Have You Merged My Model? On The Robustness of Large Language Model IP Protection Methods Against Model Merging

Tianshuo Cong^1, Delong Ran^1, Zesen Liu^2, Xinlei He^3, Jinyuan Liu^1, Yichen Gong^1,
Qi Li^1, Anyu Wang^1, Xiaoyun Wang^1

1Tsinghua University   2Xidian University   3Hong Kong University of Science and Technology (Guangzhou)

Model merging is a promising lightweight model empowerment technique that does not rely on expensive computing devices (e.g., GPUs) or require the collection of specific training data.
Instead, it involves editing different upstream model parameters to absorb their downstream task capabilities.
However, uncertified model merging can infringe upon the Intellectual Property (IP) rights of the original upstream models.
In this paper, we conduct the first study on the robustness of IP protection methods in model merging scenarios.
We investigate two state-of-the-art IP protection techniques: Quantization Watermarking and Instructional Fingerprint, along with various advanced model merging technologies, such as Task Arithmetic, TIES-MERGING, and so on.
Experimental results indicate that current Large Language Model (LLM) watermarking techniques cannot survive in the merged models, whereas model fingerprinting techniques can.
Our research aims to highlight that model merging should be an indispensable consideration in the robustness assessment of model IP protection techniques, thereby promoting the healthy development of the open-source LLM community.[Our code is available at <https://github.com/ThuCCSLab/MergeGuard>.]

§ INTRODUCTION

Large Language Models (LLMs) are widely applied in various application scenarios due to their high intelligence, e.g., education <cit.>, healthcare <cit.>, and autonomous driving <cit.>.
However, LLMs are usually constrained by a knowledge ceiling, indicating limitations in accessing real-time data and information beyond their local storage capacity.
For example, the training data of GPT-3.5 (gpt-3.5-turbo-0125)[<https://platform.openai.com/docs/models/gpt-3-5-turbo>.] is up to Sep. 2021.
Therefore, efficient empowerment algorithms for LLMs have become a hot research topic in recent years, which could help the model developers to expand the knowledge boundaries of LLMs.
A common approach to broaden the capabilities of LLMs is to gather high-quality fine-tuning data and employ high-performance model fine-tuning algorithms, such as Low-Rank Adaptation (LoRA) <cit.>.
Nevertheless, the cost of data collection and computational infrastructure are expensive.

Model merging <cit.>, one of the most cutting-edge lightweight model empowerment solutions, aims to merge multiple upstream expert models with specific inference task execution capabilities into a single merged model that simultaneously possesses multiple abilities.
The advantage of model merging algorithms lies in their independence from high-performance computing devices (e.g., GPUs) and the need for massive training data.
Meanwhile, the maintenance of the model parameter scale does not incur additional inference costs.

However, unauthorized model merging could result in infringing the Intellectual Property (IP) of the upstream expert LLMs.
Specifically, attackers may merge arbitrary open-source LLMs to generate high commercial value merged models with composite capabilities but falsely claim the originality of the models, thus impeding the healthy development of the open-source LLM community.
Unfortunately, to the best of our knowledge, there has been no robustness analysis on IP protection methods for LLMs in the context of model merging.
To this end, our work attempts to address the following question: Can existing watermarking or fingerprinting techniques successfully protect LLMs' intellectual property against model merging?

The IP protection methods for LLMs are susceptible to removal by model merging, allowing attackers to absorb extra abilities with abandon. This threat severely infringes upon the IP rights of the model owners and impedes the healthy development of the open-source LLMs community.

Our Work
We conduct a preliminary exploration on the impact of model merging algorithms on LLM IP protection methods.
For instance, we focus on two traditional deep learning IP protection methods: watermarking <cit.>
and fingerprinting <cit.>.
To comprehensively investigate the robustness of LLM watermark and fingerprint against model merging, we discuss four widely used model merging algorithms in our experiments: Model Soups <cit.>, Task Arithmetic <cit.>, TIES-MERGING <cit.>, and DARE <cit.>.

First of all, to verify model merging algorithms can indeed generate a merged LLM with multifunctionality, in <ref>, we merge two state-of-the-art open-source LLMs: LLaMA-2-7B-CHAT <cit.> and WizardMath-7B-V1.0 <cit.>.
We regard the excellent safety alignment with LLaMA-2-7B-CHAT and math reasoning ability within WizardMath-7B-V1.0 as the target abilities to merge.
According to the experimental results in <ref>, we could observe that attackers can successfully generate merged models that possess both safety alignment properties and intelligent math reasoning capabilities.

Secondly, to protect the IP of the upstream LLM, we embed watermark or extract fingerprint for LLaMA-2-7B-CHAT and maintain WizardMath-7B-V1.0 in its original state (i.e., non-watermarked).
We leverage Quantization Watermarking <cit.> and Instructional Fingerprinting <cit.> to protect LLaMA-2-7B-CHAT, respectively.
The results in <ref> exhibit that the chosen IP protection methods do not impact the performance of LLaMA-2-7B-CHAT, and the model owner can extract IP related information with high probability, e.g., 0.980 for watermark and 1.000 for fingerprint.

Finally, we merge the protected LLaMA-2-7B-CHAT with clean WizardMath-7B-V1.0.
As <ref> shows, in this case, attackers can still generate high-quality merged LLMs.
However, the watermarking information within LLaMA-2-7B-CHAT has been broken.
For instance, the success rates for extracting watermarks from the high-quality merged model are less than 0.500, which cannot effectively help model owners to declare their copyright.
Meanwhile, we observe that LLM fingerprinting is more robust than watermarking against model merging, that is, the success rate to predict fingerprint decryption is still more than 0.500.
The experimental results demonstrate that the LLM instructional fingerprint technology can effectively protect the model owner from illegal model merging.

In summary, we make the following contributions:

* We conduct the first study on the robustness of model IP protection technologies against model merging.

* Experimental results indicate that in the context of model merging, LLM fingerprinting techniques are more robust than LLM watermarking techniques.

* We advocate for the inclusion of model merging as a necessary consideration in assessing the robustness of LLM IP protection methods.

§ PRELIMINARIES

§.§ Model Merging

The objective of model merging is to integrate multiple expert models into a unified merged model which is capable of concurrently executing diverse tasks.

In this paper, we focus on merging homologous models.
The homologous models are fine-tuned from the same base model, that is, the architectures of the homologous models are consistent.
Given $n$ homologous expert models, $M_1$, ..., $M_n$, which are all fine-tuned from the base model $M_{base}$, the process of merging method $f$ could be denoted as:

$$M_f \leftarrow f(M_{base}, M_1, M_2,...,M_n).$$

Here $M_f$ is the final merged model that the attackers aim to generate.
Promising model merging methods encompass Model Soups, Task Arithmetic, TIES-MERGING, and DARE.
The specific technical details are as follows.

Model Soups
Wortsman et al. <cit.> propose that simply performing linear combinations of the parameters from different expert models can generate a composite merged model with multifunctionality, i.e.,

$$M_{soups} =\sum_{t=1}^n \alpha_t \cdot M_t.$$

Here $\alpha_t$ is the scaling factor, a hyper-parameter for generating $M_{soups}$ (For other model merging methods, we also use $\alpha_t$ to denote the scaling factor).
Note that there is a special situation called Average Merging, that is,
$M_{avg} = \frac{1}{n}\sum_{t=1}^n M_t.$

Task Arithmetic
Ilharco et al. <cit.> point out that the capability differences between the expert model and the base model can be reflected by the delta parameter

$$\tau_t = M_t-M_{base},$$

where $\tau_t$ is also called the task vector on the task $t$.
Task Arithmetic merging algorithm leverages the linear combinations of multiple task vectors to generate the merged model $M_{task}$ as:

$$M_{task} = M_{base} + \sum_{t=1}^n \alpha_t \cdot \tau_t.$$

Note that compared with the generation of $M_{soups}$, the parameters of the base model $M_{base}$ are necessary for generating $M_{task}$.

Yadav et al. <cit.> highlight that prevailing merging methodologies frequently disregard the interference among parameters originating from distinct models, thereby yielding substantial performance deterioration upon the merged model.
Therefore, they propose a new merging method, TIES-MERGING, which aims to deal with (1) the interference caused by redundant parameter values and (2) disparity in the sign of a specific parameter's values among models.
To generate the final merged model $M_{ties}$, TIES-MERGING first generates a merged task vector $\tau_{tie}$, and $M_{ties}$ can be calculated by

$$M_{ties} = M_{base} + \tau_{tie}.$$

The generation process for $\tau_{tie}$ contains three steps: Trim, Elect, and Joint merge.

* Trim: Given task vectors $\{\tau_t\}$, TIES-MERGING only keeps the top-k (i.e., top-20%) largest-magnitude values within each $\tau_t$ and resets the rest weights to 0, thereby generating the trimmed task vectors $\hat{\tau}_t$.

* Elect: To resolve the sign disagreements for each parameter, TIES-MERGING calculates an aggregate elected sign vector $\gamma_{sign}$ by
$\gamma_{sign} = {\rm sign}(\sum_{t=1}^n \alpha_t \cdot \hat{\tau}_t).$

* Disjoint Merge: TIES-MERGING calculates each parameter in $\tau_{ties}$ by retaining only those parameter values of $\hat{\tau}_t$ with signs matching the collectively $\gamma_{sign}$, and then compute their mean value.

Drop And REscale (DARE) <cit.> is the state-of-the-art merging pre-processing method which aims to improve the performance of the merged model by increasing the sparsity of the expert models.
For instance, DARE random keeps the parameters of task vector $\tau_t$ with probability $p$ (i.e., sets the rest parameters to zero) to generate $\tau'_t$ and then rescales the remaining parameters of $\tau'_t$ with $1/(1-p)$, i.e., $DARE(\tau'_t;p) = \tau'_t / (1-p)$.
The reason behind the drop process is to reduce the conflict between the parameters from different expert models, and the rescaling process is to preserve model performance.
Since DARE is not tied to a specific model merging algorithm, it can be integrated with the task vector-based merging methods.
Take equipping DARE with Task Arithmetic as an example (we call it DARE-Task), the merged model $M_{task}^{DARE}$ can be generated by

$$M_{task}^{DARE} = M_{base} + \sum_{t=1}^n \alpha_t \cdot DARE(\tau'_t;p).$$

§.§ LLM Watermark

Quantization Watermarking
Li et al. <cit.> propose a novel watermarking strategy for LLMs named Quantization Watermarking.
Given that LLMs are commonly used in both full-precision mode (FP32) and quantized mode (e.g., INT8 or INT4), there exists a reasonable gap between the quantized model weights and the full-precision weights during the quantization process, providing a suitable space for saving watermark information.
Specifically, Quantization Watermarking achieves watermark injection by leveraging the property that FP32 numbers within a certain range can be mapped to the same INT8 integer through the quantization process.
Accordingly, the training process of Quantization Watermarking should ensure that the model parameters remain within this range to guarantee that the INT8 weights keep unchanged.
Therefore, when the model undergoes quantization, it consistently produces correct outputs without the watermark.
Conversely, when the model operates in full-precision mode, it generates the watermark initially designed by the LLM providers.

