Self particle translational diffusion

In the detection of the autocorrelation functions in self-beat spectroscopy, solution polydispersity can lead to a non-exponential form. If we assume that there are no contributions to the autocorrelation function except those from translational diffusion for the different types of molecules, we can consider two simple cases a continuous distribution of solute particle sizes and several distinct components in a solution. We shall approach the two cases by determining their effect on the observed correlation function. 

Looking at translational diffusion in liquid systems, at least two elementary categories have to be taken into consideration self-diffusion and mutual diffusion [1, 2], In a liquid that is in thermodynamic equilibrium and which contains only one chemical species,3 the particles are in translational motion due to thermal agitation. The term for this motion, which can be characterized as a random walk of the particles, is self-diffusion. It can be quantified by observing the molecular displacements of the single particles. The self-diffusion coefficient Ds is introduced by the Einstein relationship... 

The self part Gs(r,f) of the van Hove correlation function represents the probability that a proton, which was at the origin at time 0, will be at position r at time t. For a particle undergoing a translational diffusion with diffusion coefficient D, Gs(r,0 therefore obeys the Fickian diffusion equation... 

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... 

MD simulation is advantageous for obtaining dynamic properties directly, since the MD technique provides not only particle positions but also particle velocities that enable us to utilize the response theory (e.g., the Kubo formula [175,176]) to calculate the transport coefficients from time-dependent correlation functions. For example, we will examine the self-diffusion process of a tagged PFPE molecular center of mass (Fig. 1.49) from the simulation to gain insight into the excitation of translational motion, specifically, spreading and replenishment. The squared displacement of the center mass of a molecule or a bead is used as a measure of translational movement. The self-diffusion coefficient D can be represented as a velocity autocorrelation function... 

Equation (6-17) can be easily adapted for the critical state, if p is substituted with the critical pressure, pc, obtained with Eq. (6-12) and if W/ in the exponent is substituted with w, e. This takes into account the absence of a pure translational energy contribution in the critical state. On the contrary, an additional negative term, the critical compression factor Zc = -w,lw, is introduced in the exponent, taking into account the decrease in diffusion velocity caused by attraction between the particles. As a result the following equation gives the coefficient of self-diffusion in the critical state ... 

The self-diffusivity of propane in the coked polycrystalline grains unveils details of coke formation in polycrystalline particles. Figure 44 shows that coking reduces the translational mobility inside the grains. The effect of intracrystalline coke deposition on the translational mobility of propane is indicated in Fig. 43. After a coking time of 1 h, only a slight increase of Di ,ra with increasing observation times occurs. After 12 h, a decrease is ob-... 

Self-diffusion is the random translational motion of ensembles of particles (molecules or ions) originating from their thermal energy. It is well known that diffusion, which is closely related to the molecular size of the diffusing species, is given by the Einstein-Smoluchowski equation, Eq. (6.1) [8] ... 

Various transport coefficients can also be related to time-correlation functions. For instance, as was shown in Section 5.9, the translational self-diffusion coefficient is proportional to the area under the time-correlation function of the velocity of the center of mass of the particle. 

To make the significance of the NMR technique as an experimental tool in surfactant science more apparent, it is important to compare the strengths and the weaknesses of the NMR relaxation technique in relation to other experimental techniques. In comparison with other experimental techniques to study, for example, microemulsion droplet size, the NMR relaxation technique has two major advantages, both of which are associated with the fact that it is reorientational motions that are measured. One is that the relaxation rate, i.e., R2, is sensitive to small variations in micellar size. For example, in the case of a sphere, the rotational correlation time is proportional to the cube of the radius. This can be compared with the translational self-diffusion coefficient, which varies linearly with the radius. The second, and perhaps the most important, advantage is the fact that the rotational diffusion of particles in solution is essentially independent of interparticle interactions (electrostatic and hydrodynamic). This is in contrast to most other techniques available to study surfactant systems or colloidal systems in general, such as viscosity, collective and self-diffusion, and scattered light intensity. A weakness of the NMR relaxation approach to aggregate size determinations, compared with form factor determinations, would be the difficulties in absolute calibration, since the transformation from information on dynamics to information on structure must be performed by means of a motional model. 

Phillies and co-workers [290, 291] studied the translational self-diffusion of well-defined colloidal spheres through polymer solutions and showed that the interpretation of the measured friction coefficient of the particles is fairly complicated. For a spherical particle that moves through a medium containing small solvent molecules, the friction coefficient is proportional to the solvent viscosity. When the solvent is replaced by a polymer solution one may naively expect that the friction coefficient is proportional to the viscosity of polymer solution. Measurements indicate that this is only true when the chains are very small compared to the size of the particle. 

Here, we want to discuss diffusion NMR experiments from a pragmatic point of view in order to show what information can be obtained and how reliable it is, focusing attention on supramolecular objects of intermediate dimensions. In particular, after recalling the principles underlying diffusion NMR spectroscopy and the measurement of the translational self-diffusion coefficient (A) (Section 2), we show how accurate hydrodynamic dimensions can be derived from A once the shape and size of the diffusing particles have been correctly taken into account (Section 3). Later on, the application of diffusion NMR to the study of supramolecular systems is described (Section 4) in terms of determination of the average hydro-dynamic dimensions and thermodynamic parameters of the self-assembly processes. 

The translational self-diffusion coefficient (Dt) is the physical observable that can be derived from diffusion NMR experiments. A accounts for the net result of the thermal motion induced by random-walk processes experienced by particles or molecules in solution, in the absence of any chemical potential gradient. 

Figure 6. Thermodynamic and structural quantities for the YK fluid with a = 3.3. Left column thermal expansion coefficient ap (units of k /e), isothermal compressibility Kp (units of o /e) and constant-pressure specific heat Cp (units of b) as a function of T along the isobar P = 2.5. For conventional liquids, oip, Kp, and Cp monotonically increase with T and ap > 0. Right column translational order parameter —sz (units of ks), bond-order parameter ge [89). and self-diffusion coefficient D ((units of cr (e/m / )), where m is the particle mass) as a function of P along the isotherm T = 0.06. For conventional liquids, —sz and ge increase with P while D decreases monotonically. Data are from Ref. [88].

Variable frequency proton Ti studies were first used to detect the characteristic dependence of Ti due to director fluctuations [6.20] in liquid crystals. It was recognized soon after that besides the director fluctuations, relaxation mechanisms, which are effective in normal liquids such as translational self-diffusion and molecular reorientation [6.24], also contribute to the proton spin relaxation in liquid crystals. Though the frequency dependences of these latter mechanisms are different from the relaxation, the precise nature of proton Ti frequency dispersion studied over a limited frequency range using commercial NMR spectrometers often may not be unambiguously identified. Furthermore, because of a large number of particles involved in collective motions, the motional spectrum has much of its intensities in the low-frequency domain far from the conventional Larmor frequencies. The suppression of director fluctuations in the MHz region due... 

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