# U3E: Unsupervised and Erasure-based Evidence Extraction for Machine Reading
Comprehension

1stSuzhe He  Beijing Institute of Technology
Beijing, China
<EMAIL_ADDRESS>2ndShumin Shi  Beijing Institute of Technology
Beijing, China
<EMAIL_ADDRESS>3ndChengHao Wu  Beijing, China
<EMAIL_ADDRESS>
###### Abstract

More tasks in Machine Reading Comprehension (MRC) require, in addition to
answer prediction, the extraction of evidence sentences that support the
answer. However, the annotation of supporting evidence sentences is usually
time-consuming and labor-intensive. In this paper, to address this issue and
considering that most of the existing extraction methods are semi-supervised,
we propose an unsupervised evidence extraction method (U3E). U3E takes the
changes after sentence-level feature erasure in the document as input,
simulating the decline in problem-solving ability caused by human memory
decline. In order to make selections on the basis of fully understanding the
semantics of the original text, we also propose metrics to quickly select the
optimal memory model for this input changes. To compare U3E with typical
evidence extraction methods and investigate its effectiveness in evidence
extraction, we conduct experiments on different datasets. Experimental results
show that U3E is simple but effective, not only extracting evidence more
accurately, but also significantly improving model performance.

###### Index Terms:

Machine Reading Comprehension, evidence extraction, unsupervised learning,
feature erasure

##  I Introduction

MRC has attracted more and more attention, and it can be roughly divided into
two categories: extractive and non-extractive. Extractive MRC requires one or
more paragraphs of text to be selected as the answer, such as SQuAD[1] and
DROP[2]. Non-extractive MRC need reasoning ability more than the former. It
includes multiple-choice MRC[3, 4, 5], unanswerable questions[6, 7],
verification MRC[8, 9]. As shown in table I, there is no doubt that evidence
sentences can not only help the MRC model to predict the correct answer, but
also help improve the interpretability of the model. [36] effectively fuse the
extracted evidence in reasoning to enhance the power of relation extraction.
[37] aIteratively performing long document segmentation enhances text
generation performance.

However, it is difficult to label evidence sentences on large-scale datasets,
especially for non-extractive MRC. Because a large number of these questions
are not just extractive (e.g. 87% of questions in RACE([24], [25])). Answering
Such questions may require more advanced reading comprehension skills such as
single-sentence or multi-sentence reasoning skills ([26], [27]).

Considering the high cost of human-labeling evidence sentences, some recent
efforts have been devoted to improving MRC by exploiting noisy evidence labels
when training evidence extractors. Some work([10, 11]) generates remote tags
using handcrafted rules and external sources, [23] apply remote supervision to
generate labels, denoising using a deep probabilistic logical learning
framework. Some studies ([29]) employ reinforcement learning (RL) to determine
the labels of evidence, but this RL approach suffers from training
instability. More distant monitoring techniques are also used to refine noisy
labels. However, improving the evidence extractor is still challenging when
golden evidence labels are not available.

So some recent works focus on weakly supervised extraction of evidence
sentences, [12] uses a small number of evidence annotations combined with a
large number of document-level labels to select evidence, and[13] uses a self-
training approach that uses automatically generated evidence labels to
supervise evidence selection in an iterative process.

TABLE I: Examples of Verified Reading Comprehension. Bold parts are evidence
sentences. Answer 0 means the document does not support the option, 1 means it
does. O: It is legal to own an ar15 in california.
---
D: … State of California. California’s 2000 Assault Weapons ban
went further and banned AR-15s made by …
A: 0
O: Ferguson jenkins is in the hall of fame.
D: … following his major league career. In 1991, Jenkins became
the first Canadian to be inducted into the National Baseball
Hall of Fame.
A: 1

In this paper, we propose an unsupervised method U3E for selecting evidence
that is more in line with human intuition. Inspired by[14], they change the
characteristics of paragraph input (called erasure) by penalizing illogical
predictions to improve the model.

Instead of the loss value used in their work, we choose to use the change in
the predicted value to reflect this memory-decay-like behavior because the
predicted value is more sensitive. We also innovatively apply it to sentence
level and the task of evidence extraction in MRC. Besides, in order to better
adapt to this task, we propose an optimal memory model selection method BMC
(balance model and changes). Experiments show that our method achieves good
results.

##  II Related Work

###  II-A Attribution Interpretation Methods

In the field of hindsight, there are ”variable importance” methods and
gradient-based methods. The ”variable importance” method[15, 16] refers to the
difference in the prediction performance of a model when the value of a
variable changes. In gradient-based methods, the magnitude of the gradient is
used as the feature importance score. Gradient-based methods are suitable for
differentiable models [33]. Erasure[14] as a ”variable importance” method, it
is model independent. The advantage of the erasure method is that it is
conceptually simple and can be optimized for well-defined objectives[34].

###  II-B Evidence Extraction

In the early MRC, some works focus on better representation of the features of
the question and the context[17], and constantly explore better fusion
matching between them[18, 4]. With the emergence of pre-training models (such
as: BERT[19]), some works want to understand the basis for the model to
predict the answer. Extracting evidences in MRC is attracting increasing
attention, although still quite challenging.

Evidence extraction aims to find evidence and relevant information for
downstream processes in the task, which arguably improves the interpretability
of the task. Evidence extraction is useful which is intuitive and becomes an
important part of fact verification ([38], [39]), multiple-choice reading
comprehension ([40]), open-domain question answering ([10]), multi-hop reading
comprehension ([41]) , natural language inference ([42]), and a wide range of
other tasks ([43]).

which can be roughly divided into two categories: one is supervised learning,
which requires a lot of resources to manually label all the evidence sentence
labels, such as: HOTPOTQA[20] Select evidence sentences on the basis of asking
to answer their specific tasks, and work on it[21] iteratively sorts the
importance of sentences to select evidence sentences, [22] decomposes the
question and becomes a single-hop MRC extracts the evidence sentence while
selecting the answer.

The second is semi-supervised learning. Because it is difficult to extract
evidence sentences in non-extractive MRC, some works use semi-supervised
methods to extract evidence, [23] use remote supervision to generate imperfect
labels, and then use deep probabilistic logic learning to remove noise. [12]
label and improve model performance by combining specific tasks with weakly
supervised evidence extraction. Finally, on the basis of weakly supervised
learning, [28] use reinforcement learning to obtain better evidence extraction
strategies.

Our method U3E tends to use an unsupervised method, complete the extraction
task in stages. Adjust the erasure method so that the importance of sentences
can be obtained explicitly. Compared with other methods, not only the cost is
small, but also the effect is remarkable.

##  III Method

The overall architecture of U3E is shown in figure 1, which consists of three
stages:
Train and Acquire (T&A): train models according to the specific task and
achieve changes.
Select and Reacquire (S&R): select the optimal memory model according to our
proposed BMC method and use the model to reacquire changes.
Apply and Retrain (A&R): extract evidence through changes and retrain
according to the evidence.
In the following specific implementation, we will explain these stages in
order.

Figure 1: The overall structure of U3E, which includes three stages: T&A, S&R,
and A&R.

###  III-A Task Definition

Assuming that each sample of the dataset can be formalized as follows: Given a
reference document consisting of multiple sentences
$D=\left\\{S_{1},S_{2},...,S_{m}\right\\}$ and a statement $O$(If there is a
question, then $O$ is represented as the concatenation of the question and the
candidate) . The model should determine whether to support this statement
according to the document, the support is marked as 1, otherwise it is marked
as 0. It can also use to extract the evidence sentence set
$E=\left\\{S_{j},S_{j+1},...,S_{j+k-1}\right\\}$, which contains $k(<m)$
sentences in $D$.

