Tracers, molecular motion

Thermal or low-energy neutron scattering experiments have been most valuable in throwing light on molecular motion in plastic crystals. These experiments measure changes in the centre of mass of a molecule. Diffusion constants obtained from neutron experiments differ widely from those obtained from tracer experiments since neutron scattering is mainly determined by rotational diffusion. The scattering function has the form... 

As regards the relative merits of deuterium and tritium as isotopes of hydrogen (protium) for tracer and mechanistic studies, it must at once be stated that these isotopes are complementary. Each has advantages for specific applications. When molecular motion in macromolecules, lipid layers, membranes. 

In laminar flow, a similar mixing process occurs when the liquid is sheared between two rotating cylinders. During each revolution, the thickness of the fluid element is reduced, and molecular diffusion takes over when the elements are sufficiently thin. This type of mixing is shown schematically in Figure 7.3 in which the tracer is pictured as being introduced perpendicular to the direction of motion. 

Theory for the self- and tracer-diffusion of a diblock copolymer in a weakly ordered lamellar phase was developed by Fredrickson and Milner (1990). They modelled the interactions between the matrix chains and a labelled tracer molecule as a static, sinusoidal, chemical potential field and considered the Brownian dynamics of the tracer for small-amplitude fields. For a macroscopically-oriented lamellar phase, they were able to account for the anisotropy of the tracer diffusion observed experimentally. The diffusion parallel and perpendicular to the lamellae was found to be sensitive to the mechanism assumed for the Brownian dynamics of the tracer. If the tracer has sufficiently low molecular weight to be unentangled with the matrix, then its motion can be described by a Rouse model, with an added term representing the periodic potential (Fredrickson and Bates 1996) (see Fig. 2.50). In this case, motion parallel to the lamellae does not change the potential on the chains, and Dy is unaffected by... 

At this point, the variables C and u are now used to connote the average (macroscopic) tracer concentration and velocity, respectively, and the overbar is dropped for convenience. Implicit in this formulation is the belief that fundamentally Lagrangian (particle) dispersion can be modeled as a continuum Eulerian phenomenon in a fashion analogous to the Fickian formulation of molecular transport by Brownian motion. This is a useful fiction for simple modeling exercises, but must be used with caution (see the next section on isopycnal diffusion). 

Molecular diffusion coefficients in water are usually determined in the laboratory by using a tracer in agar or gel to insure that the solution is totally free of turbulent motion. Values for gases and ions in pure water are presented in units of cm s in Table 9.1. The molecular diffusion coefficients and their temperature dependence for gases were calculated from the Eyring equation. 

As indicated, the flux may be expressed either in units of molecules/m2 s or in units of kg/m2 s. Here, p and n are the density and number density of air, respectively, and K is called the eddy diffusion coefficient. This quantity must be treated as a tensor because atmospheric diffusion is highly anisotropic due to gravitational constraints on the vertical motion and large-scale variations in the turbulence field. Eddy diffusivity is a property of the flowing medium and not specific to the tracer. Contrary to molecular diffusion, the gradient is applied to the mixing ratio and not to number density, and the eddy diffusion coefficient is independent of the type of trace substance considered. In fact, aerosol particles and trace gases are expected to disperse with similar velocities. 

It is worth a short detour into fluid mechanics to explore some details of this approach and how it fits into the reactor conservation equations. For moderate flow velocities the dispersion of a tracer in laminar flow will occur by axial and radial diffusion from the flow front of the tracer and, in the absence of eddy motion, this will be via a molecular diffusion mechanism. However, the net contribution of diffusion in the axial direction can be taken as small in comparison to the contribution of the flow velocity profile. This leaves us with a two-dimensional problem with diffusion in the radial direction and convection in the longitudinal direction the situation is considered in illustrated in Figure 5.7. 

In the following sections, we treat briefly or sketch the relevant observable quantities in NMR spectroscopy for the investigation of rotational motions and the examples of molecular or molecular-group rotations mentioned, as well as the tracer methods and their results for translational diffusion. 

It turns out that turbulent diffusion can be described with Fick s laws of diffusion that were introduced in the previous section, except that the molecular diffusion coefficient is to be replaced by an eddy or turbulent diffusivity E. In contrast to molecular diffusivities, eddy dififusivities are dependent only on the phase motion and are thus identical for the transport of different chemicals and even for the transport of heat. What part of the movement of a turbulent fluid is considered to contribute to mean advective motion and what is random fluctuation (and therefore interpreted as turbulent diffusion) depends on the spatial and temporal scale of the system under investigation. This implies that eddy diffusion coefficients are scale dependent, increasing with system size. Eddy diffusivities in the ocean and atmosphere are typically anisotropic, having much large values in the horizontal than in the vertical dimension. One reason is that the horizontal extension of these spheres is much larger than their vertical extension, which is limited to approximately 10 km. The density stratification of large water bodies further limits turbulence in the vertical dimension, as does a temperature inversion in the atmosphere. Eddy diffusivities in water bodies and the atmosphere can be empirically determined with the help of tracer compounds. These are naturally occurring or deliberately released compounds with well-estabhshed sources and sinks. Their concentrations are easily measured so that their dispersion can be observed readily. 

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