Diffusion in Solids (eBook)

Field Theory, Solid-State Principles, and Applications

Martin Eden Glicksman (Autor)

eBook Download: EPUB

2022 | 1. Auflage
496 Seiten
Wiley (Verlag)
978-1-394-17255-9 (ISBN)

Infos

This book offers a modern treatment of diffusion in solids, covering such core topics as the transport of mass through the lattice of a crystalline solid. Part I of the book develops basic concepts in diffusion field theory and illustrates them with several applications, while Part II focuses on key solid-state principles needed to apply diffusion theory to real materials.

Autor

Martin Eden Glicksman is the author of Diffusion in Solids: Field Theory, Solid-State Principles, and Applications, published by Wiley.

Inhalt

FIELD THEORY.

Laws of Diffusion.

Diffusion in Generalized Media.

Solutions to the Linear Diffusion Equation.

Diffusion Couple.

Diffusion Point Sources in Higher Dimensions.

Generalized Sources.

Diffusion-Reaction.

Linear Flow in Finite Systems.

Spherical Bodies.

Steady-State Diffusion.

Inverse Methods.

SOLID-STATE PRINCIPLES.

Random Walks and Diffusion.

Structure and Diffusion.

Correlation Effects in Diffusion.

Vacancy-Assisted Diffusion.

Diffusion in Dilute Alloys.

Kirkendall Effect.

Influence of Solution Ideality.

Diffusional Anelasticity.

Field-Assisted Diffusion.

Multiparticle Diffusion: Capillary Effects.

Population Dynamics.

Multicomponent Diffusion.

Multicomponent Diffusion: Profiler Program.

Appendices.

Index.

Textauszug

1
LAWS OF DIFFUSION

Diffusion is at once a venerable subject within classical physics and a thoroughly modem one. Today’s forefronts of research in science and technology are replete with fundamental developments in diffusion evolving the core subject, as well as with fresh technical applications verifying its underlying utilitarian character. The twentieth century opened with great scientific strides in establishing diffusion as a microscopic (i.e., atomic-scale) phenomenon, and closes with diffusion processes playing pivotal roles in the new age of information technology. An observer of the current science and technology scene cannot help but be impressed that applications as disparate as biodiffusion across cell membranes, geodiffusion in mineral deposits, and dopant diffusion in semiconductors are predictable from the few basic rules governing diffusion. The time and length scales involved in these applications differ enormously, but the central ideas supporting our understanding of these processes remain identical; such is the reach of this venerable subject.

Chapter 1 provides our starting point for understanding diffusion and associated phenomena in many types of materials and other condensed phases. The diffusion coefficient is defined, and the basic differential equations governing diffusion, known as Fick’s laws, are developed for several important spatial symmetries. The importance of keeping compatible physical units is discussed, and the key vector calculus operations — gradient, divergence, and the Laplacian — are reviewed for various coordinate geometries.

1.1. PRELIMINARY DEFINITIONS

Diffusion is a kinetic process that leads to the homogenization, or uniform mixing, of the chemical components in a phase. Although mixing in a fluid (liquid or gas) may occur on many length scales, as induced by macroscopic flow, diffusive mixing in solids, by contrast, occurs only on the atomic or molecular level. As time increases, the extent of homogenization by diffusion also increases, and the length scale over which chemical homogeneity persists within a phase gradually extends to macroscopic distances. Diffusion that results in the net transport of matter over such macroscopic distances is considered to be a nonequilibrium process, insofar as it ceases when the phase eventually achieves full thermodynamic equilibrium.

The distinction between mixing at molecular scales and mixing at macroscopic scales is easily appreciated. For example, one could observe the rapid dilution of a drop of dye in a stirred container of water; and then contrast it to the behavior of a quiescent layer of dye in contact with a stationary volume of water. In the former case, mixing is caused by the convective flows induced by the stirring, whereas in the latter case the dye and water exchange molecules spontaneously by diffusion, with the mixed regions restricted to layers adjacent to the original dye–water interface.

1.2. DIFFUSION IN AN ISOTROPIC MEDIUM

The laws of diffusion are mathematical relationships that relate the rate of diffusion to the concentration gradients responsible for net mass transfer. Such laws are considered to be phenomenological. Phenomenological laws are used to describe a physical effect — in this instance diffusive transport — purely on the basis of observing it, without claiming that it derives from some deeper understanding or more basic precept. Adolf Fick was the scientist who first reported the behavior of a salt-water system undergoing diffusion. His description of diffusion required the definition and first recorded use of the diffusion coefficient, D. Fick’s published research appeared in 1855 [1,2], almost a half-century after Fourier (1807) had developed an analogous description for the flow of heat [3]. The linear response observed by Fick between the applied concentration gradient and the subsequent diffusive mixing of salt and water established an empirical fact, not proof of his stated mathematical generalizations. The true power and generality of Fick’s phenomenological laws — as well as their limitations — must instead be gauged by the ability of these laws to predict the quantitative response of a system undergoing diffusion.

