# Cross-Modal Knowledge Distillation Method for Automatic Cued Speech
Recognition

###### Abstract

Cued Speech (CS) is a visual communication system for the deaf or hearing
impaired people. It combines lip movements with hand cues to obtain a complete
phonetic repertoire. Current deep learning based methods on automatic CS
recognition suffer from a common problem, which is the data scarcity. Until
now, there are only two public single speaker datasets for French (238
sentences) and British English (97 sentences). In this work, we propose a
cross-modal knowledge distillation method with teacher-student structure,
which transfers audio speech information to CS to overcome the limited data
problem. Firstly, we pretrain a teacher model for CS recognition with a large
amount of open source audio speech data, and simultaneously pretrain the
feature extractors for lips and hands using CS data. Then, we distill the
knowledge from teacher model to the student model with frame-level and
sequence-level distillation strategies. Importantly, for frame-level, we
exploit multi-task learning to weigh losses automatically, to obtain the
balance coefficient. Besides, we establish a five-speaker British English CS
dataset for the first time. The proposed method is evaluated on French and
British English CS datasets, showing superior CS recognition performance to
the state-of-the-art (SOTA) by a large margin.

Index Terms: Cued Speech, Cross-modal knowledge distillation, Teacher-student
structure, Cued Speech recognition

##  1 Introduction

Cued Speech (CS) is a visual mode of communication for hearing impaired people
proposed by Cornett [1] in 1967. It is to solve the confusion of lip reading.
CS uses the shape and position of hand to assist in lip reading to make visual
communication easier and more efficient for hearing impaired people. The hand
shape is used to encode consonants, and the hand position is used to encode
vowels (see Figure 1).

Currently, there are some automatic CS recognition works [2, 3, 4, 5, 6] based
on two small scale single-speaker datasets (one is in French [7] and the other
is in British English [8, 9]). The tandem architecture in [4] achieved 62%
accuracy on the French CS dataset with single speaker and 238 French
sentences. More recently, [5] proposed a deep sequence learning approach,
which consists of an image learner based on convolutional neural networks
(CNNs) [10] and a fully convolutional encoder-decoder. It was evaluated on the
British English CS dataset for the first time achieving 63.75% with single
speaker and 97 British English sentences. These two works were both focus on
continues CS recognition with no artificial marks [2, 3, 11, 12] but their
performance were still limited by the data scarcity, which leads to the
overfitting in a deep neural network training.

Figure 1: Chart of the French Cued Speech (from [2]). Figure 2: Teacher-
student structure for automatic CS recognition.

Knowledge distillation with the teacher-student structure can transfer and
preserve the cross-modality knowledge successfully [13, 14, 15, 16, 17, 18].
It has been successfully applied in many fields. For example, in [19], Gupta
et al. transferred the knowledge from the teacher model to the student model
relying on unlabeled paired samples involving both RGB and depth images. Zhao
et al. [20] used synchronized radio signals and camera images to transfer the
knowledge across modalities for radio-based human pose estimation. Thoker and
Gall [13] transferred the knowledge obtained from RGB videos to a skeleton-
based human action recognition model.

In this study, considering the limited size and multi-modality (including
audios and videos) of CS data, we innovatively propose a cross-modal knowledge
distillation architecture, which transfers the knowledge learned from audio
speech to visual-based CS (see Figure 2). Although the size of CS dataset is
limited, many acoustic speech datasets are publicly available. What’s more, CS
and audio speech are both phoneme-level coding thus they have the same phoneme
semantics. It motivates us to exploit cross-modal distillation in CS
recognition. Firstly, we pretrain a teacher model for phoneme recognition with
a large amount of open source audio speech data, and simultaneously pretrain
feature extractors for lips, hand shape and hand position. Then, we distill
the speech knowledge from the large teacher model into the small student model
to estimate the parameters of bi-directional long short-term memory (BiLSTM)
[21] in the student network with frame-level and sequence-level [22, 23]
distillation strategies. For frame-level distillation, the weights of losses
are learned automatically based on multi-task learning, and then they are used
to compute the balance coefficient (i.e., hyperparameters used to balance the
gradient among loss functions).

Compared with [4], we achieve significant CS phoneme recognition accuracy
improvement of 12.2% in frame-level (17% in sequence-level) for French task.
Compared with the state-of-the-art (SOTA) [5], we improve 3.32% in frame-level
(8.12% in sequence-level) for French CS and 1.15% in frame-level (7.95% in
sequence-level) for British English CS with single speaker.

In summary, the key contributions of this work are:

  1. 1.

This is the first attempt to apply cross-modal knowledge distillation method
to automatic CS recognition to overcome the problem of overfitting caused by
limited size of CS dataset.

  2. 2.

To obtain a better model automatically, we exploit multi-task learning to
weigh losses, and then compute the balance coefficient to balance the gradient
among loss functions.

  3. 3.

We establish the first five-speaker British English CS dataset with 390
English consecutive sentences.

##  2 Methodology

In this section, we introduce our method, which includes teacher network and
student network in the cross-modal knowledge distillation architecture, as
well as frame-level and sequence-level distillation strategies.

###  2.1 DeepSpeech2 teacher network

DeepSpeech2 [24] is an end-to-end deep learning approach for automatic speech
recognition (ASR) with good generalization performance. The architecture of
DeepSpeech2 we used is shown as the Teacher Model in Figure 2. It consists of
two 2D convolutional layers followed by five BiLSTM layers with batch
normalization [25] and a fully connected layer (FC). It is trained with the
connectionist temporal classification (CTC) [26] loss function to predict
audio speech transcriptions.

To use DeepSpeech2 as teacher model for CS recognition, we translate sentence
label into CS phoneme sequence. Let $x^{(i)}$ is a time-series, where each
time-slice is a vector of audio features, and $y^{(i)}$ is the corresponding
label. The inputs of the network are spectrogram of power normalized audio
clips, and the outputs are the phonemes. At each output time-step $t$, BiLSTM
makes a prediction over phonemes, $p(l_{t}|x)$ where
$l_{t}\in\\{blank,\Gamma\\}$. $\Gamma$ is the set of CS phonemes that are
different in French and British English. We represent the CTC loss function as
$L(x,y;\theta)$, where $(x,y)$ is an input-output pair, and $\theta$ is the
current parameters of the network. The network parameters are updated making
use of the derivative $\triangledown_{\theta}L(x,y;\theta)$.

###  2.2 CNN-BiLSTM student network

As shown in Figure 2, the architecture of the student model consists of two
pre-trained CNNs and one pre-trained artificial neural network (ANN) [27],
followed by two BiLSTM layers and two FC layers. CNNs act as feature
extractors. Each focuses on lips or hands separately, trained with a set of 8
visemes [28] as targets for lips and 8 shapes for hands. The details of the
CNNs and ANN network structure are the same as [4]. After extracting features
based on CNNs and ANN, the three stream features are concatenated in a single
feature vector as the input of BiLSTM.

To verify the effectiveness of knowledge distillation for CS recognition in
frame-level and sequence-level, we evaluate the student model trained by
minimizing Cross Entropy (CE) loss and CTC loss, respectively.

###  2.3 Knowledge distillation strategy

We consider two distillation strategies i.e., frame-level and sequence-level.
The standard knowledge distillation applied to frame-level prediction should
be effective for CS recognition, and a novel sequence-level knowledge
distillation strategy might further improve performance.

####  2.3.1 Frame-level distillation strategy

For frame-level distillation strategy, we try an existing general method
firstly. Considering a generalized $softmax$ function [29]:

$q_{i}=\frac{exp(z_{i}/T)}{\sum_{j}exp(z_{j}/T)},$ (1)

where $z_{i}$ is the logit from network output layer, $q_{i}$ is the class
probability corresponding to each $z_{i}$, and $T$ is the temperature
coefficient that is normally set to 1. Both the soft targets from the teacher
and the ground-truth label are of great importance for improving the
performance of the student model. Therefore, we consider Kullback-Leibler
Divergence (KL) as distillation loss and CE as student loss as follow:

$L(x_{t},x_{s};W)=\alpha KL(P_{(T)},Q_{(T)})*T^{2}+(1-\alpha)CE(Y,Q_{(1)}),$
(2)

where $x_{t}$ is the CS audio input of the pre-trained teacher model and
$x_{s}$ is the input of the student model which is concatenated feature
vector. $W$ are the parameters of the student model, and $\alpha$ is a
regulated parameter. $P_{(T)}$ and $Q_{(T)}$ are the $softmax$ output of
teacher model and student model, respectively when temperature is $T$. $Y$ is
the one-hot ground-truth label, and $Q_{(1)}$ is the $softmax$ output of
student model when the temperature is 1. For this loss function, we tune $T$
and $\alpha$ manually to find a model with higher CS phoneme recognition
accuracy.

The loss reduction curve is shown in the Figure 3 when $T$ is set to 1 and
$\alpha$ is set to 0.5. We observe that the gradient of KL is almost 0, which
means that the knowledge of the teacher model is not effectively distilled to
the student model, and there is an imbalance in the gradient between KL and
CE. To avoid tuning $T$ and $\alpha$ manually, we derive a multi-task joint
loss function as Equation (3) based on the task uncertainty.

$L_{mtl}=\frac{1}{\sigma_{1}^{2}}KL(P,Q)+\frac{1}{\sigma_{2}^{2}}CE(Y,Q)+log\sigma_{1}+log\sigma_{2},$
(3)

where $T$ and $\alpha$ are not considered anymore, and we automatically learn
the observation noise scalar $\sigma_{1}$ and $\sigma_{2}$. As reported in
[30], the task uncertainty captures the relative confidence between tasks,
reflecting the uncertainty inherent to the regression or classification task.
Therefore, in our two classification tasks (i.e., the student model learns
from the teacher model and learns from the ground-truth) of the distillation
process, uncertainty is used as a basis to weigh losses.

Further, to balance the loss gradient between these two tasks, the loss joint
function based on balance coefficient (i.e., $a$) is proposed as:

$L(x_{t},x_{s};W)=\frac{1}{2}(a\times KL(P_{(1)},Q_{(1)})+CE(Y,Q_{(1)})),$ (4)

We obtain the balance coefficient through
$a=\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}}$.

####  2.3.2 Sequence-level distillation strategy

Similarly, for sequential distillation, both the soft targets from the teacher
and the ground-truth label are of great importance for improving the
performance of the student model. Therefore, for student network, a
distillation loss that assists CTC is explored. Rather than using KL
divergence, we apply cosine similarity (cos) as the distillation loss, since
it pays more attention to the difference in the direction between the two
vectors. It is shown as follows:

$L(x_{t},x_{s};W)=\frac{1}{2}(1-cos(S_{s},S_{t})+CTC(Y,logQ)),$ (5)

where $x_{t}$, $x_{s}$, $W$ and $Q$ are the same as described in section
2.3.1. $S_{s}$ and $S_{t}$ are the frame-based logits before the $softmax$
layer of the student and teacher model, respectively. $Y$ is corresponding
transcript.

Figure 3: Comparison of gradient descent for KL, CE and their joint loss when
$T=1$ and $\alpha=0.5$ in Equation (2).

##  3 Experiments

###  3.1 Datasets

CS datasets for student model We use a public French CS dataset [7] and the
newly built British English CS dataset with 5 speakers specially for this
work. The detail of the public single-speaker French can refer to [7]. Here,
we mainly introduce the new multi-speaker British English CS dataset.

A single-speaker (named CA) British English CS dataset [8] which contains 97
sentences. Now, the dataset has been expanded with data for other four CS
interpreters named EM, KA, LD and VK, respectively. There are 390 British
English audios and videos of all five speakers in total. The RGB video image
recorded at 25fps. In the British English CS system, the 12 monophthongs are
encoded by four hand positions and the 8 diphthongs are encoded by four hand
slips, while the 24 consonants are encoded by eight hand shapes.

