Diffusion coefficient measurement

In this section, we examine some common experimental techniques for measuring the diliusion coefiicient in hquids and sohds under isothermal conditions. Both steady-state and transient conditions are encountered in these methods. Either Eq. (13.2.1) or (13.2.5) is employed for the former situation, whereas Eq. (13.2.11) is used in the latter case, because the flow terms are either identically zero or negligible. 

Example 13.2 Northrop and Anson examined the diffusion of HCl in water by means of a diaphragm cell fitted with a porous aluminum membrane this membrane separated pure water from 0.1 HCl. Over a period of 30 min, the amount of acid that diffused through the membrane was equivalent to 0.26 cm of 0.1HCl. What is the value of the membrane constant L/Al It is known that D12 equals 2.14 x 10 cm /sec. Assume that cj and cj remain unchanged over the course of the experiment. 

Solution Using Eq. (13.3.3) and noting that c, = 0, we find the following  

Although the diaphragm cell has been used by some researchers to measiue diffusion coefficients in polymer solutions, the results are likely to be influenced by the mechanical stirring of the solutions, which can cause extension and orientation of the polymer molecules. Polymer molecules can also adsorb on the membrane. As a consequence, it is preferable to use methods that do not require the fluid to be subjected to any shear stress. This can be achieved in free-diffusion experiments [15]. 

D g is obtained by comparing tbe measured concentration profile with Eq. (13.3.8). 

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Balcom, B Fischer, A Carpenter, T Hall, L, Diffusion in Aqueous Gels. Mutual Diffusion Coefficients Measured by One-Dimensional Nuclear Magnetic Resonance Imaging, Journal of the American Chemical Society 115, 3300, 1993. 

Since it was proposed in the early 1980s [6, 7], spin-relaxation has been extensively used to determine the surface-to-volume ratio of porous materials [8-10]. Pore structure has been probed by the effect on the diffusion coefficient [11, 12] and the diffusion propagator [13,14], Self-diffusion coefficient measurements as a function of diffusion time provide surface-to-volume ratio information for the early times, and tortuosity for the long times. Recent techniques of two-dimensional NMR of relaxation and diffusion [15-21] have proven particularly interesting for several applications. The development of portable NMR sensors (e.g., NMR logging devices [22] and NMR-MOUSE [23]) and novel concepts for ex situ NMR [24, 25] demonstrate the potential to extend the NMR technology to a broad application of field material testing. 

Pore shape is a characteristic of pore geometry, which is important for fluid flow and especially multi-phase flow. It can be studied by analyzing three-dimensional images of the pore space [2, 3]. Also, long time diffusion coefficient measurements on rocks have been used to argue that the shapes of pores in many rocks are sheetlike and tube-like [16]. It has been shown in a recent study [57] that a combination of DDIF, mercury intrusion porosimetry and a simple analysis of two-dimensional thin-section images provides a characterization of pore shape (described below) from just the geometric properties. 

Fig. 2.14 The scheme of the cylindrical lens method for diffusion coefficient measurement (1) the source with the horizontal slit (2) the condenser supplying a handle of parallel beams (3) the cuvette with a refraction index gradient where the beams are deflected (4) the objective lens focusing the parallel beams to a single point (5) the optical member with an oblique slit and a cylindrical lens (6) the photosensitive material...

Extrapolation, to liquid density, of thermalization time in gaseous water (also approximate) by Christophorou et al. (1975), based on drift velocity and transverse diffusion coefficient measurement, which gives tth 2.0 x 10-1+ s... 

C, calculated from measured liquid-phase diffusion coefficients, measured range -10 to 20°C,... 

Fig. 18. The dependence of the diffusion coefficient, measured at 240°C, on the gas-phase compensation ration [Jf = [B2H6]/[PH3]) (Street et al., 1987b).

Fig. 19. Hydrogen diffusion coefficient, measured at 240°C, as a function of phosphine and diborane gas phase doping level, deduced from the data in Fig. 17. The dependence on dangling bond density is indicated on the top horizontal scale (Street el al., 1987b).

Applications to fluorescent or fluorescently labeled proteins and nucleic acids, and to fluorescent lipid probes in phospholipid bilayers, have been reported. In the latter case, the diffusion coefficients measured above the chain melting temperature were found to be 10 7 cm2 s 1, which is in agreement with values obtained by other techniques. 