1.3. FICK’S FIRST LAW

The mass flux, J(x,y,z), of a diffusing component is defined as a vector quantity in all spatial settings, d = 1, 2, or 3, where d designates the dimensionality of the diffusion space. The magnitude of the flux is defined as the mass of component flowing per unit time through unit area. The direction of the flux vector is usually chosen parallel to the area’s normal vector, n. Fick found by direct observation that the magnitude of the mass flux is proportional to the magnitude of the concentration gradient at that point. In isotropic media (i.e., those lacking directional properties) the mass-flux vector remains collinear with the concentration gradient, but oppositely directed. The mass flux in three spatial dimensions can be expressed as the vector sum of its components.

In Cartesian (x,y,z) coordinates, with the unit vectors ex, ey, and ez, the mass flux may be written as a vectorial sum:

(1.1)

Figure 1.1 shows a mass-flux vector in three dimensions resolved into its components. The individual flux magnitudes appearing in eq. (1.1) may be expressed by component equations through Fick’s first law:

(1.2)

In eqs. (1.2), the symbol D denotes the diffusivity, or interdiffusion coefficient, and C is the concentration of the component at the point in question. The partial derivatives appearing in eqs. (1.2) imply that there is a concentration field C(x,y,z,t) that varies continuously and smoothly in both space and time. The individual derivatives in eq. (1.2) comprise the components of the concentration gradient vector ▽C, which is defined as

(1.3)

FIGURE 1.1 Mass-flux vector J, resolved into its Cartesian components.

The del or nabla symbol, ▽ is used throughout the book to define the vector operation on the right-hand side of eq. (1.3). The del itself is not a vector in the usual sense; rather, it represents a vector operator. Indeed, ▽ lacks specific meaning until we provide it with a function, or field, upon which to operate. Furthermore, ▽ does not multiply the function; rather, it provides an operational instruction to differentiate that function partially. With this qualification, the operator ▽ mimics the behavior of a vector in virtually every way.

The gradient operator acts on scalar fields — in this case, C(x,y,z,t) — and produces a related vector field known as the concentration gradient field. Although the gradient operator assumes a different mathematical form in each coordinate system, the gradient at any point in a concentration field is itself a well-defined vector quantity. The concentration gradient vector always points in the direction for which the concentration field undergoes the most rapid increase; moreover, its magnitude equals the maximum rate of increase of concentration at the point.

1.4. UNITS

Mass fluxes carry units (dimensions) of mass per unit area, per unit time. The mass units commonly encountered in diffusion problems appear in several forms: for example, grams, gram-moles, liters-STP (standard temperature and pressure), atoms. Diffusion coefficients, however, are most often tabulated in either cgs units (cm2/s), or in SI (System International) units (m2/s). Thus the mass flux must be expressed in units compatible with the units chosen to express the concentration and distance. As an example, if D bears the unit of cm2/s, and the concentration C is expressed in g/cm3, then eqs. (1.2) show that

(1.4)

The mass-flux vector in this case would be given in the compatible unit [g/cm2 · s].

1.5. VECTOR FORM OF FICK’S FIRST LAW

Equations (1.2) can be written using compact vector notation, as

(1.5)

Thus, as mentioned in Section 1.3, the flux vector represents a physical quantity, independent of the coordinates. The flux vector is directed parallel to, but opposite, the concentration gradient vector; its magnitude is, according to Fick’s first law, proportional to ▽C. Individual components of the flux vector, however, do depend on the coordinate system chosen to describe the concentration field. Their sum, which is a physical quantity, must be invariant to the choice of coordinates. If the magnitudes of the flux and gradient at a point are both known, Fick’s first law defines D as their ratio. In most time-dependent diffusion problems, however, both the flux and concentration fields are functions of space and time. Thus, as a practical matter, the fields C(x,y,z,t) and J(x,y,z,t) may be difficult to measure simultaneously. Additional constraints on the linear flux equations (Fick’s first law) are therefore usually needed to determine the diffusion response of a system to time-varying concentration gradients.

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Produkt-Details

• Erscheint lt. Verlag: 14.9.2022
• Sprache: englisch
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: Diffusion • Festkörper • Maschinenbau • mechanical engineering • thermodynamics • Thermodynamik
• ISBN-10: 1-394-17255-9 / 1394172559
• ISBN-13: 978-1-394-17255-9 / 9781394172559