Datasets for teacher model pre-training We train two teacher models for
distillation with the French speech dataset and English speech dataset in
Common Voice [31].

Common Voice is an open source project launched by Mozilla including French
and English speech data. The total number of verification hours of French
version is 350h and English version is 1118h. We convert the French sentence
text into French CS phonemes using Lliaphon [32]. Then British English
phonetic transcript is obtained by eSpeak (i.e., a compact open source
software speech synthesizer for English and other languages). It should be
mentioned that the English datasets only contains 8% of British English, and
the remaining 92% consists of American English and dialects.

Table 1: Performance comparison in Acc(%) in frame-level distillation. Fully Conv [5] is the SOTA. $\rm JLF1$ is the best performance obtained by minimizing the joint loss function represented by Equation (2). $\rm JLF2_{mtl}$ represents the performance of the optimal model obtained by Equation (3). $\rm JLF3$ is the performance obtained by Equation (4). Dataset |  French |  Single-speaker (British) |  Multi-speaker (British)
---|---|---|---
CNN-HMM [4] |  62% |  — |  —
Fully Conv [5] |  70.88% |  63.75% |  —
Student CE |  64.4% |  52.5% |  62.3%
$\rm JLF1$ |  72.5% ($T=5,\alpha=0.5$) |  61.3% ($T=5,\alpha=0.7$) |  65.7% ($T=5,\alpha=0.7$)
$\rm JLF2_{mtl}$ |  72.5% ($\sigma_{1}=0.37,\sigma_{2}=1.20$) |  63.1% ($\sigma_{1}=0.59,\sigma_{2}=1.02$) |  68.5% ($\sigma_{1}=0.23,\sigma_{2}=0.74$)
$\rm JLF3$ |  74.2% ($a=\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}}\approx 10$) |  64.9% ($a=\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}}\approx 3$) |  69.7% ($a=\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}}\approx 10$)

###  3.2 Protocol and metrics

In our experiments, each CS dataset for student model uses 80% of the data for
training, 10% of the data for validation, and the other 10% for testing. These
three parts of data do not overlap. All models are evaluated in CS phoneme
accuracy (Acc) which is defined as $Acc(\%)=1-PER(\%)$. PER is the phoneme
error rate.

###  3.3 Result and analysis

We evaluate the teacher model, as well as the student model trained with
distillation in frame-level and sequence-level on CS datasets.

####  3.3.1 Evaluation on teacher model

The phoneme recognition accuracy of the teacher model is 83.7% on French CS
and 70.1% on the new British English CS audio speech data. We hypothesize that
the low performance of the English teacher model is caused by the limited
British English data (only 8% in Common Voice French dataset).

####  3.3.2 Evaluation on student model

Frame-level distillation For frame-level knowledge distillation, we evaluate
three loss functions (i.e., Equation (2), (3) and (4)). In Table 1, we find
out that the knowledge distillation method bring considerable improvement
compared with training student model with baseline CE. On each CS dataset,
$\rm JLF3$ is the best result. We achieve Acc improvement of 12.2% compared
with CNN-HMM [4], and 3.32% compared with SOTA [5] in French CS task. In
single-speaker British CS task, we improve 1.15% Acc compared with SOTA. For
the new dataset with multi speakers, $\rm JLF3$ achieves Acc of 69.7%.

More balance coefficients are explored to show the effectiveness of the joint
loss function we proposed (see Equation (4)). From Figure 4, it can be
observed that the closer the gradient between the KL and CE at the beginning
of the training step, the more efficient the distillation.

Figure 4: Comparison of gradient descent among KL, CE and their joint loss
with different balance coefficients in French CS recognition task.

Sequence-level distillation As demonstrated in Table 2, in addition to multi-
speaker, as an auxiliary loss, cos performs better than training the student
model with CTC alone. The English teacher model (70.1%) is not better than the
student model trained by CTC (78.6%), so that it is not enough to play a
positive role in guiding the student model.

Table 2: Performance comparison by Acc(%) in sequence-level distillation. Fully Conv [5] is the SOTA. Dataset | French |  Single-speaker (British) |  Multi-speaker (British)
---|---|---|---
CNN-HMM | 62% | — | —
Fully Conv | 70.88% | 63.75% | —
Student CTC | 77% | 68.6% | 78.6%
KL+CTC | 75% | 68.4% | 75.2%
cos+CTC | 79% | 71.4% | 77.5%

To conclude, we verify the effectiveness of knowledge distillation, the joint
loss function based multi-task learning and the balance coefficient in CS
recognition with limited data size. Besides, the performance on the multi-
speaker CS dataset shows good generalization performance for different
speakers.

##  4 Conclusions

In this work, we proposed a cross-modal knowledge distillation method
including two structures, a teacher model and a CNN-BiLSTM student model with
two distillation strategies for CS recognition on limited size of datasets. We
highlighted the effectiveness of our method and the importance of weighing
losses based multi-task learning as well as the balance coefficient for the
frame-level distillation gradient descent. The comparative evaluation
demonstrated that the proposed method achieves new SOTA on the CS recognition
in the multi-speaker scenario. As for future work, we will be engaged in
improving the teacher and student model by language models to decrease the
insertion errors.

##  5 Acknowledgements

This work was supported by the GuangDong Basic and Applied Basic Research
Foundation (No. 2020A1515110376) and the National Natural Science Foundation
of China (grant No. 61977049). The authors would like to thank the
professional CS speakers from Cued Speech UK for the British English CS
dataset recording.

## References

  * [1] R. O. Cornett, “Cued speech,” _American annals of the deaf_ , pp. 3–13, 1967\.
  * [2] P. Heracleous, D. Beautemps, and N. Aboutabit, “Cued speech automatic recognition in normal-hearing and deaf subjects,” _Speech Communication_ , vol. 52, no. 6, pp. 504–512, 2010.
  * [3] P. Heracleous, D. Beautemps, and N. Hagita, “Continuous phoneme recognition in cued speech for french,” _in Proc. EUSIPCO_ , pp. 2090–2093, 2012.
  * [4] L. Liu, T. Hueber, G. Feng, and D. Beautemps, “Visual recognition of continuous cued speech using a tandem cnn-hmm approach.” _in Proc. INTERSPEECH_ , pp. 2643–2647, 2018.
  * [5] K. Papadimitriou and G. Potamianos, “A fully convolutional sequence learning approach for cued speech recognition from videos,” _in Proc. EUSIPCO_ , pp. 326–330, 2021.
  * [6] L. Liu, G. Feng, D. Beautemps, and X.-P. Zhang, “Re-synchronization using the hand preceding model for multi-modal fusion in automatic continuous cued speech recognition,” _IEEE Transactions on Multimedia_ , vol. 23, pp. 292–305, 2020.
  * [7] L. Liu, G. Feng, and D. Beautemps, “Automatic temporal segmentation of hand movements for hand positions recognition in french cued speech,” _in Proc. ICASSP_ , pp. 3061–3065, 2018.
  * [8] L. Liu, J. Li, G. Feng, and X.-P. S. Zhang, “Automatic detection of the temporal segmentation of hand movements in british english cued speech.” _in Proc. INTERSPEECH_ , pp. 2285–2289, 2019.
  * [9] L. Liu, G. Feng, D. Beautemps, and X.-P. Zhang, “A novel resynchronization procedure for hand-lips fusion applied to continuous french cued speech recognition,” _in Proc. EUSIPCO_ , pp. 1–5, 2019.
  * [10] Y. LeCun, Y. Bengio _et al._ , “Convolutional networks for images, speech, and time series,” _The handbook of brain theory and neural networks_ , vol. 3361, no. 10, p. 1995, 1995.
  * [11] N. Aboutabit, “Reconnaissance de la langue française parlée complété (lpc): décodage phonétique des gestes main-lèvres.” Ph.D. dissertation, Institut National Polytechnique de Grenoble-INPG, 2007.
  * [12] T. Burger, A. Caplier, and S. Mancini, “Cued speech hand gestures recognition tool,” _in Proc. EUSIPCO_ , pp. 1–4, 2005.
  * [13] F. M. Thoker and J. Gall, “Cross-modal knowledge distillation for action recognition,” _in Proc. ICIP_ , pp. 6–10, 2019.
  * [14] Z. Luo, J.-T. Hsieh, L. Jiang, J. C. Niebles, and L. Fei-Fei, “Graph distillation for action detection with privileged modalities,” _in Proc. ECCV_ , pp. 166–183, 2018.
  * [15] E. J. Crowley, G. Gray, and A. J. Storkey, “Moonshine: Distilling with cheap convolutions.” _in Proc. NeurIPS_ , pp. 2893–2903, 2018.
  * [16] W. Hao and Z. Zhang, “Spatiotemporal distilled dense-connectivity network for video action recognition,” _Pattern Recognition_ , vol. 92, pp. 13–24, 2019\.
  * [17] M.-C. Wu, C.-T. Chiu, and K.-H. Wu, “Multi-teacher knowledge distillation for compressed video action recognition on deep neural networks,” _in Proc. ICASSP_ , pp. 2202–2206, 2019.
  * [18] S. Zhang, S. Guo, L. Wang, W. Huang, and M. Scott, “Knowledge integration networks for action recognition,” _in Proc. AAAI_ , vol. 34, no. 07, pp. 12 862–12 869, 2020.
  * [19] S. Gupta, J. Hoffman, and J. Malik, “Cross modal distillation for supervision transfer,” _in Proc. CVPR_ , pp. 2827–2836, 2016.
  * [20] M. Zhao, T. Li, M. Abu Alsheikh, Y. Tian, H. Zhao, A. Torralba, and D. Katabi, “Through-wall human pose estimation using radio signals,” _in Proc. CVPR_ , pp. 7356–7365, 2018.
  * [21] A. Graves, A.-r. Mohamed, and G. Hinton, “Speech recognition with deep recurrent neural networks,” _in Proc. ICASSP_ , pp. 6645–6649, 2013.
  * [22] M. Huang, Y. You, Z. Chen, Y. Qian, and K. Yu, “Knowledge distillation for sequence model.” _in Proc. INTERSPEECH_ , pp. 3703–3707, 2018.
  * [23] R. Takashima, L. Sheng, and H. Kawai, “Investigation of sequence-level knowledge distillation methods for ctc acoustic models,” _in Proc. ICASSP_ , pp. 6156–6160, 2019.
  * [24] D. Amodei, S. Ananthanarayanan, R. Anubhai, J. Bai, E. Battenberg, C. Case, J. Casper, B. Catanzaro, Q. Cheng, G. Chen _et al._ , “Deep speech 2: End-to-end speech recognition in english and mandarin,” _in Proc. ICML_ , pp. 173–182, 2016.
  * [25] S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep network training by reducing internal covariate shift,” _in Proc. ICML_ , pp. 448–456, 2015.
  * [26] A. Graves, S. Fernández, F. Gomez, and J. Schmidhuber, “Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks,” _in Proc. ICML_ , pp. 369–376, 2006.
  * [27] R. Schalkoff, _Artificial Neural Networks_. McGraw-Hill New York, 1997.
  * [28] C. Benoit, T. Lallouache, T. Mohamadi, and C. Abry, “A set of french visemes for visual speech synthesis,” pp. 485–501, 1992.
  * [29] G. Hinton, O. Vinyals, and J. Dean, “Distilling the knowledge in a neural network,” _arXiv preprint arXiv:1503.02531_ , 2015.
  * [30] A. Kendall, Y. Gal, and R. Cipolla, “Multi-task learning using uncertainty to weigh losses for scene geometry and semantics,” _in Proc. CVPR_ , pp. 7482–7491, 2018.
  * [31] R. Ardila, M. Branson, K. Davis, M. Henretty, M. Kohler, J. Meyer, R. Morais, L. Saunders, F. M. Tyers, and G. Weber, “Common voice: A massively-multilingual speech corpus,” _arXiv preprint arXiv:1912.06670_ , 2019.
  * [32] F. Béchet, “Lia phon: un systeme complet de phonétisation de textes,” _Traitement automatique des langues_ , vol. 42, no. 1, pp. 47–67, 2001.


