Brownian diffusion isotropic

Molecular motions in low molecular weight molecules are rather complex, involving different types of motion such as rotational diffusion (isotropic or anisotropic torsional oscillations or reorientations), translational diffusion and random Brownian motion. The basic NMR theory concerning relaxation phenomena (spin-spin and spin-lattice relaxation times) and molecular dynamics, was derived assuming Brownian motion by Bloembergen, Purcell and Pound (BPP theory) 46). This theory was later modified by Solomon 46) and Kubo and Tomita48 an additional theory for spin-lattice relaxation times in the rotating frame was also developed 49>. 

Therefore, for the internal (Neel) relaxation the parameter, r m plays the same role as the fluid viscosity r in the mechanism of the external (Brownian) diffusion. Note that the density of the anisotropy energy K is not included in x. This means that xD can be considered as the internal relaxation time of the magnetic moment only for magnetically isotropic particles (where K = a = 0). The sum of the rotations—thus allowing for both the diffusion of the magnetic moment with respect to the particle and for the diffusion of the particle body relative to the liquid matrix—determines the angle ft of spontaneous rotation of the vector p at the time moment t ... 

This formula in an equivalent form (with Xg> instead of Xg because a magnetic moment diffusion inside an isotropic particle was considered) had been obtained in Section III.A.3 as Eq. (4.96). Besides that, similar formulas, with xg indeed, are well known in the theory of rotary Brownian diffusion in dipolar fluids [69]. 

Figure 6.3 Orientation dependence of the EPR spectrum of a planar copper(II) complex and lineshapes in the slow tumbling regime (simulations), (a) Geometry of the complex, molecular coordinate frame, and orientations of the magnetic field vector Bo corresponding to the spectra in (b). (b) EPR spectra at four selected orientations simulated with EasySpin 1.3 (http //www.esr.ethz.ch). (c) Slow tumbling spectra assuming isotropic Brownian diffusion with rotational correlation time Tr = 1 /(6 2 ) simulated with the Schneider-Freed suite of programs [32].

In a system that enables rotational diffusion of paramagnetic species, the anisotropy of magnetic interactions is subject to a partial or complete averaging, and this results in changes of the EPR line shape. The correlation time of rotational movements, Tr, is related to the viscosity and temperature of the medium in the case of isotropic Brownian diffusion of a spherical molecule, it is given by the equation ... 

It has been recognized for a long time that the orientation dependence of a vector fixed to a polymer chain cannot be represented by a simple isotropic rotational diffusion model. In such a model (12) the orientation is assumed to follow a vector joining the center of a sphere to a point performing a random brownian diffusion on the surface of that sphere. According to this model which describes well the orientation of spherical objects or infinitely thin rigid rods, the OACF is an exponential function (13). 

A comparison of rotational and translational diffusion results obtained in l-octyl-3-imidazolium tetrafluoroborate, [omim][BF4], and in 1-propanol and isopropyl benzene has been given for TEMPONE. Measurements at different temperatures and concentrations indicate that rotational motion can be described by isotropic Brownian diffusion only for the classical organic solvents used, but not for the IL. Simulation of the EPR spectra fit with the assumption of different rotational motion around the different molecular axes. Rotational diffusion coefficients >rot follow the Debye-Stokes-Einstein law in all three solvents, whereas the translational diffusion coefficients do not follow the linear Stokes-Einstein relation D ot versus Tlr ). The activation energy for rotational motions Ea,rot in [omim][BF4] is higher than the corresponding activation energies in the organic solvents. 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

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... 

Abstract We use Nuclear Magnetic Resonance relaxometry (i.e. the frequency variation of the NMR relaxation rates) of quadrupolar nucleus ( Na) and H Pulsed Gradient Spin Echo NMR to determine the mobility of the counterions and the water molecules within aqueous dispersions of clays. The local ordering of isotropic dilute clay dispersions is investigated by NMR relaxometry. In contrast, the NMR spectra of the quadrupolar nucleus and the anisotropy of the water self-diffusion tensor clearly exhibit the occurrence of nematic ordering in dense aqueous dispersions. Multi-scale numerical models exploiting molecular orbital quantum calculations, Grand Canonical Monte Carlo simulations, Molecular and Brownian Dynamics are used to interpret the measured water mobility and the ionic quadrupolar relaxation measurements. 

Four different models for the molecular dynamics have been tested to simulate the experimental spectra. Brownian rotational diffusion and jump type diffusion [134, 135] have been used for this analysis, both in their pure forms and in two mixed models. Brownian rotational diffusion is characterized by the rotational diffusion constant D and jump type motion by a residence time t. The motions have been assumed to be isotropic. In the moderate jump model [135], both Brownian and jump type contributions to the motion are eou-pled via the condition Dx=. ... 

Pairwise Brownian dynamics has been primarily used for the analysis of diffusion controlled reactions involving the reaction between isotropic molecules with complex reactive sites. Since its introduction by Northrup et al. [58], the pairwise Brownian dynamics method has been considerably refined and modified. Some of the developments include the use of variable time steps to reduce computational times [61], efficient calculation methods for charge effects [63], and incorporation of finite rates of reaction [58,61,62]. We review in the following sections, application of the method to two example problems involving isotropic translational diffusion reaction of isotropic molecules with a spherical reaction surface containing reactive patches and the reaction between rodlike molecules in dilute solution. 

