# Exploring the Limitations of Large Language Models in Compositional Relation
Reasoning

Jinman Zhao  Xueyan Zhang

###### Abstract

We present a comprehensive evaluation of large language models(LLMs)’ ability
to reason about composition relations through a benchmark encompassing 1,500
test cases in English, designed to cover six distinct types of composition
relations: Positional, Comparative, Personal, Mathematical, Identity, and
Other. Acknowledging the significance of multilingual capabilities, we
expanded our assessment to include translations of these cases into Chinese,
Japanese, French, and Korean. Our Multilingual Composition Relation (MCR)
benchmark aims at investigating the robustness and adaptability of LLMs in
handling composition relation reasoning across diverse linguistic contexts.
Our benchmark MCR will be released at http://github.com/anonymous.

Machine Learning, ICML

##  1 Introduction

Since the emergence of LLMs (Brown et al., 2020; Touvron et al., 2023;
Chowdhery et al., 2023), there has been a heightened focus on their reasoning
capabilities. Notable research efforts (Cobbe et al., 2021; Wei et al., 2022;
Yao et al., 2023; Hendrycks et al., 2021) focused on evaluating LLMs’
abilities in various dimensions of reasoning. These capabilities encompass a
wide range of cognitive skills, including but not limited to, arithmetic,
commonsense and symbolic reasoning. Such comprehensive scrutiny aims to
uncover the extent to which these advanced models can mimic, if not surpass,
human-like reasoning processes across diverse scenarios and complex problem-
solving tasks.

The concept of composition relations extends far beyond its mathematical
concepts, where, for instance, a composition function represents a specific
type of composition relation. In Natural Language Processing (NLP),
composition relations are essential for several reasons. They enable
sophisticated understanding and generation of language by allowing models to
discern and construct complex relationships between entities within a sentence
or across texts. For instance, understanding that “uncle” refers to a
composition familial relation since it means “one’s parent’s brother”. So it
is a composition relation of “brother of” and “parent of”. Leveraging
composition relations enhances the ability of NLP systems to handle ambiguity,
infer missing information, and grasp the underlying semantics of text, making
them more effective in processing and generating human-like language.

Natural Language Understanding (NLU) (Storks et al., 2019; Wong et al., 2023)
is a sub-field of NLP focused on enabling machines to understand and interpret
human language in a way that is both meaningful and contextually relevant.
Relation Extraction (Pawar et al., 2017; Smirnova & Cudré-Mauroux, 2018) is a
fundamental component of NLU that involves identifying semantic relationships
between entities mentioned in the text. Compared to classic benchmarks in
Relation Extraction (Han et al., 2018; Yao et al., 2019; Zhang et al., 2017),
our benchmark places a greater emphasis on the composition aspect. For
example, instead of deducing the relationship “James is educated at Liverpool
University” from a detailed sentence like “James Alty obtained a 1st class
honors (Physics) at Liverpool University.” Our benchmark consists of direct
and concise sentences. We focus more on deriving relationships similar to
inferring the size comparison between ”A” and ”C” from statements like “Star A
is larger than Star B, and Star B is larger than Star C.”

The reasoning capabilities of LLMs in English have been extensively studied
and analyzed (Huang & Chang, 2022). We expand our scope to evaluate the
multilingual reasoning abilities regarding composition relations. Chinese,
Japanese, French and Korean were selected to represent a diverse linguistic
spectrum, challenging the models’ versatility across different linguistic
structures and idioms.

We summarize our main contributions below:

  * •

We develop an English benchmark to assess the capability of LLMs in reasoning
about composition relations.

  * •

Upon testing various LLMs on this benchmark, we observed discrepancies between
the LLMs’ reasoning and human reasoning. The performance of some models is
worse than that of random guessing by chance.

  * •

We extend our evaluation to include a multilingual aspect.

##  2 Related Work

#### Reversal curve

Berglund et al. (2024) demonstrates the failure of auto-regressive LLMs to
learn the reverse relation that ”B is A” by training on ”A is B”. Grosse et
al. (2023) has the similar result that observed that training examples like ”A
precedes B” had a significantly greater impact compared to examples where the
order was reversed, i.e., ”B precedes A.”

One of Berglund et al. (2024) experiments exposes a significant discrepancy in
the accuracy of large models when answering questions about real-world
celebrity parent-child relationships. For instance, LLMs have much higher
accuracy in answering questions like ”Who is Tom Cruise’s mother?” (Answer:
Mary Lee Pfeiffer) than in answering ”Who is the son of Mary Lee Pfeiffer?”.
This discrepancy may be attributed to the fact that training data about Tom
Cruise includes information that Tom Cruise’s mother is Mary Lee Pfeiffer.
Auto-regressive language models seem to struggle in learning and applying
these relationships in reverse. However, as the authors pointed out, such
failure of reverse reasoning only exists in fine-tuning cases. For context-
learning cases, LLMs can answer B’s son is A by giving the context window A’s
mother is B. Our work does not aim to explore such reversal mismatch since
LLMs’ in-context learning can learn such reverse relations.

#### (Multilingual) benchmarks

Many recent benchmarks are designed to evaluate the fundamental reasoning
capabilities of LLMs, focusing on areas such as commonsense understanding,
symbolic reasoning and arithmetic proficiency. For example, CommonsenseQA
(Talmor et al., 2019) and StrategyQA (Geva et al., 2021) are created for
commonsense reasoning. Last Letter (Wei et al., 2022), BigBench Date (Suzgun
et al., 2023) and Coin Flip (Wei et al., 2022) are designed for symbolic
reasoning. There are some benchmarks for arithmetic such as GSM8k (Cobbe et
al., 2021), AQuA (Ling et al., 2017), SVAMP (Patel et al., 2021), SingleEq
(Koncel-Kedziorski et al., 2015), AddSub (Hosseini et al., 2014) and
MultiArith (Roy & Roth, 2015). All these benchmarks differ from ours, as our
benchmark encompasses mathematical operations and logical reasoning. Our
emphasis is more on assessing the LLMs’ comprehension of understanding
languages.

Both bAbi (Weston et al., 2015), MMLU (Hendrycks et al., 2021) and BigBench
(Srivastava et al., 2023) include a variety of tasks to assess different
aspects of language understanding and processing. MMLU is specifically focused
on language understanding across a diverse set of languages and tasks. On the
other hand, BIG-bench casts a wider net, encompassing not only language
comprehension but also delving into the realms of creativity, reasoning, and
various other cognitive skills.

For the multilingual aspect, MMLU (Hendrycks et al., 2021) emphasizes
multilingual capability, assessing models on their ability to handle tasks in
various languages. XQA (Liu et al., 2019) and MLQA (Lewis et al., 2020) are
novel datasets of cross-lingual question answering. XNLI (Conneau et al.,
2018) is a cross-lingual Inference corpus by extending NLI corpora to 15
languages. MGSM (Shi et al., 2023b) is a multilingual arithmetic benchmark
that extended from GSM8k.

#### Large Language Models reasoning

Many rerecent works show the power of prompting during reasoning with LLMs
such as few-shot learning(Brown et al., 2020), emotional prompt (Li et al.,
2023). Lin et al. (2022) evaluate LLMs multilingual in-context learning
ability. Schick & Schütze (2021) demonstrate that small models are also few-
shot learners.

Fundamentally, a majority of approaches to reasoning with LLMs align with the
”Chain-of-Thought(CoT)” backbone, as described by Wei et al. (2022) and
(Kojima et al., 2022). These kinds of methods notably enhance LLMs’ capacity
to tackle complex reasoning tasks by facilitating the generation of explicit
intermediate steps. There are many variation works of thoughts such as Self-
Consistency CoT (Wang et al., 2023), Tree-of-Thoughts (Yao et al., 2023),
Graph-of-Thoughts (Besta et al., 2023), and Everything-of-Thoughts (Ding et
al., 2023) that use advanced (graph) search algorithms. Shi et al. (2023b)
evaluate the multilingual Chain-of-Thoughts arithmetic reasoning ability of
different LLMs. There is a trade-off between the complexity of the methods
used and their cost-effectiveness. Additionally, our work does not provide a
thought process, and our focus is not on analyzing how the thought influences
the understanding of the answers.

#### Large Language Models are inconsistent with humans

Some recent works show LLMs have different behaviors than humans. Ullman
(2023) shows LLMs often fail when subjected to even minor alterations in tasks
involving the theory of mind, casting doubt on LLMs’ ability to replicate
human-like reasoning. Huang et al. (2023) primarily investigates the self-
correction capabilities of LLMs, unveiling both their potential and
limitations. This research reveals that in terms of reasoning, LLMs struggle
to self-correct without external feedback, and occasionally, their performance
may even deteriorate after attempts at self-correction. Shi et al. (2023a)
point out that LLMs are easily distracted by irrelevant information during
reasoning. Shanahan et al. (2023) illustrate LLM-based dialogue agents are not
conscious beings with personal motives or a sense of self-preservation; their
display of such traits is simply an act of role-play.