# Elementary mechanisms of shear-coupled grain boundary migration for
different complexions of a copper tilt grain boundary

Swetha Pemma  Max-Planck-Institut für Eisenforschung GmbH, Max-Planck-Straße
1, 40237 Düsseldorf, Germany  Rebecca Janisch  Interdisciplinary Centre of
Advanced Materials Simulation (ICAMS), Ruhr-Universität Bochum, 44780 Bochum,
Germany
Gerhard Dehm<EMAIL_ADDRESS>Max-Planck-Institut für Eisenforschung GmbH, Max-
Planck-Straße 1, 40237 Düsseldorf, Germany  Tobias Brink<EMAIL_ADDRESS>Max-
Planck-Institut für Eisenforschung GmbH, Max-Planck-Straße 1, 40237
Düsseldorf, Germany

###### Abstract

The migration of grain boundaries leads to grain growth in polycrystals and is
one mechanism of grain-boundary-mediated plasticity, especially in metallic
nanocrystals. This migration is due to the movement of dislocation-like
defects, called disconnections, which couple to externally applied shear
stresses. Here, we investigate a $\Sigma$19b symmetric tilt grain boundary
without pre-existing defects using atomistic computer simulations with
classical potentials. This specific grain boundary exhibits two different
atomic structures with different microscopic degrees of freedom (complexions),
called “domino” and “pearl” complexion. We show that the grain boundary
migration is affected by both the formation energy of a disconnection dipole
and the Peierls-like barrier required to move the disconnections. For the
pearl complexion, the latter is much higher, leading to a high stress required
for grain boundary migration at low temperatures. However, in absolute values,
the Peierls barrier is low and can be overcome by thermal energy even at room
temperature. Since the domino complexion has higher disconnection formation
energies, it is more resistant to migration at room temperature and above.

##  I  Introduction

Grain boundaries (GBs) influence mechanical properties of polycrystalline
materials and GB engineering is critical in material design [1]. The motion of
GBs is the key factor in the microstructure evolution of poycrystalline
materials [2, 3]. When subjected to shear stress, GBs move and can account for
part of the plastic deformation in nanocrystalline materials [4, 5, 6, 7].
Some of the GB deformation mechanisms discussed in the literature are GB
sliding [8, 9, 10], grain rotation [11, 12, 13, 14, 15], shear coupled GB
migration [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32], diffusional creep [6, 33], dislocation interaction at grain boundaries
[4, 34, 6, 33], and enhanced partial dislocation activity [4, 34, 6, 33].

Shear coupling is the migration of GBs driven by shear stress across the GB
plane [35, 19, 24]. It can lead to complex effects during grain growth in a
polycrystal such as grain rotation, stress generation, and grain growth
stagnation, which are all inter-related [15]. A shear coupling factor
$\beta=v_{\parallel}/v_{\perp}$ describes how a GB migrates: A relative shear
velocity $v_{\parallel}$ of the two grains parallel to the GB is coupled
directly to the GB migration velocity $v_{\perp}$ normal to its plane [36, 12,
19, 37]. This factor $\beta$ is influenced by parameters such as temperature,
bicrystallography, and the type of the driving force [38]. Microscopically,
shear-coupled motion is caused by the movement of disconnections, which are
line defects at the GB [39, 36, 40, 37]. Disconnections have dislocation
character insofar that they have a Burgers vector $\mathbf{b}$, which couples
to externally applied stress. They also lead to a step of size $h$ in the GB,
which results in the GB migration during disconnection nucleation and movement
[35, 41]. The formation and migration of disconnections play a vital role in
the kinetic properties of GBs [42, 43, 44, 45, 46] and pre-existing mobile
disconnections lead to a reduced stress required for GB migration [47].
Bicrystallography influences the possible disconnection modes, i.e., which
Burgers vectors can occur, and the applied stress and temperature influence
the active disconnection mode [19, 37, 38, 45, 48, 49].

At the atomic scale, however, different structures at GBs and first-order
transitions between them were observed even in pure metallic materials [50,
51, 52, 53, 54, 55, 56, 57]. These different structures can be treated as
interface phases, which can only exist in contact with the abutting
crystallites, and can be treated using a thermodynamic framework [58, 59, 60,
61, 62, 63]. They are called complexions [64, 65, 66, 67] or GB phases [68].
Complexion transitions, then, are analogous to bulk phase transitions: The GB
structure, composition, and properties change discontinuously at critical
values of thermodynamic parameters such as temperature, pressure, and chemical
potential [64, 62, 63, 68, 66, 67]. Only a few simulation studies investigated
the influence of complexion transitions on the shear coupled motion. They
found that $\beta$ can depend on the complexion present at the GB, even if the
macroscopic degrees of freedom of the GB are constant (congruent complexion
transition [66]) and only the microscopic degrees of freedom change [19, 69].
The understanding of how complexion transitions can affect shear-coupled GB
migration from a mechanistic viewpoint is thus still at its infancy.

In this paper, we report the underlying mechanisms and parameters of shear-
coupled motion for a $\Sigma 19$b $[11\overline{1}]$ $(178)$ symmetric tilt
GB. This GB can exhibit two different complexions [55]. We discuss the
disconnection mode that is active in these complexions and how differences of
disconnection formation and migration energies affect the GB migration.

##  II  Computational methods

We studied bicrystals with symmetric tilt GBs using molecular dynamics (MD)
simulations, which were performed using LAMMPS [70, 71] with the embedded atom
method (EAM) potential of Mishin et al [72]. This potential reproduces some
properties of Cu very well, comprising elastic constants, phonon frequencies,
thermal expansion, intrinsic stacking fault energy, the coherent twin boundary
energy and others. We used a time integration step of
$2\text{\,}\mathrm{f}\mathrm{s}$ for all dynamics simulations.

The bicrystals for $\Sigma$19b $[11\overline{1}]$ $(178)$ symmetric tilt GBs
(misorientation of $None$) were created by constructing two fcc crystals with
the desired crystallographic orientations $[\overline{5}3\overline{2}]$,
$[178]$, $[11\overline{1}]$ in $x$, $y$, $z$ directions for the top crystal
and $[\overline{3}52]$, $[718]$, $[11\overline{1}]$ for the bottom crystal
[Fig. 1(a)]. Here, $y$ is the GB normal and $z$ the tilt axis. The $x$ and $z$
directions were periodic and we used open boundaries in $y$ direction,
allowing us to produce differently-sized cells by repeating the unit cell
along the periodic directions. The periodic unit cell dimensions were
$L_{x}=$11.142\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ and
$L_{z}=$6.261\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$. We used $L_{y}$
ranging from $385.317\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ to
$5790.070\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ (see below). The GB
structures were formed by combining the two crystallites, sampling the
microscopic degrees of freedom by displacing the top crystal, and minimizing
the result in molecular statics ($\gamma$-surface method). We did this until
the previously reported pearl and domino complexions [55] were found (we
evaluated that structure, GB energy, and excess volume match). The minimum
energy GB structures for domino and pearl are visualized using OVITO [73] and
are shown in Fig. 2(a)–(b).

Figure 1: (a) Schematic of the bicrystallography of a $\Sigma$19b
$[11\overline{1}]$ $(178)$ symmetric tilt GB. Top and bottom fcc crystals are
joined in the indicated orientations, leading to a misorientation between the
two crystals of $None$. (b) Schematic of the construction of the disconnection
dipole with opposite Burgers vetor by repeating bicrystals A and B. Bicrystal
B is moved by a translation vector and step height compared to A. By varying
the number of B unit cells, the separation $\delta$ between disconnections can
be controlled.

For the simulation of shear-coupled GB motion, we used a simulation cell of
size $222.814\times
385.317\times$62.613\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{3}$$ ($20\times
20\times 10$ unit cells, resulting in 455,400 atoms) unless specified
otherwise.

We started by using molecular statics simulations ($T=$0\text{\,}\mathrm{K}$$)
and applying a displacement on the top boundary in $x$ direction and keeping
the bottom boundary fixed. Both boundaries were the regions at the surfaces in
$y$ direction each with a width of $55\text{\,}\AA$ [Fig. 2(c)–(d)]. To study
GB migration in domino and pearl, the shear displacement $d$ was applied
stepwise in increments of $0.05\text{\,}\AA$, while minimizing the system
after every step. We calculated the resulting shear stress by dividing the
reaction force at the boundary by the area $L_{x}\times L_{z}$ of the top
boundary.

Then, MD simulations were performed in the canonical ensemble (Nosé–Hoover
thermostat at $T=100$, 200, 300, 400, 500, $600\text{\,}\mathrm{K}$). At
finite temperatures, the system was scaled to obtain the correct lattice
constant at the desired temperature $T$ and then equilibrated for
$4\text{\,}\mathrm{n}\mathrm{s}$. In this simulation procedure, we applied a
velocity in $x$ direction to the top boundary, while keeping the bottom
boundary fixed (see Ref. [19] and Fig. 2). The top boundary was allowed to
move freely in $y$ and $z$ direction. Shear-coupled simulations were performed
for shear velocities in the range from
$0.01\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$ to
$10\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$. We typically applied a constant
shear velocity of $0.1\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$ unless
otherwise noted. Shear stress was calculated as for the molecular statics
simulations.

The desired disconnection dipoles were constructed for different disconnection
widths $\delta$ (separation distance between the two disconnections). This is
similar to the construction of disconnection as described in Refs. [74, 75]:
Bicrystals A and B were generated such that they contain the same complexion.
The GB in bicrystal B was moved by one step height compared to A. By
replicating A and B and assembling them (e.g., as AABBBAA), two disconnections
with opposite Burgers vector appear at the junctions …AB…and …BA… [Fig. 1(b)].
By varying the number of B unit cells, the separation $\delta$ between
disconnections can be controlled. We calculated the energy of the
disconnection dipoles in simulation cells of sizes ranging from
$1114.220\times
964.490\times$6.261\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{3}$$
($5.6\text{\times}{10}^{5}\text{\,}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}$)
to $1114.220\times
5790.070\times$6.261\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{3}$$
($3.4\text{\times}{10}^{6}\text{\,}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}$)
to ensure that size effects due to the open boundary condition in $y$ could be
excluded and that the results converged.

Monopoles (single disconnections) were constructed in a similar way by
combining A and B so that only one disconnection occurs (…AAABBB…) and by
leaving open boundaries in $x$ direction. Simulation cell sizes ranging from
$2228.830\times 1949\times$6.261\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{3}$$
($2.3\text{\times}{10}^{6}\text{\,}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}$)
to $3342.799\times
5790.07\times$6.261\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{3}$$
($1.0\text{\times}{10}^{7}\text{\,}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}$)
with open boundary conditions in $x$ and $y$ directions were investigated. We
computed the monopole energy in a region of size $L_{x}/2\times L_{y}/2\times
L_{z}$ in the center of the simulation cell. We varied $L_{x}$ and $L_{y}$ to
observe convergence of the monopole energies, but could not achieve any
convergence with $L_{x}$ due to the infinitely long-ranged elastic interaction
between the Burgers vector and the surfaces.