J. Srinivasalu Gupta and D. V. Khakhar, Brownian dynamics simulation of diffusion-limited polymerization of rodlike molecules Isotropic translational diffusion, J. Chem. Phys., 107 (1997) 3289-3294. 

Above 100 K, motional effects on spectrum become pronounced with increasing temperature and, above 230 K, the spectra consist of essentially an isotropic and equally spaced hyperfine triplet, but with different relative intensities. The line shape simulations were carried out by adopting a Brownian rotational diffusion model in order to evaluate the associated (average) rotational correlation time, and its degree of anisotropy, = zpy, /... 

Figure 2. Theoretical spectra of NO2 anisotropically rotating about its x axis (b), y axis (c), and z axis (d). The spectra were calculated for Brownian rotational diffusion model by using R// = 5.27 X 10 sec- Rj = 5.27 x 10 sec and T2 =3.0 G. For isotropic rotation (a), R// = Rj = 1.67 X 10 sec l and T2 l=3.0 G were used. The rotational diffusion values used through (a) to (d) correspond to a constant value of r =l x 10 sec v, 9 167 GHz...

The diffusion term on the right-hand side assumes that the flux of surfactant that is due to random Brownian motions is linear in the gradient of T and that the process is isotropic. In writing a complete mass balance for any species, it is necessary to include the possibility of a mean flux that is due to random molecular motion in the presence of a gradient in concentration - just as it was necessary to hypothesize a heat flux vector in the conservation of energy balance. The relationship... 

Consider particles of radius a Ao and assume that in the course of their motion in the liquid, they are completely entrained by turbulent pulsations that play the basic role in the mechanism of mutual approach of suspended particles. Then it can be assumed that particle transport is performed via isotropic turbulence. Since particles move chaotically in the liquid volume, their motion is similar to Brownian one and can be considered as diffusion with some effective factor of turbulent diffusion Dr. In the same manner as in the case of Brownian coagulation, it is possible to consider the diffusion flux of particles of radius U2 toward the test particle of radius Uj. The distribution of particles U2 is characterized by the stationary diffusion equation... 

In an excellent review article, Tirrell [2] summarized and discussed most theoretical and experimental contributions made up to 1984 to polymer self-diffusion in concentrated solutions and melts. Although his conclusion seemed to lean toward the reptation theory, the data then available were apparently not sufficient to support it with sheer certainty. Over the past few years further data on self-diffusion and tracer diffusion coefficients (see Section 1.3 for the latter) have become available and various ideas for interpreting them have been set out. Nonetheless, there is yet no established agreement as to the long timescale Brownian motion of polymer chains in concentrated systems. Some prefer reptation and others advocate essentially isotropic motion. Unfortunately, we are unable to see the chain motion directly. In what follows, we review current challenges to this controversial problem by referring to the experimental data which the author believes are of basic importance. 

The electrochemical character of metal and semiconductor wet etching is responsible for some special features in the etching processes. Electrochemical processes always include a characteristic series of process steps. So, ligands must first diffuse to the solid surface. Then the surface process with the electrochemical charge transition can occur. Finally, the complexes formed have to desorb from the surface and diffuse into the solution. The slowest of these three steps determines the rate of the overall process. Ideal isotropic behavior is found if one of the diffusion steps is rate determining. Diffusion is caused by the Brownian motion of particles and therefore is independent of direction. That is the reason that transport control in wet etching results in isotropic etch behavior. 

In a macroscopically disordered system such as a microcrystalline powder or a glassy frozen solution, all possible orientations p occur with weighting factors sin p. The EPR spectrum of such a disordered system depends on whether reorientation by rotational diffusion is very slow, moderate, or very fast on the EPR timescale. In the following we assume isotropic Brownian rotational diffusion with an isotropic value Riso of the diffusion tensor and a transverse relaxation time of 150 ns. 

Carper and co-workers have performed a detailed analysis of the relaxation times of both [C4mim] [25] and [Cgmim]" [26] with [PF0]", which was later extended with more detail to deconvolute the relative contributions of the various relaxation mechanisms [27]. They found that the contribution of CSA to the experimentally observed relaxation time was about half of the contribution from dipolar relaxation. This work raises doubts about the applicability of isotropic relaxation models to ionic liquids. It is important to note that the and measurements of ionic liquids in the literature show different behaviour when attached to the same ion. The random Brownian motion that occurs in most liquids leads to rapid spin diffusion between nuclei bonded to a common ion or molecule, causing them to all exhibit the same T. The lack of such behaviour is a clear indication that the dynamics of ionic liquids are... 

When the relaxation mechanism is the modulation of the magnetic g and A components due to the rotational diffusion of the paramagnetic group (mainly for the nitroxide radicals, and for 5 = y paramagnetic ions), the analysis of the spectra in the fast-slow motion regime provides the correlation time for the rotational motion. An increase in the correlation time corresponds to a decrease in mobility of the paramagnetic probe or label. The evaluation of the correlation time for the rotational motion was performed by simple methods or by computation of the spectra. Different diffusion models can be considered, such as Brownian or jump diffusion models, and the rotational mobility may be considered isotropic or anisotropic. In this latter case, for nitroxide radicals, the main information was obtained from the perpendicular component of the correlation time. Furthermore, a shift of the main rotational axis accounts for the compression of the labels due to other molecules approaching the label at the dendrimer surface. 

---