###  III-B Train and Acquire

####  III-B1 Task-specific Training

We first train according to the specific task (here is the classified task),
and then save the model $M=\\{M^{1},M^{2},...,M^{x}\\}$ under all epochs,
where x represents the largest epoch trained. The model structure during
training is pretrained model111different pretrained models used on different
datasets and linear layer. The input is in ”
$[CLS]+Option+[SEP]+Document+[SEP]$ ” format. The hidden representation of the
$[CLS]$ token goes through a linear layer for binary classification to predict
whether the document $D$ supports the sentence $O$:

Figure 2: Implementation details for achieving changes.

$\hat{y}=softmax(W_{p}h_{cls})$ (1)

where $W_{p}$ is the model parameter and $h_{cls}$ is the hidden
representation of $[CLS]$ token . The loss function is the cross entropy loss:

$L=-\sum_{i=1}^{n}(y_{i}\cdot log(\hat{y}_{i})+(1-y_{i})\cdot
log(1-\hat{y}_{i}))\quad j\in[1,N]$ (2)

where $y_{i}$ represents the label of the sample $(O_{i},D_{i})$, $\hat{y}$
represents predicted value of the model, $N$ is the total number of samples.

####  III-B2 Change Acquisition

In this part, we use the leave-one-out method[15] to simulate memory decay and
obtain the importance of each sentence. As shown in figure 2, $M^{l}$ is the
model after the $l$th epoch training is completed, $R$ represents the binary
classification222without softmax result predicted by model $M^{l}$, where
$R_{0}$ means the probability value of that document $D$ does not support
statement $O$, otherwise, $R_{1}$ means support, and $R_{1}$ marked in red
means that the correct answer for this example is support.

$D^{-j}$ represents the new document obtained after erasing the j-th sentence,
and the same $R^{-j}$ represents the result obtained after replacing $D^{-j}$
with $D$ and then input the model $M^{l}$ in the same format. The importance
of the sentence $C=\left\\{c_{1},c_{2},...,c_{m}\right\\}$ is calculated as
follows:

$c_{j}=abs(R_{y}-R_{y}^{-j})\quad j\in[1,m]$ (3)

Where $y$ represents the sample label, $abs$ represents the absolute value
function and $c_{j}$ indicates the importance of sentence $j$.

We chose this method because the large positive effect on the result after
erasure also means that it is most relevant to the statement $O$. In other
words, it is also a evidence, but it is an interference item, which can help
the model improve its reasoning ability. More importantly, the predicted value
changes can be more sensitive to the input changes.

###  III-C Select and Reacquire

####  III-C1 Optimal Model Selection

We have experimented with two optimal model selection methods as follows:

##### MTEST

This method considers the generalization of the model, and specifically
selects the model according to the maximum accuracy on the test set.

##### BMC

According to research by [30], neural reading comprehension models often
exploiting spurious correlations in the data to predict answers. In order to
prevent the model from biasing evidence sentences selection due to excessive
pursuit of specific tasks, we propose an optimal model $M_{0}$ selection
expression BMC (balance model and changes).

The core of BMC method is to select the model that is most sensitive to the
evidence sentence. So the BMC expression is as follows:

$M_{0}=\mathop{argmax}\limits_{M^{l}}(-\lambda\cdot Acc(M^{l})+SC^{l})\quad
M^{l}\in M$ (4)

$SC^{l}=\frac{1}{N}\cdot\sum_{i}^{N}{\frac{\sum C_{\sim k}(M^{l})}{\sum
C(M^{l})}}$ (5)

Where $M^{l}$ is the model after the l-th epoch is trained, $Acc$ represents
the accuracy on the testing set and $\lambda$ is a hyperparameter. $k$ is the
number of evidence sentences to be selected.

TABLE II: Results in VGaokao. | Respective Acc. |  Acc.
---|---|---
Full context | $\bm{-}$ |  64.46
Top 2 sentences by word vector | $\bm{-}$ |  63.79
Top 2 sentences by beam search | $\bm{-}$ |  63.63
Top 2 sentences by U3Emtest | 69.91 / 64.46 |  63.93
Top 2 sentences by U3EBMC | 99.64 / 64.18 |  64.54
Top 2 sentences by U3EMAX | 99.51 / 62.32 |  65.96

$\sum C_{\sim k}(M^{l})$ is the sum of the largest $k$ values in $C$, $\sum
C(M^{l})$ is the sum of all the values in $C$, the division of both represents
the sensitivity of the model to the evidence sentence, we call it salient
changes (SC).

Note that our model selection can be done simultaneously with training, so
when training is over, the optimal model has already been selected.

####  III-C2 Changes Reacquisition

The steps to reacquire the changes are similar to III-B2, except that the
model $M^{l}$ is replaced with the optimal model $M_{0}$. The final optimal
changes are $C_{0}$.

###  III-D Apply and Retrain

We choose the $k$ sentences with the largest $c_{j}$ in $C_{0}$ as the
evidence sentence set $E$ of statement $O$. Then under the same model
structure, the documents are replaced with ordered evidence
sentences333consistent with the relative order of the original sentences for
retraining.

This part is used to study whether the extracted evidence can improve the
performance of the model(IV-C), and is not needed in the evidence validity
part(IV-D).

##  IV Experiments and Analysis

###  IV-A Dataset

VGaokao[9] is a validation dataset, which comes from the Chinese native
speaker’s Gaokao Chinese test and is a public dataset. It includes documents
and statements, each statement requires at most two evidence sentences, and
the answer yes/no directly indicates whether the document supports the
statement.

C3[31] is a multiple-choice reading comprehension dataset. Here we use the C3
dataset444https://ymcui.com/expmrc/ proposed in [32], which is a competition
dataset. It is necessary to construct pseudo-evidence labels for the training
set, while the validation set has manually annotated evidence labels.

We use VGaokao to study the effect of generalization and the improvement of
model performance, and C3 to illustrate the accuracy of extracting evidence,
since it has annotated test data.

###  IV-B Baseline

####  IV-B1 Model structure

Both use the method of ”pre-training model + linear layer”, the difference is
that the latter has two additional linear layers for evidence span extraction,
they are used to predict where evidence begins and ends. The loss functions
are all cross entropy loss. The former all-process pre-trained model uses
chinese-roberta-wwm-ext, and the latter uses chinese-macbert-
base.555https://huggingface.co/ Other differences in the C3 are introduced in
IV-D1.

####  IV-B2 Implementation

For dataset VGaokao, in order to obtain a more accurate experimental
conclusion, during training, we select multiple most relevant sentences for
each statement according to the static word
vector666https://github.com/Embedding/Chinese-Word-Vectors , ensuring that the
length after splicing does not exceed the length limit of the pre-training
model(average sentence length after operation is 7.793). Then the new document
is formed after sorting according to the original position. During testing, we
forecast the original document in blocks(step 128). Later, the subsequent
experiments were conducted on the new document, and the tests were the same.

We use the following baselines.
-Word Vector: select the most relevant $k$ sentences from document $D$ as evidence sentences for retraining based on word vectors6. Here we use average pooling as sentence representation.
-Beam Search: beam search using the Hard Masking method proposed by [9], they delete the previously selected sentence information in the query, and iteratively select the evidence sentence.
-Full Context Train: in order to study the improvement of model performance, we use the original document $D$(new document after filtering) for training.

For dataset C3, in order to exclude interference, we delete the data that is
too long in the training set provided by [35]. The model structure used is
consistent with the baseline proposed in [32]. Since the test set is not
public, we evaluate it on the validation set. Here, only one sentence is
required to be extracted as a proof sentence.