Finally, in order to obtain the barriers for the migration of the
disconnections themselves, the minimum energy path for a change of the
disconnection dipole width $\delta$ was obtained by nudged elastic band (NEB)
method calculations [76, 77]. The spring constants for the parallel and
perpendicular nudging forces were both
$1.0\text{\,}\mathrm{e}\mathrm{V}\mathrm{/}\AA^{2}$. The minimization scheme
used was quickmin [78]. The initial and final states of the minimum energy
path are the GB structures with disconnection dipole widths $\delta$ varying
by $11.142\text{\,}\AA$ (equivalent to a CSL periodicity vector and thus the
distance between two local minima for the disconnections). The saddle point
observed along the minimum energy path is the required disconnection migration
barrier (difference in minimum and maximum energy observed along minimum
energy path).

Raw data for all simulations and analyses is available in the companion
dataset [79].

##  III Results and Discussion

###  III.1  Shear coupling

Figure 2:  Atomic structures of the (a) domino and (b) pearl complexions in
our computer model. Both complexions consist of two repeating structural units
(…RLRLRL…). Shear coupling simulations for $20\text{\,}\mathrm{n}\mathrm{s}$
with a shear velocity of $0.1\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$ at
$300\text{\,}\mathrm{K}$ lead to a migration distance on the order of
$2\text{\,}\mathrm{n}\mathrm{m}$ for both (c) domino and (d) pearl. Shear
coupled GB motion is highlighted by the yellow fiducial mark, which exhibits a
slope (equivalent to $\beta^{-1}$) in the region traversed by the GB. The top
grain is sheared and the bottom grain grows at the expense of the top grain.

In the present work, we investigated $\Sigma$19b $[11\overline{1}]$ $(178)$
symmetric tilt GBs in copper. For these GBs, two different complexions can
occur [55], named domino [Fig. 2(a)] and pearl [Fig. 2(b)]. These complexions
each consist of two structural units, here designated as …RLRLRL….

Figure 3:  Displacement $u_{x}$ of atoms as a function of their position along
the $y$ axis for the (a) domino and (b) pearl complexions during the shear-
coupling simulations. Note that $y=0$ corresponds to the initial GB position.
The shear coupling factor $\beta$ is the slope (red and blue lines) of these
graphs and was obtained by linear regression (Eq. 1). The regions belonging to
the bottom crystal ($y<$0\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$, zero
displacement) and to the top crystal
($y>$2.4\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$) are strain-free (constant
displacement), while the region traversed by the GB was sheared.

We simulated the shear-coupled motion of these two complexions by applying a
shear displacement. The result for the applied shear velocity of
$0.1\text{\,}\mathrm{m}\mathrm{/}\mathrm{s}$ after a simulation time of
$20\text{\,}\mathrm{n}\mathrm{s}$ at $300\text{\,}\mathrm{K}$ is shown in Fig.
2(c)–(d). Before the simulations, atoms in a vertical line were marked in
yellow. This fiducial line highlights the atomic displacements: The GB has
moved in positive $y$ direction from its original position (dashed line),
while the material was sheared in positive $x$ direction. The macroscopically
applied displacement couples thus to the GB migration. The slope of the
fiducial line provides the ratio of GB migration distance to sliding and is
therefore equivalent to the inverse of the shear coupling factor [36, 12, 19,
37]

$\beta=\frac{v_{\parallel}}{v_{\perp}},$ (1)

where $v_{\parallel}$ is the shear velocity applied to the system and
$v_{\perp}$ the velocity of GB migration. To more accurately calculate
$\beta$, we can record the displacement $u_{x}$ of the atoms and plot it as a
function of the atomic position normal to the GB (Fig. 3). The slope can be
obtained by linear regression and corresponds directly to $\beta$. Earlier
studies show that $\beta$ is a characteristic of a GB and depends on the
misorientation of top and bottom crystal [19], but can also depend on the
complexion for fixed macroscopic GB parameters [69]. In this study, $\beta$ is
observed to be 0.874 for both domino and pearl. This is independent of the
applied shear velocity or temperature, see Fig. S1 in the Supplemental
Material (SM) [80]. We note, however, that the individual displacements
$u_{x}$ of atoms at a certain $y$ position in the region traversed by the GB
depend on the complexion. We observe three distinct offsets of the
displacement $u_{x}$ of atoms in domino (with the same slope), but only one in
pearl. This hints at differences of the internal atomic shuffling during GB
migration in domino and pearl.

Figure 4:  Shear stress response to displacement at 0 K plotted for domino and
pearl. Each shear stress drop observed corresponds to a unit step GB
migration. The maximum shear stress is the critical shear stress $\tau_{c}$
required for GB migration. The dotted lines represent unloading followed by
loading in the opposite direction after the first stress drop. The inset shows
the shear deformation $d_{s}$ for GB migration by a unit step. The shear
deformation is observed to be $\approx$0.550\text{\,}\AA$$ for both domino and
pearl. The migration normal to the GB plane from the initial position obtained
from atomistic simulations is $\approx$0.647\text{\,}\AA$$ and
$\approx$0.686\text{\,}\AA$$ for domino and pearl respectively.

In these simulations, displacement was imposed and the GB migration velocity
$v_{\perp}$ only depends on the applied shear velocity $v_{\parallel}$ and
$\beta$. In reality, it is often the case that a given stress is applied, so a
relevant figure of merit is the critical stress $\tau_{c}$ required to start
GB migration. This can be obtained by monitoring the reaction forces at the
boundary where the shear is applied. We started with deformation in molecular
statics ($T=$0\text{\,}\mathrm{K}$$). As the displacement $d$ increases, the
shear stress $\tau$ increases linearly in the elastic regime (Fig. 4). At a
certain displacement, the shear stress drops as the GB migrates. The critical
shear stress $\tau_{c}$ is observed to be
$1.117\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ for pearl and
$0.849\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ for domino. As we increase the
displacement, the GB continues to move, resulting in recurring shear stress
drops at intervals of displacement characteristic to the simulation cell size.
This saw-tooth behavior is similar to earlier reported simulation studies [19,
21, 22, 27]. The pearl complexion has a significantly higher barrier for GB
migration than the domino complexion.

By unloading the system after the first stress drop down to zero shear stress,
we obtain a residual displacement of
$d_{s}\approx$0.550\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ (marked in the
inset of Fig. 4). The distance $h$ traveled by the GB during the event is
obtained from atomistic simulations by marking the equivalent position in the
structural unit (e.g., in R) along $y$ in the initial and final GB positions.
Together with the distance
$h\approx$0.647\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ that the GB traveled
during the event, we again obtain $\beta=d_{s}/h\approx 0.850$ for domino and
$h\approx$0.686\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$,
$\beta=d_{s}/h\approx 0.801$ for pearl. The difference in $\beta$ is due to
the relatively large error of determining $h$ from a single GB migration step,
and the values from Fig. 3 should be preferred. We also observe that the
structural units in both complexions switch from …RLRLRL…to …LRLRLR…, which
indicates that the migration is related to the atomic structure, but that the
basic character of the GB is preserved during migration. We will now
investigate the atomistic mechanism behind the difference in critical
stresses.

###  III.2  Atomistic mechanisms and bicrystallography

It is known that line defects with dislocation character can exist on GBs [39,
36, 40, 35, 41, 37]. These are called secondary GB dislocations or
disconnections and possess both a Burgers vector $\mathbf{b}$ (leading to the
dislocation character) and a step height $h$. Disconnections can only exist on
and move along the GB, where they also introduce a step of the GB plane. The
Burgers vector is associated with a long-range strain field. Applied shear
couples to this strain field, and the GB migrates due to the step of the
disconnections, which nucleate and propagate under the stress. Similar to a
bulk dislocation, the structure of the GB on both sides of the disconnection
is undisturbed. Thus, the Burgers vectors must be displacement shift complete
(DSC) vectors. We found that the disconnection that is active during GB
migration in both domino and pearl is
$(\mathbf{b},\textit{h})=([0.586,0,0]\,\text{\AA},$0.677\text{\,}\AA$)$,
resulting in $\beta=b_{x}/h=0.865$ (see Appendix A).

During a unit GB migration step, we found that the atoms do not jump
immediately from their initial to their final positions in the respective fcc
lattices of the top and bottom crystal, indicating that the GB has a certain
width in which the equilibrium positions do not correspond to crystallographic
sites in fcc (see Appendix B for more details). The differences in atomic
shuffling between domino and pearl, however, did not provide a conclusive
reason for the difference in critical stress $\tau_{c}$ for GB migration.

###  III.3  Disconnection monopoles and dipoles

Table 1:  GB energies and parameters $K$ and $\delta_{0}$ for the dipole energies obtained from fitting Eq. (6) to Edipole from the simulations. We list the GB energy $\gamma^{*}$ including a disconnection dipole with $\delta=L_{x}/2$, the parameter $K$ describing anisotropic crystal elasticity, and the effective disconnection core size $\delta_{0}$. The energy differences are always $\Delta E=E_{\text{domino}}-E_{\text{pearl}}$, with $\Delta E^{*}_{\text{dipole}}$ being the dipole energy difference at $\delta=L_{x}/2$, and $\Delta E_{\text{monopole}}$ being the energy difference between monopoles (single disconnection). With the latter, we can also extract the difference of the elastic interaction energies $\Delta E_{\text{elastic}}$ and the ratio of disconnection core sizes $\delta_{c}^{\text{domino}}/\delta_{c}^{\text{pearl}}$. $L_{y}$ (Å) | 964.490 | 1929.540 | 3859.800 | 5790.070
---|---|---|---|---
$\gamma^{*}_{\text{domino}}$ (J/m2) | 0.872 | 0.873 | 0.873 | 0.873
$K_{\text{domino}}$ (meV/Å3) | 41.9 | 75.6 | 77 | 76.8
$\delta_{0}^{\text{domino}}$ (Å) | 0.191 | 3.540 | 3.791 | 3.761
$\gamma^{*}_{\text{pearl}}$ (J/m2) | 0.836 | 0.837 | 0.837 | 0.837
$K_{\text{pearl}}$ (meV/Å3) | 40.6 | 75 | 76.4 | 76.4
$\delta_{0}^{\text{pearl}}$ (Å) | 0.211 | 4.067 | 4.358 | 4.366
$\Delta E^{*}_{\text{dipole}}$ (meV/Å) | 4.885 | 4.567 | 4.557 | 4.555
$2\Delta E_{\text{monopole}}$ (meV/Å) | – | 5.718 | 5.549 | 5.661
$\Delta E_{\text{elastic}}$ (meV/Å) | – | $-1.151$ | $-0.992$ | $-1.105$
$\delta_{c}^{\text{domino}}/\delta_{c}^{\text{pearl}}$ | – | 1.045 | 1.038 | 1.042
Figure 5:  Structure of the disconnection dipoles in (a) domino and (b) pearl.
On the left, the disconnection is ($\mathbf{b}=[0.586,0,0]\,\text{\AA}$,
$h=0.677$) with structural units …LRLRLLRLRL…. On the right, we have the
opposite disconnection ($\mathbf{b}=[-0.586,0,0]\,\text{\AA}$, $h=-0.677$)
with structural units …RLRLRRLRLR… The disconnection dipoles were constructed
for different disconnection dipole widths $\delta$. In the image, the ellipsis
indicates that $\delta$ is larger than visible. Burgers circuits are drawn
around the disconnections (black atoms) to verify the disconnection mode
obtained after minimization.

