Diffusion in liquids and solids

The transport of matter within a solid or liquid is described by the diffusion constant D, which is temperature dependent. In liquids, the translational diffusion constant is closely related to the viscosity coefhcient 

The range of timescales over which diffusion can be observed in Mossbauer spectroscopy is determined by the lifetime of the excited state of the Mossbauer nucleus and the Larmor precession time of the hyperfine interactions. For Fe the times involved are in the range between 5 x 10 and 5 X 10 s and the diffusional line broadening is equal to the natural linewidth when the diffusion constant D is of the order of 10 m s in a liquid. 

The concept of diffusional broadening was originally developed for the incoherent quasi-elastic scattering of slow neutrons from moving particles. The resolution in this situation is determined by the instrumentation and 

The analysis of Mossbauer spectra affected by diffusion is based on an early work by Singwi Sjolander (1960). The absorption cross-section r, is given in this theory by the space and time Fourier transform of the nuclear decay factor exp (— t/T ) times the self-correlation function G(r, t), which is the probability of finding the Mossbauer nucleus which was at the origin at time t O, at a position r at time t. 

In liquids the diffusion is isotropic and is a quasi-continuous process. A consideration of the Singwi and Sjolander theory and the quasi- 

Fick s law of diffusion is also used for problems involving liquid and solid diffusion, and the main difficulty is one of determining the value of the diffusion coefficient for the particular liquid or solid. Unfortunately, only approximate theories are available for predicting diffusion coefficients in these systems. Bird, Stewart, and Lightfoot [9] discuss the calculation of diffusion in liquids, and Jost [6] gives a discussion of the various theories which have been employed to predict values of the diffusion coefficient. The reader is referred to these books for more information on diffusion in liquids and solids. 

Diffusion in solids is complex because of the strong influence of the molecular force fields on the process. For these systems Fick s law [Eq. (11-1)] is often used, along with an experimentally determined diffusion coefficient, although there is some indication that this relation may not adequately describe the physical processes. The numerical value of the diffusion coefficient for liquids 

---

Diffusion is ubiquitous in nature whenever there is heterogeneity, there is diffusion. In liquid and gas, flow or convection is often present, which might be the dominant means of mass transfer. However, inside solid phases (minerals and glass), diffusion is the only way of mass transfer. Diffusion often plays a major role in solid-state reactions, but in the presence of a fluid dissolution and recrystallization may dominate. 

Secondly, the model of thermal diffusion does not allow one to explain the independence of the reaction rate on temperature observed for many low-temperature electron transfer processes. Indeed, the thermal diffusion of molecules in liquids and solids is known to be an activated process and its rate must be dependent on temperature. True, at low temperatures when activated processes are very slow, diffusion itself can be assumed to become a non-activated process going on via a mechanism of nuclear tunneling, i.e. by tunneling transitions of atoms over very short (less than 1 A) distances. A sequence of such transitions can, in principle, result in a diffusional approach of reagents in the matrix. Direct tunneling of the electron, whose mass is less than that of an atom by a factor of 10 or 104, can, however, be expected to proceed much faster. 

The application of NMR to the study of diffusion in zeolites involves the refinement and extension of methods originally developed to study self diffusion in liquids and low melting solids. The method is restricted to species such as hydrocarbons which contain a sufficiently high density of atoms such as H with unpaired nuclear spins. Authoritative reviews of the application of NMR to the study of adsorbed molecules have been given by Pfeifer(22,23) and only a brief outline is included here. 

In all three phases, the diffusion constant should decrease as its density is increased. At higher densities, molecules are closer to each other. In gases, they will collide more often and travel shorter distances between collisions. In liquids and solids, there will be less space for molecules to move around each other. 

Esmaeeli A, Ervin E, Tryggvason G (1994) Numerical simulations of rising bubbles. In Blake JR, Boulton-Stone JM, Thomas NH (eds) Bubble Dynamics and Interfacial Phenomena. Kluwer Academic Publishers, Dordrecht Esmaeeli A, Tryggvason G (1996) An Inverse Energy Cascade in Two-Dimensional Low Reynolds Number Bubbly Flows. J Fluid Mech 314 315-330 Fan F-S, Tsuchiya K (1990) Bubble Wake Dynamics in Liquids and Solid-Liquid Suspensions. Butterworth-Heinemann, Boston Fick A (1855) Ueber Diffusion. Ann der Physik 94 59-86 Scott Fogler H (2006) Elements of Chemical Reaction Engineering. Fourth Edition, Prentice-Hall International, Inc, New Jersey... 

The next three sections deal with steady and pulsed field gradient methods for measuring translational diffusion in liquids and plastic solids. There are other methods for measuring slower diffusion which we will not really touch upon. See, for example, Ailion, referenced in IV.C.l., and Burnett and Harmon (1972). 

This method is widely used to simulate motions in liquids and solids and to study rapid diffusions in ionic lattices. Although the application of molecular dynamics (MD) to studying ion transport in polymer systems is still in a seminal state, it is included in this chapter because of its correlation with the expansion of computing power, which will probably result in its prominence as an investigative method in this field. 

From the kinetic theory of gases, we know that the mean speed of molecules is proportional to the square root of its absolute temperature. Millions of intermolecular collisions occur per second and are proportional to the concentration of molecules. In a gas of nonuniform composition, this random interaction results in molecular diffusion until the composition is uniform throughout. Similar diffusive transport processes occur in liquids and solids as well. Diffusion is thus one aspect of the transport pillar and is the subject of this handbook. It comprises rates of chemical transport within and between media. We assert that this topic has been the poor relation in environmental science. 

This case of diffusion in polymers is described by ideas drawn from both diffusion in liquids and diffusion in solids. The theoretical development takes place in two steps. First, the binary diffusion coefficient D is corrected for the nonideal solution... 

Ternary diffusion coefficients in liquids and solids cannot be found from binary values, but only from experiments. When experiments are not available, which is usually the case, one can make estimates by assuming that the Onsager phenomenological coefficients are a diagonal matrix that is. 

---