Diffusion mechanism

The diffusion coefficient, D, is a strong function of temperature. The temperature dependence arises because some energy (typically a few electron volts, eV) is required for an atom to jump from one atomic position to another. This energy is often called the activation energy, EA. The diffusion coefficient can be written 

The diffusion of impurities into Si wafers typically is done in two steps. In the first step, dopants are implanted into the substrate to a relatively shallow depth of a few thousand angstroms. After the impurities have been introduced into the Si substrate, they are diffused deeper into the substrate to provide a suitable impurity distribution in the substrate. The solid solubility and diffusion of dopant atoms in Si are given in the top and bottom, respectively, of Fig. 9.10. 

The dopants introduced by the ion implantation step can be redistributed deeper in the substrate to lower the concentrations by a drive-in step. In the drive-in step, the total amount of dopant atoms, Q, remains fixed. The concentration profile due to the drive-in diffusion is given by 

For diffusion in a crystalline solid, consider an impurity atom located between the host atoms, Fig. 9.11a. This impurity atom, called an interstitial, can jump from one interstitial site to the next vacant interstitial site. Occasionally, an impurity atom located in a lattice site that is normally occupied by a host atom will jump to 

Several different models for this atomic motion have been proposed of these possibilities, two dominate for metallic diffusion. 

The second type of diffusion involves atoms that migrate from an interstitial position to a neighboring one that is empty. This mechanism is foimd for interdiffusion of impurities 

PigMre 5.3 Schematic representations of (a) vacancy diffusion and (b) interstitial diffusion. 

In most metal alloys, interstitial diffusion occirrs much more rapidly than diffusion by the vacancy mode, because the interstitial atoms are smaller and thus more mobile. Furthermore, there are more empty interstitial positions than vacancies hence, the prob-abihty of interstitial atomic movement is greater than for vacancy diffusion. 

Diffusion is a time-dependent process—that is, in a macroscopic sense, the quantity of an element that is transported within another is a function of time. Often it is necessary to know how fast diffusion occurs, or the rate of mass transfer. This rate is frequently expressed as a diffusion flux (/), defined as the mass (or, equivalently, the number of atoms) M diffusing through and perpendicular to a imit cross-sectional area of solid per unit of time. In mathematical form, this may be represented as 

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A diffusion mechanism is also used in dialysis as a means of separating colloids from crystalloids. The rate of diffusion of molecules in gels is practically the same as in water, indicating the continuous nature of the aqueous phase. The diffusion of gases into a stream of vapour is of considerable importance in diffusion pumps. 

Ultrafiltration separations range from ca 1 to 100 nm. Above ca 50 nm, the process is often known as microfiltration. Transport through ultrafiltration and microfiltration membranes is described by pore-flow models. Below ca 2 nm, interactions between the membrane material and the solute and solvent become significant. That process, called reverse osmosis or hyperfiltration, is best described by solution—diffusion mechanisms. 

Basic Principles of Operation RO and NF are pressure-driven processes where the solvent is forced through the membrane by pressure, and the undesired coproducts frequently pass through the membrane by diffusion. The major processes are rate processes, and the relative rates of solvent and sohite passage determine the quality of the product. The general consensus is that the solution-diffusion mechanism describes the fundamental mechanism of RO membranes, but a minority disagrees. Fortunately, the equations presented below describe the obseiwed phenomena and predict experimental outcomes regardless of mechanism. 

He studied the sintering of copper particles in the diameter range 15-100 microns and of silver particles of diameter 350 microns. The results for the larger volume fraction of copper and for silver were shown to fit the volume diffusion mechanism and yielded the results for volume self-diffusion... 

In order to answer these questions as directly as possible we begin by looking at diffusive and displacive transformations in pure iron (once we understand how pure iron transforms we will have no problem in generalising to iron-carbon alloys). Now, as we saw in Chapter 2, iron has different crystal structures at different temperatures. Below 914°C the stable structure is b.c.c., but above 914°C it is f.c.c. If f.c.c. iron is cooled below 914°C the structure becomes thermodynamically unstable, and it tries to change back to b.c.c. This f.c.c. b.c.c. transformation usually takes place by a diffusive mechanism. But in exceptional conditions it can occur by a displacive mechanism instead. To understand how iron can transform displacively we must first look at the details of how it transforms by diffusion. 

Many stainless steels, however, are austenitic (f.c.c.) at room temperature. The most common austenitic stainless, "18/8", has a composition Fe-0.1% C, 1% Mn, 18% Cr, 8% Ni. The chromium is added, as before, to give corrosion resistance. But nickel is added as well because it stabilises austenite. The Fe-Ni phase diagram (Fig. 12.8) shows why. Adding nickel lowers the temperature of the f.c.c.-b.c.c. transformation from 914°C for pure iron to 720°C for Fe-8% Ni. In addition, the Mn, Cr and Ni slow the diffusive f.c.c.-b.c.c. transformation down by orders of magnitude. 18/8 stainless steel can therefore be cooled in air from 800°C to room temperature without transforming to b.c.c. The austenite is, of course, unstable at room temperature. Flowever, diffusion is far too slow for the metastable austenite to transform to ferrite by a diffusive mechanism. It is, of course, possible for the austenite to transform displacively to give... 