When disconnections nucleate in otherwise defect-free GBs, a pair of opposite
disconnections, namely $(\textbf{b},h)$ and $(-\textbf{b},-h)$, has to be
created. This would be similar for bulk dislocations. These disconnections
form a disconnection dipole with a separation $\delta$ between disconnections.
Due to the opposite signs of the Burgers vectors, the disconnections in the
dipole have an attractive interaction. We created dipoles of different widths
in domino and pearl GBs to evaluate their energy. The smallest possible stable
dipole width is $11.142\text{\,}\AA$, corresponding to the GB unit cell size
along $\langle$532$\rangle$. To introduce the dipole, part of the GB is
translated and moved with the desired b and h and the energy of the entire
structure is minimized. Parts of the dipoles are shown in Fig. 5. We verified
that we obtained the desired defects by making a Burgers circuit around them
(black atoms, see Fig. S2 in the SM [80] for details).

Figure 6:  (a) The formation energies $E_{\text{dipole}}$ of disconnections
for different dipole widths $\delta$ of the disconnection mode
$(\mathbf{b},h)=($0.586\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,$0.677\text{\,}\mathrm{\SIUnitSymbolAngstrom}$)$
are plotted for domino and pearl complexions. (b) Zoom of the gray area in
(a). The energies of dipoles during their migration (thick lines) were
obtained by NEB between the indicated local minima (squares and circles). The
disconnection migration barrier $E_{\text{mig}}$ is highlighted by red and
blue arrows for domino and pearl.

The dipole energy is

$E_{\text{dipole}}=\frac{E_{1}-E_{0}}{L_{z}},$ (2)

where $E_{0}$ is the energy of a system with a GB but no disconnection,
$E_{1}$ is the energy of the same system with a disconnection dipole, and
$L_{z}$ is the width of the system along the disconnection lines. The dipole
energy consists of core and step energies for each disconnection and an
elastic interaction energy [81, 82, 47, 37]:

$\begin{split}E_{\text{dipole}}&=2E_{\text{core}}+2E_{\text{step}}+E_{\text{elastic}}\\\
&=2E_{\text{monopole}}+E_{\text{elastic}}.\end{split}$ (3)

The monopole energy is thus the energy of a single, isolated disconnection.
The elastic interaction energy is [82]

$E_{\text{elastic}}=Kb^{2}\ln\left(\frac{\delta}{\delta_{c}}\right),$ (4)

where $K$ is the energy coefficient describing the anisotropic crystal
elasticity [82] and $\delta_{c}$ is the disconnection core size. Eqs. 3 and 4
can be simplified by a mathematical trick: we can “hide” the monopole energies
by defining $\delta_{0}=\delta_{c}\exp(-2E_{\text{monopole}}/(Kb^{2}))$ and
writing

$E_{\text{dipole}}=Kb^{2}\ln\left(\frac{\delta}{\delta_{0}}\right).$ (5)

The length $\delta_{0}$ is now an effective core size without direct physical
meaning, but Eq. 5 can be fitted directly to the dipole energies obtained by
molecular statics simulations without knowledge of $E_{\text{monopole}}$. With
periodic boundary conditions along the $x$ and $z$ directions in a bicrystal
simulation, the energy of pair of disconnections is given by [27, 44, 37]

$E_{\text{dipole}}=Kb^{2}\ln\left(\frac{\sin(\pi\delta/L_{x})}{\sin(\pi\delta_{0}/L_{x})}\right),$
(6)

taking into account the image interactions. When $L_{x}$ is infinite, Eq. 6
reduces to Eq. 5. In our simulations, we have to consider the periodic case
with a simulation box size of $$1114.220\text{\,}\AA$\times
L_{y}\times$6.261\text{\,}\AA$$. We varied $L_{y}$ to verify that there are no
size effects. Table 1 lists the GB energies

$\gamma=\frac{E_{1}-E_{\text{fcc}}}{L_{x}L_{z}}$ (7)

for the different box sizes. Here, $E_{1}$ is the energy of a system
containing a GB with a disconnection dipole and $E_{\text{fcc}}$ is the energy
of the same system without a GB. For simplicity, we report $\gamma^{*}$, which
is the value for a dipole width $\delta^{*}=L_{x}/2$. These values converge at
$L_{y}=$1929.540\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$, which are the
results we report here. We do not use the larger box sizes in the main text of
this paper, since we also performed NEB calculations on this system, which
have convergence issues at larger system sizes. More data is provided in Figs.
S3–S4 in the SM [80]. The resulting dipole energies are plotted in Fig. 6(a)
and we list the fit parameters $K$ and $\delta_{0}$ in Table 1. These
parameters are comparable to those in a previous study on a $\Sigma 13$ tilt
GB in copper [27] (see Appendix C for details). The parameter
$K_{\text{domino}}\approx$76.8\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\SIUnitSymbolAngstrom}^{3}$$
for domino and
$K_{\text{pearl}}\approx$76.4\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\SIUnitSymbolAngstrom}^{3}$$
for pearl is the same because it is a result of long-range interactions which
are solely due to the Burgers vector. Methods were developed to directly
calculate $K$ from the stiffness tensor of the crystal, while taking into
account the crystal anisotropy [83, 84, 85, 82]. Following the methods of
Eshelby et al. [83] and Stroh [85], we also obtained
$K=$76.7\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{\SIUnitSymbolAngstrom}^{3}$$,
matching the fitted value.

Furthermore, we also calculated $E_{\text{monopole}}$ by performing
simulations with a single disconnection and open boundaries in $x$ direction
(with a monopole, periodic boundaries are not possible). We found that the
absolute monopole energy does not converge even at large $L_{x}$ (see Table S1
in the SM [80]), likely due to elastic interactions of the disconnection with
the surfaces. The monopole energy difference between domino and pearl,
however, did converge. For
$L_{x}=$3342.799\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$, we obtain a
monopole energy of
$73.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}{\AA}$ for domino and
$70.7\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}{\AA}$ for pearl,
resulting in a difference of
$2.8\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}{\AA}$.

The energy of the disconnection dipole is maximum at a disconnection dipole
width $\delta^{*}=L_{x}/2$ is

$E_{\text{dipole}}^{*}\approx
Kb^{2}\log\left(\frac{L_{x}}{\pi\delta_{0}}\right)$ (8)

for periodic boundary conditions and the approximation is valid when
$\delta_{0}/L_{x}\ll 1$. We obtained $K$ and $\delta_{0}$ from atomistic
simulations by fitting Eq. 6, and hence $E^{*}_{\text{dipole}}$ can be
calculated for any given $L_{x}$ of the simulation cell. Together with the
difference in monopole energies, we can now obtain the ratio of true core
sizes by separating $\delta_{0}^{\text{domino}}/\delta_{0}^{\text{pearl}}$
back into $\delta_{c}^{\text{domino}}/\delta_{c}^{\text{pearl}}$ and $\Delta
E_{\text{monopole}}$ (cf. Eqs. 3–5; results in Table 1). The core size of the
disconnection in domino is 4–5% larger than in pearl.

All results indicate that it is easier to form disconnection dipoles in pearl
than in domino. Our earlier results, however, were that pearl needs a higher
critical stress for GB migration. It is clear that the formation energies
cannot explain our observations.

###  III.4  Disconnection migration barriers

Apart from their formation, disconnections also need to move in order to
facilitate GB migration. In analogy to bulk dislocations, there is also a
Peierls barrier $E_{\text{mig}}$ for disconnections, requiring a critical
Peierls–Nabarro stress $\tau_{\text{mig}}$ to move [86, 87, 81, 47, 44, 45].
The atomic configuration at the saddle point of this barrier is not stable and
can therefore not be explored with simple molecular statics calculations. The
minimum energy path for the migration of a disconnection along the GB,
extending the dipole width from $\delta_{1}$ to $\delta_{2}$, was calculated
using NEB. We built two simulation boxes with dipoles that differ in spacing
by a single GB unit cell (11.142 Å), which is the smallest distance between
metastable configurations. This can be repeated for different $\delta_{1}$ and
$\delta_{2}$ to obtain a continuous path as a function of $\delta$. Figure 6
shows the results of four NEB calculations superimposed on the results from
static calculations. It is clear that $E_{\text{mig}}$ is much higher for
pearl ($7.6\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\AA$ as compared
to $1.1\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\AA$ for domino),
leading to a steeper energy landscape during disconnection migration. These
values of $E_{\text{mig}}$ are of similar magnitude as for a different,
$\Sigma 13$ tilt GB in copper, which had a barrier of
$5.2\pm$0.4\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\AA$$ [27]. In our
case, the differences between domino and pearl might be explained by the
disconnection core size differences in domino and pearl (Table 1). Similar to
dislocations [87, 81, 88], the larger disconnection core size (spread out
core) in domino compared to pearl might correlate with the lower Peierls
barrier $E_{\text{mig}}$. This steepness of the energy landscape in pearl
results in its higher GB migration stresses, despite the lower disconnection
formation energies.

###  III.5  Temperature effects and critical stresses

Figure 7:  The critical shear stress $\tau_{c}$ as a function of temperature.
At temperatures of around $200\text{\,}\mathrm{K}$ and above, the critical
stress $\tau_{c}$ of pearl drops below the one for domino. Figure 8:  (a)–(c)
Shear stress–displacement curves for GBs with pre-existing disconnection
dipoles of width $\delta^{*}=L_{x}/2$. The shear stress response to
displacement at temperatures (a) 0 K, (b) 100 K, and (c) 200 K is plotted. The
critical stress $\tau_{\text{mig}}$ for disconnection migration is indicated
by the horizontal dotted lines. (d) It reduces with increasing temperature,
dropping to close to zero at $200\text{\,}\mathrm{K}$ and above for both
domino and pearl.

We proceeded to simulate the critical stress $\tau_{c}$ at finite temperatures
up to $T=$700\text{\,}\mathrm{K}$$. With the applied shear we found that the
metastable domino phase transitions to pearl immediately at
$700\text{\,}\mathrm{K}$ and above, so that meaningful stress values cannot be
extracted. (The domino complexion is metastable over the whole temperature
range [55], but the higher temperatures accelerate its transition to pearl
even within the short simulation timescale.) We restrict ourselves therefore
to the temperature range up to $600\text{\,}\mathrm{K}$. We averaged the
critical stress over several migration events (which are each equal to a
stress drop), each time recording the maximum of the stress curve. The results
are shown in Fig. 7 and are of similar magnitude as other GBs in fcc metals
(see Appendix C). Interestingly, the critical stress for pearl is only higher
than for domino up to a point between $100\text{\,}\mathrm{K}$ and
$200\text{\,}\mathrm{K}$. If we compare the Peierls barrier $E_{\text{mig}}$
to the absolute formation energy $E_{\text{dipole}}$, however, we notice that
the former is quite small (Fig. 6). Only due to its steepness, is it connected
to a high stress. It is conceivable that the thermal energy would be
sufficient to help overcome this small barrier, so that in the end only the
formation energy matters. We tested this by starting with systems that already
have a disconnection dipole of width $\delta^{*}=L_{x}/2$ inserted before
applying shear. We can thus probe only $\tau_{\text{mig}}$, the critical
stress for disconnection migration. Figure 8(a) shows that domino has
$\tau_{\text{mig}}\approx$0\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$, while
pearl has
$\tau_{\text{mig}}\approx$0.5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$. The
difference in $\tau_{c}$ (Fig. 4) is roughly
$0.27\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, which is smaller. That is not
surprising, since there is also a higher stress connected to the nucleation of
domino. With increasing $T$, however, the disconnection migration barrier can
be overcome more and more easily, resulting in
$\tau_{\text{mig}}\approx$0\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$ for both
domino and pearl already at $T=$200\text{\,}\mathrm{K}$$ [Fig. 8(b)–(d)]. This
can explain the temperature dependence: It is easier to nucleate
disconnections in the pearl complexion, but at low temperatures these
disconnections have to cross high barriers to move. These barriers, however,
are only high compared to domino and can be overcome with thermal energy. At
room temperature and above, pearl GBs are easier to migrate since the GB
migration is limited by defect nucleation.