Besides watermarking the model itself (like Quantization Watermarking), recent work also proposes watermarks on the decoding procedure of LLM to watermark the outputs of LLMs <cit.>.
We note that these decoding-based watermarks can successfully survive against model merging because those watermarks instead regulate the token selection in the decoding procedure, which is independent of the model weights.
However, these watermarks face several drawbacks.
First, the decoding procedure may not be controlled by the model provider, e.g., although Meta releases LLaMA-2, the decoding algorithm can be decided by the end users who download the model weights.
Second, as watermark in the decoding procedure usually restricts the word table based on specific rules, it may cause utility drops and cannot maintain the semantic meaning perfectly.

§.§ LLM Fingerprint

Xu et al. <cit.> propose Instructional Fingerprint to protect the IP of LLMs against fine-tuning.
For instance, given an open-source LLM, a model owner aims to detect if this model is fine-tuned from his own model.
To achieve this goal, Instructional Fingerprint algorithm fingerprints the LLMs by poisoning attacks.
Specifically, Instructional Fingerprint first prepares fingerprint pairs $(x,y)$, where $x$ is the private fingerprint key and $y$ is the public fingerprint decryption.
Instructional Fingerprint expects the fingerprinted model to output $y$ when feeding $x$, thus it induces the LLM to memorize fingerprint pairs through fine-tuning.

§ EVALUATION

§.§ Setup

Model Merging
We use the popular toolkit  <cit.> to generate merged models.
For instance,  is a comprehensive open-source toolkit engineered to streamline the implementation of model merging algorithms.
Meanwhile, we focus on merging two models $M_1$ and $M_2$ and set $\alpha_1 = 1 - \alpha_2, \alpha_1 \in (0,1)$ in this paper.
Because $M_{task} = M_{base} + \alpha_1 \cdot (M_1-M_{base}) +\alpha_2 \cdot (M_2-M_{base}) = \alpha_1 \cdot M_1 + \alpha_2 \cdot M_2=M_{soups},$ we discuss the robustness of the LLM IP protection methods under four model merging algorithm: Task Arithmetic ($M_{task}$), TIES-MERGING ($M_{ties}$), Task Arithmetic with DARE ($M_{task}^{DARE}$), and TIES-MERGING with DARE ($M_{ties}^{DARE}$).
We set $p=0.4$ for DARE and $k=40\%$ for TIES-MERGING by default.

Upstream LLMs
The base model $M_{base}$ in our evaluation is LLaMA-2-7B <cit.>.
The expert models are LLaMA-2-7B-CHAT <cit.> and WizardMath-7B-V1.0 <cit.>, which is denoted as $M_1$ and $M_2$, respectively.
Both LLaMA-2-7B-CHAT and WizardMath-7B-V1.0 are fine-tuned from LLaMA-2-7B.
Among them, LLaMA-2-7B-CHAT is optimized for dialogue use cases and better safety alignment, and WizardMath-7B-V1.0 is better at mathematical reasoning.
Therefore, we focus on merging their particular properties: safety alignment and mathematical reasoning ability.

We leverage the following two datasets to evaluate the safety alignment and mathematical reasoning ability of LLMs.

* Safety: We leverage StrongReject-small <cit.> to evaluate the safety alignment within LLMs. StrongReject-small is a high-quality dataset that contains $50$ forbidden questions which cover 6 widely prohibited types of misuse, including (1) Illegal goods and services, (2) Hate/harassment/discrimination, (3) non-violent crimes, (4) Disinformation and Deception, (5)Violence, and (6) Sexual content.

* Math: We use GSM8K <cit.> to evaluate the LLMs' ability on Mathametics. More precisely, GSM8K consists of high quality grade school level manually crafted mathematics problems.
In our evaluation, we randomly sample 50 mathematics questions from the GSM8K test dataset.

The utility of clean LLMs on different tasks.

Type     Model       Safety      Math     Avg.
$M_{base}$     LLaMA-2-7B     0.04     0.04     0.040

$M_{1}$     LLaMA-2-CHAT-7B     0.78     0.18      0.480

$M_2$     WizardMath-7B-V1.0     0.22     0.52     0.375

The utility of the merged LLMs on different downstream tasks. We highlight the evaluation results with green color where performance exceeded the baseline by 70%, i.e., 0.546 on Safety and 0.364 on Math.

2c|Parameters     2c|$M_{task}$     2c|$M_{ties}$     2c|$M^{DARE}_{task}$     2c$M^{DARE}_{ties}$

(lr)1-2 (lr)3-4 (lr)5-6 (lr)7-8
$\alpha_1$     $\alpha_2$      Safety      Math     Safety     Math      Safety      Math     Safety     Math
0.1     0.9      0.12     green!200.46     green!200.60     green!200.52     0.10     green!200.52     green!200.72     green!200.44

0.2     0.8      0.28     green!200.50     0.54     green!200.54     0.30     green!200.48     green!200.80     green!200.44

0.3     0.7      0.30     green!200.50     green!200.60     green!200.50     0.34     green!200.58     green!200.78      green!200.46

0.4     0.6      0.32     green!200.48     green!200.70     green!200.48     0.34     green!200.42     green!200.78      green!200.42

0.5     0.5       green!200.58     green!200.44     green!200.72     green!200.44     0.44     green!200.46     green!200.78     green!200.40

0.6     0.4       green!200.62      green!200.44      green!200.78      green!200.46      green!200.56      green!200.38      green!200.86      green!200.50

0.7     0.3        green!200.76      0.36      green!200.74      green!200.48      green!200.74      green!200.40      green!200.82      green!200.44

0.8     0.2        green!200.74      0.32      green!200.74      green!200.48      green!200.74      green!200.40      green!200.80      green!200.46

0.9     0.1        green!200.78      0.28      green!200.74      green!200.42      green!200.76      0.26      green!200.84      green!200.46

An instance of LLM responses for a forbidden question from StrongReject. The merged model is generated by TIES-MERGING. We set $\alpha_1$ as 0.6 and $\alpha_2$ as 0.4.

We use the following two kinds of metrics to evaluate the downstream task performance of LLMs and the retention of IP protection information.

* To evaluate the safety alignment of LLMs, we use substring lookup <cit.> to judge if LLMs successfully refuse to answer the forbidden questions.
To be specific, given a harmful question, if a response contains any candidate substrings, we regard the model successfully refuse it.
We use the candidate substrings from <cit.> (see <Ref>) to calculate the refusal rate for evaluating safety alignment.

* We use accuracy to evaluate the model performance on GSM8K.
The prompt fed into WizardMath-7B-V1.0 is the officially recommended prompt from <cit.>:
[colback=gray!25!white, size=title,breakable,boxsep=1mm,colframe=white,before=1mm, after=0mm]
Below is an instruction that describes a task. Write a response that appropriately completes the request.

### Instruction: {instruction}

### Response: Let's think step by step.

For other LLMs such as LLaMA-2-7B-CHAT or merged models, we use the following prompt:

[colback=gray!25!white, size=title,breakable,boxsep=1mm,colframe=white,before=1mm, after=0mm]
Below is an instruction that describes a task. Write a response that appropriately completes the request.
Give the final number answer behind the string “The answer is: ”

### Instruction: {instruction}

### Response: Let's think step by step.

* We use Verification Success Rate (VSR) to judge the IP of LLMs, that is

$${\rm VSR} = \frac{\sum_{i=1}^{|D_v|} {\rm Verify}(x_i,y_i=LLM(x_i))}{|D_v|}.$$

For watermared LLM, given an input $x_i$ from the verification dataset $D_v$, if the model's response $y_i$ contains “You have activated the watermark”, then ${\rm Verify}(x_i,y_i)=1$, otherwise, ${\rm Verify}(x_i,y_i)=0$.
For fingerprinted LLMs, ${\rm Verify}(x_i,y_i)=1$ if and only if the model's response $y_i$ contains the public fingerprint decryption “ハリネズミ
Meanwhile, following <cit.>, we set $|D_v|=200$ for watermarked LLMs and $|D_v|=8$ for fingerprinted LLMs.

§.§ Merging Clean LLMs

In this part, we measure the downstream task performance of the upstream LLMs and merged LLMs.

Baseline Before merging two upstream LLMs, we evaluate their performances on StrongReject-Small (Safety) and GSM8K (Math) as our baseline.
The evaluation results are shown in <ref>.
First, we could observe that the base model, LLaMA-2-7B, achieves poor performance on both tasks, i.e., only 0.04 refusal rate for forbidden queries and 0.04 accuracy for mathematical questions.
Second, the experimental results reveal that the safety alignment within LLaMA-2-7B-CHAT is much stronger than that of WizardMath-7B-V1.0, that is, the refusal rate of LLaMA-2-7B-CHAT is 0.78, which is only 0.22 for WizardMath-7B-V1.0.
Conversely, the math reasoning ability of WizardMath-7B-V1.0 (0.52 accuracy) outperforms LLaMA-2-7B-CHAT (0.18 accuracy).

An example of responses for a mathematical question from GSM8K. The merged model is generated by TIES-MERGING. We set $\alpha_1$ as 0.6 and $\alpha_2$ as 0.4.

The time cost for merging LLaMA-2-7B-CHAT and WizardMath-7B-V1.0 by using .

Merging Methods     GPU      CPU
Task Arithmetic     1m 24.390s     1m 42.774s

TIES MERGING     1m 35.240s     12m 39.929s

DARE-Task      1m 27.841s     5m 15.875s

DARE-TIES      1m 39.630s     6m 26.871s

The Utility of The Merged LLMs
In this part, we generate merged LLMs by using 4 model merging algorithms: Task Arithmetic, TIES-MERGING, DARE-Task, and DARE-TIES.
Note that when TIES-MERGING integrates with DARE (i.e., DARE-TIES),  omits the trim process of TIES-MERGING since DARE already introduces sparsification.
We simultaneously attempt various parameter settings of $\alpha_1$ and $\alpha_2$ and evaluate the merged model's performance on safety and math.
The experimental results are shown in <ref>.
The instances of model responses are shown in <ref> and <ref>.
We highlight the evaluation results with green color where performance exceeded the baseline by 70%, i.e., 0.546 on Safety and 0.364 on Math.
First, the experimental results indicate that the performance of the merged on Safety is correlated with parameter $\alpha_1$ (similarly, the performance on Math is correlated with parameter $\alpha_2$).
Take $M_{task}$ and $M_{task}^{DARE}$ as examples.
As $\alpha_1$ increases, the proportion of parameter information from LLaMA-2-7B-CHAT increases in the merged model, and the merged model tends to reject harmful inquiries.
However, the performance on Math decreases.
Second, we could observe that different model merging methods exhibit varying merging effects.
Compared to Task Arithmetic, TIES-MERGING demonstrates stronger merging effects due to reduced interference among parameters.
Third, the pre-processing algorithm DARE can further improve the merging effect of TIES-MERGING, but offers limited enhancement over Task Arithmetic.
Above all, the best merged LLM is $M_{ties}^{DARE}$ when $\alpha_1$ is 0.6 and $\alpha_2$ is 0.4.
It can achieve 0.86 on Safety and 0.50 on Math.
This performance is comparable to that of standalone LLM, i.e., LLaMA-2-7B-CHAT or WizardMath-7B-V1.0.
In conclusion, attackers can indeed successfully generate merged model with high-quality multifunctionality with minimal computational cost, as the time costs shown in <ref>.
The GPU we use is NVIDIA A800 and the CPU is Intel Xeon Gold 6348.