###  IV-C Performance improvement

TABLE III: Results in C3. Type of PrM | Method | ANS_F1 | EVI_F1 | ALL_F1
---|---|---|---|---
Base | RAW | 72.245 | $\bm{-}$ | $\bm{-}$
WV | 71.881 | 65.533 | 49.634
| U3Emtest | 72.475 | 68.840 | 53.385
Large | RAW | 77.425 | $\bm{-}$ | $\bm{-}$
WV | 76.832 | 68.049 | 55.355
| U3Emtest | 78.812 | 76.425 | 62.720

####  IV-C1 Implementation

The results are shown in table II. Except in Full data, Acc. represents the
test set accuracy on the original data. In other experiments, Acc. represents
the accuracy on the test set during retraining after evidence extractionIII-D.
Acc. for all experiments is obtained under the same experimental settings. The
Respective Acc. represents the accuracy of different methods of selecting the
optimal model on the two dataset.

The model with the highest accuracy on the test set is selected for evidence
extraction and retraining, and the result is subscripted by ’mtest’. For BMC,
the hyperparameter $\lambda$ is set to 0.1, and the result is subscripted by
’BMC’.

Then we also recorded the optimal result in 10 epoch and the test set accuracy
at this time, which is subscripted by ’MAX’.

####  IV-C2 Result Analysis

In general, the accuracy rate of using the U3E method after retraining is
stronger than that of the general methods, indicating that U3E does have a
better effect on the task of extracting key sentences. And U3EBMC and U3EMAX
are higher than Full context, especially U3EMAX is more than 1 point higher,
which also means that our U3E method helps to improve the performance of the
model.

U3EBMC higher than U3Emtest by 0.61 indicates that the BMC optimal model
selection method is more effective than simply using the accuracy of the test
set. This also suggests that excessive pursuit of task-specific accuracy may
reduce the probability that the model predicts the answer from the evidence
sentence. This is consistent with the study of [30]. However, U3EBMC is still
more than 1 point higher than the best effect U3EMAX, indicating that the BMC
method still has room for improvement.

We observe that both U3EBMC and U3EMAX have more than 90 accuracy on the
training set, indicating that the U3E method needs to have sufficient
knowledge of the training set.

###  IV-D Evidence Validity

To further demonstrate the accuracy of our method for extracting evidence, we
conduct experiments on dataset C3.

####  IV-D1 Implementation

Considering that the candidate answers of the c dataset are not long, we do
not need to perform evidence extraction for each candidate, so we do the
splicing of ”document, question and candidate” for each candidate, and then
enter the model together.

Considering that there are two tasks, we design the total loss function as
follows:

$Loss_{total}=Loss_{ans}+\alpha\cdot Loss_{evi}\quad\alpha\in[0,1]$ (6)

Where $Loss_{ans}$ is the loss of multiple answers, and $Loss_{evi}$ is the
loss of evidence span. We record the maximum experimental results every 0.1
from 0 to 1.

In this part, we use U3E’s MTEST method for evidence extraction, which is to
extract evidence according to the best model on the validation set. Note that
here we only extract evidence for training set.

As shown in Table III, RAW represents the answer prediction accuracy without
evidence extraction task. WV means using static word vectors to extract
evidence, and U3Emtest means using U3E method to extract evidence, both of
which select evidence sentences according to the correct answers and
questions. ANS_F1 is the accuracy of multiple-choice answers, EVI_F1 is the
accuracy of evidence, and ALL_F1 is the accuracy of the combination of the
two. The evaluation method is provided by [32].

####  IV-D2 Result Analysis

As shown in Table III, the evidence obtained using our method greatly improves
the upper limit of the evidence accuracy rate, from 65.533 to 68.840, and
simultaneous training can help improve the accuracy of multiple-choice
answers, from 72.245 to 72.475. Moreover, according to the above research,
method NTtest does not use the optimal model, and a greater improvement is
expected.

Moreover, when the pre-training model is larger777
https://huggingface.co/hfl/chinese-macbert-large, we observe more significant
changes. U3E not only improves the evidence prediction by more than 8 points,
but also improves the multi choice answer prediction by more than 1 point
compared with the original training data.

TABLE IV: Evidence extraction results. WV is the static word vector,and
U3E is the U3E method. The following is the Chinese translation. Query: What
might their relationship be?
---
WV:  Woman: Yes, it’s been more than ten years since we saw each other.
U3E: Man: But you haven’t changed at all, just like when you were in college.
Query: How do animals behave when they sense low-frequency sound waves?
WV:  This friction creates a low-frequency sound wave that is lower than what
the human hearing can perceive.
U3E:  Those animals with very sensitive senses will panic when they feel this
low sound wave, and even …
Query: What kind of teacher did Zhang Liyong become?
WV:  Zhang Liyong did not become a teaching assistant in the English
Department of Tsinghua University, but many people asked him to be a teacher.
U3E:  He is an English tutor for a third-year student.

We also compare the specific effects of using word vectors and U3E, as shown
in Table IV, the former tends to select more similar sentences, which not only
fail to answer the question, but also mislead the model. The latter makes full
use of the favorable conditions of the model ”understanding the semantics of
the original text” and selects the sentence, which has the largest change in
the confidence of the correct answer after the memory declines, as the
evidence sentence. It’s more intuitive and more accurate.

##  V Conclusion

We propose an erasure-based method called U3E for unsupervised evidence
extraction. U3E build a model based on the reduced ability to process problems
due to memory loss in humans. In order to select the optimal memory model
quickly, a calculation method called BMC is proposed. The experimental results
show that U3E not only accurately extracts evidence, but also improves model
performance.

Next, we are going to mine a more precise relationship between model accuracy
and changes to quickly complete the selection of the optimal model, and apply
the U3E to other NLP tasks such as natural language inference.