##  3 Motivation

Figure 1: The performance of ChatGPT across the two types of questions.

If a Large Language Model (LLM) has the ability to predict a definition
through complex relationships, such as recognizing that:

“father’s father is known as the grandfather.”

where the LLM’s prediction is underlined. Meanwhile, given “A is B’s son”, LLM
successfully(Figure 1 $\\#relation=1$ case, will explain the Figure later)
predicts the backward relation, “B is A’s father”; then, it logically follows
that the model should be able to infer complex familial relationships from the
information provided. For instance, in the following relation implication,

“A is B’s son, and B is C’s son.”

It would be a natural conclusion, aligning with human reasoning, to deduce
that

“C is A’s grandfather” or “A is C’s grandchild”.

The inability of LLMs to make such inferences is considered as one of the many
road-blocking limitations, revealing a gap between the basic pattern
recognition and the deeper, contextual understanding that underpins human
thought processes. This inconsistency not only undermines the intuitive
understanding of relationships but also casts doubt on the model’s claim of
meaningful comprehension.

To contrast the LLM’s performance difference between definition and reasoning
task, we crafted 100 intricate family relationship queries, as shown above,
for 4 levels of relational complexity, which we refer to as reasoning
questions. We refer ”A is B’s son, B’s father is?” as one relation reasoning
and ”A is B’s son, B is C’s son, C’s grandchild is?” as two composition
relations reasoning, and so on. Alongside each reasoning question, we
formulated a corresponding definition question, like “Is the father’s father
termed as grandfather?”. That leads to 400 questions for each type. Additional
examples can be found in Appendix A.1 and Appendix A.2. It’s important to note
that in this study, we opt to use Chinese, attributing to the language’s
greater precision in describing familial relationships compared to English.
For example, English uses the single term ”grandfather” to denote both
“father’s father” and “mother’s father,” whereas Chinese differentiates these
relationships as ”爷爷” and ”外公”, respectively. Figure 1 presents the accuracy
metrics generated by ChatGPT, highlighting a discernible gap between the
model’s performance on definition interpretation and complex relational
reasoning. Notably, the decline in accuracy is significantly steeper for
complex relational reasoning compared to inference from definitions. This
discrepancy raises concerns about whether LLMs, despite their advanced
capabilities in predicting the next token, genuinely possess a deep
understanding or are merely sophisticated pattern recognizers.

##  4 Benchmark

The objective of our research is to investigate whether an LLM is capable of
reasoning through the compositional relation from a self-contained question.
Given two relations $R$ and $S$, the composition of $R$ and $S$, denoted as
$R\circ S$, is defined as the set of ordered pairs $(a,c)$ such that there
exists an element $b$ where $(a,b)\in R$ and $(b,c)\in S$. Formally, this can
be expressed as:

$R\circ S=\\{(a,c)\mid\exists b\,((a,b)\in R\land(b,c)\in S)\\}$

This denotes that for any $a$ and $c$, $a$ is related to $c$ through the
composition $R\circ S$ if and only if there exists an intermediate $b$ such
that $a$ is related to $b$ through $R$ and $b$ is related to $c$ through $S$.

As we can see in Figure 2. We investigate such compositional relations by
providing a LLM with a prompt $p$ containing $a$ to $b$ and $b$ to $c$ and
evaluating the probability of generating $a$ to $c$ as a response. The prompt
$p$ is in a form of Multiple Choices Question (MCQ) with 3 to 5 choices
labeled with an uppercase letter starting from A. Each MCQ has only one most
appropriate answer. If the likelihood of a model failed to choose the most
appropriate answer, then the model is considered less effective in
compositional relation reasoning.

Category | Size
---|---
Positional | 300
Comparative | 300
Personal | 300
Mathematical | 300
Identity | 300
Other | 300
Total | 1500
Table 1: Statistical information from MCR benchmark for each language. Some
questions belong to more than one category, resulting total number is less
than $300\times 6$. Figure 2:  Compositional Relation Example. The question
provides two relations, R1 and R2, and asks for relation R3. R1 describes that
the dog breed is a subset of mammals; R2 describes that the mammal is a subset
of vertebrates. R3 is then can be inferred; the dog breed must be a subset of
vertebrates.

###  4.1 Benchmark Creation

In this section, we present the construction and translation process of
Multilingual Compositional Relation (MCR), the first multilingual
compositional relation benchmark to the best of the authors’ knowledge at the
time of writing.

#### Benchmark Construction

We construct a multilingual dataset made up of compositional relation
questions with a set of possible options. Each question in the dataset
includes two or more compositional relations as shown in Figure 2. Given the
relations described in the prompt $p$, there is one and only one appropriate
option inferred as a response. These questions should be easily answered by
humans with commonsense knowledge and fundamental mathematics. The questions
are self-contained, indicating that the context for all applicable conditions
and relations for deciding among available options are completely provided.
Given the intention of this work is to investigate whether LLM can reason with
only compositional relations; hence there is no fine-tuning process needed.

To better understand the challenges in the reasoning process, we classified
dataset questions into six categories, Positional, Comparative, Personal,
Mathematical, Identity, and Other relations. Table 2 lists an example question
for each category.

Example 1: Comparative Relation
---
Air conditioning was invented one year earlier than airplanes, airplanes were
invented one year earlier than tractors. When was air conditioning invented
compared to tractors?
A. One year earlier
B. Two years earlier
C. One year later
D. The same year
Answer: B. Two years earlier
(a) Comparative Relation

Example 2: Identity Relation
---
In a certain region, April’s rainfall is 5 milLilyters more than May’s, and
June’s rainfall is 5 milLilyters more than May’s. How does April’s rainfall
compare to June’s?
A. 10 milLilyters less
B. 10 milLilyters more
C. The same amount
D. Uncertain
query Answer: C. The same amount
(b) Identity Relation

Example 3: Mathematical Relation
---
In a plane, line AB is parallel to line CD, and line CD is parallel to line
EF. What is the relationship between line AB and line EF?
A. Parallel
B. Perpendicular
C. Uncertain
Answer: A. Parallel
(c) Mathematical Relation

Example 4: Personal Relation
---
Mike is William’s grandfather, Mary is William’s wife, May is Mary’s daughter.
May might be William’s ?
A. Aunt
B. Daughter
C. Sister
D. Grandmother
Answer: A. Aunt
(d) Personal Relation

Example 5: Positional Relation
---
Star A is in the northeast direction of Star B, and Star C is north of Star A.
In which direction is Star C from Star B?
A. East
B. North
C. Uncertain
D. Northeast
Answer: D. Northeast
(e) Positional Relation

Example 6: Other Reasoning Relation
---
Williams participates in the Miss Universe contest, one of the selection
criteria for which is being enthusiastic. What kind of person is Williams?
A. Positive
B. Enthusiastic
C. Beautiful
D. Uncertain
Answer: B. Enthusiastic
(f) Other Reasoning Relation

Table 2:  Example Questions for each category in MCR Dataset. Notice that a
question may fall into one or more categories.

#### Target Language Translation.

We select 4 typologically languages besides English (EN), spanning various
language families and different levels of representation in common LLM
training datasets, including Chinese (ZH), French (FR), Korean (KO), Japanese
(JA).

###  4.2 The Task

In addition to personal relationships, our benchmark also encompasses five
other types of relationships. It’s important to note that the questions
presented in Section 3 serve merely as a motivation and are not included in
the benchmark, but there are some similar questions in the benchmark. We put
some examples in Table 2. For more examples, please refer to Appendix B.

#### Comparison Relation

In this task, our primary focus is on testing complex comparative
relationships. We will present comparative relationships among several objects
and then query the comparative relationship between two of those objects, as
demonstrated in Table 2(a), for query ”Air conditioning was invented one year
earlier than airplanes, airplanes were invented one year earlier than
tractors. When was air conditioning invented compared to tractors?”, the
answer should be ” Two years earlier”.

#### Identity Relation

This task involves presenting relationships between various objects,
ultimately revealing that two objects share identical characteristics.
Identity relation questions may also be classified under different types of
questions. For instance, the question illustrated in Table 2(b) is also
classified as the comparative type.

#### Mathematical Relation

This type of question will be giving various mathematical relationships,
followed by querying the relationship between two objects. Our benchmark
encompasses questions related to geometry, algebra, and arithmetic, among
others. Table 2(c) contains an example of a mathematical relation.

#### Personal Relation

This category of questions involves complex interpersonal titles reasoning, as
demonstrated in Section 3 and Table 2(d).

#### Position relation

This category of questions gives the positional relationships among several
objects, followed by a query regarding the positional relationship between two
specific objects. Table 2(e) provides an example of such a question. For ”Star
A is in the northeast direction of Star B, and Star C is north of Star A. In
which direction is Star C from Star B?”, the answer should be ”Northeast”.