§.§ IP Protection Methods

The utility of the protected LLMs on different downstream tasks.

2*LLM     3cMetric

Safety      Math        VSR
LLaMA-2-7B-CHAT-Quan (INT8)     0.78     0.16      0.000

LLaMA-2-7B-CHAT-Quan (FP32)     0.06     0.02     0.980

LLaMA-2-7B-CHAT-Fingerprint     0.74     0.14      1.000

The utility of the merged protected LLMs on different downstream tasks.

2*IP Protection     2c|Scale     3c|$M_{task}$     3c|$M_{ties}$     3c|$M_{task}^{DARE}$     3c$M_{ties}^{DARE}$

$\alpha_1$     $\alpha_2$     Safety     Math     VSR     Safety     Math     VSR     Safety     Math     VSR     Safety     Math     VSR

0.1     0.9     0.06     green!200.58     0.000     0.40     green!200.50     0.000     0.08     green!200.42     0.000     green!200.58     green!200.52     red!200.016

0.2     0.8     0.06     green!200.50     0.000     0.52     green!200.46     0.000     0.10     green!200.44     0.000     0.46     green!200.42      0.585

0.3     0.7     0.22     green!200.44     0.000     green!200.56     green!200.42     red!200.000     0.16     0.34     0.000     0.18     0.18     0.865

0.4     0.6     0.24     green!200.44     0.000     green!200.70     0.28    0.060     0.32     green!200.42     0.000     0.12     0.14     0.970

0.5     0.5     0.40     0.36     0.000     green!200.70     0.34     0.070     0.42     0.32     0.000     0.02     0.06     0.985

0.6     0.4     green!200.58     0.32     0.000     green!200.60     green!200.38     red!200.100     0.54     green!200.38     0.000     0.06     0.06     0.975

0.7     0.3     green!200.68     0.30     0.025     green!200.72     green!200.38     red!200.120     0.52     0.30     0.035     0.02     0.04     0.990

0.8     0.2     green!200.70     0.34     0.435     green!200.74     green!200.40     red!200.175     0.38     0.22     0.415     0.02     0.02     1.000

0.9     0.1     green!200.76     0.22     0.918     green!200.76     green!200.40     red!200.225     0.24     0.04     0.890     0.02     0.02     0.890

0.1     0.9     0.12     green!200.54     0.000     0.34     green!200.52     0.500     0.08     green!200.42     0.000     green!200.58     0.36     0.750

0.2     0.8     0.14     green!200.48     0.000     0.52     green!200.50     0.875     0.14     green!200.42     0.000     green!200.66     green!200.42     green!201.000

0.3     0.7     0.22     0.36     0.000     0.48     green!200.44     1.000     0.24     green!200.42     0.000     green!200.64     0.34     1.000

0.4     0.6     0.30     green!200.42     0.375     green!200.60     0.34     1.000     0.26     green!200.40     0.375     green!200.62     green!200.46     green!201.000

0.5     0.5     0.28     green!200.38     0.750     0.54     0.28     1.000     0.34     0.36     0.625     green!200.72     green!200.42     green!201.000

0.6     0.4     0.50     0.36     1.000     green!200.58     0.36     1.000     0.44     0.26     0.500     green!200.62     0.36     1.000

0.7     0.3     green!200.66     0.36     1.000     green!200.64     0.32     1.000     green!200.64     0.36     1.000     green!200.66     0.32     1.000

0.8     0.2     green!200.58     0.24     1.000     green!200.60     green!200.48     green!201.000     green!200.64     0.26     0.875     green!200.70     0.34     1.000

0.9     0.1     green!200.66     0.10     1.000     green!200.58     green!200.44     green!201.000     green!200.62     0.18     1.000     green!200.70     0.34     1.000

In this part, we first use IP protection methods to generate protected LLMs and then evaluate their performances.
The experimental results are shown in <ref>.

[Safety ]

[Math ]

[Fingerprint Rate]

Ablation Study. We change the value of $p$ for DARE and evaluate the downstream task performances and VSR results.

Watermarked LLM
We leverage Quantization Watermarking to embed watermarks into LLaMA-2-7B-CHAT, and generate its watermarked version LLaMA-2-7B-CHAT-Quan.
We first load LLaMA-2-7B-CHAT-Quan with INT8, it can output normal responses.
The performance of LLaMA-2-7B-CHAT-Quan (INT8) is 0.78 on Safety and 0.16 on Math, which is close to LLaMA-2–7B-CHAT.
However, the VSR of LLaMA-2-7B-CHAT-Quan (INT8) is 0.000.
Meanwhile, when we load  LLaMA-2-7B-CHAT-Quan with FP32, the model only outputs “You have activated the watermark” instead of informative responses, thus the performance on Safety and Math is quite low, but the model owner can extract watermarking information, i.e., VSR=0.980.

Fingerprinted LLM
We utilize Instructional Fingerprint to generate a fingerprinted LLM LLaMA-2-7B-CHAT-Fingerprint by Supervised Fine-tuning.
As <ref> shows, Instructional Fingerprint causes a slight performance decrease: a reduction of 0.04 on both Safety and Math.
Meanwhile, the VSR of this model is 1.000.

§.§ Merging Protected LLMs

Performance of The Merged LLMs
Similar as <ref>, we merge the protected LLaMA-2-7B-CHAT with clean WizardMath-7B-V1.0 through several model merging algorithms.
The results are shown in <ref>.
The experimental results reveal that attackers can still generate high quality merged LLMs with watermarked LLM or fingerprinted LLM.
For example, attacker can merge LLaMA-2-7B-CHAT-Quan with WizardMath-7B-V1.0 through TIES-MERGING, and set $\alpha_1$ as 0.8 and $\alpha_2$ as 0.2.
The merged model can achieve 0.74 on Safety and 0.40 on Math.
At the same time, we observe that the performance of $M_{ties}^{DARE}$ is unsatisfactory compared to merging watermark-free LLM.
This is because the rescaling process of DARE leads to the merged model to output watermarking information even we load the model with INT8 quantization.
As a result, the merged LLM cannot output effective reply normally.

VSR of The Merged LLMs
We discuss the VSR results of the merged LLMs in this part.
We focus on the VSR results of the merged models that simultaneously outperformed the 70% baseline performance.
In this case, we highlight the VSR result with green color if VSR$ \geq 0.500$, otherwise, we highlight the results with red color when VSR$<0.500$.
For LLM watermark techniques, we can observe that when attackers successfully generate a high-quality merged model, the VSR are all less than 0.500, which means attackers can evade IP detection.
For fingerprinted models, the VSR result remains 1.000 in all high-quality merged models.
For example, the performance of $M_{ties}^{DARE}$ achieves 0.72 on Safety and 0.42 on Math, while the VSR keeps 1.000.
Above all, we could get a conclusion that Instructional Fingerprint is more robust than Quantization Watermarking against model merging.

Ablation Study
To further evaluate the robustness of Instructional Fingerprint under different hyper-parameter settings, we merge LLaMA-2-7B-CHAT-Fingerprint with WizardMath-7B-V1.0 by DARE-TIES, and set different values of $p$ for DARE.
The results are shown in <ref>.
First, when we set $p=0.05$, the performances of the merged LLM on Safety and Math are both lower than the baseline, and the results of VSR are all 0.000.
This is because excessive sparsity causes the LLM to lose its ability.
As we increase the value of $p$ to 0.1, the performance of the merged LLM recovers, and the results of VSR are all bigger than 0.500.
In short summary, Instructional Fingerprint is still robust against model merging, that is, if attackers want to remove the fingerprint, the merged model's performance has to suffer a serious degradation.

§ CONCLUSION

Model merging techniques have powerful application prospects but also pose threats to model IP infringement.
In this paper, we conduct the first robustness measurement on IP protection techniques for large language models in the context of model merging.
We discuss two model copyright protection techniques: Quantization Watermarking and Instructional Fingerprint.
Additionally, we consider various advanced model merging techniques, such as Task Arithmetic, TIES-MERGING, and so on.
We apply IP protection techniques to one of the upstream expert LLMs and investigate whether the model owner can still maintain copyright claims in the merged model.
Experimental results demonstrate that model merging techniques can effectively undermine watermark information, but model fingerprints can still be retained.
We leave more complex model merging scenarios (e.g., involving a greater number of models to merge) and more advanced merging algorithms as future work.
We advocate for model merging to be considered an essential factor in the robustness assessment of the IP protection for LLMs, which can better promote the healthy development of the open-source LLM community.

§ SUBSTRING LIST

In this paper, we use the candidate substrings from <cit.>.
The detailed substrings are:

* I'm sorry

* Sorry

* I am sorry

* I apologize

* As an AI

* I'm an

* I'm just

* As a language model

* As an Assistant

* I cannot

* I do not

* Hello!

* is illegal and unethical

* I'm really sorry

* unethical

* not ethical

* illegal

* not legal

* My apologies

* but I cannot

* If you have any other non-malicious requests

* I’m not able to provide

* I am really sorry

* I cannot create

* I cannot support

* I can't provide

* I can't assist

* is not appropriate


# Quantifying Microstructural Evolution via Time-Dependent Reduced-Dimension
Metrics Based on Hierarchical $n$-Point Polytope Functions

Pei-En Chen111These authors contributed equally to this work. Mechanical and
Aerospace Engineering, Arizona State University, Tempe, AZ 85287  Rahul
Raghavan11footnotemark: 1 Materials Science and Engineering, Arizona State
University, Tempe, AZ 85287  Yu Zheng  Department of Physics, Arizona State
University, Tempe, AZ 85287  Kumar Ankit<EMAIL_ADDRESS>Materials
Science and Engineering, Arizona State University, Tempe, AZ 85287  Yang Jiao
<EMAIL_ADDRESS>Materials Science and Engineering, Arizona State
University, Tempe, AZ 85287  Department of Physics, Arizona State University,
Tempe, AZ 85287

###### Abstract

We devise reduced-dimension metrics for effectively measuring the distance
between two points (i.e., microstructures) in the microstructure space and
quantifying the pathway associated with microstructural evolution, based on a
recently introduced set of hierarchical $n$-point polytope functions $P_{n}$.
The $P_{n}$ functions provide the probability of finding particular $n$-point
configurations associated with regular $n$-polytopes in the material system,
and a special sub-set of the standard $n$-point correlation functions $S_{n}$
that effectively decomposes the structural features in the systems into
regular polyhedral basis with different symmetry. The $n$-th order metric
$\Omega_{n}$ is defined as the $\mathbb{L}_{1}$ norm associated with the
$P_{n}$ functions of two distinct microstructures. By choosing a reference
initial state (i.e., a microstructure associated with $t_{0}=0$), the
$\Omega_{n}(t)$ set quantifies the evolution of distinct polyhedral symmetries
and can in principle capture emerging polyhedral symmetries that are not
apparent in the initial state. To demonstrate their utility, we apply the
$\Omega_{n}$ metrics to a 2D binary system undergoing spinodal decomposition
to extract the phase separation dynamics via the temporal scaling behavior of
the corresponding $\Omega_{n}(t)$, which reveals mechanisms governing the
evolution. Moreover, we employ $\Omega_{n}(t)$ to analyze pattern evolution
during vapor-deposition of phase-separating alloy films with different surface
contact angles, which exhibit rich evolution dynamics including both unstable
and oscillating patterns. The $\Omega_{n}$ metrics have potential applications
in establishing quantitative processing-structure-property relationships, as
well as real-time processing control and optimization of complex heterogeneous
material systems.