##  IV  Conclusion

We used MD simulations to investigate the nucleation and migration of
disconnections as the elementary mechanisms of shear coupled motion in
$\Sigma$19b symmetric tilt GBs in Cu. In our clean GBs under peridodic
boundary conditions, the migration is a cycle of a disconnection dipole
nucleating, growing, and annihilating. These GBs exhibit two complexions,
domino and pearl, which migrate via the same disconnection mode
($\mathbf{b}=[0.586,0,0]\,\text{\AA}$,
$h=$0.677\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$). This mode corresponds to
the shortest possible DSC vector, which also lies in the GB plane, leading to
conservative motion of the GB. However, the critical shear stress $\tau_{c}$
required to move the GBs differs between the two complexions. At low
temperatures, we found by comparing the disconnection dipole formation and
migration energies of both the complexions, that the difference in shear
stress is predominantly due to the steeper Peierls-like barrier in pearl. This
is caused by the differences in atomic structure, in particular the core
structure of the disconnections. We observed that pearl becomes easier to move
than domino at around $200\text{\,}\mathrm{K}$ and above. By investigating the
stress required to move a pre-existing disconnection dipole, we found that
this stress becomes close to zero at these temperatures, indicating that the
GB migration barrier is then dominated by the dipole nucleation. We found that
disconnections in domino have higher formation energies, but very low
disconnection migration barriers. This explains why the critical shear stress
for pearl changes steeply from being much higher than for domino to lower than
domino between $0\text{\,}\mathrm{K}$ and $300\text{\,}\mathrm{K}$. Future
work will have to consider imperfect GBs and the interaction with dislocations
and nonconservative disconnections to get a full picture of GB migration.

###### Acknowledgements.

The authors thank Nicolas Combe and Anupam Neogi for helpful discussions. This
project has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 787446; GB-CORRELATE).

##  Appendix A  Disconnection modes from bicrystallography

Figure 9:  (a) Dichromatic pattern of the $\Sigma$19b $[11\overline{1}]$
$(178)$ symmetric tilt GB in the $xy$ plane. White and black symbols represent
the atoms from the two different grains. Circle, triangle, and square indicate
different layers along $z$. Four unique DSC vectors of low magnitude are
marked by arrows. The DSC vector $\mathbf{b}_{-15/-15}$ is marked with a red
arrow and lies within the (178) GB plane, while $\mathbf{b}_{-6/-9}$,
$\mathbf{b}_{-7/-8}$, and $\textbf{b}_{1/-1}$ have components normal to the GB
plane. (b) Pattern after moving the top crystal by $\mathbf{b}_{-15/-15}$. If
we imagine that the original GB plane passed through the coincidence site
marked by a gray circle, the new GB plane must also pass through a coincidence
site (red circle) and has thereby moved by $h$. The same is true for an
initial GB plane at any other position.

The possible disconnections modes for a given GB—consisting of a set of
Burgers vectors and step heights (b,h)—can be obtained from its
bicrystallography. For a certain b, there are multiple possible step heights
h. Figure 9 shows the dichromatic pattern of our $\Sigma$19b
$[11\overline{1}]$ $(178)$ GB. To describe b, the notation $\mathbf{b}_{j/k}$
is used [89], where $j$ and $k$ are integers indicating the ending ($j$) and
starting ($k$) plane of the vector. The planes are counted from a coincidence
site, as demonstrated for $\mathbf{b}_{1/-1}$ in Fig. 9, where the dashed
lines indicate the starting plane ($-1$) and the ending plane ($1$). The
integers $j$ and $k$ can be multiplied with the unit step height
$h_{0}=a/(2|178|)$ to obtain the absolute length of the component normal to
the GB. Four examples of Burgers vectors of low magnitude, i.e.,
$\textbf{b}_{-15/-15}$, $\textbf{b}_{-6/-9}$, $\textbf{b}_{-7/-8}$, and
$\textbf{b}_{1/-1}$, are marked in Fig. 9. There are six Burgers vectors in
total with the same length as $\textbf{b}_{-15/-15}$ and $\textbf{b}_{-6/-9}$,
which have a magnitude of $0.586\text{\,}\AA$. There are another six Burgers
vectors of the same length as $\textbf{b}_{-7/-8}$ and $\textbf{b}_{1/-1}$,
which shift the top grain position from a given $[11\overline{1}]$ plane to an
adjacent $[11\overline{1}]$ plane, which have a magnitude of
$2.114\text{\,}\AA$. The in-plane shifts move identical symbol types, while
$\mathbf{b}_{j/k}$ with out-of-plane component move any symbol on one of a
different type. The Burgers vector $\textbf{b}_{-15/-15}$ is the shortest
vector that lies in the GB plane (pointing in $\langle 532\rangle$ direction).
This is favorable for the mobility of the corresponding disconnection, because
its glide plane lies in the GB plane. The disconnections $\textbf{b}_{-6/-9}$,
$\textbf{b}_{-7/-8}$, and $\textbf{b}_{1/-1}$ have components normal to the GB
plane and the disconnection motion would require climb in addition to glide.
This can be understood by considering that the glide plane of the
disconnection would no longer be equal to the GB plane, but that the
disconnection can only exist on the GB plane. The resulting requirement of
climb involves diffusion of vacancies in and out of the disconnection and is
therefore slow. The disconnections $\textbf{b}_{-7/-8}$ and
$\textbf{b}_{1/-1}$ are longer than $\textbf{b}_{-15/-15}$ and
$\textbf{b}_{-6/-9}$ and likely have a higher core energy. It is thus likely
that the shear-coupled motion is driven by $\mathbf{b}_{-15/-15}$
disconnections.

We can get the step height caused by a Burgers vector from the the dichromatic
pattern. When the top crystal moves by $\mathbf{b}_{-15/-15}$, the original
coincidence site (gray circle) shifts to a new position (red circle) along
$\langle 178\rangle$. The difference in old and new position of coincidence
site along $\langle 178\rangle$ is the step height (GB migration distance) $h$
of $0.677\text{\,}\AA$, as shown in Fig. 9(b). This disconnection is thus
represented as disconnection mode
$(\mathbf{b},\textit{h})=([0.586,0,0]\,\text{\AA},$0.677\text{\,}\AA$)$. The
ratio of shear displacement (equal to $b_{x}$) to GB migration distance (equal
to $h$) is thus $\beta=b_{x}/h=0.865$. This corresponds to the value obtained
from the simulations in Sec. III.1. Together with $\mathbf{b}_{-15/-15}$ being
the shortest Burgers vector lying in the GB plane, this provides strong
evidence that this is the active disconnection mode during shear-coupled GB
migration in domino and pearl.

Figure 10:  Atom positions in (a) domino and (b) pearl before (black) and
after (red/blue) shear-coupled GB motion. These images are after $n=20$ unit
steps of GB migration, which correspond to a GB migration distance of
$n\mathit{h}=$13.540\text{\,}\AA$$ along $y$ and a shear displacement
$n|\mathbf{b}|=20d_{s}=$11.720\text{\,}\AA$$ along $x$. In the traversed
region, an image similar to the dichromatic pattern appears due to overlaying
atoms from before the migration, which belong to the top crystal, and after
the migration, which now belong to the bottom crystal. An additional offset
between the atoms in the pattern is due to the microscopic degrees of freedom,
i.e., the top and bottom crystal are shifted against each other depending on
the complexion. Furthermore, the pattern at the start and end of the migration
region is somewhat smeared out, indicating that the atomic jumps during the GB
migration do not necessarily go from the initial to the final position, but
can also occupy intermediary positions. Figure 11:  Displacements of the atoms
in the GB during a single GB migration step, plotted by (a) $u_{x}$ and
$u_{y}$ components and (b) $u_{x}$ and $u_{z}$ components. Here, we define GB
atoms as those atoms that were not identified as fcc either before or after
the migration event by the polyhedral template matching method [90] in OVITO
[73]. The displacements are symmetric around
$|\mathbf{b}|/2=$0.293\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$ in $x$
direction.

##  Appendix B  Atomic shuffling during GB migration

The perfect GBs in our simulations thus went through a cycle of nucleating,
propagating and annihilating the $\mathbf{b}_{-15/-15}$ disconnection. Since
we seem to observe the same disconnection mode for domino and pearl (same
$\beta$), the differences in shear stress required to move the disconnection
can also be due to the differences in the atomic shuffling during GB migration
for both the complexions [49, 91].

The atomic shuffling during GB migration for both the complexions was observed
after 20 shear stress drops in the molecular statics calculations
(corresponding to $n=20$ unit steps of GB migration). The mean GB plane moved
a total distance of $n\mathit{h}=$13.540\text{\,}\AA$$ along $y$ and the shear
displacement $20d_{s}=n|\mathbf{b}|=$11.720\text{\,}\AA$$ along $x$ for both
the complexions. In Fig. 10, the initial positions of the atoms in the top and
bottom grain before GB migration are plotted in black and their final
positions after migration are marked in red and blue for domino and pearl,
respectively. At first glance, we can see that the dichromatic pattern appears
in the traversed region. This is as expected, due to the rearrangement of
atoms in this regions from the crystallography of the top crystal (black,
before) to the one of the bottom crystal (red/blue, after) [92, 93]. On closer
inspection, it can be seen that there are no true coincidence sites, which is
a result of the microscopic degrees of freedom of the GB: The dichromatic
pattern is always plotted such that coincidence sites exist, but in reality
the top crystal can always be translated arbitrarily against the bottom
crystal [63, 94, 57]. Furthermore, a more complex pattern arises above/below
the traversed region. It appears that atoms do not directly jump from their
initial to their final position during one GB migration step.

Hence we analyzed the atomic displacements during a single step of GB
migration. A simulation cell of size $11.142\times
192.291\times$6.261\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{3}$$ ($1\times
10\times 1$ units cells,
$1137\text{\,}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}$) was used
for this. We only considered atoms that were not identified as fcc atoms
either before or after the GB migration step, utilizing the polyhedral
template matching structure identification method [90] as implemented in OVITO
[73]. The results are shown in Fig. 11. The jump vectors are symmetric around
$\lvert\mathbf{b}\rvert/2=$0.586\text{\,}\mathrm{\SIUnitSymbolAngstrom}$/2=$0.293\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$
in $x$ direction. The average of all jumps has to be
$\lvert\mathbf{b}\rvert/2$ because the displacement for atoms with $y$
coordinates below the GB has to be zero, while the displacement above the GB
has to be $\lvert\mathbf{b}\rvert$. The additional symmetry of the jump
vectors is due to the symmetry of the GB. The atomic displacements during a
single GB migration step do not correspond to DSC vectors. In our simulations,
atoms thus transition from one crystallite to the GB region and only then to
the second crystallite during GB migration. The non-DSC nature of the jump
vectors is due to the internal degrees of freedom for the atomic positions of
the domino and pearl complexions.

We furthermore probed the effort required to effect those jumps by calculating
the $L^{2}$-norm of a combined vector of the displacement components for the
GB atoms $i$ as $\sqrt{\sum_{i=1}^{n}{(x_{i}^{2}+y_{i}^{2}+z_{i}^{2})}}$. The
atomic jump lengths is evaluated to be
$3.018\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ for domino and
$2.877\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ for pearl. The difference in
jump lengths is small and seems to be unlikely to explain the differences in
$\tau_{c}$ for the two complexions. It is therefore necessary to calculate the
exact energy cost of introducing the disconnections as in Sec. III.3.

