Diffusion

Abstract

Material transport by migration of atoms or molecular entities is a fundamental process of nature. Its understanding is, for example, of crucial importance for control of the microstructure of materials. Continuum and atomistic approaches to diffusion are discussed. Possible diffusion mechanisms are presented. The diffusion coefficient is a tool to characterize the diffusion process, via the activation energy. A corresponding, broad discussion of experimental results and their interpretation is given for various materials. It is shown to be necessary to distinguish the overall material flow from the intrinsic diffusion processes, as given by the individual atomic jumps. This culminates in extensive discussion of the Kirkendall effect and the Darken treatment. Via the “driving force” for diffusion, a link can be made from the thermodynamics to the kinetics. Especially along the latter route the so-called intrinsic diffusion coefficients, to be distinguished from the chemical or interdiffusion diffusion coefficient, can be determined, as illustrated experimentally. Fundamental and practical problems are touched upon. Many material systems are subjected to a state of stress, e.g., thin films. A treatment of diffusion in a state of (nonhydrostatic) stress is still not possible in a rigorous way on the basis of boundary conditions compatible with reality. Yet, some interesting considerations and results are presented. Diffusion experiments can be very revealing about the defects in the structural (atomic) arrangement of a material. This is demonstrated here for in particular the role of grain boundaries, including moving grain boundaries.

Appendix: Concentration-Depth Profiles in Thin Layer Systems

With a view to the great practical importance of thin film systems, as in the microelectronic industry, a number of solutions to Fick’s second law for a variety of thin layer systems is summarized at the end of this chapter in this appendix. Various cases with different initial and boundary conditions can be considered for different diffusion stages. It will be assumed that the diffusion coefficient can be taken as concentration independent.

Case 1 (Fig. 8.23a): A bilayer (AB) or multilayer (ABABAB …) for which the thickness of each sublayer is much larger than the diffusion length, $\sqrt {Dt}$, corresponding to a very early diffusion stage for the (multi)layer.

Fig. 8.23

Concentration-depth profiles in thin film systems. a Bilayer or multilayer with the thickness of each sublayer much larger than the diffusion length. b Trilayer or multilayer with the thickness of each sublayer A much smaller than that of sublayers B and the diffusion length much smaller than the thickness of sublayers B. c Trilayer with thickness of each sublayer of the order of the diffusion length. d Multilayer with the thickness of each sublayer of the order of the diffusion length

The solution for the bilayer AB (possibly as part of the multilayer ABABAB…) with the A/B interface at z = z_i and initial conditions C = C₀ for z ≤ z_i and C = 0 for z > z_i is given by (Crank 1975):

$$C(z,t) = \frac{{C_{0} }}{{2\sqrt {\pi Dt} }}\int\limits_{{z - z_{i} }}^{\infty } {\exp \left( { - \frac{{\xi^{2} }}{4Dt}} \right){\text{d}}\xi } = \frac{1}{2}C_{0} {\text{erfc}}\frac{{z - z_{i} }}{{2\sqrt {Dt} }}$$

(8.64)

where erfc (=1 − erf) denotes the error-function (erf) complement (cf. Eq. 8.16). Note that, evidently, C = C₀/2 at z = z_i for all t > 0.

Case 2 (Fig. 8.23b): A trilayer (BAB) or multilayer (BABABA…) for which the thickness of sublayers A is much smaller than that of the sublayers B, i.e. d_A ≪ d_B, and $\sqrt {Dt}$ ≪ d_B.

For the trilayer BAB (possibly as part of the multilayer BABABA…) with z = z_i at the centre plane of sublayer A and initial conditions C = C₀ for z_i − d_A/2 ≤ z ≤ z_i + d_A/2 and C = 0 for z < z_i − d_A/2 and z > z_i + d_A/2, the concentration profile is obtained as (cf. Eq. 8.64)

$$\begin{aligned} C(z,t) & = \frac{{C_{0} }}{{2\sqrt {\pi Dt} }}\int\limits_{{z - (z_{i} + d_{A} /2)}}^{{z - (z_{i} - d_{A} /2)}} {\exp \left( { - \frac{{\xi^{2} }}{4Dt}} \right){\text{d}}\xi } \\ & = \frac{1}{2}C_{0} \left[ {{\text{erf}}\frac{{d_{A} /2 + (z - z_{i} )}}{{2\sqrt {Dt} }} + {\text{erf}}\frac{{d_{A} /2 - (z - z_{i} )}}{{2\sqrt {Dt} }}} \right] \\ \end{aligned}$$

(8.65)

It is clear that the system can be cut in half by a plane at z = z_i without affecting the distribution, which is symmetrical about z = z_i. Therefore, Eq. (8.65) also holds for a bilayer system composed of a sublayer A of thickness of d_A/2, with the surface or a diffusion barrier at z = z_i, on top of the semi-infinite sublayer B (substrate, see Fig. 8.23b). Redefining d_A/2 as h and taking z_i = 0, it follows for the diffusion-induced concentration profile in this case

$$C(z,t) = \frac{{C_{0} }}{2}\left[ {{\text{erf}}\frac{h + z}{{2\sqrt {Dt} }} + {\text{erf}}\frac{h - z}{{2\sqrt {Dt} }}} \right]$$

(8.66)

Case 3 (Fig. 8.23c): A trilayer (BAB) for which the thickness of each sublayer is not much larger than the diffusion length $\sqrt {Dt}$, which represents a relatively advanced diffusion stage for the trilayer.

