Correlation function concentration

In the real space the correlation function (6) exhibits exponentially damped oscillations, and the structure is characterized by two lengths the period of the oscillations A, related to the size of oil and water domains, and the correlation length In the microemulsion > A and the water-rich and oil-rich domains are correlated, hence the water-water structure factor assumes a maximum for k = k 7 0. When the concentration of surfac-... 

The most important result of this work is that despite two different SRO patterns, we have found concentration independent EPI. The evolution of the diffuse intensity with composition is thus mainly due to the sensitivity of the equilibrium state (i.e. the correlation function) to the concentration. 

Mw = 2.1 x 106g/mol) in water, which is denoted Cw(t) in the original work [44]. The subscript indicates that both the incoming beam and the scattered light are vertically polarized. The correlation function was recorded for a solution with a concentration of c = 0.005 g/L at a scattering vector of q = 8.31 x 106m-1. The inset shows the distribution function of the relaxation times determined by an inverse Laplace transformation. 

The systems undergoing phase transitions (like spinodal decomposition) often exhibit scaling phenomena [ 1—4] that is, a morphological pattern of the domains at earlier times looks statistically similar to a pattern at later times apart from the global change of scale implied by the growth of L(f)—the domain size. Quantitatively it means, for example, that the correlation function of the order parameter (density, concentration, magnetization, etc.)... 

Here scalar order parameter, has the interpretation of a normalized difference between the oil and water concentrations go is the strength of surfactant and /o is the parameter describing the stability of the microemulsion and is proportional to the chemical potential of the surfactant. The constant go is solely responsible for the creation of internal surfaces in the model. The microemulsion or the lamellar phase forms only when go is negative. The function/(<))) is the bulk free energy and describes the coexistence of the pure water phase (4> = —1), pure oil phase (4> = 1), and microemulsion (< ) = 0), provided that/o = 0 (in the mean-held approximation). One can easily calculate the correlation function (4>(r)(0)) — (4>(r) (4>(0)) in various bulk homogeneous phases. In the microemulsion this function oscillates, indicating local correlations between water-rich and oil-rich domains. In the pure water or oil phases it should decay monotonically to zero. This does occur, provided that g2 > 4 /TT/o — go- Because of the < ), —<(> (oil-water) symmetry of the model, the interface between the oil-rich and water-rich domains is given by... 

The p.c.s. measurements were carried out using a Malvern multibit correlator and spectrometer together with a mode stabilized Coherent Krypton-ion laser. The resulting time correlation functions were analysed using a non-linear least squares procedure on a PDP11 computer. The latex dispersions were first diluted to approximately 0.02% solids after which polymer solution of the required concentration was added. 

We define an "i-th nearest neighbour complex to be a pair of oppositely charged defects on lattice sites which are i-th nearest neighbours, such that neither of the defects has another defect of opposite charge at the i-th nearest neighbour distance, Rit or closer. This corresponds to what is called the unlike partners only definition. A different definition is that the defects be Rt apart and that neither of them has another defect of either charge at a distance less than or equal to R. This is the like and unlike partners definition. For ionic defects the difference is small at the lowest concentrations the definition to be used depends to some extent on the problem at hand. We shall consider only the first definition. It is required to find the concentration of such complexes in terms of the defect distribution functions. It should be clear that what is required is merely a particular case of the specialized distribution functions of Section IV-D and that the answer involves pair, triplet, and higher correlation functions. In fact this is not the procedure usually employed, as we shall now see. 

At this time diffraction data for ion-ion distributions in aqueous solutions of moderate concentration are beginning to become available. In aqueous NiCl2 solutions very refined neutron diffraction studies indicate that the Ni2+-Cl pair correlation function has a peak near 3.l8 under conditions in which the Cl does not penetrate the Ni(H20)g2+ unit. (J+2 ) It is reported that EXAFS studies give the same result. (1 3) While the information is most welcome it is puzzling because a geometrical calculation indicates that the closest center to center distance for the Ni2+ and a Cl that does not penetrate the hydration shell is closer to 3.98. (7)... 