## References

  * [1] P. Rajpurkar, J. Zhang, K. Lopyrev, and P. Liang, “Squad: 100,000+ questions for machine comprehension of text,” _arXiv preprint arXiv:1606.05250_ , 2016\.
  * [2] D. Dua, Y. Wang, P. Dasigi, G. Stanovsky, S. Singh, and M. Gardner, “Drop: A reading comprehension benchmark requiring discrete reasoning over paragraphs,” _arXiv preprint arXiv:1903.00161_ , 2019.
  * [3] B. Wang, T. Yao, Q. Zhang, J. Xu, and X. Wang, “Reco: A large scale chinese reading comprehension dataset on opinion,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 34, no. 05, 2020, pp. 9146–9153.
  * [4] Q. Ran, P. Li, W. Hu, and J. Zhou, “Option comparison network for multiple-choice reading comprehension,” _arXiv preprint arXiv:1903.03033_ , 2019.
  * [5] P. Zhu, Z. Zhang, H. Zhao, and X. Li, “Duma: Reading comprehension with transposition thinking,” _IEEE/ACM Transactions on Audio, Speech, and Language Processing_ , vol. 30, pp. 269–279, 2021.
  * [6] P. Rajpurkar, R. Jia, and P. Liang, “Know what you don’t know: Unanswerable questions for squad,” _arXiv preprint arXiv:1806.03822_ , 2018.
  * [7] Z. Zhang, J. Yang, and H. Zhao, “Retrospective reader for machine reading comprehension,” _arXiv preprint arXiv:2001.09694_ , vol. 1, pp. 1–9, 2020\.
  * [8] C. Clark, K. Lee, M.-W. Chang, T. Kwiatkowski, M. Collins, and K. Toutanova, “Boolq: Exploring the surprising difficulty of natural yes/no questions,” _arXiv preprint arXiv:1905.10044_ , 2019.
  * [9] C. Zhang, Y. Lai, Y. Feng, and D. Zhao, “Extract, integrate, compete: Towards verification style reading comprehension,” _arXiv preprint arXiv:2109.05149_ , 2021.
  * [10] Y. Lin, H. Ji, Z. Liu, and M. Sun, “Denoising distantly supervised open-domain question answering,” in _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , 2018, pp. 1736–1745.
  * [11] S. Min, V. Zhong, R. Socher, and C. Xiong, “Efficient and robust question answering from minimal context over documents,” _arXiv preprint arXiv:1805.08092_ , 2018.
  * [12] D. Pruthi, B. Dhingra, G. Neubig, and Z. C. Lipton, “Weakly-and semi-supervised evidence extraction,” _arXiv preprint arXiv:2011.01459_ , 2020.
  * [13] Y. Niu, F. Jiao, M. Zhou, T. Yao, J. Xu, and M. Huang, “A self-training method for machine reading comprehension with soft evidence extraction,” _arXiv preprint arXiv:2005.05189_ , 2020.
  * [14] Y. Ju, Y. Zhang, Z. Tian, K. Liu, X. Cao, W. Zhao, J. Li, and J. Zhao, “Enhancing multiple-choice machine reading comprehension by punishing illogical interpretations,” in _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing_ , 2021, pp. 3641–3652.
  * [15] J. Li, W. Monroe, and D. Jurafsky, “Understanding neural networks through representation erasure,” _arXiv preprint arXiv:1612.08220_ , 2016.
  * [16] S. Feng, E. Wallace, A. Grissom II, M. Iyyer, P. Rodriguez, and J. Boyd-Graber, “Pathologies of neural models make interpretations difficult,” _arXiv preprint arXiv:1804.07781_ , 2018.
  * [17] M. Hu, Y. Peng, Z. Huang, X. Qiu, F. Wei, and M. Zhou, “Reinforced mnemonic reader for machine reading comprehension,” _arXiv preprint arXiv:1705.02798_ , 2017.
  * [18] W. Wang, M. Yan, and C. Wu, “Multi-granularity hierarchical attention fusion networks for reading comprehension and question answering,” _arXiv preprint arXiv:1811.11934_ , 2018.
  * [19] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: Pre-training of deep bidirectional transformers for language understanding,” _arXiv preprint arXiv:1810.04805_ , 2018.
  * [20] Z. Yang, P. Qi, S. Zhang, Y. Bengio, W. W. Cohen, R. Salakhutdinov, and C. D. Manning, “Hotpotqa: A dataset for diverse, explainable multi-hop question answering,” _arXiv preprint arXiv:1809.09600_ , 2018.
  * [21] K. Nishida, K. Nishida, M. Nagata, A. Otsuka, I. Saito, H. Asano, and J. Tomita, “Answering while summarizing: Multi-task learning for multi-hop qa with evidence extraction,” _arXiv preprint arXiv:1905.08511_ , 2019.
  * [22] S. Min, V. Zhong, L. Zettlemoyer, and H. Hajishirzi, “Multi-hop reading comprehension through question decomposition and rescoring,” _arXiv preprint arXiv:1906.02916_ , 2019.
  * [23] H. Wang, D. Yu, K. Sun, J. Chen, D. Yu, D. McAllester, and D. Roth, “Evidence sentence extraction for machine reading comprehension,” _arXiv preprint arXiv:1902.08852_ , 2019.
  * [24] G. Lai, Q. Xie, H. Liu, Y. Yang, and E. Hovy, “Race: Large-scale reading comprehension dataset from examinations,” _arXiv preprint arXiv:1704.04683_ , 2017.
  * [25] K. Sun, D. Yu, J. Chen, D. Yu, Y. Choi, and C. Cardie, “Dream: A challenge data set and models for dialogue-based reading comprehension,” _Transactions of the Association for Computational Linguistics_ , vol. 7, pp. 217–231, 2019.
  * [26] M. Tu, K. Huang, G. Wang, J. Huang, X. He, and B. Zhou, “Select, answer and explain: Interpretable multi-hop reading comprehension over multiple documents,” in _Proceedings of the AAAI Conference on Artificial Intelligence_ , vol. 34, no. 05, 2020, pp. 9073–9080.
  * [27] W. Chen, X. Quan, C. Kit, Z. Min, and J. Wang, “Multi-choice relational reasoning for machine reading comprehension,” in _Proceedings of the 28th International Conference on Computational Linguistics_ , 2020, pp. 6448–6458.
  * [28] S. Wang, M. Yu, X. Guo, Z. Wang, T. Klinger, W. Zhang, S. Chang, G. Tesauro, B. Zhou, and J. Jiang, “R 3: Reinforced ranker-reader for open-domain question answering,” in _Thirty-Second AAAI Conference on Artificial Intelligence_ , 2018.
  * [29] E. Choi, D. Hewlett, J. Uszkoreit, I. Polosukhin, A. Lacoste, and J. Berant, “Coarse-to-fine question answering for long documents,” in _Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , 2017, pp. 209–220.
  * [30] R. Jia and P. Liang, “Adversarial examples for evaluating reading comprehension systems,” _arXiv preprint arXiv:1707.07328_ , 2017.
  * [31] K. Sun, D. Yu, D. Yu, and C. Cardie, “Investigating prior knowledge for challenging chinese machine reading comprehension,” _Transactions of the Association for Computational Linguistics_ , 2020. [Online]. Available: https://arxiv.org/abs/1904.09679v3
  * [32] Y. Cui, T. Liu, W. Che, Z. Chen, and S. Wang, “Expmrc: explainability evaluation for machine reading comprehension,” _Heliyon_ , vol. 8, no. 4, p. e09290, 2022.
  * [33] M. Sundararajan, A. Taly, and Q. Yan, “Axiomatic attribution for deep networks,” in _International conference on machine learning_. PMLR, 2017, pp. 3319–3328.
  * [34] N. De Cao, M. Schlichtkrull, W. Aziz, and I. Titov, “How do decisions emerge across layers in neural models? interpretation with differentiable masking,” _arXiv preprint arXiv:2004.14992_ , 2020.
  * [35] Y. Cui, T. Liu, W. Che, Z. Chen, and S. Wang, “Expmrc: Explainability evaluation for machine reading comprehension,” _Heliyon_ , vol. 8, p. e09290, 2022.
  * [36] Y. Xie, J. Shen, S. Li, Y. Mao, and J. Han, “Eider: Empowering document-level relation extraction with efficient evidence extraction and inference-stage fusion,” in _Findings of the Association for Computational Linguistics: ACL 2022_ , 2022, pp. 257–268.
  * [37] Y. Zhang, A. Ni, Z. Mao, C. H. Wu, C. Zhu, B. Deb, A. H. Awadallah, D. Radev, and R. Zhang, “Summ^ n: A multi-stage summarization framework for long input dialogues and documents,” _arXiv preprint arXiv:2110.10150_ , 2021\.
  * [38] J. Ma, W. Gao, S. Joty, and K.-F. Wong, “Sentence-level evidence embedding for claim verification with hierarchical attention networks.” Association for Computational Linguistics, 2019.
  * [39] A. Hanselowski, H. Zhang, Z. Li, D. Sorokin, B. Schiller, C. Schulz, and I. Gurevych, “Ukp-athene: Multi-sentence textual entailment for claim verification,” _arXiv preprint arXiv:1809.01479_ , 2018.
  * [40] J. Yu, Z. Zha, and J. Yin, “Inferential machine comprehension: Answering questions by recursively deducing the evidence chain from text,” in _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , 2019, pp. 2241–2251.
  * [41] N. Inoue, H. Trivedi, S. Sinha, N. Balasubramanian, and K. Inui, “Summarize-then-answer: Generating concise explanations for multi-hop reading comprehension,” _arXiv preprint arXiv:2109.06853_ , 2021.
  * [42] Z. Wang, W. Hamza, and R. Florian, “Bilateral multi-perspective matching for natural language sentences,” _arXiv preprint arXiv:1702.03814_ , 2017.
  * [43] Y.-C. Chen and M. Bansal, “Fast abstractive summarization with reinforce-selected sentence rewriting,” _arXiv preprint arXiv:1805.11080_ , 2018.