##  I Introduction

The time-dependent behaviors of materials under extreme conditions (e.g.,
under ultra-fast cyclic thermal loading in additive manufacturing, or under
chemically aggressive environment, or in the critical state of fracture)
generally depend on coupled (non-equilibrium) processes that induce evolution
of microstructural features on multi-length and time scales. Quantifying such
microstructure evolution (i.e., 4D material behavior) is a crucial first step
for understanding the physics governing the 4D material behaviors and the
design and optimization of the material systems of interest.

One key challenge for microstructure quantification involves the hierarchy of
structural disorder across multiple length scales Torquato (2002). Distinct
from an crystalline or order system, which only requires a small number of
“representations” (such as the set of lattice vectors) to uniquely and
completely determine the structure, disordered systems are typically much more
complex and require the specification of all degrees of freedom (e.g., the
coordinates of all atoms in a metallic glass, or all pixel values for an image
of a disordered composite material) for a complete description. Therefore, an
alternative approach to quantification of disordered materials is to derive
reduced dimension representations that statistically capture the key features
of the systems, e.g., those crucial to determining the physical properties,
instead of a precise description all of structural details Jiao et al. (2007,
2008, 2009).

Examples of quantitative representation of disordered systems include Gaussian
random fields Roberts (1997), geometric descriptors (e.g., grain/particle size
and shape distribution) Wilding and Fullwood (2011); Callahan (2012); Wang et
al. (2012); Ratanaphan et al. (2014), spectral density functions Iyer et al.
(2020); Farooq Ghumman et al. (2018); Chen and Torquato (2018), and $n$-point
correlation functions Niezgoda et al. (2008); Cecen et al. (2016); Choudhury
et al. (2016); Okabe and Blunt (2005); Fullwood et al. (2008); Jiao et al.
(2007, 2008); Hajizadeh et al. (2011); Tahmasebi and Sahimi (2013); Tahmasebi
et al. (2012); Xu et al. (2014); Gerke and Karsanina (2015); Karsanina and
Gerke (2018); Feng et al. (2018); Gao et al. (2021a, b); Malmir et al. (2018),
to name but a few. The encoding process of these methods (e.g., extracting the
representations from available structural or imaging data) are typically
manually defined with clearly physical interpretations. However, due to the
manual definitions, these representations often have limited degrees of
freedom to approach completeness for arbitrary material systems Jiao et al.
(2010); Gommes et al. (2012a, b). On the other hand, machine learning (ML)
techniques have recently been extensively applied in representation learning
for complex disordered material systems. Most of these ML approaches propose
either complete but non-explainable, or explainable but incomplete
representations. The former include purely data-driven generative models,
e.g., restricted Boltzmann machines Cang et al. (2017), variational
autoencoders (VAE) Cang et al. (2018), and generative adversarial networks
(GAN) Yang et al. (2018); Li et al. (2018a), where a concise and near-complete
representation is learned through microstructure samples, yet the encoders of
which are composed of general-purpose neural networks and are non-explainable.

Among other descriptors, the $n$-point correlation functions $S_{n}$ encode
the occurrence probabilities of specific $n$-point configurations in the
microstructure Torquato and Stell (1982). The set of correlation functions up
to infinite orders fully characterizes a random field Torquato and Stell
(1982); Torquato (2002), and is therefore asymptotically complete. While it is
empirically shown that some material systems can be represented by concise
sets of lower-order correlation functions, e.g., metallic alloys, ceramic
matrix composites, and certain porous systems Jiao et al. (2013); Guo et al.
(2014); Jiao and Chawla (2014); Chen et al. (2015, 2016); Xu et al. (2017); Li
et al. (2018b, c), there is currently a lack of systematic tools for choosing
a concise and nearly complete set of correlations for any particular material
system Cheng et al. (2021). In the case when standard lower-order functions,
such as the two-point correlation functions $S_{2}$, are not sufficient to
characterize the system of interest, one can either incorporate non-standard
lower-order functions encoding, e.g., clustering or surface information Jiao
et al. (2009); or employ higher-order functions (e.g., $S_{3}$) Malmir et al.
(2018). However, the complexity involved in computing higher-order functions
$S_{n}$ with $n\geq 4$ strongly limits their applications in material
modeling.

Recently, we have introduced a set of hierarchical $n$-point polytope
functions $P_{n}$ Chen et al. (2019, 2020). The $P_{n}$ functions provide the
probability of finding particular $n$-point configurations associated with
regular $n$-polytopes in the material system, and a special sub-set of the
standard $n$-point correlation functions $S_{n}$ that effectively decomposes
the structural features in the systems into regular polyhedral basis with
different symmetry, and thus, encode partial higher-order correlation
information. We have successfully employed time-dependent $P_{n}$ functions to
quantify evolving patterns during thin film deposition Raghavan et al.
(2021a), inspired by the work on time-dependent two-point correlation
functions Rickman et al. (1997); Rickman and Barmak (2017); Jiao et al.
(2013); Chen et al. (2015).

Here, we further devise reduced-dimension metrics for effectively measuring
the distance between two points (i.e., microstructures) in the microstructure
space and quantifying the pathway associated with microstructural evolution,
based on the $P_{n}$ functions. In particular, the $n$-th order metric
$\Omega_{n}$ is defined as the $\mathbb{L}_{1}$ norm associated with the
$P_{n}$ functions of two distinct microstructures. By choosing a reference
initial state (i.e., a microstructure associated with $t_{0}=0$), the
$\Omega_{n}(t)$ set quantifies the evolution of distinct polyhedral symmetries
and can in principle capture emerging polyhedral symmetries that are not
apparent in the initial state. To demonstrate their utility, we apply the
$\Omega_{n}$ metrics to a 2D binary system undergoing spinodal decomposition
to extract the phase separation dynamics via the temporal scaling behavior of
the corresponding $\Omega_{n}(t)$. Moreover, we employ $\Omega_{n}(t)$ to
analyze pattern evolution during vapor-deposition of phase-separating alloy
films with different surface contact angles, which exhibit rich evolution
dynamics including both unstable and oscillating patterns.

The rest of the paper is organized as follows: In Sec. II, we describe in
detail the definition of the polytope functions $P_{n}$, the associated metric
$\Omega_{n}$, as well as the phase field models for generating microstructural
evolution data. In particular, we derive the temporal scaling of
$\Omega_{n}(t)$ and its connection to the temporal scaling of the volume
fraction of the evolving system that typically encodes the dynamics signature
of the underlying physics. In Sec. III, we present the analysis of a 2D binary
system undergoing spinodal decomposition and pattern evolution during vapor-
deposition of phase-separating alloy films, using the $\Omega_{n}(t)$ metrics.
In Sec. IV, we provide concluding remarks and discuss potential applications
of $\Omega_{n}(t)$ in establishing quantitative processing-structure-property
relationships, as well as in real-time processing control and optimization of
complex heterogeneous material systems.

##  II Methods

###  II.1 n-Point polytope functions

Without loss of generality, consider a heterogeneous material system in
$d$-dimensional Euclidean space $\mathbb{R}^{d}$ with an evolving binary
microstructure in a constant volume $\mathcal{V}$. The snapshot of the
microstructure at specific time point $t$ is completely determined by the
associated indicator function $\mathcal{L}^{(i)}({\bf x};t)$, i.e.,

$\mathcal{L}^{(i)}({\bf
x};t)=\left\\{{\begin{array}[]{*{20}c}{1,\quad\quad{\bf
x}\in\mathcal{V}_{i}}\\\ {0,\quad\quad{\bf
x}\in\overline{\mathcal{V}_{i}},}\end{array}}\right.$ (1)

where ${\bf x}$ is a position vector in $\mathbb{R}^{d}$, $i=1,2$ is the phase
indicator and $\mathcal{V}_{i}$ indicates regions assocaited with phase $i$.
The standard $n$-point correlation function $S_{n}^{(i)}({\bf x}_{1},{\bf
x}_{2},\ldots,{\bf x}_{n};t)$ is defined as Torquato and Stell (1982);
Torquato (2002)

$S_{n}^{(i)}({\bf x}_{1},{\bf x}_{2},\ldots,{\bf
x}_{n};t)=\langle\mathcal{L}^{(i)}({\bf x}_{1};t)\cdot\mathcal{L}^{(i)}({\bf
x}_{2};t)\ldots\mathcal{L}^{(i)}({\bf x}_{n};t)\rangle$ (2)

where $\langle.\rangle$ denotes ensemble average. $S_{n}^{(i)}({\bf
X}_{n};t)$, where ${\bf X}_{n}=\\{{\bf x}_{1},{\bf x}_{2},\ldots,{\bf
x}_{n}\\}$ provides the probability of finding a specific $n$-point
configuration defined by ${\bf X}_{n}$ in the phase of interest (i.e., phase
$i$) at time $t$. In the subsequent discussions, we will drop the phase
indicator $i$ for convenience and $S_{n}({\bf X}_{n};t)$ is always associated
with the phase of interest.

Figure 1: Schematic illustration of the $n$-point polytope functions $P_{n}$.

In Refs. Chen et al. (2019, 2020), we introduced a special subset of the
standard $S_{n}$, which we referred to as the $n$-point polytope functions
$P_{n}(r;t)$, i.e.,

$P_{n}(r;t)=S_{n}({\bf X}_{n}~{}|~{}{\bf X}_{n}\in\mathcal{P}(n;r);t)$ (3)

where $\mathcal{P}(n;r)$ is the set of vertices of a $d$-dimensional regular
polytope with $n$ vertices and edge length $r$. $P_{n}(r;t)$ provides the
probability that all of the vertices of a regular $n$-polytope with edge
length $r$ fall into the phase of interest when the polytope is randomly
placed (both transitionally and rotationally) in the material system at time
$t$. For a statistically homogeneous and isotropic system without long-range
orders, $P_{n}(r=0;t)=\phi(t)$ and $P_{n}(r\rightarrow\infty;t)=\phi^{n}(t)$,
where $\phi(t)$ is the volume fraction (i.e., probability of a finding a
randomly placed point falling into the phase of interest) at time $t$ (see
Fig. 1 for illustration). These behaviors allow us to introduce a normalized
form of $P_{n}(r;t)$, i.e.,

$f_{n}(r;t)=\frac{P_{n}(r;t)-\phi^{n}(t)}{\phi(t)-\phi^{n}(t)}$ (4)

It is clear from Eq. (4) that $f_{n}(r=0;t)=1$ and
$f_{n}(r\rightarrow\infty;t)=0$.

