Translational diffusion coefficient self particle

In a liquid that is in thermodynamic equilibrium and which contains only one chemical species, 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 is introduced by the Einstein relationship... 

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... 

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. 

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... 

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. 

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].

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 ... 

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. 

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