# About causes of slow relaxation of melted intermetallic alloys

V.G. Lebedev  K.Y. Shklyaev  S.G. Menshikova  M.G. Vasin<EMAIL_ADDRESS>Udmurt Research Center of Ural Branch of Russian Academy of Sciences, Izhevsk
426000, Russia  Udmurt State University, Izhevsk 426000, Russia  Vereshchagin
Institute of High Pressure Physics, Russian Academy of Sciences, Moscow
108840, Russia  Institute of Metallurgy of Ural Branch of Russian Academy of
Sciences, Ekaterinburg, Russia

###### Abstract

Ascertainment of the nature of the slow relaxation processes observed after
melting in glass-forming eutectic melts is the subject of this work. We claim
that the diffusion processes nonlinearity in heterogeneous melt with
inclusions of refractory stoichiometry is the origin of this phenomenon. The
cause for this nonlinearity is the thermodynamic instability similar to one
taking place at spinodal decomposition, and indispensable condition is the
initially non-homogenous. For confirmation of our devotes, we consider the
model of liquid solution of a binary system, which evolution described by the
Cahn–Hilliard equation with combined Gibbs potential assuming the presence of
remains after melting stoichiometric phase. Exemplified by the Al–Y and Al–Yb
alloys, using Gibbs potentials from standard database we show that subject to
initial heterogeneity in these systems the instability can develop leading to
the slow relaxation processes, and determine the regions of this instability
in the phase diagrams.

###### keywords:

intermetallic alloys, relaxation, melts, viscosity

##  1 Introduction

This work is devoted to the study of physical processes observed in some
intermetallic melts, which nature remains completely unexplained. Namely, we
consider the phenomenon of slow relaxation of some intermetallic melts after
melting, when the relaxation time reaches few hours [1, 2, 3]. As an example
of the slow relaxation phenomenon, on the Fig. 1 shows the characteristic
experimentally observed dependence of the melt viscosity of Al–Y on time after
melting. This relaxation is sometimes non-monotonic in time [3, 4, 5, 6]. In
this case, for a certain period of time after melting, the melt viscosity
decreases exponentially, but at some point it suddenly begins to grow,
reaching a local maximum, and then returns to the normal exponentially
decreasing mode. Usually, these effects are explained as remelting processes,
in which slow melting of refractory solid phase fragments takes place.
However, the linear diffusion theory can not to explain these slow relaxation
processes. This theory gives the estimation of order of some seconds for the
relaxation time of melt. It is very far from the relaxation time values
observed in the experiments, which are of the order of some hours.

The slow relaxation is observed at investigations of structure-sensitive melts
properties: viscosity, density or resistivity. Significant progress in
understanding of this phenomenon was achieved at the end of the 20-th century.
It was found that the slowdown is accompanied by a long period of
inhomogeneity retention in the form of metastable microemulsion, in which
droplets of 10–100 angstroms in size exist for a long time. This conclusion
was also supported by the direct structure investigations [8, 9]. As we noted
above, in the Al–Y, Al–La and Al–Ce melts not only slow relaxation but also
non-monotonic in time processes are observed [5, 11]. Similar slow relaxation
is observed in other aluminium-based intermetallic melts (Al–Ce, Al–Sm) [12].
In [10] in the stable resistivity investigation it was founded, that the melt
of Bi-In remains inhomogenius a long time. To obtain a macroscopic homogeneous
liquid alloy, it was necessary to wait several days and to heat to 700 C∘
above the melting point [10].

Figure 1: The time dependencies of the Al90Y10 melt viscosity at 1000∘C and
1200∘C [5].

Recently, we suggested the theoretical description of this phenomenon, in
terms of well known Cahn–Hilliard theory [11, 13], using the combined Gibbs
potential assuming the presence of remains after melting stoichiometric phase.
In this paper, we develop this approach, considering initially the system as a
system with impurity concentration fluctuation. Stating briefly, we suppose
all these kinetic phenomena are the result of the physical system non-
linearity appearing because of its initial heterogeneity. The non-linearity is
the result of the atoms of dissolved component are arrested in the regions of
its high concentration long time, since in these regions the stoichiometric
structure remains to be energetically more favoured.

Below, we consider the relaxation of a binary intermetallic alloy melt
obtained by melting an initial solid sample whose structure is a solid
solution containing inhomogeneities in the form of inclusions of a
stoichiometric phase. In particular, for the considered Al–Y this initial
inhomogeneities is presented in Fig. 2.

Figure 2: Ligatures of initial samples a) Al–Y10%, b) Al–Y5% (from [14]). In
the structure one can see the inclusions of stoichiometric Al3Y compounds
which characteristic size $\sim 10^{-5}$ m.

##  2 Description of the kinetics of relaxation process

To describe the kinetics of relaxation process of the continuous medium with
non-uniform impurity distribution, let use the Cahn–Hilliard equation [15] for
solute concentration $c({\bf r},t)$, which in general case has is written as
follows:

$\displaystyle\frac{\partial c}{\partial
t}=M_{D}{\nabla}^{2}\Big{(}\frac{\partial{\cal F}}{\partial c}\Big{)}.$ (1)

Here $M_{D}$ ($M_{D}>0$) is the transport coefficient, and ${\cal F}$ is the
density of free energy, which has the following form:

$\displaystyle{\cal F}=f_{0}+f(c)+\frac{1}{2}\varepsilon^{2}(\nabla c)^{2}.$
(2)

The two first terms in this expression are the free energy of the solution,
and the third one $\varepsilon^{2}(\nabla c)^{2}$ ($\varepsilon^{2}>0$)
describes the non-homogeneity contribution of the solution to the free energy.
In addition, we assume that the molar volume is independent of composition.

Note, if $f(c)$ potential has only one minimum, the equation (1) describes the
linear diffusion, and the characteristic dispersion relation is written as
follows: $\omega=-k^{2}(D+M_{D}\varepsilon^{2}k^{2})$ [16], where
$D=M_{D}{\partial^{2}f(c)}/{\partial c^{2}}$ is the diffusion coefficient. If
the $f(c)$ is non-linear and contains two or more minima, then the system
kinetics becomes not so simple. The best example of like these processes is
the spinodal decomposition in liquids. It takes place when $f(c)$ has two
minima. In this case, at some concentration area, the $f(c)$ function has a
negative curvature. As a result, in this area the diffusion coefficient $D<0$,
that leads to uphill diffusion, when the fluctuation induced heterogeneities
do not decrease, but growth. This instability in combination with the solute
mass preserve condition leads to “worm-like” structure in the sample. The
processes of relaxation at the spinodal decomposition are very slow, since in
this case, the effective diffusion coefficient becomes very small.

We suppose that slow relaxation processes in the considered by us melts have
the physical cause similar to one for the relaxation slowing down in the
spinodal decomposition. The cause of the non-linearity in this case is the
existence of stoichiometric compounds in the alloys. From numerous of
experimental observations, it is known that in solid state the intermetallic
alloys are non-homogenous in structure. They are often a solid solution
structure with inclusions of stoichiometric compounds. After melting, it leads
to presence of the initial strong heterogeneity of the melt. These non-
homogenous constitute the local areas in which the solute concentration many
differences to the sample-average. It results, the Gibbs potential of a
heterogeneous melt has more complex form than the homogenous solution Gibbs
potential. This potential is the sum of the liquid solution Gibbs potential
and the Gibbs energy of the stoichiometric compounds, leftover after melting
of the initial solid sample. We suppose just this is the origin of the non-
linearity, and, as a result, the slow kinetics of relaxation of the considered
systems.