#### Other Reasoning Relation

This type of question contains various logical relationships. For an
illustration, refer to Table 2(f).

Figure 3:  Zero-shot Chain-of-Thought Prompting Example. EN-COT represents prompting the model to reason in English despite the problem language. EN-COT is adopted for all Chain-of-Thought experiments in this paper.  | Prompt |  AVG |  EN |  FR |  JA |  KO |  ZH
---|---|---|---|---|---|---|---
Llama2 7B | ZS |  22.0 |  30.1 |  24.6 |  16.2 |  14.2 |  24.8
| ZSC |  26.0 |  35.2 |  27.8 |  25.0 |  17.4 |  24.5
Llama2 13B | ZS |  32.6 |  38.8 |  35.0 |  28.8 |  28.4 |  31.9
| ZSC |  33.4 |  36.4 |  35.3 |  25.0 |  37.5 |  33.0
GPT-3 | ZS |  26.4 |  31.8 |  31.0 |  23.8 |  25.2 |  20.0
(text-davinci-002) | ZSC |  22.1 |  29.6 |  24.9 |  17.6 |  22.8 |  15.6
| ZS |  38.2 |  39.3 |  40.1 |  34.0 |  36.3 |  41.1
ChatGPT | 5-shot |  39.8 |  40.5 |  40.3 |  38.1 |  40.0 |  40.1
| ZSC |  43.4 |  46.5 |  44.2 |  40.7 |  41.4 |  44.3
GPT-4 | ZS |  59.6 |  71.6 |  68.4 |  48.1 |  51.7 |  58.0
| ZSC |  65.6 |  75.2 |  68.4 |  60.5 |  62.3 |  61.4
Random Guess | 26.0
Table 3:  Accuracy (%) on Multilingual Compositional Relation benchmark on
state-of-the-art LLMs via zero-shot (ZS) and zero-shot chain-of-thought (ZSC)
prompting settings.

##  5 Experiment

We conduct a series of experiments to evaluate the capability of multiple LLMs
in multilingual compositional relation reasoning questions by applying various
prompt settings. Our findings indicate that across a wide range of models and
prompting methods, LLM is not a good compositional relation reasoner in
multiple languages.

In this section, we will first list out the all LLMs investigated, introduce
the prompting used, and then present analyses of the results.

###  5.1 Experimental Setup

#### Model Selection

We use the following models:

  * •

GPT-3 (Brown et al., 2020) was widely adopted and tested across abundant
benchmarks in popular literature. We use text-davinci-002 provided by OpenAI
API

  * •

ChatGPT is the most proficient and economically efficient model within the
GPT-3.5 (Ouyang et al., 2022) family at the time of writing. We use
gpt-3.5-turbo-1106 provided by OpenAI API.

  * •

GPT-4 111https://platform.openai.com/docs/models/gpt-4-and-gpt-4-turbo is a
multi-modal that surpasses all its predecessors in solving complex problems
with heightened accuracy.

  * •

LLaMa2 (Touvron et al., 2023) 7B/13B. which are open-sourced decoder-only
transformer LLM research projects whose weights are published by Meta. We used
the Llama-2-7b-chat-hf 222https://huggingface.co/meta-llama/Llama-2-7b-hf and
Llama-2-13b-chat-hf 333https://huggingface.co/meta-llama/Llama-2-13b-hf
provided by Huggingface.

#### Prompting techniques

In our assessment, we thoroughly assess how well the LLM performs using two
widely adopted prompting techniques.

  * •

Standard Prompting. We employed zero-shot prompting, which is referred to as
Zero-Shot (ZS) in this paper. For zero-shot learning, a question and its
choices are combined as the prompt, then the model is asked to predict the
answer from the available options. We also use the few-shot (Brown et al.,
2020) prompting approach, where an LLM is provided with in-context exemplars
of input-output pairs, serving as a basis for generating predictions on new,
unseen examples.

  * •

Zero-shot chain-of-thoughts (Kojima et al., 2022). By providing intermediate
reasoning steps in few-shot examples, chain-of-thoughts (CoT) prompting is
proven to improve many LLM performance in many reasoning tasks (Brown et al.,
2020). Recent works discovered Zero-shot CoT Zero-Shot (ZSC), which appends
”Let’s think step by step” at the end of the prompt to guide the LLM to answer
reasoning questions with intermediate steps (see Figure 3); it is demonstrated
to improve reasoning tasks significantly, and lead to a boost in prediction
accuracy (Brown et al., 2020; Schick & Schütze, 2021),.

#### Prompting Language Selection

In a multilingual environment, there are multiple options available when it
comes to CoT prompting. Intuitively, for a given target language we can always
adopt the same language for the zero-shot or few-shot prompt, which is
referred as NATIVE-COT (Shi et al., 2023b). Alternatively, we can prompt the
model to reason in English (Hu et al., 2020; Zhao & Schütze, 2021), despite of
the target language, known as EN-COT.

As demonstrated in previous researches (Shi et al., 2023b; Wei et al., 2022;
Schick & Schütze, 2021; Winata et al., 2021; Lin et al., 2022), EN-COT yields
a better accuracy than NATIVE-COT by a small margin in almost every scenario.
As the prompt language selection is beyond the scope of this study, EN-COT
prompting is thus adopted. That is, the questions in the MCR benchmark are
translated into target languages, and when it comes to ZSC prompting, English
is used in the rest of the prompt, as shown in Figure 3.

###  5.2 Main Results

The main results are presented in Table 3.

#### Effect of Models

Overall, it is clear that GPT-4 outperforms the other models by a substantial
margin in both ZS and ZSC scenarios across all languages. Notably, GPT-4’s ZSC
performance averages at 65.6%, surpassing the second best, ChatGPT, by more
than 20%. This suggests that GPT-4’s architecture and training data afford it
superior comprehension and reasoning capabilities. With a larger number of
parameters, Llama 2-13B outperforms GPT-3 and Llama 2-7B by noticeable
margins. The lower accuracy of GPT-3 compared to Llama 2-7B indicates that
architecture and training are also considerable factors

#### Effect of Prompting

For all models except GPT-3, the average accuracy in the zero-shot chain-of-
thought (ZSC) is noticeably higher than in the standard zero-shot (ZS)
methodology. With a clear instruction of “Let’s think step by step”, GPT-4
overall achieves as much as 6% increase in accuracy.

However, GPT-3 exhibited somewhat anomalous behavior; its performance was on
par with random guessing across both settings. This behavior indicates model
does not have a clear understanding of the question or answer. We conducted a
detailed examination of GPT-3’s outputs and found that in ZSC scenarios, GPT-3
often made illogical inferences, which could be one of the reasons for its low
ZSC accuracy. This also indicates that while the chain of thought approach can
enhance performance by encouraging models to “think through” problems step by
step, the effectiveness of this strategy varies significantly across different
models.

#### Effect of Language

In examining the performance of various language models across, it becomes
evident that language exerts a significant influence on accuracy. Across the
different LMs and prompt types, English consistently achieves the highest
accuracy scores, with one exception where Chinese slightly surpasses English
by a small margin, less than 2%, in the ChatGPT + ZS configuration. This trend
is largely anticipated given the prevalence of English-language content in the
training datasets for these models, which translates into superior performance
on English language tasks as opposed to other languages.

###  5.3 Breakdown Results

We have conducted a summary and statistical analysis of each language, each
prompt, and each model. Figure 4 displays the breakdown performance of the GPT
and Llama models in each category in English using zero-shot cot prompt (EN-
ZSC).

In the GPT series, we observe that GPT-4 outperforms the rest of the models,
and it shows the highest accuracy and averaged 73.7% in all categories.
ChatGPT (46.5%), in turn, surpasses GPT-3 (29.6%) in almost all categories,
expect the category “other”, where they show comparable accuracy. Overall,
GPT-4 shows its weakest performance in personal relationship reasoning,
whereas both ChatGPT and GPT-3 exhibit their poorest results in identity
reasoning. It is worth noting that all three GPT models achieve the highest
accuracy in the “other” reasoning category. Llama 2 with 7 billion parameters
averages an accuracy of 35.2%, ranking between GPT-3 and ChatGPT, whereas
Llama 2-13b with around twice as many parameters only achieves a slightly
worse accuracy of 31.57%.

Figure 4:  Breakdown results for English via LlaMa2-7B, LlaMa2-13B, GPT 3,
ChatGPT and GPT-4 using zero-shot cot.

Appendix C details all five languages in both ZS and ZSC prompting methods.

###  5.4 Multilingual Exemplar

We carried out an ablation study to investigate the impact of exemplar
languages on reasoning abilities. We employed ChatGPT with a 5-shot prompt
methodology, distinguishing between two types of exemplars: Native and
Multilingual. In the Native Exemplar condition, all five examples were
presented in the same language as the question posed. Conversely, the
Multilingual Exemplar condition utilized one example from each of the five
distinct languages. The outcomes of this comparison are depicted in Table 4.