##  Appendix C  Shear coupled motion in other GBs of fcc metals

The energy of disconnections depends on both their core and step energy and
their elastic interaction energy, and is therefore best described by the
parameters $K$ and $\delta_{0}$ (see Sec. III.3). Here, $K$ encodes the
elasticity of the crystal lattice and $\delta_{0}$ the properties of the
disconnection core. Only a limited number of studies list such values, and for
copper GBs we just found Ref. [27], in which as $\Sigma 13$ $[001]$ $(320)$
symmetric tilt GB is simulated. They obtained a value of
$K=$30.468\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\AA^{3}$$, whereas
we found $K=$76.7\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\AA^{3}$$
(Sec. III.3). The difference could be a result of the anisotropy of copper and
the different GB planes. The paper reported
$E_{\text{monopole}}=$5.3\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\AA$$
and $\delta_{c}=$3.615\text{\,}\AA$$. The latter was chosen arbitrarily and we
therefore combined these values into $\delta_{0}=$2.556\text{\,}\AA$$ (see Eq.
5 and surrounding discussion), which is of the same order of magnitude as our
values of $\delta_{0}=3.8$–$4.4\text{\,}\AA$ (Table 1). The migration barrier
$E_{\text{mig}}$ was reported as
$5.2\pm$0.4\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\AA$$, which is in
the same range as our values of
$E_{\text{mig}}=1.1$–$7.6\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\mathrm{/}\AA$
(Sec. III.4).

Previously, the critical shear stress $\tau_{c}$ was calculated for various
GBs in fcc metals. Values lie in the range of
1–$4\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$. At $0\text{\,}\mathrm{K}$, shear
stress in $\Sigma 13$ $[001]$ $(320)$ and $\Sigma 17$ $[001]$ $(410)$
symmetric tilt Cu GBs is observed to be
$1.4\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ and
$2.1\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, respectively [27, 46]. Likewise,
shear stress in the $\Sigma 41$ $[001]$ $(540)$ Al GB is noted to be
$2.85\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ [95]. This is in the same range
as our $0\text{\,}\mathrm{K}$ values, which are
$1.117\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ for pearl and
$0.849\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$ for domino. Shear stress as a
function of temperature is reported for complexions in Cu $\Sigma 5$ $[001]$
$(210)$ GB [69]. At $500\text{\,}\mathrm{K}$, shear stress is observed to be
$\approx$0.95\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$ and
$\approx$0.58\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$ for split kite and
filled kite complexions, respectively. At the same temperature, the critical
shear stress in our domino and pearl complexions is
$\approx$0.5\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$ and
$\approx$0.4\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$, respectively. The shear
stress difference between the complexions can be due to the activation of
different disconnection modes in, e.g., split kites and filled kites contrary
to domino and pearl.

