# MMAC-Copilot: Multi-modal Agent Collaboration Operating System Copilot

Zirui Song111Equal contribution Yaohang Li22footnotemark: 2 Meng Fang
Corresponding Authot. Email<EMAIL_ADDRESS>Zhenhao Chen  Yuan
Huang  Zhecheng Shi  Ling Chen  University of Technology Sydney  University of
Liverpool  Mohamed bin Zayed University of Artificial Intelligence
Northeastern University

###### Abstract

Autonomous virtual agents are often limited by their singular mode of
interaction with real-world environments, restricting their versatility. To
address this, we propose the Multi-Modal Agent Collaboration framework (MMAC-
Copilot), a framework utilizes the collective expertise of diverse agents to
enhance interaction ability with operating systems. The framework introduces a
team collaboration chain, enabling each participating agent to contribute
insights based on their specific domain knowledge, effectively reducing the
hallucination associated with knowledge domain gaps. To evaluate the
performance of MMAC-Copilot, we conducted experiments using both the GAIA
benchmark and our newly introduced Visual Interaction Benchmark (VIBench).
VIBench focuses on non-API-interactable applications across various domains,
including 3D gaming, recreation, and office scenarios. MMAC-Copilot achieved
exceptional performance on GAIA, with an average improvement of 6.8% over
existing leading systems. Furthermore, it demonstrated remarkable capability
on VIBench, particularly in managing various methods of interaction within
systems and applications. These results underscore MMAC-Copilot’s potential in
advancing the field of autonomous virtual agents through its innovative
approach to agent collaboration.

1888

##  1 Introduction

Figure 1: Illustrates the sequence of actions undertaken by MMAC-Copilot in
response to the user’s request ”Play ¡Love Story¿ on Spotify”. (A) Represents
the initial state where the ”Planner” formulates preliminary plans. . (B)
”Programmer” launches the application using a bash command. (C) ”Viewer” click
the search box and enter ¡Love Story¿. (D) ”Viewer” click the play button.

Large Language Models (LLMs) [achiam2023gpt, anil2023palm,
anthropic2023claude2, driess2023palme, ouyang2022training, touvron2023llama2,
touvron2023llama] has demonstrated unprecedented capabilities to tackle
complex tasks that require human-like reasoning, decision-making, and
collaborative efforts [ding2023everything, hao2023reasoning,
park2023generative, qian2023communicative, xi2023rise]. Soon afterwards, Large
Vision Models (LVMs) [cai2023benchlmm, li2024llava, liu2023improved,
liu2024visual, luo2023wizardcoder, zhou2023lima] expand beyond traditional
text-based processing by integrating a visual dimension into LLMs. The
applications of LVMs have reached diverse domains, with one notable example
being their pivotal usage in autonomous virtual agents [wu2024oscopilot].

Usually, autonomous virtual agents require a unified interaction interface and
a high degree of generalization to adapt to the challenges posed by the
numerous different applications within an operating system. Although previous
works [tan2024towards, wu2024oscopilot, zhang2024ufo] that utilize scripts and
the planning abilities of LLMs have demonstrated success in some applications
there are still some challenges: First, the acquisition of real-world
environmental information relies on a single text modality which lead to a
limited range of applications. Second, the hallucinations caused by the
knowledge domain gap could lead to incorrect planning by virtual agents,
resulting in extremely severe consequences.

To engage with the initial challenge, we introduce MMAC-Copilot, a framework
designed to promote information exchange and collaboration among multi-modal
agents. Specifically, we created five key agents: Planner, an expert in making
plans, strategic thinking and resource allocation,ensuring that the
collaborative efforts of all agents are directed to achieve the target;
Librarian, proficient in managing question-answering tasks and API in
retrieving information; Programmer, specializing in coding tasks, particularly
skilled at executing Bash commands and Python programs; Viewer, focused on
understanding image content for visual information and execute click
interaction; and Video Analyst, dedicated to analyzing video content and
extracting key events. In addition, there is an auxiliary agents: the Mentor,
supervising system operations and agent interactions to ensure smooth
functionality and integrity. When receiving a user query, Planner formulates
an initial plan based on the user query’s requirements. It then engages with
other agents through a team collaboration chain, allowing for a dynamic
exchange of information and insight. As the plan progresses, it is
continuously adapted based on the insights of each specialized agent.