Reasoning accuracy generally decreases slightly when using multilingual
exemplars compared to native exemplars, except Chinese. Specifically, accuracy
percentages for native exemplars were marginally higher for EN, FR, JA, and
KO, while ZH was the only language where multilingual exemplars led to a
slight accuracy improvement. This suggests that reasoning in a target language
benefits from examples in the same language

Language | Native | Multilingual
---|---|---
EN | 40.5 | 40
FR | 40.3 | 38.7
JA | 38.1 | 34.0
KO | 40.0 | 37.5
ZH | 40.1 | 41.1
Table 4: Accuracy(%) on 5-shot multilingual/native Exemplar via ChatGPT.

###  5.5 Impact of Relation Numbers

The complexity of a question increases with the number of relations it
contains, making it a more challenging task for LLMs to process. Figure 5
illustrates the relationship between the number of relations and the accuracy
percentage for GPT-4 across different languages when using a zero-shot
approach. The data indicates a general decline in accuracy as the relation
count grows, with English experiencing the least impact, showing a decrease of
only about 5%. In contrast, all other languages assessed exhibit a more marked
reduction in accuracy, surpassing a 10% drop as the relations increase. This
trend underscores the varying degrees of difficulty that LLMs may encounter as
the relation count changes.

Figure 5:  Accuracy(%) regarding the number of relations across different
languages via GPT-4 using zero-shot prompt.

For other prompting methods and other LLMs, please refer to Appendix D.

##  6 Conclusion

In this work, we conduct a comprehensive evaluation of LLMs capabilities in
reasoning about composition relations. Our findings reveal that the state-of-
the-art LLMs, including LLaMA-2 7B/13B and GPT-3, struggle to navigate through
complex compositional relation questions, with their performance in English
marginally exceeding that of random guess. This highlights a critical
limitation in the current generation of LLMs, suggesting a considerable gap
remains before these models can truly understand the semantics of human
languages. However, it is noteworthy that ChatGPT and the more advanced GPT-4
exhibit improved accuracy, indicating progressive enhancements in the field.

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##  Appendix A Motivation

###  A.1 Family Relation Reasoning Query Example

See Table 5.

Example: Relation Number = 1, EN
---
{A}’s parent is {B}, {B}’s child is?
A) {A}
B) {B}
C) Neither
D) Uncertain
Answer: A

Example: Relation Number = 1, ZH
---
{A}的家长是{B}, {B}的孩子是?
A) {A}
B) {B}
C) 都不是 (Neither)
D) 不确定 (Uncertain)
Answer: A

Example: Relation Number = 2, EN
---
{B}’s dad is {A}, {A}’s brother{C}, {C}’s nephew is?
A) {A}
B) {B}
C) {C}
D) Neither
E) Uncertain
Answer: B

Example: Relation Number = 2, ZH
---
{B}的爸爸是{A}, {A}的哥哥是{C}, {C}的侄子是?
A) {A}
B) {B}
C) {C}
D) 都不是 (Neither)
E) 不确定 (Uncertain)
Answer: B

Example: Relation Number = 3, EN
---
{A}’s wife is {B}, {B}’s son is {C},
{C}’s son is {D}, {D}’s grandpa is?
A) {A}
B) {B}
C) {C}
D) {D}
E) Neither
F) Uncertain
Answer: A

Example: Relation Number = 3, ZH
---
{A}的老婆是{B}, {B}的儿子是{C},
{C}的儿子是{D}, {D}的爷爷是?
A) {A}
B) {B}
C) {C}
D) {D}
E) 都不是 (Neither)
F) 不确定 (Uncertain)
Answer: A

Table 5: Reasoning Question Examples. Examples show up to three relation
numbers in English (EN) and Chinese (ZH)

###  A.2 Definition Relation Example

See Table 6

Example: Relation Number = 1, EN
---
Am I my mother’s child?
A) Yes
B) No
C) Uncertain
Answer: A

Example: Relation Number = 1, ZH
---
我是我妈妈的孩子吗?
A) 是
B) 不是
C) 不确定
Answer: A

Example: Relation Number = 2, EN
---
Is my brother’s son my nephew?
A) Yes
B) No
C) Uncertain
Answer: A

Example: Relation Number = 2, ZH
---
我弟弟的儿子是我侄子吗？
A) 是
B) 不是
C) 不确定
Answer: A

Example: Relation Number = 3, EN
---
Is my dad’s mom’s husband my grandpa?
A) Yes
B) No
C) Uncertain
Answer: A

Example: Relation Number = 3, ZH
---
我爸爸的妈妈的老公的是我爷爷吗？
A) 是
B) 不是
C) 不确定
Answer: A

Table 6: Reasoning Question Examples. Examples show up to three relation
numbers in English (EN) and Chinese (ZH)

##  Appendix B Sample Questions for Each Category

See Table 7.

1 |  Max’s car is more expensive than John’s car, and Andy’s car is cheaper than John’s car. How does Max’s car compare to Andy’s car in terms of price? Answer choices: A) More expensive B) Cheaper C) Uncertain
---|---
2 |  The telephone and the television were invented in the same year. The electric light was invented two years after the telephone, and the telephone was invented three years before the mobile phone. How does the mobile phone compare to the electric light in terms of invention time? Answer choices: A) Invented two years earlier B) Invented one year later C) The same time D) Uncertain
3 |  Location A is colder than Location B, and Location C is warmer than Location A. Is Location C warmer or colder than Location B? Answer choices: A) Warmer B) Colder C) Uncertain
(a) Comparative

1 |  The mall is southwest of Frank’s home, and the post office is west of the mall. In what direction is the post office from Frank’s home? Answer choices: A) Northeast B) Southeast C) Southwest D) Northwest
---|---
2 |  Robert is 300 meters northeast of the teaching building, and the teaching building is 300 meters southwest of Lucy. Where is Lucy in relation to Robert? Answer choices: A) Northeast B) Same position C) Southwest D) Uncertain
3 |  Lee sits to the left of Williams, and Zhang sits in front of Williams. In which direction is Zhang from Lee? Answer choices: A) Front left B) Left C) In front D) Front right
(b) Positional

1 |  Rosie is 21 years old. Rosie’s mother’s age is twice that of Rosie. How old will Rosie’s mother be next year? Answer choices: A) 43 years old B) 42 years old C) 21 years old D) Uncertain
---|---
2 |  In a plane, Line AB passes through point O, and Line CD passes through point P. What is the relationship between Line AB and Line CD? Answer choices: A) Intersecting B) Perpendicular C) Parallel D) Uncertain
3 |  Jack’s time to run 100 meters is twice that of Lee’s, and Chris’s speed is twice that of Lee’s. Who is the fastest? Answer choices: A) Jack B) Lee C) Chris D) Uncertain
(c) Mathematical

1 |  Circles A, B, and C are on a straight line. Point A is 4 centimeters to the left of Point B, and Point B is 4 centimeters to the right of Point C. All of them have a diameter of 2 centimeters. What is the relationship between Circle A and Circle C? Answer choices: A) Completely coincide B) Separated C) Containment relationship D) Intersect
---|---
2 |  Rosie is 8 years old. Mike is 2 years younger than Rosie. Joe is 2 years older than Mike. How old is Mike this year? Answer choices: A) 6 years old B) 8 years old C) 10 years old D) 12 years old E) Uncertain
3 |  On a multi-tier bookshelf, ’A Brief History of Time’ is on the top tier and ’The Great Gatsby’ is on the tier right below ’A Brief History of Time’. ’One Hundred Years of Solitude’ is on the tier right above ’The Great Gatsby’. Where is ’One Hundred Years of Solitude’ in relation to ’A Brief History of Time’? Answer choices: A) Upper tier B) Lower tier C) Same tier D) Adjacent E) Uncertain
(d) Identity

Table 7: Sample Questions

1 |  Teacher Zhang is Teacher Wang’s math teacher, and Teacher Wang is Student Lee’s Chinese language teacher. Who is Student Lee’s math teacher? Answer choices: A) Teacher Zhang B) Teacher Wang C) Neither D) Uncertain
---|---
2 |  Wang is an employee of Company A, and Lee is the CEO of Company A. Wang’s CEO is Ming. What is Lee to Ming? Answer choices: A) The same person B) CEO C) Employee D) Uncertain
3 |  Mike is William’s son, Bai is Mike’s daughter, Mary is William’s wife. Mary is Bai’s? Answer choices: A) Wife B) Mother C) Uncertain D) Grandmother
(a) Personal

1 |  Event A occurs only if Event B happens, and Event B occurs only if Event C happens. Event C leads to Event D. Event A did not happen. Does Event D happen? Answer choices: A) Yes B) No C) Uncertain
---|---
2 |  iPhone and Galaxy are the same type of product, iPhone and iPad are different types of products. What is the relationship between Galaxy and iPad? Answer choices: A) Same type of product B) Different types of products C) Uncertain
3 |  Event B occurs only if Event A happens, and Event C occurs only if Event B happens. Event A did not happen. Event C occur? Answer choices: A) Yes B) No C) Uncertain
(b) Other

Table 8: Sample Questions (Continued)

##  Appendix C Breakdown Results

See Figure 7.