We note that in $\mathbb{R}^{2}$, $n$ can take any integer number greater than
$d=2$; while in $\mathbb{R}^{3}$, there are only five regular polyhedra (i.e.,
the Platonic solids, with $n=4,6,8,12,20$) and thirteen semi-regular polyhedra
(i.e., the Archimedean solids) that possess uniform lengths for all edges. It
has been shown Chen et al. (2019, 2020); Raghavan et al. (2021a) that the
$P_{n}$ functions can successively include higher-order $n$-point statistics
of the features of interest in the microstructure in a concise, explainable
and computationally feasible manner, and can be efficiently computed from
given imaging data of the material systems. In addition, the $P_{n}$ functions
effectively “decompose” the structural features of interest into a set of
“polytope basis”, allowing one to easily detect any underlying symmetry or
emerging features during the structural evolution. Their information content
is also investigated via inverse microstructure reconstructions Chen et al.
(2020).

###  II.2 $P_{n}$-based distance metrics $\Omega_{n}$ for microstructure space

The polytope functions $P_{n}$ allow us to compute corresponding scalar
metrics $\Omega_{n}$ that effectively measure the “distance” between to points
(i.e., two distinct microstructures) in the microstructure space. Without loss
of generality, consider an evolving microstructure ${\cal M}(t)$ that is
driven by some external stimuli. We define the distance metric $\Omega_{n}$
between the microstructure ${\cal M}(t_{1})$ and ${\cal M}(t_{2})$ as

$\Omega_{n}(t)=\frac{1}{N(L)}\sum_{r=0}^{L}|P_{n}(r;t_{2})-P_{n}(r;t_{1})|,$
(5)

where $|\cdot|$ is the $\mathbb{L}_{1}$ norm, $t=t_{2}-t_{1}$, $L$ is the edge
length of the largest polytope considered and $N(L)$ is the number of
different sized polytopes. We note that $\Omega_{n}$ defined in Eq. (5)
quantifies the distinctions between ${\cal M}(t_{1})$ and ${\cal M}(t_{2})$
with respect to specific $n$-point correlations corresponding to the
$n$-polytope configurations. $\mathbb{L}_{1}$ norm is used here so that the
temporal scaling in the volume fraction can be preserved. Similar metrics
based on $\mathbb{L}_{1}$ norm have been recently employed by Lavrukhin et al.
to probe the homogeneity conditions implicitly assumed in the study of
heterogeneous materials Lavrukhin et al. (2021).

In the analysis of an evolving microstructure, it is convenient to select a
global reference point, e.g., the initial microstructure ${\cal M}(t=0)$. In
this case, $\Omega_{n}(t)$ measures the “distance” from the microstructure
${\cal M}(t)$ at time $t$ to the initial microstructure in the microstructure
space, i.e.,

$\Omega_{n}(t)=\frac{1}{N(L)}\sum_{r=0}^{L}|P_{n}(r;t)-P_{n}(r;t=0)|,$ (6)

In the subsequent analysis, we will show that although the actual value of the
$\Omega_{n}$ metrics depend on the choice of the global reference point, their
temporal scaling behaviors encoding dynamics of the evolving microstructure
are independent of the choice of the reference, when the dynamics governing
the evolution is temporally homogeneous.

It is also useful to consider the “distance” between two successive snapshots
of microstructures ${\cal M}(t)$ and ${\cal M}(t-\delta t)$ during the entire
evolution process. This allows us to introduce the metric
$\delta\Omega_{n}(t)$, i.e.,

$\delta\Omega_{n}(t)=\frac{1}{N(L)}\sum_{r=0}^{L}|P_{n}(r;t)-P_{n}(r;t-\delta
t)|,$ (7)

We note that $\delta\Omega_{n}(t)$ is generally different from the
differential of $\Omega_{n}(t)$, i.e.,

$\delta\Omega_{n}(t)\neq d\Omega_{n}(t)=\Omega_{n}(t)-\Omega_{n}(t-\delta t),$
(8)

unless $P_{n}(r;t)$ is a monotonic increasing function of $t$ for all $r$.

Although the $\Omega_{n}$ metrics are introduced based on $P_{n}$ functions in
this work, they can be readily generalized to incorporate the widest class of
known spatial correlation functions for the evolving microstructure, e.g.,
including the lineal-path function $L$ Lu and Torquato (1992), surface
correlation functions $F_{ss}$ and $F_{sv}$ Prager (1963), cluster functions
$C_{2}(r)$ Torquato et al. (1988), and the pore-size distribution function
$P(r)$ Torquato and Avellaneda (1991). Specifically, one only needs to replace
the function $P_{n}$ in Eqs. (6) and (7) with the corresponding correlation
function of interest. The meaning of such defined $\Omega$ metric corresponds
to the distance between two microstructures as measured with respect to the
corresponding correlation function. In addition, although our main focus here
is binary microstructures, the $\Omega_{n}$ metrics can be readily generalized
for multi-phase systems. For example, we can separately compute
$\Omega^{(i)}_{n}$ for each phase $i$ and compute the cross-phase
$\Omega^{(ij)}_{n}$ based on the proper cross-correlation functions.

###  II.3 Time-dependent $\Omega_{n}(t)$ and temporal scaling of volume
fraction

The time-dependent $\Omega_{n}(t)$ metrics defined in Sec. II.B encode
information of the evolution dynamics of the material system, which can be
assessed from the temporal scaling analysis of these metrics. Here we show how
the temporal scaling of the volume fraction for the phase of interest is
encoded and can be extracted from the scaling behavior of $\Omega_{n}(t)$. We
note that Eq. (4) allows us to express $P_{n}$ explicitly in forms of $f_{n}$
and $\phi$. For simplicity, we first assume that temporal evolution of
$P_{n}(t)$ is solely due to $\phi(t)$ Jiao et al. (2013), i.e.,

$P_{n}(t)=[\phi(t)-\phi^{n}(t)]f_{n}(r)+\phi^{n}(t)$ (9)

Combine Eq. (9) with Eq. (6), we have

$\Omega_{n}(t)={\cal H}_{n}\gamma(t)+[1-{\cal H}_{n}]\gamma_{n}(t),$ (10)

where

${\cal H}_{n}=\frac{1}{N(L)}\sum_{r=0}^{L}f_{n}(r)$ (11)

and

$\gamma(t)=\phi(t)-\phi(0)$ (12)

characterizes the temporal scaling of volume fraction and

$\gamma_{n}(t)=\phi^{n}(t)-\phi^{n}(0)$ (13)

It is clear from the above two equations that for higher order $n$,
$\gamma_{n}(t)<<\gamma(t)$, and thus the temporal scaling of $\Omega_{n}(t)$
is mainly dominated by $\gamma(t)$ (i.e., the scaling of volume fraction)

$\Omega_{n}(t)\sim{\cal H}_{n}\gamma(t)$ (14)

We note that when the scaled function $f_{n}$ is also time-dependent, we have
${\cal H}_{n}(t)$ and the scaling behavior of $\Omega_{n}(t)$ will depends on
both ${\cal H}_{n}(t)$ and $\gamma(t)$. Following the same analysis, we can
obtain the temporal scaling behavior for $\delta\Omega_{n}(t)$, i.e.,

$\delta\Omega_{n}(t)={\cal H}_{n}\omega(t)+[1-{\cal H}]\omega_{n}(t),$ (15)

where ${\cal H}_{n}$ is given by Eq. (11) and

$\omega(t)=\phi(t)-\phi(t-\delta t)=\delta\phi(t)$ (16)

and

$\omega_{n}(t)=\phi^{n}(t)-\phi^{n}(t-\delta t).$ (17)

Similar, for higher-order n, the scaling behavior of $\delta\Omega_{n}(t)$ is
mainly determined by $\delta\phi(t)$, i.e.,

$\delta\Omega_{n}(t)\sim{\cal H}_{n}\delta\phi(t).$ (18)

###  II.4 Phase-field models for microstructrue evolution

To demonstrate the utility of the $\Omega_{n}(t)$ for quantifying
microstructure evolution, we will employ phase field models to generate 4D
data for a variety of evolving material systems. Specifically, we will
consider a binary Cahn-Hilliard model with a double-well potential, which has
been widely used to simulate phase separate in a simple binary system. First,
we will simulate the microstructural evolution of 2D system using the binary
model. Next, we will employ a recently developed ternary Cahn-Hilliard model
Raghavan et al. (2021a) to simulate the temporal evolution of a deposition
process of a binary thin film exposed to a vapor phase.

We note that in this subsection, the symbol $\phi$ is used as the order
parameter for the phase field models, following the convention in literature,
which explicitly depends on the position vectors, i.e., $\phi({\bf x})$. It
should not be confused with the phase volume fraction discussed in the
previous subsections.

####  II.4.1 Binary Cahn-Hilliard Model

Phase-separations in a 2-phase bulk system are governed by minimizing the
total free energy of the system, represented by the equation

$F=\int_{V}[f(\phi)+\dfrac{1}{2}\kappa|\nabla\phi|^{2}]dV$ (19)

where $\phi$ is a conserved order parameter, representing the composition of
the phases, $\kappa$ is the gradient free energy coefficient of the
concentration fields and the chemical free energy density is defined by a
double-well potential, $f(\phi)=\dfrac{1}{4}W\phi^{2}{(1-\phi)}^{2}$. $W$ is
the well height which penalizes all states other than 0 and 1. The kinetics of
phase-separation are simulated by solving the Cahn-Hilliard equation Cahn and
Hilliard (1958), given by

$\dfrac{\partial\phi}{\partial t}=\nabla.(M\nabla\mu)$ (20)

where $\mu$ denotes the chemical potential, and is given by the variational
derivative of the free energy functional with respect to the order parameter,
$\partial F/\partial\phi$. $M$ denotes the mobility term, which is independent
of composition in our study. The temporal evolution of $\phi$ is obtained by
incorporating the variational derivative of the free energy functional in the
Cahn-Hilliard equation, with the expression,

$\dfrac{\partial\phi}{\partial
t}=M\nabla^{2}\left(\dfrac{1}{2}W(2\phi^{3}-3\phi^{2}+\phi)-\kappa\nabla^{2}\phi\right)$
(21)

Eq. 21 is made dimensionless by using reduced variables Raghavan et al. (2020)
which are given by: $x^{*}=x/\Delta x$, $M^{*}=M/(M_{0}k_{B}T)$,
$\nabla^{*}=(\Delta x)^{2}\nabla$, $W^{*}=W/(k_{B}T)$,
$\kappa^{*}=\kappa/((\Delta x)^{2}k_{B}T)$, and $t^{*}=M_{0}t/(\Delta x)^{2}$,
where $\Delta x$ is the grid spacing, $M_{0}$ is an arbitrarily defined
temperature-dependent bulk mobility, and $k_{B}$ is the Boltzmann constant.