## References

  * Randle [2010] V. Randle, Grain boundary engineering: an overview after 25 years, Materials Science and Technology 26, 253 (2010).
  * Simpson  _et al._ [1971] C. J. Simpson, K. T. Aust, and W. C. Winegard, The four stages of grain growth, Metallurgical Transactions 2, 987 (1971).
  * Riontino  _et al._ [1979] G. Riontino, C. Antonione, L. Battezzati, F. Marino, and M. C. Tabasso, Kinetics of abnormal grain growth in pure iron, Journal of Materials Science 14, 86 (1979).
  * Swygenhoven  _et al._ [2001] H. V. Swygenhoven, A. Caro, and D. Farkas, Grain boundary structure and its influence on plastic deformation of polycrystalline FCC metals at the nanoscale: a molecular dynamics study, Scripta Materialia 44, 1513 (2001).
  * Shan  _et al._ [2004] Z. Shan, E. A. Stach, J. M. K. Wiezorek, J. A. Knapp, D. M. Follstaedt, and S. X. Mao, Grain boundary-mediated plasticity in nanocrystalline nickel, Science 305, 654 (2004).
  * Meyers  _et al._ [2006] M. Meyers, A. Mishra, and D. Benson, Mechanical properties of nanocrystalline materials, Progress in Materials Science 51, 427 (2006).
  * Lohmiller  _et al._ [2014] J. Lohmiller, M. Grewer, C. Braun, A. Kobler, C. Kübel, K. Schüler, V. Honkimäki, H. Hahn, O. Kraft, R. Birringer, and P. A. Gruber, Untangling dislocation and grain boundary mediated plasticity in nanocrystalline nickel, Acta Materialia 65, 295 (2014).
  * Swygenhoven and Derlet [2001] H. V. Swygenhoven and P. M. Derlet, Grain-boundary sliding in nanocrystalline fcc metals, Physical Review B 64, 10.1103/physrevb.64.224105 (2001).
  * Langdon [2006] T. G. Langdon, Grain boundary sliding revisited: Developments in sliding over four decades, Journal of Materials Science 41, 597 (2006).
  * Wang  _et al._ [2022] L. Wang, Y. Zhang, Z. Zeng, H. Zhou, J. He, P. Liu, M. Chen, J. Han, D. J. Srolovitz, J. Teng, Y. Guo, G. Yang, D. Kong, E. Ma, Y. Hu, B. Yin, X. Huang, Z. Zhang, T. Zhu, and X. Han, Tracking the sliding of grain boundaries at the atomic scale, Science 375, 1261 (2022).
  * Murayama  _et al._ [2002] M. Murayama, J. M. Howe, H. Hidaka, and S. Takaki, Atomic-level observation of disclination dipoles in mechanically milled, nanocrystalline fe, Science 295, 2433 (2002).
  * Cahn and Taylor [2004] J. W. Cahn and J. E. Taylor, A unified approach to motion of grain boundaries, relative tangential translation along grain boundaries, and grain rotation, Acta Materialia 52, 4887 (2004).
  * Upmanyu  _et al._ [2006] M. Upmanyu, D. Srolovitz, A. Lobkovsky, J. Warren, and W. Carter, Simultaneous grain boundary migration and grain rotation, Acta Materialia 54, 1707 (2006).
  * Wang  _et al._ [2014] L. Wang, J. Teng, P. Liu, A. Hirata, E. Ma, Z. Zhang, M. Chen, and X. Han, Grain rotation mediated by grain boundary dislocations in nanocrystalline platinum, Nature Communications 5, 10.1038/ncomms5402 (2014).
  * Thomas  _et al._ [2017] S. L. Thomas, K. Chen, J. Han, P. K. Purohit, and D. J. Srolovitz, Reconciling grain growth and shear-coupled grain boundary migration, Nature Communications 8, 10.1038/s41467-017-01889-3 (2017).
  * Winning  _et al._ [2001] M. Winning, G. Gottstein, and L. Shvindlerman, Stress induced grain boundary motion, Acta Materialia 49, 211 (2001).
  * Winning  _et al._ [2002] M. Winning, G. Gottstein, and L. Shvindlerman, On the mechanisms of grain boundary migration, Acta Materialia 50, 353 (2002).
  * Suzuki and Mishin [2005] A. Suzuki and Y. M. Mishin, Atomic mechanisms of grain boundary motion, Materials Science Forum 502, 157 (2005).
  * Cahn  _et al._ [2006] J. W. Cahn, Y. Mishin, and A. Suzuki, Coupling grain boundary motion to shear deformation, Acta Materialia 54, 4953 (2006).
  * Gianola  _et al._ [2006] D. Gianola, S. V. Petegem, M. Legros, S. Brandstetter, H. V. Swygenhoven, and K. Hemker, Stress-assisted discontinuous grain growth and its effect on the deformation behavior of nanocrystalline aluminum thin films, Acta Materialia 54, 2253 (2006).
  * Mishin  _et al._ [2007] Y. Mishin, A. Suzuki, B. P. Uberuaga, and A. F. Voter, Stick-slip behavior of grain boundaries studied by accelerated molecular dynamics, Physical Review B 75, 10.1103/physrevb.75.224101 (2007).
  * Ivanov and Mishin [2008] V. A. Ivanov and Y. Mishin, Dynamics of grain boundary motion coupled to shear deformation: An analytical model and its verification by molecular dynamics, Physical Review B 78, 10.1103/physrevb.78.064106 (2008).
  * Gorkaya  _et al._ [2009] T. Gorkaya, D. A. Molodov, and G. Gottstein, Stress-driven migration of symmetrical $\langle 100\rangle$ tilt grain boundaries in al bicrystals, Acta Materialia 57, 5396 (2009).
  * Caillard  _et al._ [2009] D. Caillard, F. Mompiou, and M. Legros, Grain-boundary shear-migration coupling. II. Geometrical model for general boundaries, Acta Materialia 57, 2390 (2009).
  * Mompiou  _et al._ [2009] F. Mompiou, D. Caillard, and M. Legros, Grain boundary shear–migration coupling—I. In situ TEM straining experiments in Al polycrystals, Acta Materialia 57, 2198 (2009).
  * Mompiou  _et al._ [2011] F. Mompiou, M. Legros, and D. Caillard, Direct observation and quantification of grain boundary shear-migration coupling in polycrystalline Al, Journal of Materials Science 46, 4308 (2011).
  * Rajabzadeh  _et al._ [2013a] A. Rajabzadeh, F. Mompiou, M. Legros, and N. Combe, Elementary mechanisms of shear-coupled grain boundary migration, Physical Review Letters 110, 10.1103/physrevlett.110.265507 (2013a).
  * Li and Soh [2013] J. Li and A. Soh, Toughening of nanocrystalline materials through shear-coupled migration of grain boundaries, Scripta Materialia 69, 283 (2013).
  * Homer  _et al._ [2013] E. R. Homer, S. M. Foiles, E. A. Holm, and D. L. Olmsted, Phenomenology of shear-coupled grain boundary motion in symmetric tilt and general grain boundaries, Acta Materialia 61, 1048 (2013).
  * Rajabzadeh  _et al._ [2014] A. Rajabzadeh, F. Mompiou, S. Lartigue-Korinek, N. Combe, M. Legros, and D. Molodov, The role of disconnections in deformation-coupled grain boundary migration, Acta Materialia 77, 223 (2014).
  * Zhu  _et al._ [2019] Q. Zhu, G. Cao, J. Wang, C. Deng, J. Li, Z. Zhang, and S. X. Mao, In situ atomistic observation of disconnection-mediated grain boundary migration, Nature Communications 10, 10.1038/s41467-018-08031-x (2019).
  * Wei  _et al._ [2021] J. Wei, B. Feng, R. Ishikawa, T. Yokoi, K. Matsunaga, N. Shibata, and Y. Ikuhara, Direct imaging of atomistic grain boundary migration, Nature Materials 20, 951 (2021).
  * Dao  _et al._ [2007] M. Dao, L. Lu, R. Asaro, J. De Hosson, and E. Ma, Toward a quantitative understanding of mechanical behavior of nanocrystalline metals, Acta Materialia 55, 4041 (2007).
  * Yamakov  _et al._ [2002] V. Yamakov, D. Wolf, S. R. Phillpot, A. K. Mukherjee, and H. Gleiter, Dislocation processes in the deformation of nanocrystalline aluminium by molecular-dynamics simulation, Nature Materials 1, 45 (2002).
  * Hirth and Pond [1996] J. Hirth and R. Pond, Steps, dislocations and disconnections as interface defects relating to structure and phase transformations, Acta Materialia 44, 4749 (1996).
  * Ashby [1972] M. Ashby, Boundary defects, and atomistic aspects of boundary sliding and diffusional creep, Surface Science 31, 498 (1972).
  * Han  _et al._ [2018] J. Han, S. L. Thomas, and D. J. Srolovitz, Grain-boundary kinetics: A unified approach, Progress in Materials Science 98, 386 (2018).
  * Chen  _et al._ [2019] K. Chen, J. Han, S. L. Thomas, and D. J. Srolovitz, Grain boundary shear coupling is not a grain boundary property, Acta Materialia 167, 241 (2019).
  * Bollmann [1970] W. Bollmann, General geometrical theory of crystalline interfaces, in  _Crystal Defects and Crystalline Interfaces_ (Springer Berlin Heidelberg, 1970) pp. 143–185.
  * Hirth and Balluffi [1973] J. Hirth and R. Balluffi, On grain boundary dislocations and ledges, Acta Metallurgica 21, 929 (1973).
  * Hirth  _et al._ [2006] J. Hirth, R. Pond, and J. Lothe, Disconnections in tilt walls, Acta Materialia 54, 4237 (2006).
  * Rajabzadeh  _et al._ [2013b] A. Rajabzadeh, M. Legros, N. Combe, F. Mompiou, and D. Molodov, Evidence of grain boundary dislocation step motion associated to shear-coupled grain boundary migration, Philosophical Magazine 93, 1299 (2013b).
  * Combe  _et al._ [2016] N. Combe, F. Mompiou, and M. Legros, Disconnections kinks and competing modes in shear-coupled grain boundary migration, Physical Review B 93, 10.1103/physrevb.93.024109 (2016).
  * MacKain  _et al._ [2017] O. MacKain, M. Cottura, D. Rodney, and E. Clouet, Atomic-scale modeling of twinning disconnections in zirconium, Physical Review B 95, 10.1103/physrevb.95.134102 (2017).
  * Chen  _et al._ [2020a] K. Chen, J. Han, and D. J. Srolovitz, On the temperature dependence of grain boundary mobility, Acta Materialia 194, 412 (2020a).
  * Combe  _et al._ [2021] N. Combe, F. Mompiou, and M. Legros, Multiple coupling modes to relax shear strain during grain boundary migration, Acta Materialia 218, 117222 (2021).
  * Khater  _et al._ [2012] H. Khater, A. Serra, R. Pond, and J. Hirth, The disconnection mechanism of coupled migration and shear at grain boundaries, Acta Materialia 60, 2007 (2012).
  * Chen  _et al._ [2020b] K. Chen, D. J. Srolovitz, and J. Han, Grain-boundary topological phase transitions, Proceedings of the National Academy of Sciences 117, 33077 (2020b).
  * Chesser  _et al._ [2021] I. Chesser, E. Holm, and B. Runnels, Optimal transportation of grain boundaries: A forward model for predicting migration mechanisms, Acta Materialia 210, 116823 (2021).
  * Frolov  _et al._ [2013] T. Frolov, D. L. Olmsted, M. Asta, and Y. Mishin, Structural phase transformations in metallic grain boundaries, Nature Communications 4, 10.1038/ncomms2919 (2013).
  * Aramfard and Deng [2018] M. Aramfard and C. Deng, Mechanically enhanced grain boundary structural phase transformation in Cu, Acta Mater. 146, 304 (2018).
  * Zhu  _et al._ [2018] Q. Zhu, A. Samanta, B. Li, R. E. Rudd, and T. Frolov, Predicting phase behavior of grain boundaries with evolutionary search and machine learning, Nature Communications 9, 10.1038/s41467-018-02937-2 (2018).
  * Frolov  _et al._ [2018] T. Frolov, Q. Zhu, T. Oppelstrup, J. Marian, and R. E. Rudd, Structures and transitions in bcc tungsten grain boundaries and their role in the absorption of point defects, Acta Mater. 159, 123 (2018).
  * Yang  _et al._ [2020] C. Yang, M. Zhang, and L. Qi, Grain boundary structure search by using an evolutionary algorithm with effective mutation methods, Comput. Mater. Sci. 184, 109812 (2020).
  * Meiners  _et al._ [2020] T. Meiners, T. Frolov, R. E. Rudd, G. Dehm, and C. H. Liebscher, Observations of grain-boundary phase transformations in an elemental metal, Nature 579, 375 (2020).
  * Langenohl  _et al._ [2022] L. Langenohl, T. Brink, R. Freitas, T. Frolov, G. Dehm, and C. H. Liebscher, Dual phase patterning during a congruent grain boundary phase transition in elemental copper, Nature Communications 13, 3331 (2022).
  * Brink  _et al._ [2023] T. Brink, L. Langenohl, H. Bishara, and G. Dehm, Universality of grain boundary phases in fcc metals: Case study on high-angle [111] symmetric tilt grain boundaries, Physical Review B 107, 054103 (2023).
  * Gibbs [1948] J. W. Gibbs,  _The Collected Works of J. Willard Gibbs, Volume 1: Thermodynamics_ (Yale University Press, New Haven, CT, USA, 1948).
  * Hart [1968] E. W. Hart, Two-dimensional phase transformation in grain boundaries, Scr. Metall. 2, 179 (1968).
  * Cahn [1982] J. W. Cahn, Transitions and phase equilibria among grain boundary structures, Le Journal de Physique Colloques 43, C6 (1982).
  * Rottman [1988] C. Rottman, Theory of phase transitions at internal interfaces, J. Phys. Colloques 49, C5 (1988).
  * Frolov and Mishin [2012a] T. Frolov and Y. Mishin, Thermodynamics of coherent interfaces under mechanical stresses. I. Theory, Physical Review B 85, 10.1103/physrevb.85.224106 (2012a).
  * Frolov and Mishin [2012b] T. Frolov and Y. Mishin, Thermodynamics of coherent interfaces under mechanical stresses. II. Application to atomistic simulation of grain boundaries, Physical Review B 85, 10.1103/physrevb.85.224107 (2012b).
  * Tang  _et al._ [2006] M. Tang, W. C. Carter, and R. M. Cannon, Diffuse interface model for structural transitions of grain boundaries, Physical Review B 73, 10.1103/physrevb.73.024102 (2006).
  * Dillon  _et al._ [2007] S. J. Dillon, M. Tang, W. C. Carter, and M. P. Harmer, Complexion: A new concept for kinetic engineering in materials science, Acta Materialia 55, 6208 (2007).
  * Cantwell  _et al._ [2014] P. R. Cantwell, M. Tang, S. J. Dillon, J. Luo, G. S. Rohrer, and M. P. Harmer, Grain boundary complexions, Acta Materialia 62, 1 (2014).
  * Cantwell  _et al._ [2020] P. R. Cantwell, T. Frolov, T. J. Rupert, A. R. Krause, C. J. Marvel, G. S. Rohrer, J. M. Rickman, and M. P. Harmer, Grain boundary complexion transitions, Annual Review of Materials Research 50, 465 (2020).
  * Frolov and Mishin [2015] T. Frolov and Y. Mishin, Phases, phase equilibria, and phase rules in low-dimensional systems, The Journal of Chemical Physics 143, 044706 (2015).
  * Frolov [2014] T. Frolov, Effect of interfacial structural phase transitions on the coupled motion of grain boundaries: A molecular dynamics study, Applied Physics Letters 104, 211905 (2014).
  * Plimpton [1995] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, Journal of Computational Physics 117, 1 (1995).
  * Thompson  _et al._ [2022] A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. in 't Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton, LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales, Computer Physics Communications 271, 108171 (2022).
  * Mishin  _et al._ [2001] Y. Mishin, M. J. Mehl, D. A. Papaconstantopoulos, A. F. Voter, and J. D. Kress, Structural stability and lattice defects in copper:$ab$ $initio$, tight-binding, and embedded-atom calculations, Physical Review B 63, 10.1103/physrevb.63.224106 (2001).
  * Stukowski [2009] A. Stukowski, Visualization and analysis of atomistic simulation data with OVITO–the open visualization tool, Modelling and Simulation in Materials Science and Engineering 18, 015012 (2009).
  * Winter  _et al._ [2022a] I. Winter, T. Oppelstrup, T. Frolov, and R. Rudd, Characterization and visualization of grain boundary disconnections, Acta Materialia 237, 118067 (2022a).
  * Winter  _et al._ [2022b] I. Winter, R. Rudd, T. Oppelstrup, and T. Frolov, Nucleation of grain boundary phases, Physical Review Letters 128, 10.1103/physrevlett.128.035701 (2022b).
  * Henkelman and Jónsson [2000] G. Henkelman and H. Jónsson, Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points, The Journal of Chemical Physics 113, 9978 (2000).
  * Henkelman  _et al._ [2000] G. Henkelman, B. P. Uberuaga, and H. Jónsson, A climbing image nudged elastic band method for finding saddle points and minimum energy paths, The Journal of Chemical Physics 113, 9901 (2000).
  * Sheppard  _et al._ [2008] D. Sheppard, R. Terrell, and G. Henkelman, Optimization methods for finding minimum energy paths, The Journal of Chemical Physics 128, 134106 (2008).
  * Pemma  _et al._ [2023] S. Pemma, R. Janisch, G. Dehm, and T. Brink, Dataset for “Elementary mechanisms of shear-coupled grain boundary migration for different complexions of a copper tilt grain boundary”, Zenodo (2023).
  * [80] See Supplemental Material at [link to be inserted by publisher] for additional data.
  * Nabarro [1952] F. Nabarro, Mathematical theory of stationary dislocations, Advances in Physics 1, 269 (1952).
  * Hirth  _et al._ [1983] J. P. Hirth, J. Lothe, and T. Mura, Theory of dislocations (2nd ed.), Journal of Applied Mechanics 50, 476 (1983).
  * Eshelby  _et al._ [1953] J. Eshelby, W. Read, and W. Shockley, Anisotropic elasticity with applications to dislocation theory, Acta Metallurgica 1, 251 (1953).
  * Foreman [1955] A. Foreman, Dislocation energies in anisotropic crystals, Acta Metallurgica 3, 322 (1955).
  * Stroh [1958] A. N. Stroh, Dislocations and cracks in anisotropic elasticity, Philosophical Magazine 3, 625 (1958).
  * Peierls [1940] R. Peierls, The size of a dislocation, Proc. Phys. Soc. 52, 34 (1940).
  * Nabarro [1947] F. R. N. Nabarro, Dislocations in a simple cubic lattice, Proc Phys Soc 59, 256 (1947).
  * Courtney [2000] T. H. Courtney,  _Mechanical Behavior of Materials_, McGraw-Hill series in materials science and engineering (McGraw-Hill, 2000).
  * Serra and Bacon [1996] A. Serra and D. J. Bacon, A new model for 1012 twin growth in hcp metals, Philosophical Magazine A 73, 333 (1996).
  * Larsen  _et al._ [2016] P. M. Larsen, S. Schmidt, and J. Schiøtz, Robust structural identification via polyhedral template matching, Modelling and Simulation in Materials Science and Engineering 24, 055007 (2016).
  * Chesser  _et al._ [2022] I. Chesser, B. Runnels, and E. Holm, A taxonomy of grain boundary migration mechanisms via displacement texture characterization, Acta Materialia 222, 117425 (2022).
  * Zhang and Srolovitz [2006] H. Zhang and D. J. Srolovitz, Simulation and analysis of the migration mechanism of $\sigma$5 tilt grain boundaries in an fcc metal, Acta Materialia 54, 623 (2006).
  * Race  _et al._ [2015] C. P. Race, R. Hadian, J. von Pezold, B. Grabowski, and J. Neugebauer, Mechanisms and kinetics of the migration of grain boundaries containing extended defects, Physical Review B 92, 10.1103/physrevb.92.174115 (2015).
  * Frolov  _et al._ [2021] T. Frolov, D. L. Medlin, and M. Asta, Dislocation content of grain boundary phase junctions and its relation to grain boundary excess properties, Physical Review B 103, 10.1103/physrevb.103.184108 (2021).
  * Larranaga  _et al._ [2020] M. Larranaga, F. Mompiou, M. Legros, and N. Combe, Role of sessile disconnection dipoles in shear-coupled grain boundary migration, Physical Review Materials 4, 10.1103/physrevmaterials.4.123606 (2020).