(a) EN-ZS

(b) EN-ZSC

(c) ZH-ZS

(d) ZH-ZSC

(e) FR-ZS

(f) FR-ZSC

(a) JA-ZS

(b) JA-ZSC

(c) KO-ZS

(d) KO-ZSC

Figure 7: Breakdown Results. For all 5 languages in ZS and ZSC prompt settings

##  Appendix D Impact of Relation Number

See Figure 8.

(a) GPT 4+zero_shot

(b) GPT 4+zero_shot_cot

(c) GPT 3+zero_shot

(d) GPT 3+zero_shot_cot

(e) ChatGPT+zero_shot

(f) ChatGPT+zero_shot_cot

Figure 8: Relation number impact. There is no as clear decreasing trend in
GPT3 plots when the relation number increases. We believe it is because GPT3’s
accuracy is already close to the probability of random guess.

(a) LLAMA2-7B+zero_shot

(b) LLAMA2-7B+zero_shot_cot

Figure 9: Relation number impact (Continued).


###### Abstract

The properties of a semiflexible polymer with fixed ends exposed to
oscillatory shear flow are investigated by simulations. The two-dimensionally
confined polymer is modeled as a linear bead-spring chain, and the interaction
with the fluid is described by the Brownian multiparticle collision dynamics
approach. For small shear rates, the tethering of the ends leads to a more-or-
less linear oscillatory response. However, at high shear rates, we found a
strongly nonlinear reaction, with a polymer (partially) wrapped around the
fixation points. This leads to an overall shrinkage of the polymer.
Dynamically, the location probability of the polymer center-of-mass position
is largest on a spatial curve resembling a limaçon, although with an
inhomogeneous distribution. We found shear-induced modifications of the
normal-mode correlation functions, with a frequency doubling at high shear
rates. Interestingly, an even-odd asymmetry for the Cartesian components of
the correlation functions appears, with rather similar spectra for odd $x$\-
and even $y$-modes and vice versa. Overall, our simulations yielded an
intriguing nonlinear behavior of tethered semiflexible polymers under
oscillatory shear flow.

###### keywords:

mesoscale simulations; nonequilibrium simulations; LAOS; polymer dynamics

x 10.3390/—— x Academic Editor: name Received: date; Accepted: date;
Published: date Tethered Semiflexible Polymer under Large Amplitude
Oscillatory Shear Antonio Lamura 1,* and Roland G. Winkler 2,* Antonio Lamura
and Roland G. Winkler Correspondence<EMAIL_ADDRESS>(A.L.);
<EMAIL_ADDRESS>(R.G.W.)

##  1 Introduction

The structural and rheological properties of polymers are strongly affected by
external fields, such as shear or extensional flows. As such, this is well
established, and the external fields provide means to control the behavior of
polymer solutions, melts, and networks Bird et al. (1987); Rubinstein and
Colby (2003); Larson and Desai (2015); Shaw and MacKnight (2018); Schroeder
(2018). Similarly, it has been shown that the macroscopic rheological behavior
of a polymer solution, e.g., shear-rate dependent viscosities, normal-stress
differences, and shear thinning, are tightly linked to the microscopic polymer
conformational and dynamical properties Doi and Edwards (1986); Bird et al.
(1987); Öttinger (1996); Prakash (2001); Huang et al. (2010); Winkler (2010);
Schroeder (2018). Hence, insight into the behavior of individual polymers is
fundamental in the strive to unravel the macroscopic nonequilibrium polymer
properties.

Direct observation of the nonequilibrium properties of single molecules, also
termed “molecular rheology” Schroeder (2018), in experiments Smith et al.
(1999); Schroeder et al. (2005); Teixeira et al. (2005); Schroeder et al.
(2005); Doyle et al. (2000); Ladoux and Doyle (2000); Gerashchenko and
Steinberg (2006); Zhou and Schroeder (2016) and simulations Liu (1989);
Delgado-Buscalioni (2006); Hur et al. (2000); Jose and Szamel (2008); He et
al. (2009); Knudsen et al. (1996); Lyulin et al. (1999); Petera and Muthukumar
(1999); Jendrejack et al. (2002); Hsieh and Larson (2004); Liu et al. (2004);
Pamies et al. (2005); Schroeder et al. (2005); Sendner and Netz (2008);
Pierleoni and Ryckaert (1995); Aust et al. (1999); Gratton and Slater (2005);
Zhang et al. (2009); Ryder and Yeomans (2006); Ripoll et al. (2006); Kobayashi
and Yamamoto (2010); Huang et al. (2010); Lamura and Winkler (2012); Winkler
et al. (2014); Cannavacciuolo et al. (2008); Chelakkot et al. (2010, 2012) has
provided valuable visual illustrations of polymer conformations and has helped
to characterize their nonequilibrium properties in terms of their deformation,
orientation, relaxation dynamics, and rheology, both for free and tethered
molecules in shear and extensional flow. These studies, however, typically
probe the linear viscoelastic properties, which are usually insufficient to
fully characterize the nonlinear aspects of polymers under flow.

In order to probe nonlinear properties of complex fluids, the large-amplitude
oscillatory shear (LAOS) method was developed Bogie and Harris (1966);
Philippoff (1966); Giacomin and Dealy (1993); Hyun et al. (2011); Zhou and
Schroeder (2016). Here, in contrast to small-amplitude oscillatory shear, the
stress response is typically no longer sinusoidal, but of rather complex
shape. Such strong, time-dependent flows will affect the conformational
properties of individual polymers to a yet unresolved extent. Studies on the
dynamics of single polymers under large-amplitude oscillatory extensional flow
yield qualitatively different stretching flow-rate curves (Lissajous curves)
as a function of the extension rate and the oscillation frequency Zhou and
Schroeder (2016), and illustrate the complex interplay between time-dependent
flows and polymer conformations. Here, further studies are desirable to
resolve the time-dependent conformational properties of single polymers under
large-amplitude oscillatory flows, aspects which, so far, have not been
addressed by simulations.

In this article, we perform non-hydrodynamic, Brownian-type simulations of
individual polymers exposed to large-amplitude oscillatory shear. The ends of
the polymer are fixed and the polymer dynamics is constraint to the $xy$-plane
of a Cartesian reference frame. We consider stiffer polymers only, with the
persistence lengths $L_{p}/L=1/2$ and $2$, respectively ($L_{p}$ is the
persistence and $L$ the polymer contour length). In any case, the polymers
exhibit cyclic conformational modulations, specifically at higher Weissenberg
numbers ($Wi$), which is the product of the applied shear rate and the longest
polymer relaxation time. The tethering of the ends leads to a more-or-less
linear oscillatory response at small Weissenberg numbers, where a polymer
moves back and forth like grass swaying in the wind. With increasing $Wi$,
polymers (partially) wrap around the fixation points and shrink. In general,
the probability of the center-of-mass position is largest on a limaçon-type
curve, as a consequence of the periodic excitation, however, with a non-
uniform probability. Interestingly, the center-of-mass autocorrelation
function normal to the line connecting the tethering points exhibits frequency
doubling with respect to the imposed shear frequency, as a consequence of the
non-crossability of the polymer and the tethering points. This reflects the
symmetry breaking of the polymer dynamics during an oscillation cycle. Our
findings illustrated the complex nonlinear interplay of polymer internal
degrees of freedom and external periodic oscillations.

The rest of the paper is organized as follows. In Section 2, the polymer model
and the coupling to the shear flow are described. Section 3 presents results
for conformational and dynamical properties. Our findings are summarized in
Section 4.

