For self-diffusion in polycrystalline copper with a lattice diffusion coefficient $D_L$, grain boundary diffusion coefficient $D_{GB}$, and surface diffusion coefficient $D_S$, the correct relationship is
Diffusion Coefficients Relationship in Polycrystalline Copper
Understanding Diffusion Mechanisms
Self-diffusion in solids occurs through different paths, each offering varying resistance to atomic movement:
- Lattice Diffusion ($D_L$): Atoms move through the bulk crystal lattice, typically via vacancies or interstitial mechanisms. This path has the most ordered structure and highest resistance.
- Grain Boundary Diffusion ($D_{GB}$): Atoms move along the interfaces between adjacent crystal grains. Grain boundaries are regions of atomic mismatch and disorder, offering less resistance than the lattice.
- Surface Diffusion ($D_S$): Atoms move along the free surface of the material. Surfaces are the most disordered regions with the least constraint on atomic movement, offering the least resistance.
Deriving the Diffusion Coefficient Relationship
The rate of diffusion is inversely related to the resistance of the path. Therefore, the diffusion coefficient is highest for the path with the least resistance.
- Surface diffusion occurs along the most disordered path, hence it is the fastest.
- Grain boundary diffusion occurs along a less ordered path than the surface but more ordered than the lattice.
- Lattice diffusion occurs through the most ordered path and is the slowest.
Based on this, the relationship between the diffusion coefficients is:
$ D_S \gt D_{GB} \gt D_L $
This indicates that surface diffusion is significantly faster than grain boundary diffusion, which in turn is faster than lattice diffusion in polycrystalline copper.
Final Answer
The correct relationship is $D_S > D_{GB} > D_L$.
Important Questions from Diffusion Fick's Second Law Concentration Profile
During carburizing of a steel, the surface concentration is kept constant at 1.4 wt.% carbon. Diffusivity of carbon for the steel at 950 $^\circ$C is $6.25 \times 10^{-11}$ m$^2$/s. At 950 $^\circ$C, the time required to carburize the steel with an initial composition of 0.2 wt.% carbon to 0.8859 wt.% carbon at a depth of 0.2 mm is ______________ seconds (approximate to the nearest integer).
Use the nearest value of the error function from the table given below for your calculation.
z erf (z) 0.3 0.3268 0.4 0.4284 0.5 0.5205 - What is the depth (in $µm$) from the surface of the specimen at which a composition of 0.4 wt.% C is obtained after carburizing at $870^\circ C$ for 10 h?
- The concentration $C$ of a solute (in units of atoms$\cdot\text{mm}^{-3}$) in a solid along $x$direction (for $x > 0$) follows the expression
$C = a_1x^2 + a_2x$
where $x$ is in mm, $a_1$ and $a_2$ are in units of atoms$\cdot\text{mm}^{-5}$ and atoms$\cdot\text{mm}^{-4}$,respectively. Assuming $a_1= a_2= 1$, the magnitude of flux at $x = 2 \text{ mm}$ is________ $\times 10^{-3} \text{ atoms} \cdot \text{mm}^{-2} \cdot \text{s}^{-1}$ (answer rounded off to the nearest integer).
Given: diffusion coefficient of the solute in the solid is $3 \times 10^{-3} \text{ mm}^2 \cdot \text{s}^{-1}$. - Determine the correctness or otherwise of the following Assertion [a] and the Reason [r]
Assertion [a]: The rate of homogenization in a dilute substitutional solid solution of B in A is controlled by the diffusivity of B.
Reason [r]: Atomic migration cannot occur along dislocations and grain boundaries. - A species can diffuse through the lattice (diffusion coefficient, $D_L$), along grain boundaries (diffusion coefficient, $D_{GB}$), and along free surfaces (diffusion coefficient, $D_S$). Which of the following relations is CORRECT?
