Macroscopic diffusion

From this consideration one can derive the macroscopic diffusion equation for the concentration c(x, t) of the chemical component as... 

FIG. 14 Illustration of the vapor pressures measured in an a measurement. The vapor pressure in both the water and the food containers is established by macroscopic diffusion of water out of the sample (pure water or food). 

MACROSCOPIC DIFFUSION CONTROL MICROSCOPIC DIFFUSION CONTROL Magic acid,... 

MICROSCOPIC DIFFUSION CONTROL MACROSCOPIC DIFFUSION CONTROL MICROSCOPIC REVERSIBILITY CHEMICAL REACTION DETAILED BALANCING, RRINCIRLE OF CHEMICAL KINETICS MICROTUBULE ASSEMBLY KINETICS BIOCHEMICAL SELF-ASSEMBLY ACTIN ASSEMBLY KINETICS HEMOGLOBINS POLYMERIZATION... 

The macroscopic diffusion coefficient ) is defined in terms of the mean jump distance a and mean time between jumps r as ... 

There are two arenas for describing diffusion in materials, macroscopic and microscopic. Theories of macroscopic diffusion provide a framework to understand particle fluxes and concentration profiles in terms of phenomenological coefficients and driving forces. Microscopic diffusion theories provide a framework to understand the physical basis of the phenomenological coefficients in terms of atomic mechanisms and particle jump frequencies. 

Macroscopic treatments of diffusion result in continuum equations for the fluxes of particles and the evolution of their concentration fields. The continuum models involve the diffusivity, D, which is a kinetic factor related to the diffusive motion of the particles. In this chapter, the microscopic physics of this motion is treated and atomistic models are developed. The displacement of a particular particle can be modeled as the result of a series of thermally activated discrete movements (or jumps) between neighboring positions of local minimum energy. The rate at which each jump occurs depends on the vibration rate of the particle in its minimum-energy position and the excitation energy required for the jump. The average of such displacements over many particles over a period of time is related to the macroscopic diffusivity. Analyses of random walks produce relationships between individual atomic displacements and macroscopic diffusivity. 

In general, a particle migrates in a material by a series of thermally activated jumps between positions of local energy minima. Macroscopic diffusion is the result of all the migrations executed by a large ensemble of particles. The spread of the ensemble due to these migrations connects the macroscopic diffusivity to the microscopic particle jumping. 

A relationship between the macroscopic diffusivity, D, of a component i and the mean-square displacement, (R2 (Nr)), can be obtained from the behavior of ci(x,t) as it evolves from an initial point source at the origin. Using the solution for diffusion from an instantaneous point source in three dimensions in Table 5.1, the distribution of particles after a time r will be given by... 

If a particle moves by a series of displacements, each of which is independent of the one preceding it, the particle moves by a random walk. Random walks can involve displacements of fixed or varying length and direction. The theory of random walks provides distributions of the positions assumed by particles such distributions can be compared directly to those predicted to result from macroscopic diffusion. Furthermore, the results from random walks provide a basis for understanding non-random diffusive processes. 

Equation 7.48 relates the macroscopic diffusivity and microscopic jump parameters for uncorrelated diffusion in one dimension. 

Equation 7.52 is of central importance for atomistic models for the macroscopic diffusivity in three dimensions (see Chapter 8). For isotropic diffusion in a system of dimensionality, d, the generalized form of Eq. 7.52 is... 

The development of the miscibility gap for W < 0 and the antiphases ( Tjeq) for W > 0 have entirely different kinetic implications. For decomposition, mass flux is necessary for the evolution of two phases with differing compositions. Furthermore, interfaces between these two phases necessarily develop. The evolution of ordered phases from disordered phases (i.e., the onset of nonzero structural order parameters) can occur with no mass flux macroscopic diffusion is not necessary. Because the 77+q-phase is thermodynamically equivalent to the 7/iq-phase, the development of 77+q-phase in one material location is simultaneous with the evolution of r lq-phase at another location. The impingement of these two phases creates an antiphase domain boundary. These interfaces are regions of local heterogeneity and increase the free energy above the homogeneous value given by Eq. 17.14. The kinetic implications of macroscopic diffusion and of the development of interfaces are treated in Chapter 18. 

