# Visualization of dislocation dynamics in colloidal crystals

Zach Wissner-Gross (April 14, 2009)

## Information

Visualization of dislocation dynamics in colloidal crystals

Peter Schall, Itai Cohen, David A. Weitz, and Frans Spaepen

Science, 2004, 305, 1944-1948

## Soft matter keywords

Crystal defects (dislocations), colloidal crystals, continuous and discrete models

## Summary

Figure 1: Laser diffraction microscopy images showing growth of dislocations, represented by thin black lines where Bragg diffraction is disrupted.

Crystal defects have physical consequences at multiple length scales: the structure of the dislocation itself depends on interatomic potentials; interactions between dislocations results in forces on the mesoscopic scale; and finally, macroscopic crystal deformation depends on the dislocations as well. The authors point out the difficulty in being able to observe all these length scales simultaneously, to gain a more complete understanding of crystal defects.

Figure 2: Reconstruction of a colloidal crystal using laser scanning confocal microscopy to generate the 3D image. Dislocations, defined by regions in which colloidal particles have a different number of nearest neighbors than they would in an face-centered cubic model, are shown in red.

But why not simply scale everything up a few orders of magnitude, using colloidal crystals (with colloids as the repeating units, rather than atoms)? While the interactions between colloids may not be exactly the same as those between atoms, the authors state that colloid crystals can make a good model for crystals in general, and go on to study dislocations in colloid crystals at multiple length scales.

Using a laser scanning confocal microscope to determine the positions of individual colloid particles (Figure 1), along with laser diffraction microscopy to detect dislocation structure and growth (Figure 2), the authors present several new findings, most notable of which is that the continuum model for crystal defects yields good physical approximations, even for crystals too thin to be considered good candidates for the model.

## Soft matter discussion

Not being particularly familiar with dislocation mechanics, I will admit I had a difficult time understanding the authors' discussion of Burgers vectors and Shockley partials. But what struck me is that the authors' analysis (using the continuum model) of the critical crystal height at which dislocations begin to nucleate is just an energy minimization similar to what we've seen in class.

The two energies we need to consider in a thin crystal film with dislocations and deformations are: (1) the elastic energy stored in the strained film, $U_{\text{el}}$ and (2) what the authors call "the energy cost per unit area associated with the misfit dislocations," $U_{\text{l}}$. While there are many variables in this analysis, including various moduli, angles, and particle sizes, the important two parameters are the height of the film $h$ and the density of dislocations $\Lambda^{-1}$. Elastic energy linearly depends on height and quadratically with dislocation density:

$U_{\text{el}}=\frac{1}{2}(\epsilon_0-\Lambda^{-1}b\cos{\alpha})^2Eh$

while dislocation energy only linearly depends on dislocation density:

$U_{\text{l}}=\Lambda^{-1}\frac{\mu b^2\ln{\frac{R}{r_c}}}{4\pi(1-\nu)}$

The total energy per area of the crystal film is simply $U_{\text{el}}+U_{\text{l}}$. The authors minimize this energy by taking its derivative with respect to $\Lambda^{-1}$ and setting it equal to zero, obtaining the result:

$(\epsilon_0-\Lambda^{-1}b\cos{\alpha})b\cos{\alpha}Eh=\frac{\mu b^2\ln{\frac{R}{r_c}}}{4\pi(1-\nu)}$

And here the authors cleverly point out that the critical height $h_{\text{c}}$ at which deformations start to form is the height at which the dislocation density $\Lambda^{-1}$ just exceeds zero. Setting $\Lambda^{-1}=0$ gives:

$h_{\text{c}}=\frac{\mu b\ln{\frac{R}{r_c}}}{4\pi(1-\nu)\epsilon_0 E\cos{\alpha}}$

Plugging in the various parameters for the colloidal crystal into this formula gives a critical height of 22 [itex]\mu>m. The authors report that dislocations began forming between heights of 22 and 31 [itex]\mu>m, consistent with this continuum model.