In resolving the second dilemma, we introduce the team collaboration chain,
allowing participating agents to adapt the initial plan crafted based on their
domain expertise. Refer to the scenario depicted in Figure 3, where the user
request is ”Open the Discord application and send Hi to Dylan Li”. Planner
firstly delineates five sub-tasks: ”Open Discord”, ”Navigate to friends”,
”Search for Dylan Li”, ”Open Chat with Dylan Li”, and ”Send a Message”, all of
which are assigned to Programmer. Upon completion of the first subtask by the
Programmer, Mentor observes that Discord has been successfully launched and
notes that ”Dylan is already present in the application.” Planner, upon
receiving this feedback, revises the initial plan to ”Open chat with Dylan”
and ”Send the message”, and then reassigns it to Programmer. In this instance,
Planner overlooks the fact that programmer needs to specifically integrate
Discord’s proprietary API to accomplish these tasks. Adapting to the
proprietary interfaces of thousands of applications to achieve universality is
clearly impractical. Furthermore, unforeseen scenarios such as pop-up
notifications or advertisements may disrupt task execution. Programmer fails
to execute and returns the message, ”Sorry, I don’t know how to do this.” Upon
receiving this message, Planner reallocates the task to Viewer. Viewer
decomposes the coarse-grained task into finer-grained tasks: ”click on Dylan”,
”type Hi”, and ”return the message”, which are then successfully executed.
Additionally, each subtask completion is reviewed by Mentor, and upon
approval, the process advances to the next round. The process concludes after
all rounds are completed.

We assess MMAC-Copilot using two benchmarks: GAIA [mialon2023gaia] and our
newly introduced Visual Interaction Benchmark (VIBench). GAIA is designed to
test the abilities of general artificial intelligence assistants. The MMAC-
Copilot performed in level-1, level-2, and level-3 tasks with scores of 45.16,
20.75, and 6.12, respectively. It outperforms previous methods by 6.8%.
VIBench, specifically developed for this study, tests the framework’s
performance in non-API-interactable applications across diverse domains such
as 3D gaming, recreation, and office environments. The results exhibits
outstanding adaptability, handling interfaces with a high degree of accuracy,
enhancing interaction within digitally environments.

In summary, our main contributions are as follow:

  * •

We propose MMAC-Copilot, a collaborative framework involving specialized
agents designed to navigate and perform general operating system tasks. MMAC-
Copilot enhances the interaction capabilities of autonomous virtual agents
with operating systems by leveraging multi-modality in processing to tasks.
The framework consists of distinct modal agents: ”Librarian” for question-
answering and information retrieval, ”Programmer” for executing code and bash
commands, ”Viewer” for interpreting visual information, and ”Video Analyst”
for analyzing video content. By leveraging the collaboration of modal agents,
MMAC-copilot can handle a diverse range of tasks with enhanced accuracy.

  * •

To mitigate the incorrect planning issue caused by LLMs’ hallucination, we
introduce the team collaboration chain which allows participating agents to
adjust the initial plan based on their domain expertise into MMAC-Copilot. It
has demonstrated adaptability in handling a variety of case.

  * •

We develop the Visual Interaction Benchmark (VIBench), specifically designed
to assess the system’s performance in non-API-interactable applications across
diverse domains such as 3D gaming, recreation, and office scenarios. VIBench
complements existing benchmarks by focusing on the agents’ ability to interact
with dynamic and visually complex interfaces, thus providing a more strict
test of their practical interaction capabilities.

##  2 Related work

In this section, we review some related works, with a focus on the domains of
LLM-based agents and Autonomous virtual agents.

Figure 2: An overview of MMAC-Copilot, below is the concrete definition of
each agent.

###  2.1 LLM-based agents

The sunrise field of LLM-based agents, highlighted by projects such as AutoGPT
[autogpt] and TaskWeaver [qiao2023taskweaver], marks a significant milestone
in the field of LLM research. These agents adopt a human-like approach to
problem-solving by decomposing complex tasks into manageable subtasks. The
capability demonstrates a profound advancement in planning, observation, and
responsive actions, closely mirroring human cognitive processes
[touvron2023llama, wang2024survey, xi2023rise].