Therefore we expect Df, identified as the fast diffusion coefficient measured in dynamic light-scattering experiments, in infinitely dilute polyelectrolyte solutions to be very high at low salt concentrations and to decrease to self-diffusion coefficient D KRg 1) as the salt concentration is increased. The above result for KRg 1 limit is analogous to the Nernst-Hartley equation reported in Ref. 33. The theory described here accounts for stmctural correlations inside poly electrolyte chains. 

For small chains in solution the translational diffusion significantly contributes to the overall decay of Schain(Q>0- Therefore precise knowledge of the centre of mass diffusion is essential. Combing dynamic light scattering (DLS) and NSE revealed effective collective diffusion coefficients. Measurements at different concentrations showed that up to a polymer volume fraction of 10% no concentration dependence could be detected. All data are well below the overlap volume fraction of (p =0.23. Since no -dependence was seen, the data may be directly compared with the Zimm prediction [6] for dilute solutions ... 

The diffusion coefficient of TcO " ascertained conductometrically has been found to be 1.48 X 10 cm xs whereas the self-diffusion coefficient measured with the capillary method in the presence of the supporting electrolytes 1 M... 

While D issuing from these experiments is not strictly the diffusion coefficient of water per se, but rather that of H throughout the ensemble of environments in the hydration microstructure, these authors rationalized that it could in fact be identified with D at both high and low water contents. It should be appreciated that self-diffusion coefficients measured in this way reflect fundamental hopping events on a molecular scale. 

The self-diffusion coefficient measurements were made using the pulsed-gradient spin-echo (PGSE) method which has recently been reviewed (JLL). For a nucleus in an isotropic solution, the PGSE-NMR normalized signal intensity, 1/1q is given by ... 

Figure 6.29 Diffusion coefficient measured directly (open circles) and calculated from electrical conductivity data (closed circles) for Na+ in sodium chloride. From W. D. Kingery, H. K. Bowen, and D. R. Uhhnann, Introduction to Ceramics. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc.

NMR spectroscopic data in CD2C12 solution verified the encapsulation of cobaltocenium inside a capsule composed of six molecules of resorcinarene 4. From NMR diffusion coefficient measurements, we concluded that the resorcinarene molecular capsules were preformed in CD2CI2 solution51 in the absence of cobaltocenium or any other cationic species. Similar results in CDC13 solution have been reported by Cohen and coworkers.52 53 It is now clear that cationic species are not required to seed the self-assembly of the capsules. 

Since the study of diffusion-limited reactions in solution seeks to discover more about the nature of the reaction path, the nature of the encounter pair, the energetics of the reaction and possibly the rate of reaction of the encounter pair, ftact, it is to be recommended that experimentalists actively seek to measure the diffusion coefficients of the reactants (or similar species), as well as any other parameters which may have an important bearing on the rate coefficient. By so doing, some of the uncertainty in estimating encounter distance may be removed and inconsistencies between diffusion coefficients measured independently and those obtained from an analyses of rate coefficient time dependence may provide valuable insight into the nature of the diffusion process at short distances. 

Fluorescein-labeled proteins are also used to measure the translational mobility of proteins and lipids by the Fluorescence Recovery After Photo-bleaching technique [54-59]. The uniformly labeled fluorescent sample is flashed with an intense light source to bleach a spot, thus producing a concentration gradient. The rate of recovery of fluorescence in that bleached area is measured and used to calculate the diffusion coefficient of the probe dye into the bleached zone. Such diffusion coefficient measurements have been used to determine the association constants of proteins in cells [60], to measure the exchange of tubulin between the cytoplasm and the microtubules [61,62], to study the polymerization-depolymerization process of actin [63-65] and to monitor the changes that occur upon cell maturation [66,67]. 

Besides kinetic applications, which are still to be fully realized, hydro-dynamic modulation is useful for Schmidt number and diffusion coefficient measurements not only in Newtonian fluids but also in viscoelastic polymer solutions (Ostwald fluids) [291]. 

The solid lines in Figs. 25 to 28 are the result of model predictions of metal deposition based on porphyrin reaction pathways. Curves are generated using intrinsic kinetic rate parameters and effective diffusion coefficients for the metal species on the order of 10 6 cm2/sec. These values are similar to diffusion coefficients measured in the independent studies referenced. 

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