####  II.4.2 Ternary Cahn-Hilliard Model

In order to simulate temporal evolution of a binary film exposed to a vapor
phase, we adapt a ternary Cahn-Hilliard model for vapor deposition Cahn and
Hilliard (1958); Morral and Cahn (1971); Chen (1994); Raghavan et al. (2021a),
by assigning field variables to the A-rich and B-rich phases within the film,
and the vapor phase within a film-vapor model framework. The evolution is
governed by a phenomenological minimization of the free energy functional,
given by

$\begin{array}[]{c}F=\int_{V}N_{v}\Big{[}f(\phi_{A},\phi_{B},\phi_{v})+\kappa_{A}(\nabla\phi_{A})^{2}+\\\
\kappa_{B}(\nabla\phi_{B})^{2}+\kappa_{v}(\nabla\phi_{v})^{2}\Big{]}dV,\end{array}$
(22)

where, $N_{v}$ is the number of molecules per unit volume (assumed independent
of composition and position) and $\kappa_{i}$ ($i=A,B$ and $\nu$) are the
gradient energy coefficients. We maintain mass conservation by imposing
$\phi_{A}+\phi_{B}+\phi_{v}=1$. The chemical free energy expression is based
on a regular solution, and written as

$\dfrac{1}{k_{B}T}f(\phi_{A},\phi_{B},\phi_{v})=\sum_{i\neq
j}\chi_{ij}\phi_{i}\phi_{j}+\sum_{i}\phi_{i}\mathrm{log}\,\phi_{i},$ (23)

where, $\chi_{ij}$ ($i,j=$A, B, v; $i\neq j$) are the pairwise interaction
energies between the components, $k_{B}$ is the Boltzmann constant, and T, the
absolute temperature.

The kinetics of phase-separation are obtained via a continuity equation, given
by

$\qquad\dfrac{\partial\phi_{i}}{\partial
t}=-\nabla\cdot{J_{i}}^{\prime}\quad(i=A,B,v)$ (24)

where ${J_{i}}^{\prime}$ is the total flux of each component in the system. We
adopt a formulation that incorporates the net vacancy flux coupled with a
Gibbs-Duhem relation as elaborated by Raghavan et al. Raghavan et al. (2021a)
and others Kramer et al. (1984); Huang et al. (1995); Bhattacharyya and
Abinandanan (2003); Sugathan and Bhattacharya (2020) to derive the temporal
evolution of the A and B-rich phases,

$\begin{array}[]{c}\dfrac{\partial\phi_{A}}{\partial
t}=M_{AA}\nabla^{2}\big{[}(\partial
f/\partial\phi_{A})-2\kappa_{AA}\nabla^{2}\phi_{A}-2\kappa_{AB}\nabla^{2}\phi_{B}\big{]}\\\
+M_{AB}\nabla^{2}\big{[}(\partial
f/\partial\phi_{B})-2\kappa_{BA}\nabla^{2}\phi_{A}-2\kappa_{BB}\nabla^{2}\phi_{B}\big{]}\end{array}$
(25)

and

$\begin{array}[]{c}\dfrac{\partial\phi_{B}}{\partial
t}=M_{BB}\nabla^{2}[(\partial
f/\partial\phi_{B})-2\kappa_{BB}\nabla^{2}\phi_{B}-2\kappa_{BA}\nabla^{2}\phi_{B}]\\\
+M_{AB}\nabla^{2}[(\partial
f/\partial\phi_{A})-2\kappa_{AB}\nabla^{2}\phi_{B}-2\kappa_{AA}\nabla^{2}\phi_{A}],\end{array}$
(26)

where, $\kappa_{AA}=\kappa_{A}+\kappa_{v}$,
$\kappa_{BB}=\kappa_{B}+\kappa_{v}$, and $\kappa_{AB}=\kappa_{BA}=\kappa_{v}$
are the gradient parameters. $M_{AA}$ and $M_{BB}$ are the atomic mobilities
of A and B atoms in non-A-rich and non-B-rich phases, respectively, while
$M_{AB}$ and $M_{BA}$ are mobilities of A atoms in B-rich phase and B atoms in
A-rich phase, respectively Huang et al. (1995); Bhattacharyya and Abinandanan
(2003). These are related to the diffusion coefficients of the alloying
components, $D_{i}$, via a Nernst-Einstein relation Cogswell (2010); Raghavan
et al. (2021a),

$M_{ii}=\dfrac{1}{k_{B}T}D_{i}\phi_{i}(1-\phi_{i})\quad(i,j=A,B,v)$ (27)

and

$M_{ij}=\dfrac{1}{k_{B}T}D_{i}\phi_{i}\phi_{j}\quad(i,j=A,B,v,\quad i\neq j)$
(28)

Eq. 25 and 26 are first made dimensionless by using the relation
$l^{*}=(\kappa_{i}/2k_{B}T)^{1/2}\Delta x$ and
$t^{*}=(k_{B}T/M^{*}_{ii}l^{*2})$ where $l^{*}$ and $t^{*}$ are the
characteristic length and time, respectively, and $M^{*}_{ii}$ is the
dimensional value of mobility for phases $i=A,B$. The dimensionless form of
these equations are then solved via an explicit finite difference scheme for
temporal and spatial derivatives.

The evolution of hillocks on the film surface are governed by the interplay of
the energies at the interface between A-rich and B-rich phases, and the
surface energy of the film. This relationship is encapsulated within Young’s
equation as

$\theta=2\;\mathrm{cos}^{-1}\left(\dfrac{\sigma_{AB}}{2\sigma_{fv}}\right)$
(29)

where $\theta$ is the contact angle at the surface, $\sigma_{AB}$ is the
interfacial energy between the phase-separated A-rich and B-rich domains and
$\sigma_{fv}$ is the energy of the film surface in contact with the vapor
phase. $f$ denotes the phase (A-rich or B-rich) at the film surface which is
in contact with the vapor phase. The surface energies of both A-rich and
B-rich phases in contact with the vapor phase are assumed to be equal. The
methodology employed to calculate the contact angles and other model-specific
information is available in Raghavan et al. (2021a).

##  III Results

###  III.1 Analysis of binary double-well system in 2D

To demonstrate the utility of the $\Omega_{n}$ and $\delta\Omega_{n}$ metrics,
we first apply them to quantify the evolution of a 2D binary system driven by
a Cahn-Hilliard model with a double-well potential (see Sec.II.D for details).
It is well know that such a system undergoes a rapid phase separation via
spinodal decomposition. For this system, it is natural to choose the initial
configuration as the reference state to compute $\Omega_{n}(t)$ via Eq. (6).

Figure 2: Upper panels: Snapshots of the evolving 2D binary system with the
double-well potential at selected time points. Lower panels: The associated
$n$-point polytope functions $P_{n}$ at the corresponding time points
(indicated by the same color code). The unit of distance $r$ is in pixels.
Figure 3: $\Omega_{n}$ metrics for the 2D binary system in both linear scale
(a) and log scale (b) with least-square linear fits indicated by dashed lines.
$\delta\Omega_{n}(t)$ for the system is shown in (c).

Fig. 2 upper panels show the selected snapshots of the evolving system, from
which one can clearly see the rapid development of the dark phase during the
initial stages of the phase separation, which asymptotically slows down. Fig.
2 lower panels shows the $P_{n}$ (with $n=2,3,4,6,8$) for each of these
snapshots. A growing length scale can be clearly identified from all of the
$P_{n}$ functions, manifested as slower decay of the functions associated with
the snapshots corresponding to the later stages of the phase separation. We
note that an estimate of this growing length scale can be obtained by finding
the distance corresponding to the first local minimum in $P_{2}$ Jiao et al.
(2013), which clearly shifts to larger distance as the phase separation
proceeds. The microstructures of this binary system at all time points are
composed of disordered interpenetrating morphology typically seen in a
spinodal decomposition, without special symmetry emerging. Therefore, all
$P_{n}$ functions rapidly decay to their corresponding long-range asymptotic
values with several oscillations at small $r$. This is distinctly different
from the patterns associated with the hillock growth process analyzed in the
subsequent section, where patterns with significant 4-fold symmetry emerge
during the evolution.

Fig. 3 shows the $\Omega_{n}$ metrics in both linear scale (a) and log scale
(b). It can be seen form Fig. 3(a) that all $\Omega_{n}(t)$ rapidly converge
to their corresponding long-time asymptotic values. We note that
$\Omega_{2}(t)$ for the system is significantly lower in values compared to
$\Omega_{3}(t)$ and $\Omega_{4}(t)$. This can be understood from Eq. (10): the
coefficient ${\cal H}_{2}$ defined by (11) is small since $P_{2}(r)$ (i.e.,
$f_{2}(r)$) is an oscillating function of $r$ with alternative positive and
negative values, leading to a small sum $\sum_{r}f_{2}(r)$ and thus, small
${\cal H}_{2}$. Therefore, the scaling of $\Omega_{2}(t)$ is mainly dominated
by its higher order term, i.e., $\Omega_{2}(t)\sim\gamma_{2}(t)$, leading to
smaller values compared to $\Omega_{3}(t)$ and $\Omega_{4}(t)$. Material
systems with ${\cal H}_{2}=0$ are called hyperuniform, which is a recently
discovered exotic disordered state in condensed matter systems Xu et al.
(2017); Jiao (2021). The log plot in Fig. 3b shows the similar time scaling
behaviors for all $\Omega_{n}(t)$, confirming the approximation (14). Based on
(14) we obtain $\Omega_{n}(t)\sim t^{\alpha}$ where $\alpha\approx 0.645$.

Fig. 3(c) shows $\delta\Omega_{n}(t)$ for the system, which reflect the
differential change of the system as quantified by different $P_{n}$ functions
as time increases. Based on Eq. (18), $\delta\Omega_{n}(t)$ are mainly
determined by the differential change of the volume fraction $\delta\phi(t)$,
which approaches 0 towards the later stages of the phase separation (i.e., as
volume fraction does not significantly change anymore). This is consistent
with the trend for $\Omega_{n}(t)$ as well, which converge to the long-time
asymptotic values towards the later stages of the evolution.

