in an electrolyte from generalized Debye-Hueckel theory. Phys. Rev. Lett., 1996, 76, p. 2906-9. 

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

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