By Helmut Mehrer
Diffusion is an important subject in solid-state physics and chemistry, actual metallurgy and fabrics technological know-how. Diffusion procedures are ubiquitous in solids at increased temperatures. a radical realizing of diffusion in fabrics is important for fabrics improvement and engineering. This publication first supplies an account of the important facets of diffusion in solids, for which the mandatory heritage is a direction in reliable kingdom physics. It then offers quick access to special information regarding diffuson in metals, alloys, semiconductors, ion-conducting fabrics, glasses and nanomaterials. numerous diffusion-controlled phenomena, together with ionic conduction, grain-boundary and dislocation pipe diffusion, are regarded as well.
Graduate scholars in solid-state physics, actual metallurgy, fabrics technology, actual and inorganic chemistry or geophysics will reap the benefits of this publication as will physicists, chemists, metallurgists, fabrics engineers in educational and business learn laboratories.
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Additional info for Diffusion in Solids: Fundamentals, Methods, Materials, Diffusion-Controlled Processes
2. The quantity 2 Dt is a characteristic diﬀusion length, which occurs frequently in diﬀusion problems. Salient properties of Eq. 9) are the following: 1. The diﬀusion process is subject to the conservation of the integral number of diﬀusing particles, which for Eq. 9) reads +∞ −∞ x2 M √ exp − dx = 4Dt 2 πDt +∞ M δ(x)dx = M . 11) −∞ 2. C(x, t) and ∂ 2 C/∂x2 are even functions of x. ∂C/∂x is an odd function of x. 40 3 Solutions of the Diﬀusion Equation Fig. 1. Gaussian √ solution of the diﬀusion equation for various values of the diﬀusion length 2 Dt Fig.
22) The complementary error function deﬁned in Eq. 17) has the following asymtotic properties: erfc(−∞) = 2, erfc(+∞) = 0, erfc(0) = 1 . , in [4, 9–11]. Detailed calculations cannot be performed just relying on tabular data. For advanced computations and for graphing one needs, instead, numerical estimates for the error function. Approximations are available in commercial mathematics software. In the following, we mention several useful expressions: 1. For small arguments, |z| < 1, the error function is obtained to arbitrary accuracy from its Taylor expansion  as 2 z5 z7 z3 erf (z) = √ z − + − + ...
For inﬁnitesimal size of the test volume Eq. 3) can be written in compact form by introducing the vector operation divergence ∇·, which acts on the vector of the diﬀusion ﬂux: −∇ · J = ∂C . 4) is denoted as the continuity equation. 3 Fick’s Second Law – the ‘Diﬀusion Equation’ Fick’s ﬁrst law Eq. 4) can be combined to give an equation which is called Fick’s second law or sometimes also the diﬀusion equation: ∂C = ∇ · (D∇C) . 5) ∂t From a mathematical viewpoint Fick’s second law is a second-order partial diﬀerential equation.
Diffusion in Solids: Fundamentals, Methods, Materials, Diffusion-Controlled Processes by Helmut Mehrer