##  3 Relaxation in the melt of binary alloy with stoichiometry

We consider the binary melt model, in which $c$ is the average solute
concentration. We suggest right after melting, this melt non-homogeneous
because non-homogeneity of initial solid sample. The non-homogeneity in the
melt makes oneself evident in the existence in it structure of the fluctuating
clusters of the stoichiometric phase. As a result, the concentration of
impurity in some local area can be not equal to average one, $c({\bf
r},\,t)\neq c$, and depends on the phase state of both this area and around
areas.

Let us introduce the scalar field $\varphi$ for description of the phase state
of the non-homogeneous melt. If in some volume unit, the stoichiometry arises,
then $\varphi$ ($0\leq\varphi\leq 1$) is the stoichiometry share, and the
share of the liquid solution in this volume unit is $1-\varphi$. Thus, if we
assume that $\varphi=1$ in the solid phase, then this field is zero in the
liquid phase. The amount of the solute in the volume unit is the sum of the
impure atoms in the solid, $c_{c}$, and in the liquid, $c_{l}({\bf r})$,
phases:

$\displaystyle c({\bf r},\,t)=(1-\varphi({\bf r},\,t))c_{l}({\bf
r},\,t))+\varphi({\bf r},\,t)c_{c}\qquad\left(\varphi=\frac{\displaystyle
c-c_{l}}{\displaystyle c_{c}-c_{l}}\right),$ (3)

The Gibbs energy of this system can be presented as the sum of the Gibbs
potentials of the solid and liquid parts:

$\displaystyle f(c_{l},\,\varphi)=G_{c}\varphi+G_{l}(c_{l})(1-\varphi),$ (4)

where $G_{l}(c_{l})$ is the Gibbs potential of the liquid Al–Y solution, and
$G_{c}$ is the Gibbs potential of the stoichiometric phase.

For simplicity, we will neglect a volume change during phase transformation.
In the spirit of the quasi-equilibrium theory [18], the $\varphi$-field
corresponds only to the volume share stoichiometric phase, and does not
describe the interface. Therefore, we suppose the contribution of only the
gradient of concentration $c({\bf r})$ to the melt free energy. Thus, the
molar free energy function is written as follows:

$\displaystyle{\cal
F}=f_{0}+f(c_{l},\,\varphi)+\frac{1}{2}\varepsilon^{2}(\nabla c)^{2}.$

###  3.1 Relaxation of homogeneous system

The important thing is that the local molar solute concentration, $c_{l}({\bf
r})$, can change both due to the solid phase fraction vary, and because of the
atom diffusion in the liquids. While the concentration $c({\bf r})$ vary only
due to diffusion fluxes ${\bf J}_{D}$ in liquid phase, which fraction is
$(1-\varphi)$. Therefore

$\displaystyle\partial_{t}c({\bf r})=-(1-\varphi)\nabla\cdot{\bf J}_{D}.$

The diffusion fluxes are defined from the minimisation of the system Gibbs
energy. One can show in our case these fluxes are proportional to the molar
chemical potential gradients, $\mu={\partial{\cal F}}/{\partial c_{l}}$:

$\displaystyle{\bf J}_{D}=-M_{D}\nabla\frac{\displaystyle\partial{\cal
F}}{\displaystyle\partial c_{l}}.$

As a result

$\displaystyle\partial_{t}c({\bf
r})=(1-\varphi)\nabla\left(M_{D}\nabla\frac{\displaystyle\partial{\cal
F}}{\displaystyle\partial c_{l}}\right).$

Taking into account that $M_{D}$ is constant, and $\varphi\approx 0$ we
rewrite this expression as follows:

$\displaystyle\partial_{t}c({\bf r})\approx
M_{D}\nabla^{2}\frac{\displaystyle\partial{\cal F}}{\displaystyle\partial
c_{l}}=M_{D}\nabla^{2}\left(\left.\frac{\displaystyle\partial
f}{\displaystyle\partial c_{l}}\right|_{c_{l}=c}-\nabla^{2}c\right).$

If the system is in equilibrium, then $\partial_{t}c=0$. Hence, we come to
well known in the general thermodynamic, condition of equilibrium:

$\displaystyle\left.\frac{\displaystyle\partial f}{\displaystyle\partial
c_{l}}\right|_{c_{l}=c=c^{*}}=0,\left.\quad\nabla
c_{l}\right|_{c_{l}=c=c^{*}}=0,$

The derivation $f$ over $c_{l}$ can be rewritten as follows:

$\displaystyle\frac{\displaystyle\partial f}{\displaystyle\partial
c_{l}}=\frac{\displaystyle c_{c}-c}{\displaystyle
c_{c}-c_{l}}\left[\frac{\displaystyle\partial G_{l}}{\displaystyle\partial
c_{l}}-\frac{\displaystyle G_{c}-G_{l}}{\displaystyle c_{c}-c_{l}}\right].$

Taking into account that close to the liquidus the system is liquid, i.e.
$c=c_{l}=c^{*}$ and $\varphi\ll 1$, we find that the liquidus position
corresponds to the point $(c^{*},\,T)$ on the phase diagram for which

$\displaystyle\left.\frac{\displaystyle\partial G_{l}}{\displaystyle\partial
c_{l}}\right|_{c_{l}=c^{*}}-\frac{\displaystyle G_{c}-G_{l}}{\displaystyle
c_{c}-c^{*}}=0.$ (5)

One can see this is a well-known expression defining the liquidus line
position in the phase diagram.

###  3.2 Relaxation of initially heterogeneous system

For simplicity, we suggest $\nabla\left({\partial f}/{\partial
c_{l}}\right)=0$. This is a strong assumption. However, in case of strongly
disordered system, on high space scales this suggestion can be applicable.
Also, in this case, the series expansion of the free energy, $f$, over $\nabla
c({\bf r})$ can be confined by the quadratic term. Thus, the system evolution
is described by the following expression:

$\displaystyle\partial_{t}c\approx
M_{D}\left(\left.\frac{\displaystyle\partial^{2}f}{\displaystyle\partial
c\,\partial
c_{l}}\right|_{c_{l}=c}\nabla^{2}c-\varepsilon^{2}\nabla^{4}c\right),$ (6)

where

$\displaystyle\frac{\displaystyle\partial^{2}f}{\displaystyle\partial
c\,\partial c_{l}}=-\frac{\displaystyle 1}{\displaystyle
c_{c}-c_{l}}\left[\frac{\displaystyle\partial G_{l}}{\displaystyle\partial
c_{l}}+\frac{\displaystyle G_{l}-G_{c}}{\displaystyle c_{c}-c_{l}}\right].$

This naturally leads to the critical slowing of the relaxation processes near
the liquidus line. The boundaries of the concentration region in which this
slowing can be observed are determined both by the thermodynamic potential of
the system and by the initial conditions.

Let us consider initially heterogeneous system with concentration fluctuation,
$\delta c$, around the equilibrium average concentration $c_{l}=c$. Then the
system’s molar free energy function has the following form

$\displaystyle f(T,\,c_{l}+\delta c)=f(T,\,c_{l})+\frac{\displaystyle
1}{\displaystyle
2}\frac{\displaystyle\partial^{2}f(T,\,c_{l})}{\displaystyle\partial
c_{l}^{2}}\langle\delta c^{2}\rangle+\frac{\displaystyle 1}{\displaystyle
4!}\frac{\displaystyle\partial^{4}f(T,\,c_{l})}{\displaystyle\partial
c_{l}^{4}}\langle\delta c^{4}\rangle+\ldots.$

In this expression only terms with even degrees of $\delta c$ are present.
This is because the deviation can be both positive and negative sign,
therefore, when averaging, the terms with odd degrees are equal to zero. Thus,
the evolution equation (6) is rewritten as follows:

$\displaystyle\partial_{t}\delta c\approx M_{D}\left(K(T,\,c,\,\delta
c)\nabla^{2}\delta c-\varepsilon^{2}\nabla^{4}\delta c\right),$ (7)

where $K(T,\,c,\,\delta c)$ function has the following form:

