Role of defects in the thermodynamic stability of grain boundary phases at asymmetric tilt boundaries in copper

TL;DR

This study addresses how asymmetry in grain boundary tilt, specifically Σ37c [11-1] boundaries in Cu, alters the thermodynamics of grain boundary (GB) phases or complexions. It combines molecular dynamics simulations, structure-search methods, quasi-harmonic free-energy estimates, and STEM experiments to map phase stability across inclinations and temperatures. The key finding is that geometrically necessary line defects and their defect energies strongly influence GB phase stability, with pearl-like defect configurations often stabilizing near small inclinations and domino-like steps limited to very small tilts; at larger inclinations, faceting into adjacent symmetric planes becomes prevalent. These results, supported by Cu and Al experiments, demonstrate that a complete understanding of GB thermodynamics requires explicit accounting for GB line defects in addition to the symmetric GB plane energies, with implications for predicting diffusion, mobility, and mechanical behavior in polycrystals.

Abstract

Grain boundaries can exist as different grain boundary phases (also called complexions) with individual atomic structures. The thermodynamics of these defect phases in high-angle grain boundaries were studied mostly with atomistic and phase field computer simulations, but almost exclusively for special, symmetric boundaries. Here, we use molecular dynamics simulations combined with structure search methods, as well as scanning transmission electron microscopy experiments to take a step towards understanding more general grain boundaries. Using the example of $Σ$37c $[11\overline{1}]$ tilt boundaries in Cu, we show how the grain boundary phase transition on a symmetric boundary plane is changed by the geometrically necessary defects introduced in inclined, asymmetric boundaries. We analyze the disconnections - which are dislocation-like line defects of grain boundaries - both in the simulations, as well as in experimental Cu and Al samples. A main finding is that defect energies can have a major influence on the stability of grain boundary phases, even at small inclinations. Furthermore, some defects are not able to effect large inclinations. At that point, defective asymmetric GB phases compete with grain boundaries faceting into the adjacent symmetric GB phases.

Figures (14)

• Figure 1: Bicrystallography of $\Sigma37$c $[11\overline{1}]$ tilt GBs in Cu. (a) Part of the dichromatic pattern. The blue line represents the quasi-symmetric $\{1~10~11\}$ GB planes, while the green line represents the symmetric $\{347\}$ GB planes. Asymmetric GBs, with inclinations $\phi$ between these two cases are the subject of the present study (yellow line). The coordinate system in the bottom left represents the corresponding crystal directions in the upper crystallite. (b) The zipper GB phase at $\phi=\ang{30}$ depicted with the crystal orientations as indicated by the coordinate axes to the right of the image. The vertical black lines represent the periodic unit cell of the GB structure. (c) An asymmetric GB at intermediate $\ang{0} < \phi < \ang{30}$ obtained as a mix of zipper (b) and domino (d) structures. (d)--(f) The domino and pearl phases on the $\{1~10~11\}$ GB planes. Note the difference in the purple structure for pearl #1 and #2. An asymmetric pearl phase is not depicted and will be studied in this work.
• Figure 2: Obtaining disconnection properties from a dichromatic pattern, here exemplarily for $\Sigma$7 $[11\overline{1}]$ tilt GBs (a). Blue and orange arrows show two possible DSC vectors $\pm \mathbf{b}$. Applying $+\mathbf{b}$ or $-\mathbf{b}$ displacements to the crystallites, yields the new patterns in (b) and (c), respectively. New coincidence sites lie halfway along the arrows indicated in (a). The arrows numbered 1--4 in (b) and (c) show the distance from the old to the new coincidence sites. Each number corresponds to a different type of defect, whose step height is the component of the arrow normal to the GB plane.
• Figure 3: Ground state energies $\gamma_0$ of GBs found via structure search, construction of low-energy domino/zipper GBs, or by annealing simulations. The data is presented as a function of the GB inclination $\phi$ from the symmetric reference plane (1 10 11). The second symmetric plane, (437), is located at $\phi = \ang{30}$. The energies of the symmetric GB phases are indicated with horizontal lines.
• Figure 4: Excess free energy calculations for pearl and domino GB phases at some representative inclinations $\phi$. The predicted GB phase transitions and stability ranges are marked. For the asymmetric pearl GBs (blue lines in (b),(c)), different ways to subtract the surface free energy (see Eqs. \ref{['eq:gamma-asymm-1']} and \ref{['eq:gamma-asymm-2']}) yield slightly different results. This is likely due to the limited simulation cell size, where strain fields of GB and surface slightly overlap. The estimated range of possible values is indicated by the blue shaded areas. The vertical marks represent the resulting uncertainty of the transition temperature.
• Figure 5: The GB phases present in asymmetric tilt GBs after equilibrating at temperatures between $300K$ and $400K$ for $t = 100ns$. The topmost row shows the fraction of GB phases in the starting structure, which were structures obtained with GRIP that contain both pearl and domino/zipper. For $\phi = \ang{0}$, we constructed a box with one big region each for pearl and domino. Results with additional starting structures are provided in Supplemental Fig. \ref{['fig:suppl:phase-fractions']}. For each inclination and temperature, the pie chart shows the fraction of the two GB phases. Starting from 16, there are gray bars around the data points. These indicate the maximum possible fractions of pearl phase in faceted GBs (see Sec. \ref{['sec:analyze-defects:high-incl']}). The dark gray region is for symmetric pearl facets. The light gray region indicates how much longer the pearl facets can become if they are asymmetric ($\phi = \ang{10.89}$).