The LangChain Agent [topsakal2023creating] further extends these capabilities
by enabling LLMs to sequence actions, thus expanding the potential for
automation across various domains such as customer service, logistics, and
content creation. This strategic utilization of LLMs underscores a move
towards more sophisticated, context-aware AI systems.

Developments in multi-agent systems, such as AutoGen [wu2023autogen], MetaGPT
[hong2023metagpt], and AutoAgents [chen2023autoagents], introduce a novel
dimension to the field by enabling collaboration and competition among agents.
This architecture leverages the unique strengths of individual agents,
enhancing system-wide efficiency and adaptability in complex scenarios. Such
collaborative frameworks are proving essential for large-scale applications in
sectors like healthcare [dhatterwal2023multi, Jones2022Role], finance
[gautier2023risk], and urban planning [bae2022hybrid].

The versatility and efficacy of these LLM-based agents are highlighted in a
range of application domains: In robotics, agents assist in both navigational
tasks and complex manipulations, contributing to more autonomous and adaptive
robotic systems [driess2023palme]. Web manipulation and data extraction have
seen significant enhancements, enabling better data-driven decision-making and
user interaction models [yao2022webshop, zhou2023webarena]. The gaming
industry benefits from more dynamic AI opponents and game management systems,
which improve user engagement and game design [fan2022minedojo,
tan2024towards]. Automated data analysis tools powered by LLM agents offer
sophisticated insights and predictions, facilitating more effective data
handling in business and research [wang2023voyager, zhang2023data].

###  2.2 Autonomous virtual agents

The application of LVM systems for interfacing with Graphical User Interfaces
(GUIs) in digital applications have been a key area of research. Wonderland
[yan2023gpt] harnessed the capabilities of GPT-4V which delineated by Yang et
al. [yang2023dawn] to navigate through mobile application interfaces by
analyzing screenshots. Concurrently, the AppAgent [yang2023appagent] utilizes
GPT-4V to simulate the actions of mobile users, facilitation the execution of
tasks in mobile applications through the analysis of device snapshots.
Moreover, the MobileAgent incorporates Optical Character Recognition (OCR)
technology to enhance the capabilities of of GPT-4V within a similar mobile
agent architecture. This strategic enhancement allows MobileAgent to attain
task completion rates on par with human operators. In contrast, CogAgent
[hong2023cogagent] constructs a specifically engineered LVM for interpreting
and navigating GUIs. UFO [zhang2024ufo] utilizes the windows inspect tool to
acquire component and control information from system applications, providing
it as context to GPT-4V to fully leverage its planning capabilities.

##  3 MMAC-Copilot Framework

The workflow of MMAC-Copilot is designed to leverage the specialized
capabilities of each agent in a sequential and collaborative manner. Here, we
present an algorithmic overview in Algorithm 1. In section 3.1, we discuss the
specialization of agents within the framework, each equipped with distinct
roles and tools. In section 3.2, we delve into communications protocols which
bolster accuracy in agents collaborations. In section 3.3, we introduce the
team collaboration chain, dynamic framework that utilizes continuous feedback
and iterative improvements to adapt framework strategies in real-time.

Algorithm 1 Workflow of MMAC-Copilot

1:User request $R$

2:Task completion that satisfies $R$

3:Initialize system state $S_{0}$

4:Planner formulates initial plan $P_{0}$ based on $R$

5:for each subtask $s$ in $P_{0}$ do

6: Assign $s$ to appropriate agent $A$

7: $A$ executes $s$, updates system state to $S_{i}$

8: Mentor checks $S_{i}$; provides feedback $F_{i}$

9: if $F_{i}$ indicates adjustment needed then

10: Planner revises $P_{i}$ based on $F_{i}$

11: Go to step 4 with revised $P_{i}$

12: end if

13:end for

14:Verify if $S_{i}$ satisfies $R$

15:if not satisfied then

16: Retry

17:else

18: Break loop and finalize task

19:end if

###  3.1 Specialization of agents

Empirical studies suggest that diversity within human groups contributes to a
variety of perspectives, which in turn enhances the group’s performance across
various tasks [bransford1993ideal, williams1998demography,
woolley2015collective]. Also, contemporary research indicates that assigning
specific roles to autonomous agents can improve their effectiveness
[li2023camel, qian2023communicative, salewski2024context]. Thus, in our
framework, we have defined six agents, namely: Planner, Librarian, Programmer,
Viewer, Video Analyst, and Mentor, as shown in Figure 2. The followings are
detailed introductions to each agent:

  * •

Planner: Employs GPT-4 to strategically manage and allocate tasks among
agents, optimizing workflow efficiency. In contrast to other frameworks, our
Planner is not required to decompose tasks in to indivisible atomic subtasks.
We recognize Planner operating solely in the textual modality may not
effectively integrate visual information such as UI elements, potentially
leading to hallucination. Therefore, when Planner identifies a subtask which
necessitates the incorporation of visual information, it provides only a
coarse outline of the subtask. It allows each agent to leverage its own
expertise, enhancing the adaptability of the overall framework.

  * •

Librarian: Utilizes GPT-4 and API for information retrieval, enabling it to
answer queries and provide foundational knowledge. It possesses the capability
to rapidly process vast amounts of information, thereby supporting user
requests and providing related context with other agent.

  * •

Programmer: Responsible for coding and executing scripts, including Python and
Bash commands, to interact with software environments directly. The
Programmer’s role is crucial in implementing precise operating systems
interactions that adapt to diverse requirements. Key to the Programmer’s
efficacy is its ability to refine existing code through a mechanism designed
to diagnose and resolve issues effectively. When code errors are identified,
Programmer analyzes the source of failure, applies necessary modifications,
and maintains the original code structure as much as possible. This refined
process includes a detailed explanation of identified issues, enhancing the
clarity and functionality of the code without introducing new exception, and
improving the handling of existing ones. Furthermore, Programmer can generate
accurate Python invocation statements. By deeply understanding the context of
the provided Python class, analyzing task descriptions, and interpreting
method parameter details to ensure the correct application. The result is a
syntactically correct Python statement that calls the methods with appropriate
parameters. Lastly, Programmer’s capabilities are complemented by a critical
evaluation mechanism. This mechanism assesses the refined and executed code
against the user’s specified task requirements. It evaluates the code’s
functionality, alignment with the task, and feedback provided by the
environment, culminating in a comprehensive reasoning process. The
interactions within the Programmer’s mechanism can be mathematically
represented as follows:

$C_{\text{init}}\xrightarrow[R]{E}C_{\text{mod}}\xrightarrow[X]{\delta}\text{Out}\xrightarrow[V]{T}(\text{Judge},\text{Score})$

where:

    * -

$C_{\text{init}}$ is the initial code.

    * -

$E$ represents the error analysis and feedback.

    * -

$R$ is refinement mechanism.

    * -

$C_{\text{mod}}$ is the modified code.

    * -

$\delta$ includes environmental variables such as the current working
directory or system configurations.

    * -

$X$ represents execution function.

    * -

Out includes the runtime output and any runtime errors.

    * -

$T$ is the original task requirements.

    * -

$V$ represents the evaluation mechanism.

    * -

Judge is a boolean indicating if the task is accomplished.

    * -

Score is an integer rating the code’s generality and robustness. If the score
is above the specified threshold, the code for this run is retained.

  * •

Viewer: Integrates GPT-4V for interpreting complex visual information from
screenshots. Coupled with SeeClick [cheng2024seeclick] model, Viewer could
translates visual analysis into actionable commands, such as click on
identified UI elements, bridging the gap between visual understanding and
operating systems interaction.

  * •

Video Analyst: Uses Gemini Vision to process and analyze video content,
extracting critical visual data. It is significant for subtasks which depend
on video-based information, enhancing the model’s ability to generalize across
different modal. Video Analyst interprets queries, retrieves relevant details
from video streams, and delivers accurate responses. These capabilities are
crucial for providing contextual insights, assisting other agents in
understanding the current statement. In summary, Video Analyst is adept at
handling complex scenarios where video data is key.

