21.4: Dislocation Width
- Page ID
- 31769
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)For a given atomic configuration, we can work out the total energy and then try different values of w until we find the minimum. Practically (see graph below), this can be done by summing the energy over planes from n=-1000 to n=+1000 either side of the “extra plane of atoms”, since the effects of increasing the number of planes beyond this is negligible.
As w increases, the decrease in energy associated with localising the in-plane strains (which is why we said a dislocation should never form) is offset by the increase in misalignment energy, which increases with w. There is therefore a minimum in the misfit energy, which gives the dislocation width.
The following graph shows the sums of the misfit energies (in-plane, misalignment and total) resulting from the displacements of planes either side of the initial “extra half plane of atoms”. We can determine the dislocation width (w/b) for given parameters by considering the minimum of the total misfit energy curve.
If we look at the value of 0.25Gb2 for the four materials given, it is similar to the energy value calculated by the model. This shows an approximate agreement with the linear elastic solution.
What determines w/b?
- What we are interested in is the ratio of d/b. As d/b increases, w/b increases. d/b is determined by crystal structure.
- b is the Burger's vector
- d is the spacing of the close packed planes
- We also need to take into consideration partial dislocations. For example, copper is ccp so the slip system is {111} <110>. But the dislocations dissociate into two partial dislocations\(\frac{a}{6}\)<112>; hence the magnitude of b will be \(\frac{a}{\sqrt{6}}\) where a is the lattice parameter.
- Therefore, the ratio of d/b in copper will be \(\frac{a}{\sqrt{3}} \div \frac{a}{\sqrt{6}}=\sqrt{2}\)
- Poisson ratio (ν) affects the in-plane component of the misfit energy. A higher Poisson ratio means the width of the dislocation is greater.
What determines the total energy?
- The total energy is affected by the magnitude of Gb2, where G is the shear modulus and b is the Burger's vector.
The following clip shows the free surface of a Cd single crystal subject to tensile testing. Slip is occurring on a particular set of planes, and the set of ridges that form on the free surface are created by the arrival of sets of dislocations.