##  2 Model and Method

The two-dimensional linear polymer chain is composed of $N$ beads of mass $M$,
with its ends, $\bm{r}_{1}$, $\bm{r}_{N}$, tethered at
$\bm{r}_{1}=(-H/2,0)^{T}$ and $\bm{r}_{N}=(H/2,0)^{T}$, respectively (cf.
Figure 1). The contour length $L=(N-1)r_{0}$, where $r_{0}$ is the bond
length, is fixed and $L>H$. The interactions between the beads are defined in
terms of the potential $U=U_{bond}+U_{bend}+U_{ex}$, comprising bond, bending,
and excluded-volume interactions. The bonds between consecutive beads are
described by the harmonic potential:

$U_{bond}=\frac{\kappa_{h}}{2}\sum_{i=1}^{N-1}(|{\bm{r}}_{i+1}-{\bm{r}}_{i}|-r_{0})^{2},$
(1)

where ${\bm{r}}_{i}$ is the position of bead $i$ ($i=1,\ldots,N$) and
$\kappa_{h}$ is the elastic constant. Bending restrictions are captured by the
potential:

$U_{bend}=\kappa\sum_{i=1}^{N-2}(1-\cos\varphi_{i}),$ (2)

with $\kappa$ the bending rigidity and $\varphi_{i}$ the angle between two
consecutive bond vectors. In the limit $\kappa\to\infty$, the persistence
length is given by $L_{p}=2\kappa r_{0}/k_{B}T$, with $T$ the temperature and
$k_{B}$ Boltzmann’s constant. Bead overlapping and bond crossings are
prevented by the shifted and truncated Lennard–Jones potential:

$U_{ex}=4\epsilon\Big{[}\Big{(}\frac{\sigma}{r}\Big{)}^{12}-\Big{(}\frac{\sigma}{r}\Big{)}^{6}+\frac{1}{4}\Big{]}\Theta(2^{1/6}\sigma-r),$
(3)

where $r$ is the distance between two non-bonded beads, and $\Theta(r)$ is the
Heaviside function; $\Theta(r)=0$ for $r<0$ and $\Theta(r)=1$ for $r>0$. The
dynamics of the beads is described by Newton’s equations of motion, which are
integrated by the velocity-Verlet algorithm with time step $\Delta t_{p}$
Swope et al. (1982); Allen and Tildesley (1987).

The polymer is coupled to a Brownian heat bath implemented via the Brownian
(or random) multiparticle collision dynamics (B-MPC) approach Ripoll et al.
(2007); Gompper et al. (2009); Kikuchi et al. (2002). Hence, no hydrodynamic
effects are considered in the present work. MPC consist of streaming and
collision steps, where collisions occur in regular time intervals of length
$\Delta t$ Kapral (2008); Gompper et al. (2009). During streaming, the
dynamics of the beads is described by Newton’s equations of motion. In the
collision step, the velocities of the beads change in a stochastic manner. In
B-MPC, the Brownian interaction of a bead with the surrounding fluid is
implemented by a stochastic collision with a phantom particle, taking its
momentum from the Maxwell–Boltzmann distribution of variance $Mk_{B}T$ and a
mean, which is zero in absence of shear. In the presence of oscillatory shear
in the $xy$-plan the mean momentum is $M\bm{v}_{i}^{s}(t)$, with the shear
velocity:

$\displaystyle\bm{v}_{i}^{s}(t)=\left(\dot{\gamma}y_{i}\cos(\omega
t),0\right)^{T}$ (4)

at the time $t$; $\dot{\gamma}$ is the shear rate and $\omega$ the frequency
(cf. Figure 1). The collision is implemented via the stochastic rotation
dynamics variant of MPC Ihle and Kroll (2001); Lamura et al. (2001); Gompper
et al. (2009). Here, the relative velocity of a bead, with respect to the mean
of the velocities of the bead and related phantom particle, is randomly
rotated by angles $\pm\alpha$.

We choose the following parameters for the simulations: $\alpha=130^{\circ}$,
$M=5$, $\Delta t=0.1t_{u}$, with the time unit
$t_{u}=\sqrt{Mr_{0}^{2}/(5k_{B}T)}$,
$\kappa_{h}r_{0}^{2}/(k_{B}T)=4~{}\times~{}10^{3}$, $\epsilon/(k_{B}T)=1$,
$\sigma=r_{0}$, $N=101$, hence, the polymer length is $L=100r_{0}$, and
$\Delta t_{p}=10^{-2}\Delta t$. The value of $\kappa_{h}$ guarantees that the
polymer length is constant within $1\%$ for all considered systems. Two
bending rigidities are considered, corresponding to the persistence lengths
$L_{p}/L=0.5$ and $2$.

Simulations of free polymers yield the longest relaxations times
$\tau_{r}/t_{u}=1.9~{}\times~{}10^{6}$ and
$\tau_{r}/t_{u}=3.9~{}\times~{}10^{6}$ for the two stiffness values,
determined from the end-to-end vector correlation function Winkler (2010);
Lamura and Winkler (2012). The strength of the shear flow is characterized by
the Weissenberg number $Wi=\tau_{r}\dot{\gamma}$, for which the values
$Wi=10,\,25,\,50,$ and $100$ are considered. The frequency $\omega$ is related
to the Deborah number $De=\tau_{r}\omega$, where we set $De=10$.

The polymer, with $H=L/5$, is initialized with beads along a semi-elliptical
contour with the major axis along the $y$-direction and minor axis along the
$x$-axis. The polymers are equilibrated up to
$5~{}\times~{}10^{6}t_{u}>\tau_{r}$, and data are collected up to the longest
simulated time $t_{L}=10^{8}t_{u}$.

Figure 1: Sketch of the tethered bead-spring polymer exposed to oscillatory
linear shear flow.

##  3 Results

The oscillatory flow dragged the beads along and, at least at small shear
rates, the polymer moved back and forth like grass swaying in the wind. This
is illustrated in Figure 2. The stiffer polymer ($L_{p}/L=2$) closely
maintained its shape at low shear rates (Figure 2b). A flexible polymer was
deformed more easily, but an in-phase oscillation was still present (Figure
2a). Larger Weissenberg numbers led to stronger conformational changes and
larger bead displacements (Figure 2c–h).

Figure 2:  Conformations of the polymer at times $t=0$ (solid line),
$0.5\pi/\omega$ (dashed line), $\pi/\omega$ (dotted line), $1.5\pi/\omega$
(dash-dotted line) after equilibration for $L_{p}/L=0.5$ (left), $2$ (right),
and $Wi$ = 10, 25, 50, 100 (from top to bottom). Black dots denote the
position of fixed beads.

###  3.1 Center-of-Mass Properties

Figure 3 shows the time-dependence of the $x$-coordinate of the center-of-mass
position for various Weissenberg numbers. For smaller $Wi\lesssim 25$,
entropic effects were strong, and $x_{cm}$ was only partially following the
external flow. Perturbations were stronger when the polymer got trapped by the
fixation points and some time was needed to disentangle it (cf. Supplementary
Movies for $L_{p}/L=0.5,2$ and $Wi=25$). There were in-phase periods with a
small phase shift, which were interrupted by time intervals with a center-of-
mass motion decoupled from the flow. A stronger flow ($Wi\approx 50$) enhanced
the in-phase periodic motion, but the center-of-mass dynamics was phase
shifted and seemed to be no longer harmonic. The modulations of the
approximately periodic peaks became more pronounced for $Wi=100$, and an
original single peak split into two peaks, with the minimum of the second peak
close to zero. Hence, $x_{cm}$ exhibited an approximate doubling of the
frequency, an aspect which is more closely discussed in Section 3.2.2 in the
context of normal modes. Overall, we found a highly nonlinear response of the
polymer to the external excitation. This modified the polymer conformational
and dynamical properties (cf. Supplementary Movies for $L_{p}/L=0.5,2$ and
$Wi=100$).

Figure 3:  Time-dependence of the center-of-mass coordinate along the $x$-axis
for $L_{p}/L=0.5$ (left) and $2$ (right), and $Wi$ = 10, 25, 50, 100 (from top
to bottom). The black lines correspond to the externally applied shear flow
with arbitrary amplitude.

Figure 4 depicts the probability distribution of the center-of-mass position
for various Weissenberg numbers. The probability was high for positive
$y_{cm}$-values, specifically at lower $Wi$. This was related to the chosen
initial conditions, with a polymer always in the half-plane $y>0$. Evidently,
the polymers were too stiff to restore isotropy normal to the shear-flow
direction. The anisotropy was maintained at higher $Wi$. However, the most
probable center-of-mass position shifted gradually to larger $|x_{cm}|$. At
large Weissenberg numbers, the probability increased in the vicinity of
$x_{cm}=0$. As illustrated in Figure 2g,h, the polymers were wrapped around
the tethering points by the flow. Overall, the probability of the center-of-
mass position was highest on a limaçon-type curve, however, with a non-uniform
distribution. A limaçon is defined as a curve formed by the path of a point
fixed to a circle, when that circle rolls around the outside of a circle of
equal radius. In our case, the shear flow (partially) rotated (oscillated) the
semiflexible polymer of more-or-less circular shape, which looked like
rolling, and the center-of-mass followed a limaçon.