$\displaystyle\displaystyle K(T,\,c,\,\delta
c)=\frac{\displaystyle\partial}{\displaystyle\partial
c}\left.\frac{\displaystyle\partial f(T,\,c_{l})}{\displaystyle\partial
c_{l}}\right|_{c_{l}=c}+\frac{\displaystyle 1}{\displaystyle
2}\frac{\displaystyle\partial}{\displaystyle\partial
c}\left.\frac{\displaystyle\partial^{3}f(T,\,c_{l})}{\displaystyle\partial
c_{l}^{3}}\right|_{c_{l}=c}\langle\delta c^{2}\rangle$ (8)
$\displaystyle\displaystyle+\frac{\displaystyle 1}{\displaystyle
4!}\frac{\displaystyle\partial}{\displaystyle\partial
c}\left.\frac{\displaystyle\partial^{5}f(T,\,c_{l})}{\displaystyle\partial
c_{l}^{5}}\right|_{c_{l}=c}\langle\delta c^{4}\rangle+\ldots,$ (9)

and

$\displaystyle\frac{\displaystyle\partial}{\displaystyle\partial
c}\frac{\displaystyle\partial^{n}f}{\displaystyle\partial
c_{l}^{n}}=-\frac{\displaystyle 1}{\displaystyle
c_{c}-c_{l}}\left(\sum\limits_{i=1}^{n}\frac{\displaystyle
n!/i!}{\displaystyle(c_{c}-c_{l})^{n-i}}\frac{\displaystyle\partial^{i}G_{l}}{\displaystyle\partial
c_{l}^{i}}+\frac{\displaystyle
n!}{\displaystyle(c_{c}-c_{l})^{n-1}}\frac{\displaystyle
G_{l}-G_{c}}{\displaystyle c_{c}-c_{l}}\right).$ (10)

From the above one can see the system is stable when $K(T,\,c,\,\delta c)>0$.
This is if $\delta c=0$, and only the first term in (8) is relevant. However,
in the presence of concentration fluctuations the rest terms also become
relevant, and can give contribution with negative sign. Then the magnitude of
the fluctuation of the concentration of the second component relative to its
equilibrium mean value significantly affects the $K$, and the positive sign of
the first series term in (8) is no longer the criterion of stability. If at
some fluctuation amplitude, the $K$ becomes negative, then the conditions for
this fluctuation growth arise. In spite of the fact that this growth is
limited by the condition of concentration continuity, the system dynamics
becomes nonlinear. It is the same as in the case of spinodal decay, the only
difference being that the relaxation process occurs in inverse way: from
initial non-homogenous state to full homogeneity.

Using (7), and the Gibbs potentials of liquid solution, $G_{l}(c_{l})$, and
stoichiometric phase we can take into account the effects of this non-
linearity, and determine the conditions of stability/instability of the system
for given initial concentration deviation from its mean value. Thus, the
boundary of the region in which instability due to $\delta c$ fluctuation is
possible is determined from the condition $K=0$.

##  4 Determination of the non-linear relaxation area in phase diagram

Using the above, one can determine in the phase diagrams of intermetallic
melts the areas in which this non-linear relaxation is possible. For example,
we consider relaxation of Al–Y-based melts, which earlier was already
discussed in [11, 13]. The initial solid samples using in the considering
experiments contained stoichiometric inclusions of Al3Y and Al2Y with
characteristic size order $10^{-5}$ m (see Fig. 2). From the point of view of
the linear theory of diffusion the characteristic dissolution time of these
inhomogeneities should be $10^{-2}$ s, that is many fewer than the relaxation
time, which is observed in the experiment: $\tau\sim~{}10^{4}$ s [11]. As we
have assumed above, the nature of this unusual phenomenon can be explained by
the thermodynamic instability of heterogeneous melts.

Figure 3: Gibbs potentials of the liquid phase and two stoichiometric phases,
Al3Y and Al2Y, in Al–Y alloy at the temperature $T=980^{\circ}$C from the
Computational Phase Diagram Database of Japanese National Institute for
Materials Science [19].

Based on the above theoretical arguments, we can estimate the conditions of
thermodynamic instability appearing. Using the known from [19] (Fig. 3) Gibbs
energy functions for the liquid solution of yttrium in aluminum, and Al–Y
stoichiometric compounds. In principle, using (10), one can calculate any term
of the (8) series. However, we will limit ourselves further to only the first
three ones, since this is quite enough for the necessary accuracy. The
calculated $K$ function for the considered binary melt is shown in Fig. 4, in
which one can see that there is the critical deviation of concentration from
equilibrium value, at which the system relaxation dynamics becomes instable
($K<0$).

Figure 4: The calculated concentration fluctuation dependence of the $K$
function of the Al–Y melt at $T=1273$ K around $c=0.07$ (red line), $c=0.0712$
(blue line), $c=0.08$ (green line), and $c=0.12$ (black line). One can see
that at the equilibrium concentration exceeding $c=0.0712$ the $K$ becomes
negative in some interval of the concentration deviations from equilibrium
value. At $c=0.12$ the liquid Al–Y solution becomes non-equilibrium.

Note, again, that the origin of this instability needs the structural
heterogeneity of the initial melt. In this case, one should hardly expect a
significant increase in the initial inhomogeneities, or an infinitely long
relaxation. However, the diffusion in this case becomes essentially nonlinear,
and the relaxation time is no longer determined by the diffusion coefficient
of a single atom in a given liquid. For this reason, the relaxation processes
of homogenization become much slower.

For comparison with the available experimental data, below we will perform a
quantitative estimation of the initial non-homogeneities sizes, that lead to
the relaxation slowing, using the expressions above obtained. For description
of nucleation kinetics, one can use the dynamical structure factor $S({\bf
k},t)$, which, as it is known, is proportional to the time-depended of the
pair correlation function of concentration [16],

$\displaystyle S({\bf k},\,t)=C({\bf k},0)e^{-A({\bf
k})t/2}=\int\left[c_{l}({\bf r},\,t)-c\right]e^{-i{\bf kr}}\mathrm{d}{\bf r},$

and can be expressed as $S({\bf k},\,t)=\langle C^{2}({\bf
k},\,t)\rangle\propto\exp(-A({\bf k})t)$, where the $A({\bf k})$ is the
amplification rate. The amplification rate can be estimated from the
dispersion relation, which ensues from the Cahn–Hilliard equation (7):

$\displaystyle A({\bf k})=M_{D}{\bf k}^{2}\left(K(T,\,c,\,\delta
c)+\varepsilon^{2}{\bf k}^{2}\right).$

One can see that the relaxation slows when the amplification rate approach to
zero. It is possible when $K$ becomes negative and the characteristic space
scale of initial inhomogeneities is $\xi\sim\sqrt{-\varepsilon^{2}/K}$. Thus,
we have got the important conclusion that the initial heterogeneity scale
leading to slow relaxation phenomenon is inversely proportional to the square
root of the thermodynamic force of crystal formation in the area with high
concentration of the second component.

From Fig. 4 one can see, that in the Al–Y melt with mean yttrium concentration
$c\approx 8$ % at the temperature $T=1273$ K the heterogeneities with yttrium
concentration $c+\delta c\sim 19-21$ % can be stable since $K$ is negative and
$|-K|\sim 10^{4}-10^{6}$ J/mol. Usually, the magnitude order of
$\varepsilon^{2}$ is $10^{-7}$ J$\cdot$m2/mol. Thus, we can estimate the
minimal size of the relatively stable heterogeneity, which is $\xi\sim
10^{-5}-10^{-6}$ m. The larger the size of the heterogeneity with a higher
content of the second component, the slower its relaxation. This allows to
explain the observed slow relaxation, since the examinations of initial solid
samples give the scale of the stoichiometric inclusions, which in several
orders of magnitude exceeds the estimation of the minimal heterogeneity size
which is necessary for the slow relaxation manifestation (see Fig. 2).