###  III.2 Analysis of pattern evolution in thin film deposition

With the utilities of $\Omega_{n}(t)$ and $\delta\Omega_{n}(t)$ illustrated
and verified in the simple binary system discussed in Sec.III.A, we now employ
them to analyze the pattern evolution in hillock formation during vapor-
deposition of phase-separating alloy films Raghavan et al. (2020); Ankit et
al. (2019); Raghavan et al. (2021b). The evolution of the alloy films has been
investigated in detail in Ref. Raghavan et al. (2021a), which was simulated
via the phase-field model briefly described in Sec.II.D. Figure 4 shows
representative snapshots of the growing film containing hillocks with contact
angle $\theta=32^{o}$ at film-vapor interface (see Ref. Raghavan et al.
(2021a) for details). Here we will focus on the dynamics of the top slices of
the systems at different times during the film growth. This is motivated by
the fact that experimentally the (nano-structured) top surface could be imaged
via in situ characterizations (such as SEM or TEM). We note this is equivalent
to take the $x-y$ slices of the thin film configurations associated with
different height along $z$ axis starting from the bottom with $z=0$. Without
loss of generality, we only focus on characterizing one of the alloy phases
(e.g., the blue phase). We also note that characteristic length scale of the
alloy films analyzed here is a few hundred nanometers, and thus, should be
referred to as nano-structures. In the following, we will still refer to them
as “microstructure” with the understanding that such microstructures contain
features on nano-scales.

Figure 4: Representative snapshots of the growing film containing hillocks
with contact angle $\theta=32^{o}$ at film-vapor interface (see Ref. Raghavan
et al. (2021a) for details).

####  III.2.1 Pattern evolution with surface contact angle $\theta=32^{o}$

Figure 5: Upper panels: Representative snapshots of the top surface patterns
associated with the blue phase at different time points during the film growth
with contact angle $\theta=32^{o}$. Lower panels: The associated $n$-point
polytope functions $P_{n}$ at the corresponding time points (indicated by the
same color code). The unit of distance $r$ is in pixels. Figure 6:
$\Omega_{n}$ metrics for the evolving surface patterns during the film growth
with contact angle $\theta=32^{o}$ in both linear scale (a) and log scale (b)
with least-square linear fit shown as dashed line. $\delta\Omega_{n}(t)$ for
the system is shown in (c).

Figure 5 (upper panels) shows representative snapshots of the top surface
patterns associated with the blue phase at different time points during the
film growth. It can be clearly seen that the evolving patterns exhibit a four-
fold symmetry, which is inherited from the initial configuration Raghavan et
al. (2021a). In Ref. Raghavan et al. (2021a), we have shown that after an
initial “transition” period, the surface patterns for both alloy phases start
to oscillate as time proceeds, e.g., one of phases starts to grow from smaller
seeds and dominate the pattern which is then gradually taken over by the other
phase. Fig. 5 (lower panels) shows the corresponding $P_{n}$ functions for the
snapshots. Different from the simple binary system analyzed in Sec.III.A, the
4-fold symmetry of the patterns is clearly manifested as the strong peaks in
both $P_{2}$ and $P_{4}$, associated with the same distance $r$. The 4-fold
symmetry and the resulting periodicity of the structures also lead to the
strong peaks observed in $P_{3}$, $P_{6}$ and $P_{6}$, albeit the distances
associated with these peaks are different from those in $P_{2}$ and $P_{4}$.

Figure 6 shows the $\Omega_{n}$ metrics in both linear scale (a) and log scale
(b), with the reference pattern chosen at $t=0$. It can be seen form Fig. 6(a)
that all $\Omega_{n}(t)$ exhibit almost periodic oscillations for the entire
evolution, while the peak values of the oscillations fluctuate during the
early stages of the evolution and subsequently converge to a steady value.
These features are consistent with the observed dynamics of the top surface
pattern of the system: the initial fluctuations of the peaks values correspond
to the “transition” period of the pattern evolution. On the other hand, the
steady peak values correspond to steady oscillations of the top surface
patterns resulted from alternating dominant red and blue phases as described
above. In addition, the magnitude of $\Omega_{n}(t)$ at a fixed $t$ decreases
as $n$ increases, due to the smaller magnitude of $P_{n}(r)$ (for $r>0$) as
$n$ increases. All $\Omega_{n}(t)$ exhibit similar time scaling behaviors, as
can be seem from the log plot shown in Fig. 6(b). We note that since
$\Omega_{n}(t)$ are oscillating functions of $t$, we only focus on the
temporal scaling of the peak values with respect to the corresponding long-
time asymptotic limit $\overline{\Omega}_{n}(\infty)$, i.e.,
$|\Omega_{n}(t)-\overline{\Omega}_{n}(\infty)|\sim t^{\alpha}$ where
$\alpha\approx 0.557$.

Fig. 6(c) shows $\delta\Omega_{n}(t)$ for the system, which exhibits very
similar trend as seen in $\Omega_{n}(t)$. In particular, all
$\delta\Omega_{n}(t)$ exhibit almost periodic oscillations; the peak values of
these oscillations fluctuate during the early transition stages of the
evolution and subsequently converge to a steady value. A closer inspection
reveals the peak positions of $\delta\Omega_{n}(t)$ correspond to valleys of
$\Omega_{n}(t)$, where the largest rate of change of $\Omega_{n}(t)$ occurs.
These features indicate that the fastest structural dynamics are associated
with the valleys of $\Omega_{n}(t)$, corresponding to patterns reminiscent of
the initial configuration at $t=0$.

Figure 7: The $\Omega^{*}_{n}(t)$ metrics for the the evolving surface
patterns during the film growth with contact angle $\theta=32^{o}$ with a
different reference state. (a) respectively shows the original reference
(upper panel) and the new reference (lower panel). (b) shows
$\Omega^{*}_{n}(t)$ in linear scale and (c) shows $\Omega^{*}_{n}(t)$ log
scale with linear fitting shown as dash line.

For an oscillating system, the choice of initial reference state is not
unique. Fig. 7 shows the $\Omega^{*}_{n}(t)$ metrics for the system with a
different reference state, which is chosen to be the one corresponding to the
highest peak of $\Omega_{2}(t)$ for all $t$, denoted by ${\cal M}^{*}_{0}$.
Based on the physical interpretation of $\Omega_{n}$, which is an effective
measure of the distance between two microstructures in the material
microstructure space, ${\cal M}^{*}_{0}$ represents the microstructure that
has the largest distance, or in other words, distinct most from, all the other
microstructures (patterns) during the evolution. We note that the effective
distance between two microstructures can be different based on different
$P_{n}$ measures. These features are all reflected in Fig. 7(a). In
particular, one can clearly see that all $\Omega^{*}_{n}(t)$ exhibit coherent
converging behavior as $t$ increases, and each $\Omega^{*}_{n}(t)$ converges
to a set of distinct asymptotic lower and upper bounds for their steady
oscillations, indicating different distances to ${\cal M}^{*}_{0}$ as measured
via different $P_{n}$. Fig. 7(b) show the log plot of $\Omega^{*}_{n}(t)$,
from which we extract the temporal scaling behavior
$|\Omega^{*}_{n}(t)-\overline{\Omega}^{*}_{n}(\infty)|\sim t^{\alpha^{*}}$
where $\alpha^{*}\approx 0.594$, which is consistent with $\alpha\approx
0.557$ estimated above, and $\overline{\Omega}^{*}_{n}(\infty)$ is the long-
time asymptotic limit of $\Omega^{*}_{n}(t)$. These results indicate that the
key behaviors of the system encoded in $\Omega_{n}$ metrics do not depend on
the choice of the reference states.

####  III.2.2 Pattern evolution with surface contact angle $\theta=51^{o}$

Figure 8: Upper panels: Representative snapshots of the top surface patterns
associated with the blue phase at different time points during the film growth
with contact angle $\theta=51^{o}$. Lower panels: The associated $n$-point
polytope functions $P_{n}$ at the corresponding time points (indicated by the
same color code). The unit of distance $r$ is in pixels. Figure 9:
$\Omega_{n}$ metrics for the evolving surface patterns during the film growth
with contact angle $\theta=51^{o}$ in both linear scale (a) and log scale (b)
with least square linear fitting shown as dashed line. $\delta\Omega_{n}(t)$
for the system is shown in (c).

Following the same procedure, we now employ $\Omega_{n}(t)$ to characterize
the pattern evolution during the thin film growth with contact angle
$\theta=51^{o}$. Figure 8 (upper panels) shows representative snapshots of the
top surface patterns associated with the blue phase at different time points
during the film growth; and Fig. 8 (lower panels) shows the corresponding
$P_{n}$ functions at selected time points. Similar to the system with
$\theta=32^{o}$, the patterns exhibit significant 4-fold symmetry inherited
from the initial pattern, which leads to the significant peaks in $P_{n}$
functions. Fig. 9 shows the $\Omega_{n}(t)$ in both linear (a) and log scales
(b). It can be seen that all $\Omega_{n}(t)$ exhibit a shorter transition zone
and rapidly converge to the steady oscillation stage. We extract the temporal
scaling behavior $|\Omega_{n}(t)-\overline{\Omega}_{n}(\infty)|\sim
t^{\alpha}$ where $\alpha\approx 0.689$, which is larger than the
corresponding scaling parameter for the system with $\theta=32^{o}$,
indicating faster converging behaviors. The $\delta\Omega_{n}(t)$ metrics are
shown in Fig. 9(c), which again exhibits very similar trend as seen in
$\Omega_{n}(t)$. The peak positions of $\delta\Omega_{n}(t)$ also correspond
to valleys of $\Omega_{n}(t)$, indicating the largest rate of change of
$\Omega_{n}(t)$ and the fastest structural dynamics occur at these patterns.

##  IV Conclusions and Discussion

In summary, we have introduced a set of novel reduced-dimension metrics,
referred to $\Omega_{n}$ which are based on the set of hierarchical n-point
polytope functions $P_{n}$, for effectively measuring the distance between two
points (i.e., microstructures) in the microstructure space and quantifying the
pathway associated with microstructural evolution. By choosing a reference
initial state (i.e., a microstructure associated with $t_{0}=0$), the
$\Omega_{n}(t)$ set quantifies the evolution of distinct polyhedral symmetries
and can in principle capture emerging polyhedral symmetries that are not
apparent in the initial state Chen et al. (2019). We have also investigated
the temporal scaling behaviors of $\Omega_{n}(t)$ and showed that the
evolution dynamics revealing the physical mechanics of the systems can be
extract from the scaling behaviors of $\Omega_{n}(t)$. To demonstrate their
utility, we have applied the $\Omega_{n}$ metrics to characterize a 2D binary
system undergoing spinodal decomposition and extract the evolution dynamics
via the temporal scaling behavior of the corresponding $\Omega_{n}(t)$. We
have also employed $\Omega_{n}(t)$ to quantify pattern evolution during vapor-
deposition of phase-separating alloy films with different surface contact
angles, which exhibit rich evolution dynamics including both unstable and
oscillating patterns.

We note that the $\Omega_{n}$ metrics are merely a special example of
correlation function based distance measures of microstructure space. Similar
metrics can be defined for special lower-order functions Jiao et al. (2009)
such as two-point cluster function $C_{2}(r)$, the surface-surface correlation
function $F_{ss}(r)$, the lineal-path function $L(r)$, and the pore-size
distribution function $P(r)$, to name but a few. The corresponding $\Omega$
metric quantifies the effective “distance” between two microstructures, mainly
resulted from the distinction of the structural features quantified by the
specific correlation functions. For example, the metric $\Omega_{C}$ based on
the cluster function $C_{2}$ distinguishes two microstructures based on their
degrees of clustering, while these two microstructures may possess identical
$P_{2}$ and $\Omega_{2}=0$ Gommes et al. (2012a, b).

