Autocorrelation function Brownian motion

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

P. Espanol and I. Zuniga, Force autocorrelation function in Brownian motion theory, J. Chem. Phys. 98, 574 (1993). 

We should point out here the great analogy between and the friction coefficient studied in the Brownian motion problem of Section IV (see Eq. (242)) instead of having the time autocorrelation function of the force F , we now have the time correlation function between F and Fe. 

For a single fluorescent species undergoing Brownian motion with a translational diffusion coefficient Dt (see Chapter 8, Section 8.1), the autocorrelation function, in the case of Gaussian intensity distribution in the x, y plane and infinite dimension in the z-direction, is given by... 

Since the velocity relaxation time, m/J, is typically 0.1 ps, t is rather shorter than that estimated from the decay of the velocity autocorrelation function. As an operational convenience, rrel — mjl can be deduced from the decay time re of the velocity autocorrelation functions. However, this procedure still does not entirely adequately describe the details of Brownian motion of particles over short times. The velocity relaxes in a purely exponential manner characteristic of a Markovian process. Further comments on the reduction of the Fokker—Planck equation to the diffusion equation have been made by Harris [526] and Tituiaer [527]. 

In the previous section, the phenomenological description of Brownian motion was presented. The Langevin analysis leads to a velocity autocorrelation function which decays exponentially with time. This is characteristic of a Markovian process, as Doobs has shown (see ref. 490). Since it is known heyond question that the velocity autocorrelation function is far from such an exponential function, the effect that the solvent structure has on the progress of a chemical reaction cannot be assessed very reliably by means of phenomenological Langevin description. Since the velocity of a solute is correlated with its velocity a while before, a description which fails to consider solute and solvent velocities can hardly be satisfactory. Necessarily, the analysis requires a modification of the Langevin or Fokker—Plank description. In this section, some comments are made on this new and exciting area of research. 

Many excellent introductions to quasi-elastic light scattering can be found in the literature describing the theory and experimental technique (e.g. 3-6). The use of QELS to determine particle size is based on the measurement, via the autocorrelation of the time dependence of the scattered light, of the diffusion coefficients of suspended particles undergoing Brownian motion. The measured autocorrelation function, G<2>(t), is given by... 

Photon Correlation. Particles suspended in a fluid undergo Brownian motion due to collisions with the liquid molecules. This random motion results in scattering and Doppler broadening of the frequency of the scattered light. Experimentally, it is more accurate to measure the autocorrelation function in the time domain than measuring the power spectrum in the frequency domain. The normalized electric field autocorrelation function g(t) for a suspension of monodisperse particles or droplets is given by ... 

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). 

In order to avoid bias due to number fluctuations, it is necessary that there is at least 1000 particles present in the measuring volume and, for a typical value of the scattering volume of 10 cm, effects of number fluctuations are to be expected for particle diameters greater than around 0.5 pm. Number fluctuations lead to an additional time decaying term in the autocorrelation function. Since the characteristic decay time of this additional term is usually much slower than the decay attributed to Brownian motion, the average particle size, which is proportional to the average decay time, will be overestimated if the effect of number fluctuations is neglected [277]. 

Dynamically raised processes in the dispersion, such as Brownian molecular motion, cause variations in the intensities of the scattered light with time, which is measured by PCS. Smaller the particle, higher the fluctuations by Brownian motion. Thus, a correlation between the different intensities measured is only possible for short time intervals. In a monodisperse system following first-order kinetics, the autocorrelation function decreases rather fast. In a half logarithmic plot of the auto correlation function, the slope of the graph enables the calculation of the hydrodynamic radius by the Stokes-Einstein equation. With the commercial PCS devices the z-average is determined, which corresponds to the hydrodynamic radius. 

To describe the Brownian motion of a particle in a fluid, Langevin assumed [250,251] that a random force, affecting the particle having its origin in the particles of the fluid exists and that its autocorrelation function is... 

The calculation of orientational autocorrelation functions from the free rotator Eq. (14) which describes the rotational Brownian motion of a sphere is relatively easy because Sack [19] has shown how the one-sided Fourier transform of the orientational autocorrelation functions (here the longitudinal and transverse autocorrelation functions) may be expressed as continued fractions. The corresponding calculation from Eq. (15) for the three-dimensional rotation in a potential is very difficult because of the nonlinear relation between and p [33] arising from the kinematic equation, Eq. (7). 

The orientation autocorrelation function P2[cos 0(t)] is given by r(t) and reflects the motion undergone by the fluorescent chromophore (2,14). A number of models for Brownian motion have been proposed (14) but in the simple case of a rigid sphere, r(t) is described by a single exponential decay where Tf., the rotational correlation time is related to the hydrodynamic volume of the sphere and the viscosity of the medium through the Stokes-Einstein relation (14,16). More complex motions of fluorophores necessitate the development of models which fit the functional form of r(t) experimentally obtained (14). 

The intensity I of the light scattered from a dilute macromolecular or supra-molecular solution is a fluctuating quantity due to the Brownian motion of the scattering particles. These fluctuations can be analysed in terms of the normalised autocorrelation function y1 ( t ) of the scattered electrical field Es, which contains information about the structure and the dynamics of the scattering particles [80]. 

Other workers have adopted a different corpuscular model for quasielastic light scattering from a gel (122,123,124). The gel is treated as an assembly of identical, independent, harmonically bound particles, each in Brownian motion about a stationary mean position. The analysis of Garlson and co-workers (123,124) formally includes the presence of static interference scattering components resulting from spatial structuring of the polymer chains and from the consequent constraints in the diffusive motions of the chains. This formalism leads to the prediction of nonexponential scattered intensity autocorrelation functions. 

Consider now the intensity, /, of scattered light from suspended particles to be analogous to p. It is necessary to correlate I with the ftequency at which the particles exhibit Brownian motion. A measure of the amount of correlation is the autocorrelation function which is defined by ... 

Fig. 5 Broadening of cross rrelatton toction, n the correlation function with Brownian motion I esence of Brownian motioiL (a) Autocorrelation...

Micro-PIV-Based Diffusometry, Fig. 1 Comparison of (a) autocorrelation function, (b) cross-correlation function without any Brownian motion, and (c) cross-correlation function with Brownian motion... 

A random motion of nanoparticles in aqueous suspension changes the time intensity of the scattered light and the fluctuating signal is processed by forming the autocorrelation function G(t). For a monodisperse suspension of globular particles in Brownian motion (Dahneke 1983, Chu 1991, Brown 1993, Hunter 1993), it can be written as follows ... 

Espanol, Pep. Zuniga, Ignacio. Force autocorrelation function in brownian motion theory. The Journal of Chemical Chem. Physics Phys., 1993, 98(1), 574-580. Yuan-Hui, Li. Gregory, Sandra. Diffusion of ions in sea water and in deep-sea sediments. Geochimica et Cosmochimica Acta, 1974, 38(5), 703-714. 

---