Self-diffusion coefficients translational order

The translational mobility of an ion is different by one to two orders of magnitude between moving freely in a solution and diffusing with an entity of colloidal dimensions such as a micelle. This allows counterion self-diffusion coefficients to be used for characterization of counterion distribution, for example, in terms of a counterion association degree. 

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. 

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].

Measurement of diffusion using pulsed field gradient NMR (PFG-NMR) is a powerful analytical tool because it combines the high specificity and information content of NMR spectroscopy with the size selectivity of diffusion coefficients. PFG-NMR employs timescales of tens of ms and has a displacement sensitivity of the order of 100 nm. PFG-NMR can determine molecular self-diffusion coefficients in liquid phases down to a lower limit of 10 " m s Due to the combination of experimental convenience and straightforward interpretation, PFG-NMR has become the method of choice for studying translational diffusion. PFG-NMR experiments have been reported using H, H, Li, C, F and other nuclei. The time A over which PFG-NMR measurements are possible is limited. 

For the case of an equilibrium surfactant self-assembly, as in microemulsions, discrete aggregates or droplets, if they exist, have extensions much smaller than the distance monitored in a diffusion experiment. Therefore, the experiment is not sensitive to the molecular displacement within droplets, and the only translation of the droplet component monitored is that of the entire droplet. This is much slower than the diffusion of the same component in the neat solvent or of the other solvent component of the microemulsion. For a droplet-type microemulsion, the diffusion coefficients of the two solvents differ dramatically, typically by two orders of magnitude and sometimes even more. 

The procedure of evaluating self-diffusion data in terms of microstructure is to calculate the reduced or normalized diffusion coefficient, D/Dq, for the two solvents. Do being the neat solvent value under the appropriate conditions. Here we also have to account for reductions in D resulting from factors other than microstructure, mainly solvation effects. As discussed above, solvation will lead to a reduction of solvent diffusion that is proportional to the surfactant concentration. Normally the correction has been empirical and based on diffusion studies for cases of established structure, notably micellar solutions. We need to distinguish between corrections due to polar head-water and alkyl chain-oil interactions. The latter have often been considered insignificant, but a closer analysis (either experimental or theoretical) is lacking. However, it is probably reasonable to assume, for example, that the resistance to translation is not very different in the lipophilic part of the surfactant film and in an alkane solution. (This is supported by observations of molecular mobilities of surfactant allQ l chains on the same order of magnitude as for a neat hydrocarbon.)... 

Here r is the distance from the cylinder axis and is the coherence length. The distance that the molecule migrates in the iso-tropic phase during a NMR measurement VD/(5v is in the above case of the order of the cylinder diameter R and the motional averaging by translational self-diffusion must be taken into account. The diffusion coefficient in the surface layer was estimated as Dg 10 ° cm s" and is much smaller than the bulk value. The time a molecule resides at the surface is then t IqID = QT s. In the fast exchange regime the effective spin-spin relaxation rate can be expressed as... 

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