Temporal autocorrelation functions

Fluorescence intensity detected with a confocal microscope for the small area of diluted solution temporally fluctuates in sync with (i) motions of solute molecules going in/out of the confocal volume, (ii) intersystem crossing in the solute, and (hi) quenching by molecular interactions. The degree of fluctuation is also dependent on the number of dye molecules in the confocal area (concentration) with an increase in the concentration of the dye, the degree of fluctuation decreases. The autocorrelation function (ACF) of the time profile of the fluorescence fluctuation provides quantitative information on the dynamics of molecules. This method of measurement is well known as fluorescence correlation spectroscopy (FCS) [8, 9]. 

The autocorrelation function, G(x), of the temporal fluctuation of the fluorescence intensity at the confocal volume is analytically represented by the following equation [8, 9] ... 

Fig. 5.2.8 (a) Standard deviation map, and temporal autocorrelation functions for data recorded at a gas velocity of 27 mm s arid liquid velocities of (b) 0.40 and (c) 0.79 mm s ]. The temporal autocorrelation functions shown by the dashed and solid line styles are... 

The temporal evolution of P(r,t 0,0) is determined by the diffusion coefficient D. Owing to the movement of the particles the phase of the scattered light shifts and this leads to intensity fluctuations by interference of the scattered light on the detector, as illustrated in Figure 9. Depending on the size of the polymers and the viscosity of the solvent the polymer molecules diffuse more or less rapidly. From the intensity fluctuations the intensity autocorrelation function... 

In contrast, in dynamic light scattering (DLS) the temporal variation of the intensity is measured and is represented usually through what is known as the intensity autocorrelation function. The diffusion coefficients of the particles, particle size, and size distribution can be deduced from such measurements. There are many variations of dynamic light scattering, and... 

For a theoretical calculation of relaxation times one. must write the temporal autocorrelation functions of several functions Fn of the interparticle coordinates riS(t), 0y(O, and interparticle distance and where 0,/O and external magnetic field Ho (here particle refers to magnetic nuclei and atoms). The relaxation rates are proportional to the Fourier intensities of these autocorrelation functions at selected frequencies. For example, Torrey (16) has written for this autocorrelation function the equivalent ensemble average... 

This latter expression has been used to simplify KD(t)- Note that the time dependences of the linear and angular momentum autocorrelation functions depend only on interactions between a molecule and its surroundings. In the absence of torques and forces these functions are unity for all time and their memories are zero. There is some justification then for viewing these particular memory functions as representing a molecule s temporal memory of its interactions. However, in the case of the dipolar correlation function, this interpretation is not so readily apparent. That is, both the dipolar autocorrelation function and its memory will decay in the absence of external torques. This decay is only due to the fact that there is a distribution of rotational frequencies, co, for each molecule in the gas phase. In... 

A Tektronix T921 oscilloscope is used to display the temporal decay of the autocorrelation function. 

Because of the highly scattered temporal distributions of the individual loadings of the elements, the multivariate autocorrelation function was computed as described in Section 6.6.3. The results are demonstrated in Fig. 7-1. 

Fig. 36. (a) The binary-gated map derived from the SD map shown in Fig. 34b. The pixels identified as being associated with local pulsing are identified as white pixels associated with constant gas-liquid distribution are identified as black, (b) The temporal autocorrelation function...

Fig. 37. (a) standard deviation map, and (b) temporal autocorrelation functions for data recorded at liquid and gas velocities of 2.0 and 275mm/s, respectively. The grey scale varies between lowest (black) and highest (white) standard deviation values calculated. The temporal autocorrelation functions are shown for regions (i) and (ii) by solid and dashed lines, respectively. Reprinted from Lim et al. (2004), with permission from Elsevier. Copyright (2004). 

The random Brownian motion of colloidal particles creates temporal fluctuations in the intensity of the scattered light. The fluctuating intensity signal cannot be readily interpreted because it contains too much detail. Instead, the fluctuations are commonly quantified by constructing an intensity autocorrelation function (ACF) [41J. For this reason, DLS often goes by the name photon correlation spectroscopy (PCS). 

Figure 1 a) The temporal profile of OHD-OKE on nitrobenzene. In the insert are shown the short-time behavior of the OKE transient solid line) and the intensity autocorrelation function of the laser pulse (dashed line) b) The light scattering spectrum of nitrobenzene measured by the double monochromator and the tandem interferometer (insert). 

The relaxation time of a phonon mode is related to the temporal decay of the autocorrelation function of its energy components (potential and kinetic) [71, 72]. The total energy of each mode i for a classical system, under the harmonie approximation, is given by [73] ... 

The real-time single-turnover trajectories also enabled Xie and coworkers to analyze the time-dependent activity of each enzyme molecule. They found that individual COx molecules show temporal activity fluctuations (i.e., dynamic disorder in activity), attributable to the slow conformational dynamics of the enzyme. The timescale of the activity fluctuation is the timescale of the conformational dynamics that are longer than the catalytic turnovers and can be obtained from the autocorrelation function of the waiting times (Figure 1(d)), which shows an exponential decay behavior versus the index of turnovers (m) and whose decay constant is the fluctuation timescale. This conformational dynamics-coupled enzyme catalysis is fundamental to enzyme catalysis and extremely challenging to study with traditional methods measuring the average behaviors of a population of molecules. 

Fluorescence correlation spectroscopy analyses the temporal fluctuations of the fluorescence intensity by means of an autocorrelation function from which translational and rotational diffusion coefficients, flow rates and rate constants of chemical processes of single molecules can be determined. For example, the dynamics of complex formation between /3-cyclodextrin as a host for guest molecules was investigated with singlemolecule sensitivity, which revealed that the formation of an encounter complex is followed by a unimolecular inclusion reaction as the rate-limiting step.263... 

In this section we shall see that the incorporation of polarization into the formulation for PCS enables the shape of the particles to be inferred. This is achieved by a modest augmentation of PCS equipment that facihtates the measurement of the cross-correlation of the scattered intensity that has been resolved into different polarization states. The temporal decay of the polarized intensity cross-correlation function is identical to that of the intensity autocorrelation function used in PCS, however it decays from value less than 2, and the difference from 2 is related to the particles departure from a sphere. Before demonstrating precisely how this occurs, it is first necessary to see how polarization generalizes the Siegert relation. 

Eq. (28). This information is highly singular near the real-energy axis, where distributions appear for the values k 5, (k = 0, 1, 2,...). In order to obtain the first terms of the temporal evolution of the autocorrelation function from the initial time f = 0, E(z) is expanded in Fourier series. Assuming h = 1,... 

The theoretical treatment of PCS considers the persistence of temporal information in a fluctuation trace through the construction of a temporal autocorrelation function, which can then be fit using models based on physical processes which may be present in the sample and which cause the fluctuations. Por a more detailed review the reader is referred elsewhere [28-34 ]. 

Time domain autocorrelation analysis provides a measure of the self-similarity of a time series signal and the decay of the autocorrelation function describes the temporal persistence of information carried by it. The normalized fluorescence correlation function of a fluctuating signal F t) can be written as [29],... 

The hydrodynamic radius of colloidal particles can be obtained from dynamic light scattering (DLS), also known as photon correlation spectroscopy (PCS). Here, the temporal fluctuations of scattered light intensity are measured to provide the autocorrelation function, analysis of which provides the translational diffusion coefficient. Then the Stokes-Einstein equation (Eq. 1.9) is used to determine a hydrodynamic radius. This method is described further in Section 1.9.2. 

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