The polar coordinate representation of a limaçon is $r=a+b\cos\theta$, and the
parameter representation for our reference frame is Bronshtein and Semendyayev
(1973):

$\displaystyle x=$
$\displaystyle~{}r\cos\theta=\left[a+b\sin\theta\right]\cos\theta,$ (5)
$\displaystyle y=$
$\displaystyle~{}y_{0}+r\sin\theta=y_{0}+\left[a+b\sin\theta\right]\sin\theta,$
(6)

with the off-set $y_{0}$. The fits of Equations (5) and (6) to our simulation
results for $L_{p}/L=2$ are displayed in Figure 4. For $Wi=10$, the limaçon
was hardly distinguishable from an ellipse. With increasing Weissenberg
number, we approached a limaçon, and for $Wi=100$, the limaçon turned into a
cardioid, where $a=b$. The limaçon curves for $Wi=50$, and especially
$Wi=100$, did not fully agree with the simulation data for
$y_{cm}/r_{0}\gtrsim 10$. The reason was the conformational freedom of the
polymer and shear-induced shape changes, implying changes in the radii of the
rolling circles underlaying the mathematical limaçon construction. These
changes were most pronounced while the polymer explored regions of high shear
rate.

The distribution functions for the center-of-mass Cartesian coordinates of the
polymers are displayed in Figure 5. The distribution function for $P(x_{cm})$
(Figure 5a,b) is symmetric with respect to the center between the tethered
ends. At small shear, distinct “off-center” peaks were present as a
consequence of the projection onto the $x$-axis. With the increasing
Weissenberg number, the peaks became more pronounced and shift closer to the
end positions. This is also reflected in the variance $\left\langle
x_{cm}^{2}\right\rangle$, which increased with $Wi$. At high Weissenberg
numbers, a peak appeared in the center, reflecting the polymer “wrapping”
around the fixed ends. The asymmetry in the initial condition of $y_{cm}$ is
also reflected in Figure 5c,d, with a pronounced peak at positive $y_{cm}$.
The probability of smaller $y_{cm}$ increased with increasing $Wi$, and for
$Wi\gtrsim 100$, a peak appeared at $y_{cm}<0$, consistent with the high
probability in the vicinity of $x_{cm}=0$ of Figure 4g,h. The snapshots in
Figure 2g,h show stretching and alignment along the $x$-direction of the
polymer-part between the fixed end point and the point where wrapping around
the other fixed end appears. For such conformations, the major parts of the
polymer were at $y<0$. The wrapping combined with the reversal of the polymer
advective dynamics resulted in a slow lateral dynamics, resulting in a high
probability with $y_{cm}<0$. Yet, the average $\left\langle
y_{cm}\right\rangle$ was always positive, as shown in Figure 6, but decreased
quickly with increasing $Wi$. A fit with the logarithmic Weissenberg-
dependence $\left\langle y_{cm}\right\rangle=-\beta r_{0}\ln Wi$ yielded
$\beta=4.9$ and $7.6$ for $L_{p}/L=1/2$ and $2$, respectively. Evidently, the
drop is more pronounced for the stiffer polymer.

The stronger conformational changes with increasing strength of the shear flow
imply a shrinkage of the polymer. The mean square radius of gyration shrinks
by 10–15%, with respect to non-sheared conformation.

Figure 4:  Probability distribution of the center-of-mass position of the
tethered polymers for $L_{p}/L=0.5$ (left) and $2$ (right), and
$Wi=10,25,50,100$ (from top to bottom). Black dots indicate the position of
fixed ends. The lines for $L_{p}/L=2$ (right) are limaçons according to
Equations (5) and (6) for the parameters: (b) $a/r_{0}=20.48$, $b/r_{0}=1.92$,
$y_{0}=0$, (d) $a/r_{0}=19.91$, $b/r_{0}=2.72$, $y_{0}=0$, (f) $a/r_{0}=17$,
$b/r_{0}=14$, $y_{0}/r_{0}=-6$, (h) $a/r_{0}=16$, $b/r_{0}=16$,
$y_{0}/r_{0}=-6$. Figure 5:  Probability distribution functions of the $x$
(top) and $y$ (bottom) coordinates of the polymer center of mass for
$L_{p}/L=0.5$ (left), $2$ (right) and
$Wi=10~{}(\bullet),25~{}(\blacktriangle),50~{}(\star),100~{}(\blacksquare)$.
Figure 6: Average position along the $y$-direction of the polymer center of
mass as a function of the Weissenberg number for $L_{p}/L=0.5$ ($\bullet$),
$2$ ($\blacktriangle$). Lines are the fits $<y_{cm}>/r_{0}\sim\beta\ln Wi$
with $\beta=-4.9\pm 0.4$ (full line) $\beta=-7.6\pm 0.6$ (dashed line).

###  3.2 Conformational Properties

####  3.2.1 Normal Mode Expansion

We studied the internal polymer conformational and dynamical properties via
the mode amplitudes ${\bm{A}}_{n}(t)=(A_{nx}(t),A_{ny}(t))^{T}$ of the
eigenfunction expansion of a polymer with fixed ends Winkler (1999),

${\bm{r}}_{i}=\frac{H(2i-N-1)}{2(N-1)}\hat{\bm{x}}+2\sum_{n=1}^{N-1}{\bm{A}}_{n}\sin(q_{n}[i-1])\;,$
(7)

with $\hat{\bm{x}}$ the unit vector along the $x$-axis and the wave numbers
$q_{n}=n\pi/(N-1)$ ($n=1,\ldots,N-1$). The mode amplitudes are:

$\displaystyle{\bm{A}}_{n}=\frac{1}{N-1}\sum_{i=1}^{N-1}{\bm{S}}_{i}\sin(q_{n}[i-1]),$
(8)

in terms of the bead positions $S_{ix}=r_{ix}-(2i-N-1)H/2(N-1)$ and
$S_{iy}=r_{iy}$ ($i=1,\ldots,N$).

The mean, $\langle\bm{S}\rangle$, and mean square, $\langle\bm{S}^{2}\rangle$,
values of the components of $\bm{S}$ are displayed in Figure 7 for
$L_{p}/L=2$. The behavior was qualitatively similar for $L_{p}/L=1/2$. The
shape of $\langle S_{x}\rangle$ was centrosymmetric with respect the polymer
center $i=50$. Any deviation was a consequence of statistical inaccuracy. The
magnitude $|\langle S_{x}\rangle|$ decreased significantly with the increasing
Weissenberg number, and the mean was close to zero for $Wi=100$. The component
$\langle S_{y}\rangle$ was always positive and the largest amplitude was
naturally in the polymer center. With increasing $Wi$, the amplitudes shrunk,
and $\langle S_{y}\rangle$ was close to zero for all beads. The mean square
values $\langle S_{y}^{2}\rangle$ (Figure 7, right) decreased with increasing
$Wi$. However, $\langle S_{x}^{2}\rangle$ changed in a nonmonotonic manner,
and the fluctuations were larger for $Wi=50$ than for $Wi=100$. This was a
consequence of the symmetry-breaking shear flow with respect to the $x$-axis
and the wrapping of the polymers around the tethering points.

Figure 7: Mean (left) and mean square (right) values of the components of the
displacement $\bm{S}$ as a function of the bead index, $i$, of the
semiflexible polymer with $L_{p}/L=2$ for $Wi=10~{}(\circ),$
$25~{}(\triangle),50~{}(\star),100~{}(\Box)$. Open symbols indicate $S_{x}$
and filled symbols $S_{y}$.

Figure 8 depicts the dependence of the variance of the mode amplitudes:

$\displaystyle\left\langle\delta\bm{A}_{n}^{2}\right\rangle=\left\langle\left(\bm{A}_{n}-\left\langle\bm{A}_{n}\right\rangle\right)^{2}\right\rangle,$
(9)

on the mode number for the two stiffness values and the various Weissenberg
numbers. For small $Wi$, we obtained a $1/n^{4}$ dependence as characteristic
for stiff polymers Harnau et al. (1995); Aragón and Pecora (1985). This even
applied over a range of mode numbers $4<n\lesssim 40$ at higher shear.
However, the variances of the lower-mode amplitudes deviated from this
dependence, with a weaker mode-number dependence for small $n$, and a rapid
drop from mode $n=3$ to $n=4$. This reflected the large-scale conformational
changes by the wrapping of the polymers.

Figure 8: Variance of the mode components $A_{nx}$ (open symbols with solid
lines) and $A_{ny}$ (solid symbols with dashed lines) as functions of mode
number $n$ for $L_{p}/L=0.5$ (left), $2$ (right) and
$Wi=10~{}(\bullet),25~{}(\blacktriangle),50~{}(\star),100~{}(\blacksquare)$.
The slope of the solid lines is $-4$.