The outer surfaces of both B sublayers are barriers for mass transport. Hence,

$$\frac{\partial C}{{\partial z}} = 0\quad {\text{at}}\;\;z_{1} = z_{i} - \left( d_{B} + \frac{d_{A}}{2} \right)\;\;{\text{and}}\;\;z_{2} = z_{i} + \left(d_{B} + \frac{d_{A}}{2} \right)$$

The resulting concentration profile can be constructed by the superposition (reflection) principle as follows (cf. Crank 1975). The concentration profile given by Eq. (8.65) is reflected at the plane at boundary z₂, and this reflected profile is obtained by replacing z_i in Eq. (8.65) by z_i + (2 d_B + d_A). This firstly reflected curve is reflected at the plane at z_i, and the secondly reflected profile is obtained by replacing z_i in Eq. (8.65) by z_i − (2d_B + d_A). Then, the secondly reflected profile is reflected again at the boundary z₂ (z_i → z_i + 2(2d_B + d_A)) and at z_i(z_i → z_i − 2(2d_B + d_A)) and so on. Therefore, the complete solution as the result of such successive reflections is given by

$$\begin{aligned} C(z,t) & = \frac{1}{2}C_{0} \sum\limits_{n = - \infty }^{\infty } {\left[ {{\text{erf}}\frac{{d_{A} /2 - n(2d_{B} + d_{A} ) + z - z_{i} }}{{2\sqrt {Dt} }}} \right.} \\ & \left. {\quad + {\text{erf}}\frac{{d_{A} /2 + n(2d_{B} + d_{A} ) - z + z_{i} }}{{2\sqrt {Dt} }}} \right] \\ \end{aligned}$$

(8.67)

Case 4 (Fig. 8.23d): A multilayer (ABABAB…) for which the thickness of each sublayer is not much larger than the diffusion length $\sqrt {Dt}$, which represents a relatively advanced diffusion stage for the multilayer.

Considering the initial conditions: C = C₀ for z_i − d_A/2 ≤ z ≤ z_i + d_A/2 and C = 0 for z_i − (d_A/2 + d_B) < z < z_i − d_A/2 and z_i + d_A/2 < z < z_i + (d_A/2 + d_B) with z_i denoting the centre plane of sublayer A, and recognizing that for t ≥ 0 ∂C/∂z = 0 at the centre plane of the sublayers B, the total concentration profile for the trilayer BAB in the multilayer is obtained by the superposition (reflection) principle (see also Case 3) as follows

$$\begin{aligned} &C(z,t) \\&= \frac{1}{2}C_{0} \sum\limits_{n = - \infty }^{\infty } {\left[ {{\text{erf}}\frac{{d_{A} /2 - n(d_{B} + d_{A} ) + z - z_{i} }}{{2\sqrt {Dt} }} + {\text{erf}}\frac{{d_{A} /2 + n(d_{B} + d_{A} ) - z + z_{i} }}{{2\sqrt {Dt} }}} \right]} \end{aligned}$$

(8.68)

The diffusion-induced concentration profiles as given by the Eqs. (8.64)–(8.68) have been derived considering volume (bulk) diffusion. In polycrystalline thin films the role of grain-boundary diffusion is often dominant, recognizing the high grain-boundary density and the usually applied relatively low diffusion annealing temperatures (as compared to the melting point of the components). Often columnar microstructures occur in thin films; i.e. the grain boundaries are oriented more or less perpendicular to the film surface and sublayer interfaces. Then, if the volume diffusion length, (D_bt)^1/2, is much smaller than the grain-boundary diffusion length, (D_gbt)^1/2, with D_b and D_gb as the volume and grain-boundary diffusion coefficients, respectively, it can be shown (see Wang and Mittemeijer 2004) that the laterally averaged concentration profile for Cases 1–4, as induced by grain-boundary diffusion, are also given by Eqs. (8.64)–(8.68), provided C is identified with $\overline{C}$, the laterally averaged concentration, D is identified with the grain-boundary diffusion coefficient D_gb, and C₀ is identified with C₀δη (δ is grain-boundary width and η is grain-boundary length per unit area for the plane parallel to the surface.