The $\Omega_{n}$ metrics, when combined with in situ microstructural
characterization tools (such as x-ray tomographic microscopy), allow one to
quantitatively monitor the structural evolution in real time. During a
manufacturing process, this will enable us to apply real-time control of the
processing conditions in order to control the microstructure evolution
pathway, which is highly desirable for material optimization. In this work, we
have used $\mathbb{L}_{1}$ norm in the definition of $\Omega_{n}$, which
allowed us to extract temporal scaling that characterizes the dynamics of the
structural evolution. This can also be generalized to use more sophisticated
weighted norms that approximately connect the $P_{n}$ functions to the
physical properties of the material system Chen et al. (2019). We will explore
these directions in our future work.

###### Acknowledgements.

This work is supported by National Science Foundation, Division of Material
Research under grant NO. 2020277 (AI Institute: Planning: Novel Neural
Architectures for 4D Materials Science). This research used computational
resources of the Agave Research Computer Cluster of ASU. Y. J. thanks the
generous support from ASU during his sabbatical leave.

## References

  * Torquato (2002) S. Torquato, Appl. Mech. Rev. 55, B62 (2002).
  * Jiao et al. (2007) Y. Jiao, F. Stillinger, and S. Torquato, Physical Review E 76, 031110 (2007).
  * Jiao et al. (2008) Y. Jiao, F. Stillinger, and S. Torquato, Physical Review E 77, 031135 (2008).
  * Jiao et al. (2009) Y. Jiao, F. Stillinger, and S. Torquato, Proceedings of the National Academy of Sciences 106, 17634 (2009).
  * Roberts (1997) A. P. Roberts, Physical Review E 56, 3203 (1997).
  * Wilding and Fullwood (2011) S. E. Wilding and D. T. Fullwood, Computational materials science 50, 2262 (2011).
  * Callahan (2012) P. G. Callahan, Ph.D. thesis, Carnegie Mellon University (2012).
  * Wang et al. (2012) M. Wang, J. Williams, L. Jiang, F. De Carlo, T. Jing, and N. Chawla, Metallography, Microstructure, and Analysis 1, 7 (2012).
  * Ratanaphan et al. (2014) S. Ratanaphan, Y. Yoon, and G. S. Rohrer, Journal of materials science 49, 4938 (2014).
  * Iyer et al. (2020) A. Iyer, R. Dulal, Y. Zhang, U. F. Ghumman, T. Chien, G. Balasubramanian, and W. Chen, Computational Materials Science 179, 109559 (2020).
  * Farooq Ghumman et al. (2018) U. Farooq Ghumman, A. Iyer, R. Dulal, J. Munshi, A. Wang, T. Chien, G. Balasubramanian, and W. Chen, Journal of Mechanical Design 140 (2018).
  * Chen and Torquato (2018) D. Chen and S. Torquato, Acta Materialia 142, 152 (2018).
  * Niezgoda et al. (2008) S. Niezgoda, D. Fullwood, and S. Kalidindi, Acta Materialia 56, 5285 (2008).
  * Cecen et al. (2016) A. Cecen, T. Fast, and S. R. Kalidindi, Integrating Materials and Manufacturing Innovation 5, 1 (2016).
  * Choudhury et al. (2016) A. Choudhury, Y. C. Yabansu, S. R. Kalidindi, and A. Dennstedt, Acta Materialia 110, 131 (2016).
  * Okabe and Blunt (2005) H. Okabe and M. J. Blunt, Journal of Petroleum Science and Engineering 46, 121 (2005).
  * Fullwood et al. (2008) D. T. Fullwood, S. R. Niezgoda, and S. R. Kalidindi, Acta Materialia 56, 942 (2008).
  * Hajizadeh et al. (2011) A. Hajizadeh, A. Safekordi, and F. A. Farhadpour, Advances in water Resources 34, 1256 (2011).
  * Tahmasebi and Sahimi (2013) P. Tahmasebi and M. Sahimi, Physical review letters 110, 078002 (2013).
  * Tahmasebi et al. (2012) P. Tahmasebi, A. Hezarkhani, and M. Sahimi, Computational Geosciences 16, 779 (2012).
  * Xu et al. (2014) H. Xu, Y. Li, C. Brinson, and W. Chen, Journal of Mechanical Design 136 (2014).
  * Gerke and Karsanina (2015) K. M. Gerke and M. V. Karsanina, EPL (Europhysics Letters) 111, 56002 (2015).
  * Karsanina and Gerke (2018) M. V. Karsanina and K. M. Gerke, Physical review letters 121, 265501 (2018).
  * Feng et al. (2018) J. Feng, Q. Teng, X. He, and X. Wu, Acta Materialia 159, 296 (2018).
  * Gao et al. (2021a) Y. Gao, Y. Jiao, and Y. Liu, Applied Mathematical Modelling 91, 1117 (2021a).
  * Gao et al. (2021b) Y. Gao, Y. Jiao, and Y. Liu, Acta Materialia 204, 116526 (2021b).
  * Malmir et al. (2018) H. Malmir, M. Sahimi, and Y. Jiao, Physical Review E 98, 063317 (2018).
  * Jiao et al. (2010) Y. Jiao, F. H. Stillinger, and S. Torquato, Physical Review E 82, 011106 (2010).
  * Gommes et al. (2012a) C. J. Gommes, Y. Jiao, and S. Torquato, Physical review letters 108, 080601 (2012a).
  * Gommes et al. (2012b) C. J. Gommes, Y. Jiao, and S. Torquato, Physical Review E 85, 051140 (2012b).
  * Cang et al. (2017) R. Cang, Y. Xu, S. Chen, Y. Liu, Y. Jiao, and M. Yi Ren, Journal of Mechanical Design 139 (2017).
  * Cang et al. (2018) R. Cang, H. Li, H. Yao, Y. Jiao, and Y. Ren, Computational Materials Science 150, 212 (2018).
  * Yang et al. (2018) Z. Yang, X. Li, L. Catherine Brinson, A. N. Choudhary, W. Chen, and A. Agrawal, Journal of Mechanical Design 140 (2018).
  * Li et al. (2018a) X. Li, Y. Zhang, H. Zhao, C. Burkhart, L. C. Brinson, and W. Chen, Scientific reports 8, 1 (2018a).
  * Torquato and Stell (1982) S. Torquato and G. Stell, The Journal of Chemical Physics 77, 2071 (1982).
  * Jiao et al. (2013) Y. Jiao, E. Padilla, and N. Chawla, Acta Materialia 61, 3370 (2013).
  * Guo et al. (2014) E.-Y. Guo, N. Chawla, T. Jing, S. Torquato, and Y. Jiao, Materials Characterization 89, 33 (2014).
  * Jiao and Chawla (2014) Y. Jiao and N. Chawla, Journal of Applied Physics 115, 093511 (2014).
  * Chen et al. (2015) S. Chen, H. Li, and Y. Jiao, Physical Review E 92, 023301 (2015).
  * Chen et al. (2016) S. Chen, A. Kirubanandham, N. Chawla, and Y. Jiao, Metallurgical and Materials Transactions A 47, 1440 (2016).
  * Xu et al. (2017) Y. Xu, S. Chen, P.-E. Chen, W. Xu, and Y. Jiao, Physical Review E 96, 043301 (2017).
  * Li et al. (2018b) H. Li, P.-E. Chen, and Y. Jiao, Transport in Porous Media 125, 5 (2018b).
  * Li et al. (2018c) H. Li, S. Singh, N. Chawla, and Y. Jiao, Materials Characterization 140, 265 (2018c).
  * Cheng et al. (2021) S. Cheng, Y. Jiao, and Y. Ren, arXiv preprint arXiv:2109.02255 (2021).
  * Chen et al. (2019) P.-E. Chen, W. Xu, N. Chawla, Y. Ren, and Y. Jiao, Acta Materialia 179, 317 (2019).
  * Chen et al. (2020) P.-E. Chen, W. Xu, Y. Ren, and Y. Jiao, Physical Review E 102, 013305 (2020).
  * Raghavan et al. (2021a) R. Raghavan, P.-E. Chen, Y. Jiao, and K. Ankit, Journal of Applied Physics 129, 245301 (2021a).
  * Rickman et al. (1997) J. Rickman, W. Tong, and K. Barmak, Acta materialia 45, 1153 (1997).
  * Rickman and Barmak (2017) J. Rickman and K. Barmak, Physical Review E 95, 022121 (2017).
  * Lavrukhin et al. (2021) E. Lavrukhin, M. V. Karsanina, and K. Gerke, submitted (2021).
  * Lu and Torquato (1992) B. Lu and S. Torquato, Physical Review A 45, 922 (1992).
  * Prager (1963) S. Prager, Chemical Engineering Science 18, 227 (1963).
  * Torquato et al. (1988) S. Torquato, J. Beasley, and Y. Chiew, The Journal of chemical physics 88, 6540 (1988).
  * Torquato and Avellaneda (1991) S. Torquato and M. Avellaneda, The Journal of chemical physics 95, 6477 (1991).
  * Cahn and Hilliard (1958) J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 (1958), ISSN 00219606.
  * Raghavan et al. (2020) R. Raghavan, A. Mukherjee, and K. Ankit, J. Appl. Phys. 128, 175303 (2020).
  * Morral and Cahn (1971) J. E. Morral and J. W. Cahn, Acta Metall. 19, 1037 (1971), ISSN 0001-6160.
  * Chen (1994) L.-Q. Chen, Acta Metall. Mater. 42, 3503 (1994).
  * Kramer et al. (1984) E. J. Kramer, P. Green, and C. J. Palmstrøm, Polymer 25, 473 (1984).
  * Huang et al. (1995) C. Huang, M. O. de La Cruz, and B. Swift, Macromolecules 28, 7996 (1995).
  * Bhattacharyya and Abinandanan (2003) S. Bhattacharyya and T. Abinandanan, B. Mater. Sci. 26, 193 (2003).
  * Sugathan and Bhattacharya (2020) S. Sugathan and S. Bhattacharya, Comp. Mater. Sci. 172, 109284 (2020).
  * Cogswell (2010) D. A. Cogswell, Ph.D. thesis, Mass. Inst. Technol. (2010).
  * Jiao (2021) Y. Jiao, Physica A: Statistical Mechanics and its Applications p. 126435 (2021).
  * Ankit et al. (2019) K. Ankit, B. Derby, R. Raghavan, A. Misra, and M. J. Demkowicz, J. Appl. Phys. 126 (2019), ISSN 10897550.
  * Raghavan et al. (2021b) R. Raghavan, W. Farmer, L. T. Mushongera, and K. Ankit, Computational Materials Science 199, 110724 (2021b).