By way of example, Volume 26 in Group III (Crystal and Solid State Physics) is devoted to Diffusion in Solid Metals and Alloys, this volume has an editor and 14 contributors. Their task was not only to gather numerical data on such matters as self- and chemical diffusivities, pressure dependence of diffusivities, diffusion along dislocations, surface diffusion, but also to exercise their professional judgment as to the reliability of the various numerical values available. The whole volume of about 750 pages is introduced by a chapter describing diffusion mechanisms and methods of measuring diffusivities this kind of introduction is a special feature of Landolt-Bornstein . Subsequent developments in diffusion data can then be found in a specialised journal. Defect and Diffusion Forum, which is not connected with Landolt-Bdrnstein. 

The above discussion has tacitly assumed that it is only molecular interactions which lead to adhesion, and these have been assumed to occur across relatively smooth interfaces between materials in intimate contact. As described in typical textbooks, however, there are a number of disparate mechanisms that may be responsible for adhesion [9-11,32]. The list includes (1) the adsorption mechanism (2) the diffusion mechanism (3) the mechanical interlocking mechanism and (4) the electrostatic mechanism. These are pictured schematically in Fig. 6 and described briefly below, because the various semi-empirical prediction schemes apply differently depending on which mechanisms are relevant in a given case. Any given real case often entails a combination of mechanisms. 

Fig. 6. Four mechanisms of adhesion, (a) The adsorption mechanism (contact adhesion), (b) The diffusion mechanism (diffusion interphase adhesion), (c) The mechanical interlocking mechanism. (d) The electrostatic mechanism.

Although the diffusion mechanism can be seen as mechanical but occurring at molecular dimensions, van der Waals intermolecular interactions and conformational entropic energy provide an additional mechanism that increases adhesion [62]. It is interesting to note the analogy that exists between this mechanism at the molecular level with the adherence, adhesion and viscoelastic deformations concept applied for a macroscopic adhesive. 

The skin receives heat from the core by passive conduction and active skin blood flow (Table 5.3). It transfers this heat to the surroundings by convection, radiation, and evaporative (perspiration and diffusion) mechanisms. All of these mechanisms are unregulated or passive except evaporation from sweating. The sweating process is actively controlled by the humarrs thermoregulatory center where the rate of sweat secretion is proportional to eleva tions in core and skin temperature from respective set point temperatures (Table 5.3). 

Whatever method is used to provide an adequate supply of diffusing metal, the diffusion mechanism remains identical for any given solute/solvent system, and it is appropriate to discuss the diffusion aspect in the first place. 

The exchange process in a chelating resin is generally slower than in the ordinary type of exchanger, the rate apparently being controlled by a particle diffusion mechanism. 

Before any chemistry can take place the radical centers of the propagating species must conic into appropriate proximity and it is now generally accepted that the self-reaction of propagating radicals- is a diffusion-controlled process. For this reason there is no single rate constant for termination in radical polymerization. The average rate constant usually quoted is a composite term that depends on the nature of the medium and the chain lengths of the two propagating species. Diffusion mechanisms and other factors that affect the absolute rate constants for termination are discussed in Section 5.2.1.4. 

However, while it is generally accepted that the rate of radical-radical reaction is dependent on how fast the radical centers of the propagating chains (Pp and Pj ) come together, there remains some controversy as to the diffusion mechanism(s) and/or what constitutes the rate-determining step in the diffusion process. The steps in the process as postulated by North and coworkers30 3" arc shown conceptually in Scheme 5.5. 

Center of mass or translational diffusion is believed to be the rate-determining step for small radicals33 and may also be important for larger species. However, other diffusion mechanisms are operative and are required to bring ihe chain ends together and these will often be the major term in the termination rate coefficient for the case of macromolecular species. These include ... 

However, it has been pointed out that the value of kXo in the expressions eqs. 25-23 should not be confused with the small radical A, rather, the value of kt0 represents the termination rate constant of a single unit chain if the implied diffusion mechanism was the rate-determining process. 

The concept of reaction diffusion (also called residual termination) has been incorporated into a number of treatments.7 7 Reaction diffusion will occur in all conversion regimes. However at low and intermediate conversions the process is not of great significance as a diffusion mechanism. At high conversion long chains are essentially immobile and reaction diffusion becomes the dominant diffusion mechanism (when i and j are both "large" >100). The termination rate constant is determined by the value of kp and the monomer concentration. In these circumstances, the rate constant for termination k - should be independent of the chain lengths i and j and should obey an expression of the form 75... 

More detailed consideration of the sensitivities of dehydration rates to reaction conditions (PH2o> temperature) are given in the articles cited reported values of E, at various PH2Q, are summarized in Fig. 13. From kinetic observations, it was concluded that between 353 and 383 K, the dehydration of CaS04 2 H20 [590] involved nucleation ( in Fig. 13) and boundary control (o) but for 383—425 K, a diffusion mechanism ( ) operated. The kinetics of dehydration of a-CaS04 ] H20 [590] (X,+)... 

The kinetics of resole cure reactions monitored via FTIR suggest that a diffusion mechanism dominates below 140°C. The cure above 140°C exhibits a homogeneous first-order reaction rate. The activation energy of the cure reaction was -"-49.6 kJ/mole.66... 

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