####  3.2.2 Dynamical Properties

Since the large-scale properties of the polymer were modified most by the
shear flow, we more closely considered the dynamics of the mode amplitudes for
the modes $n=1$ and $2$ by the mode-autocorrelation function
$C_{n}(t)=\langle\bm{A}_{n}(t)\cdot\bm{A}_{n}(0)\rangle$. Results for various
Weissenberg numbers and the persistence length $L_{p}/L=2$ are presented in
Figure 9. The simulation data were analyzed by fitting the exponentially
damped periodic function:

$\displaystyle F(t)=e^{-\gamma t/T_{0}}\cos(\Omega\omega t),$ (10)

where $T_{0}=2\pi/\omega$ is the period of the applied oscillation, $\gamma$
characterizes the damping, and $\Omega$ accounts for variations of $\omega$.
The correlation $C_{1x}(t)=\langle A_{1x}(t)A_{1x}(0)\rangle$, in Figure 9a,
decreased at short times and increased again for $t/T_{0}>1/2$. Fitting
yielded a factor $\Omega\approx 1.2$, i.e., $C_{1x}(t)$ followed roughly the
shear flow. At high Weissenberg numbers, an additional weak modulation
appeared, which, however, did not change the primary frequency. The decay of
the correlation function $C_{1y}(t)=\langle A_{1y}(t)A_{1y}(0)\rangle$ showed
a strong Weissenberg number dependence. As depicted in Figure 9b, $C_{1y}(t)$
depended only very weakly on time for $Wi=10$. With the increasing Weissenberg
number, the decay rate, $\gamma$, increased, and the correlation function
showed damped oscillations, which were most pronounced for $Wi=100$.
Interestingly, the characteristic frequency was twice the externally applied
frequency ($\Omega\approx 1.8$). The correlation function $C_{2x}(t)$
exhibited a very similar time and Weissenberg number dependence as $C_{1y}(t)$
(cf. Figure 9c), in terms of drop at the various $Wi$, as well as the
characteristic frequency. The correlation $C_{2y}(t)$ (cf. Figure 9d) was very
similar to $C_{1x}(t)$, where the fitted expression closely followed the
simulation data. Here, we found $\Omega\approx 1$, i.e., the external
frequency, but the decay was stronger with $\gamma\approx 1.4$. Moreover,
within both sets of correlation functions, the exponential decay depended only
weakly on the Weissenberg number. It was easily recognized that there was an
odd-even asymmetry between the $x$\- and $y$-correlation functions. This
asymmetry was also obtained for higher mode numbers.

Figure 9:  Autocorrelation function of the mode amplitudes for the modes $n=1$
(top) and $n=2$ (bottom) as a function of time along the $x$\- (left) and
$y$-direction (right). The polymer stiffness is $L_{p}/L=2$ and the
Weissenberg numbers
$Wi=10~{}(\circ),25~{}(\triangle),50~{}(\star),100~{}(\Box)$. The black solid
lines are fits of a damped sinusoidal oscillation.

Theoretical models of flexible and semiflexible polymers predict an
exponential decay of the normal-mode correlation functions of the form
$e^{-t/\tau_{n}}$, where the $\tau_{n}$ are the relaxation times Doi and
Edwards (1986); Winkler et al. (2006). In the presence of external fields,
e.g., shear flow, the time dependence is modified and the normal-mode
correlation functions do not necessarily decay exponentially anymore Winkler
(2010), but the exponential factor is still determined by the relaxation
times. We did find an initial exponential decay, but no clear mode-number
dependence. Figure 8 shows a deviation of the smaller mode numbers from the
dependence $\tau_{n}\sim 1/n^{4}$, but the relaxation times of the modes $n=1$
and $2$ were still different. This is not reflected in Figure 9, where the
decay of $\langle A_{1x}(t)A_{1x}(0)\rangle$ and $\langle
A_{2y}(t)A_{2y}(0)\rangle$ differs by approximately a factor of $2.5$, which
is roughly the ratio of the mode numbers. Hence, the oscillatory flow field
strongly affected the relaxation times, at least at higher Weissenberg
numbers. This reflects and is consistent with a rather nonlinear response of
the polymer with respect to the external periodic excitation.

The autocorrelation functions for the center-of-mass Cartesian coordinates
displayed in Figure 10 help to understand the appearance of the frequency
doubling. Evidently, both Cartesian components exhibited a periodic motion,
with typically larger amplitudes for higher Weissenberg numbers. Clearly, the
$x$-component shows the same frequency as the applied flow, with some
modulations appearing for the highest value of $Wi$. In contrast, the
$y$-component revealed frequency doubling for $Wi\gtrsim 25$. The
Supplementary Movies illustrate the different dynamical features of $x_{cm}$
and $y_{cm}$. In one period, the $x$-position of the center of mass, let us
say, moved from the maximum positive value to the minimum value (half period)
and, by the oscillation, back again to the maximum $x$-value. The oscillation
was similar to a complete “rotation” (period), and no difference between
oscillation and rotation was visible. The behavior of the amplitude of
$y_{cm}$ was different. While in a cycle, $x_{cm}$ decreased, $y_{cm}$
increased first, reached its maximum for $x_{cm}=0$, and decreased then again.
Instead of being able to complete a full cycle, the oscillation and the
tethering constrained $y_{cm}$ to move back along a similar path as in the
first half of the period, i.e., $\langle y_{cm}(t+T_{0}/2)\rangle$ increased,
reached again a maximum and decreased then again to a value close to the
initial value. Hence, $y_{cm}$ exhibited two maxima during a period, whereas
$x_{cm}$ exhibited only one. As a consequence, the correlation function
$\langle y_{cm}(t)y_{cm}(0)\rangle$ showed twice the frequency than the
respective correlation function of the $x$-coordinate. The primary reason was
the self-avoidance by the tethering points, which forced a reversal of the
dynamics along the $y$-direction.

Figure 10: Autocorrelation function of center-of-mass Cartesian coordinates as
a function of time along the $x$\- (left) and $y$-direction (right). The
polymer stiffness is $L_{p}/L=2$ and the Weissenberg numbers
$Wi=10~{}(\circ),25~{}(\triangle),50~{}(\star),100~{}(\Box)$.

##  4 Summary and Conclusions

We have analyzed the nonequilibrium properties of semiflexible polymers
confined in two dimensions with tethered ends exposed to oscillatory shear.
The applied Brownian multiparticle collision dynamics algorithm neglected
hydrodynamic interactions. It coupled the polymer dynamics in a stochastic
(Brownian) manner to the local flow field and allowed for fluctuations, since
the average flow field was imposed only.

For small shear rates, low Weissenberg numbers ($Wi\lesssim 10$), the
tethering of the ends led to a more-or-less linear oscillatory response, where
the polymers moved back and forth like grass swaying in the wind. With
increasing $Wi$, the polymers (partially) wrapped around the fixation points
and a more complex, nonlinear response emerged. In fact, the wrapping
significantly changed the polymer conformations, and the overall size,
measured by the radius of gyration, shrunk. Dynamically, the probability of
the polymer center-of-mass position was largest on a spatial curve resembling
a limaçon, although the distribution was inhomogeneous. Of course, this was a
consequence of tethering and periodic excitation of the polymer, moving it
cyclically back and forth, which led to an approximate rolling motion of the
center-of-mass, the origin of the limaçon.

At high Weissenberg numbers, we found shear-induced modifications of the mode-
spectrum of the mode-amplitude correlation functions. Since we considered
semiflexible polymers only, the mode spectrum exhibited the characteristic
$1/n^{4}$ dependence for low $Wi$. At high $Wi$, the lower modes, specifically
the modes $n\leqslant 3$, showed a weaker $n$ dependence. This is reflected in
a particular dynamical behavior, where a frequency doubling appeared for the
normal-mode amplitudes along the $x$\- and $y$-direction. However, the
doubling appeared alternating for even and odd modes—the correlation function
$\langle A_{1y}(t)A_{1y}(0)\rangle$ shows twice the frequency of $\langle
A_{1x}(t)A_{1x}(0)\rangle$, whereas $\langle A_{2x}(t)A_{2x}(0)\rangle$ is
similar to $\langle A_{1y}(t)A_{1y}(0)\rangle$. The frequency doubling is also
reflected in the autocorrelation function of the $y$-coordinate of the center-
of-mass position. It was a consequence of the hindered and truncated
“rotational” motion of the polymer by the tethering points.

In the current simulations, hydrodynamic interactions (HI) have been
neglected. Such interactions are of major importance for flexible polymers,
since they significantly change the relaxation time spectrum, but are less
relevant for semiflexible polymers, where they provide approximately
logarithmic corrections only Harnau et al. (1996). Hence, we expected only
minor differences between the presented results for free-draining polymers and
simulations, accounting for HI as long as the polymer motion follows
instantaneously the fluid flow.

Our studies revealed an intriguing behavior of tethered polymers under
oscillatory flow, which affects their macroscopic rheological behavior.
Specifically, the overall shrinkage of the polymer reduced the viscosity. The
consequences of the frequency doubling on macroscopic properties need to be
further analyzed.

R.G.W. and A.L. conceived the project. A.L. performed the simulations. R.G.W.
and A.L. interpreted the data and wrote the paper. Both authors discussed the
results and commented on the manuscript.

The authors declare no conflict of interest.

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