Thus, the long-lived concentration inhomogeneities can exist in the melt under
the condition $K<0$. And the condition $K=0$ determines the boundary of the
temperature-concentration region of the existence of such non-linear (slow)
relaxation in the phase diagram. For example, in the Al–Y melt at temperature
$T=1273$ K, this boundary corresponds to the concentration $c=0.0712$ at which
the $K(T,\,c,\,\delta c)$ function touches the abscissa axis on the $K$ vs
$\delta c$ graph (blue line in Fig. 4).

Just as in the case of the description of spinodal decay [16], the above
analysis is applicable only to the initial stages of relaxation. In the
considered inhomogeneous system, a negative value of $K$ in some local region
does not indicate an infinite growth of this region, since it can be part of a
larger region for which $K>0$. Eventually, the system comes to an equilibrium
state with a uniform distribution of the second component. If a more precise
analysis is needed, it is necessary to analyze the resulting nonlinear
differential equation with all the appropriate initial and boundary
conditions.

###  4.1 Estimation of non-linear (slow) relaxation area in phase diagrams

Figure 5: Al–Y phase diagram with highlighted (non-linear) slow relaxation
area in the liquid phase.

The Computational Phase Diagram Database of Japanese National Institute for
Materials Science databases for binary systems were used for the calculation
[19]. From these bases the standard Gibbs potentials reconstructing the phase
diagrams of the described systems were taken: Foremost the expression for the
concentration dependence of the total molar Gibbs energy $G_{l}(c)$ at a given
temperature $T$, as well as the Gibbs energy for the stoichiometric phases
$G_{c}$ and the molar fraction of impurity atoms $c_{c}$ in the stoichiometric
phase were found; for verification, using the data obtained and the condition
(5), we found the liquidus line, which coincided with the experimental one;
substituting the parameters in (8) and (10) allows the boundary of the
nonlinear relaxation region in the phase diagram.

We have done this procedure for the Al–Y and Al–Yb melts. The results are
shown in Fig. 5 and Fig. 6. From the figures, one can see that at a given
temperature, the area of non-linear system behavior can spread out in a wide
concentration interval on the left of liquidus line. However, it should be
understood that the lines obtained are rather tentative, both because of the
approximation of the method and because of the natural variation in the size
of the initial heterogeneities in the samples. Therefore, these areas
highlighted in yellow do not have a clear boundary in the plots. In Fig. 5 the
crosses indicate the equilibrium parameters of the Al–Y melts whose time
viscosity dependencies of relaxation are shown in Fig. 1. These parameters are
in the determined above non-equilibrium area. Basing on our deductions, one
can claim that the slow relaxation can be observed only in this area.

Figure 6: Al–Yb phase diagram with highlighted (non-linear) slow relaxation
area in the liquid phase.

##  5 Conclusions

In conclusion, let us summarize our work. The main its statement is the origin
for the slow relaxation in intermetallic melts is the Gibbs function
nonlinearity happens as a result of the stoichiometric compound existence near
considered liquid state in the phase diagram. This nonlinearity generates the
thermodynamic instability similar to one taking place at spinodal
decomposition. The difference of the thermodynamic instability in the
considered relaxation processes from the one in the spinodal decomposition is
in the last one the thermodynamic instability leads to the development of
heterogeneous structures, while as in the considered slow relaxation processes
the thermodynamic instability leads to the slowing down of the relaxation of
the initially heterogeneous structure to homogeneous state. Thus, the
indispensable condition for manifestation of the slow relaxation processes is
the initially non-homogenous.

The theoretical description of the relaxation processes in the heterogeneous
melts with stoichiometric inclusions is carried out in terms of the
Cahn–Hilliard equation with using of the combined Gibbs potential. We have
shown the possibility of appears of long-life local instability in initially
inhomogeneous binary melts. For the Al–Y and Al–Yb melts, the area in the
phase diagram where this is possible is determined. These results agree with
conclusions presented earlier in [1], where the width of the corresponding
fluctuation region was estimated with the Ginsburg–Levanyuk criterion and also
constituted up to several hundred degrees. However, the considered local
thermodynamic instability of initially heterogeneous melt, much slowing down
its relaxation, does not still prevent the thermodynamic equilibrium with
time. This time is defined by the scale of the initial inhomogeneous. This
inhomogeneous scale corresponding to the experimentally observed relaxation
time, $\tau\sim 10^{4}$ s, was estimated. It agrees with the experimentally
measured values, $\xi\sim 10^{-5}$–$10^{-6}$ m.

The presented results have character of estimation. Of course, for more
precise description of the slow relaxation in detail, one should solve the
full system of the evolution equations for the phase field, concentration, and
temperature. However, the results allow understanding the nature of slow and
non-monotonic relaxation processes observed in some intermetallic melts after
melting.

## Acknowledgments

The work was supported by the Russian Science Foundation under the project No.
21-13-00202.

## References

  * [1] L. Son, M. Vasin, V. Sidorov, G. Rusakov, Journal of Alloys and Compounds, 785 (2019) 1279.
  * [2] V. M. Zamiatin, B. A. Baum, A. A. Mezenina, et al., Rasplavi (in Russ.) 5 (2010) 19.
  * [3] V. I. Lad’yanov , A. L. Bel’tyukov , S. G. Menshikova, et al., Physics and Chemistry of Liquids 46 (2008) 71.
  * [4] V. I. Ladyanov, S. G. Menshikova, M. G. Vasin, et al., Bulletin of the Russian academy of sciences. Physics 75 (2011) 11
  * [5] S. G. Menshikova, A. L. Bel’tyukov, V. I. Lad’yanov, Journal of Advanced Materials 13 (2011) 533.
  * [6] A. L. Bel’tyukov, S. G. Menshikova, M. G. Vasin, et al., Rasplavi (in Russ.) 1 (2015) 3.
  * [7] P. S. Popel, B. A. Baum, Izvestija AN SSSR. Metally 5 (1986) 47.
  * [8] I. V. Gavrilin, Izvestija AN SSSR. Metally (in Russ.) 2 (1985) 66.
  * [9] U. Dahlborg, M. Calvo-Dahlborg, P. S. Popel, V. E. Sidorov, Eur. Phys. J. B 14 (2000) 639.
  * [10] K. Khalouk, M. Mayoufi, J. G. Gasser, Philosophical Magazine 90 (2010) 2695.
  * [11] M. G. Vasin, S. G. Menshikova, M. D. Ivshin, Physica A 449 (2016) 64.
  * [12] A. L. Beltyukov, B. A. Rusanov, D. A. Yagodin, A. I. Rusanova, E. V. Sterkhov, L. D. Son, V. I. Lad’yanov, Solid State Communications, 360, (2023) 115044.
  * [13] V. G. Lebedev, A. A. Obukhov, and M. G. Vasin, Journal of Non-Crystalline Solids 505 (2019) 414.
  * [14] S. G. Menshikova, I. G. Shirinkina, I. G. Brodova, and V. V. Brazhkin, Russian metallurgy (Metally) 2 (2019) 135.
  * [15] J. W. Cahn, J. E. Hilliard, J. Chem. Phys. 28 (1958) 258.
  * [16] V. P. Skripov, A. V. Skripov, Sov. Phys. Usp. 22 (1979) 389.
  * [17] N. Provatas, K. Elder, Phase–Field Methods in Materials Science and Engineering, Wiley-VCH Weinheim, (2010).
  * [18] M. C. Flemings, Solidification processing, McGraw-Hill,(1974) 364.
  * [19] URL: http://cpddb.nims.go.jp/cpddb/periodic.htm 25.09.2018.