The photoinduced absorbance anisotropy in a TPD experiment relaxes according to the same correlation function as in Eq. (4.16).(29) Effects of spatial variations in the excitation and probe beams, and chromophore concentration, have been treated and shown not to alter the final result.(29) NMR dipolar relaxation rates are expressed in terms of Fourier transforms of the correlation functions, 4ji< T2m[fi(0)] T2m[i2(f)]>> where fl(f) denotes the orientation of a particular internuclear vector. In view of Eq. (4.7), these correlation functions are independent of the index m, hence formally the same as in Eq. (4.16). For the analysis of NMR relaxation data, it is necessary also to evaluate Fourier transforms of the correlation functions. Methods to accomplish this in the case of deformable DNAs have been developed and applied to analyze a variety of data.(81 83)... 

This effective Q,t-range overlaps with that of DLS. DLS measures the dynamics of density or concentration fluctuations by autocorrelation of the scattered laser light intensity in time. The intensity fluctuations result from a change of the random interference pattern (speckle) from a small observation volume. The size of the observation volume and the width of the detector opening determine the contrast factor C of the fluctuations (coherence factor). The normalized intensity autocorrelation function g Q,t) relates to the field amplitude correlation function g (Q,t) in a simple way g t)=l+C g t) if Gaussian statistics holds [30]. g Q,t) represents the correlation function of the fluctuat-... 

Typically, scaling approaches are employed to explain the behavior in the semidilute regime. By examining static correlations near the temperature, Daoud and Jannick( ) have expressed the density-density correlation function in terms of a correlation length that is inversely proportional to concentration. Since the diffusion coefficient is inversely proportional to the correlation length it is directly proportional to the concentration. 

A chemical relaxation technique that measures the magnitude and time dependence of fluctuations in the concentrations of reactants. If a system is at thermodynamic equilibrium, individual reactant and product molecules within a volume element will undergo excursions from the homogeneous concentration behavior expected on the basis of exactly matching forward and reverse reaction rates. The magnitudes of such excursions, their frequency of occurrence, and the rates of their dissipation are rich sources of dynamic information on the underlying chemical and physical processes. The experimental techniques and theory used in concentration correlation analysis provide rate constants, molecular transport coefficients, and equilibrium constants. Magde" has provided a particularly lucid description of concentration correlation analysis. See Correlation Function... 

Correlation functions are powerful tools in statistical physics, and in the above example they permit one to examine the behavior of a fluctuating system from a reference time back to previous times. Such fluctuations can occur in the concentration of two (or more) interconverting chemical species in dynamic equilibrium, and the technique of concentration correlation analysis permits one to determine the forward and reverse rate constants for their interconversion. See Concentration Correlation Analysis... 

The plume structure data introduced in the previous sections provide the opportunity to assess whether multiple sensors that are spatially separated can rapidly acquire useful information [5], The correlation function for the instantaneous concentration acquired by two sensors at locations p and q, separated in the transverse direction, is defined as... 

The computer-reconstructed catalyst is represented by a discrete volume phase function in the form of 3D matrix containing information about the phase in each volume element. Another 3D matrix defines the distribution of active catalytic sites. Macroporosity, sizes of supporting articles and the correlation function describing the macropore size distribution are evaluated from the SEM images of porous catalyst (Koci et al., 2006 Kosek et al., 2005). Spatially 3D reaction-diffusion system with low concentrations of reactants and products can be described by mass balances in the form of the following partial differential equations (Koci et al., 2006, 2007a). For gaseous components ... 

The pair correlation function g describes the distribution of molecular centers in the solution. In concentrated systems at rest, gxl. Flow alters g, and it is this change which gives rise to the drag forces. For sufficiently slow shearing flows,... 

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