Thermal diffusivity a and thermal conductivity l strongly depend on the composition (Fig. 27). Materials containing a high residual a content show a very low thermal diffusivity (88 vol% a a = 5 mm2 s-1) whereas the same composition after complete a// transformation has 13 mm2 s-1 at RT. ass ceramics also show low thermal diffusivity (Fig. 27). The intrinsic anisotropic thermal diffusivity inside the individual grains is without connection with the macroscopic diffusivity, as a consequence of the dispersion of grain orientations in the ceramics [375]. 

The term macroscopic diffusion control has been used to describe processes in which the rate of reaction is determined essentially by the rate of mixing of the reactant solutions. The nitration of toluene in sulpholane by the addition of a solution of nitronium fluoroborate in sulpholane appears to fall into this class (Ridd, 1971a). Obviously, if a reaction is subject to microscopic diffusion control when the reactants meet in a homogeneous solution, it must also be subject to macroscopic diffusion control when preformed solutions of the same reactants are mixed. However, the converse is not true. The difficulty of obtaining complete mixing of solutions in very short time intervals implies that a reaction may still be subject to macroscopic diffusion control when the rate coefficient is considerably below that for reaction on encounter. The mathematical treatment and macroscopic diffusion control has been discussed by Rys (Ott and Rys, 1975 Rys, 1976), and has been further developed recently (Rys, 1977 Nabholtz et al, 1977 Nabholtz and Rys, 1977 Bourne et al., 1977). It will not be considered further in this chapter. 

Developing approach has given an explanation at what conditions the matrices with macroscopic non-uniformities (e.g., lamellar systems, heterogeneous inclusion phases, etc.) and with effective macroscopic diffusion coefficients in the latter case, a matrix is considered as a homogeneous one and when the phenomenological model of double sorption, which operates with sorption sites of two types could be used [190-192]. 

The diffusion pumps principle has been known for decades and these pumps were widely used till the 1970s. It relies on momentum transfer of a condensable vaporized fluid to the molecules to be pumped. Macroscopic diffusion pumps can generate vacuum down to 10-6 Pa. 

Chapter 7 of this text details the use of NMR imaging methods to determine structure and function of crosslinked rubbers. In this section we review the NMR imaging of rubbers swollen with small molecules. Two sets of experiments are described, namely the study of diffusion of small molecules into a previously unswollen rubber (macroscopic diffusion), and the measurement of images of the solvent and/or polymer after equilibrium has been achieved. 

Macroscopic Diffusion of Small Molecules in Swollen Rubbers... 

The axial voidage distribution resulting from mixing of dissimilar particles, as shown in Fig. 26, will now be examined in the light of the dynamics of these particles at the interfacial boundary between the properly juxtaposed phases of a binary particle mixture (Lou, 1964 Kwauk, 1973). Both the lighter particles at the top and the heavier particles at the bottom share the same tendency of invading the region occupied by the other. This behavior conforms to the concept of random walk, for which Fick s law can be adapted to describe the macroscopic diffusion flux of the smaller particles 2 ... 

Sometimes, this term is also used for analysis of relationships between the macroscopic diffusion coefficients... 

Macroscopic diffusion model is based on underlying microscopic dynamics and should reflect the microscopic properties of the diffusion process. A single diffusion equation with a constant diffusion coefficient may not represent inhomogeneous and anisotropic diffusion in macro and micro scales. The diffusion equation from the continuity equation yields... 

Cage effect dynamics or kinetics of geminate recombination was observed for the first time under photodissociation of aC C dimer of aromatic radicals in a viscous media. A suggestion has been made that at least in a number of smdied cases the mumal diffusion coefficient of radicals in the pair is approximately 10 times lower than the sum of macroscopic diffusion coefficients of the individual species. In other words, a geminate recombination proceeds considerably longer than expected. 

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