  * •

Mentor: Powered by GPT-4V, Mentor provides strategic oversight and
troubleshooting support after the execution processes. To be specific, Mentor
firstly analyzes screenshots of the current application window to provide
detailed description of visible state. And then Mentor identified whether the
intended change or commands have successfully impacted the application as
expected. Additionally, Mentor generates insights which are derived from a
deep analysis of the outcomes and are structured as feedback to be
incorporated into the next round.

###  3.2 Communication Protocols

Following established multi-agent frameworks [hong2023metagpt, li2023camel,
park2023generative, zhang2023building, zhuge2023mindstorms], MMAC-Copilot
adopts Structured Communication Interfaces, avoiding the traps of
unconstrained nature language communication. Each agent communicates through a
defined schema appropriate for their role.

In specific, we define the output format of the agents as JSON, and stipulate
the keys which should be output according to the different roles of each
agent. Meanwhile, each agent only receives the content of the required key.
For example, ”observation” key is exclusive output by Mentor and Viewer, as it
is specifically to their role in emphasizing visual observations of
screenshots. Apart from themselves, only the Planner receives the content of
”observation” key, both when formulating initial plans and when revising them.
These protocols are designed to prevent common issues such as data ambiguity
in complex multi-agent interactions. By standardizing the communication format
and defining clear protocols for message acknowledgment, MMAC-Copilot
minimizes the risks of miscommunication and ensures which tasks are executed
reliably. Additionally, the use of JSON allows for easy integration of new
agents as the system scales, supporting the framework’s adaptability and
expansion.

Figure 3: Example of MMAC-Copilot. User query is ”Open the Discord and Send a
message to Dylan Li”. Red represents the agent that will take the next action
in the messaging process. Brown indicates the current agent performing the
task. Green color represents state feedback.

###  3.3 Team Collaboration Chain

Team collaboration chains leverage the strengths of dispersed management
structures to facilitate enhanced cooperation among team members across
different domains. By distributing responsibilities and decision-making power,
these chains enable a more agile and responsive organizational environment.
Such chain has been successfully implemented across multiple domains
[bang2005samarbejde, huh2019xr, selvanathan2018comparative,
yatsuka2020collaboration].

In the MMAC-Copilot framework, team collaboration chain is essential for
managing difficult tasks. This section delves into the collaborative
mechanisms that enable dynamic planning and the iterative refinement of tasks,
ensure effective execution and continuous improvement. A concrete
demonstration of the collaboration chain in operation can be seen in Figure 3.

####  3.3.1 Dynamic Planning and Subtask Refinement

During the planning phase, Planner uses input from the Viewer to formulate an
initial plan. Initial plan often outline the subtasks in a coarse manner,
addressing the dynamic nature of visual environments and the inherent
limitations of planning based on textual information alone. This initial
planning stage sets the foundation for task execution but requires further
refinement to address the specifics of the visual statement.

As the system transitions to the execution phase, agents equipped with
specialized modalities, such as the Viewer and Video Analyst, assume pivotal
roles. These agents reassess the current visual context and refine the
coarsely defined subtasks into detailed, atomic subtasks. This step is crucial
for adapting to real-time changes in the visual interface, ensuring that each
task is executed with precision based on the latest visual information. This
process showcases the strength of team collaboration, where different agents
contribute uniquely to the refinement and execution of tasks.

####  3.3.2 Continuous Feedback and Iterative Improvement

Mentor plays a fundamental role in the team collaboration chain by
establishing a continuous feedback loop with the Planner. After the execution
of subtasks, Mentor evaluates the current state of the system through detailed
analysis of screenshots and the visible state of the application. This
evaluation determines whether the intended changes or commands have
successfully impacted the application as expected.

If discrepancies or incomplete tasks are identified, Mentor generates insights
and structured feedback based on a deep analysis of the outcomes. This
feedback is crucial for revising the current plans and is incorporated into
the next round of planning. This feedback loop between the Mentor and the
Planner not only ensures high-quality task completion but also facilitates
adaptive learning, allowing the MMAC-Copilot system to evolve its strategies
based on real-world interactions and outcomes.

##  4 Experiments

Figure 4: Task Completion under VIBench with MMAC-Copilot. Displayed is a
sequential illustration of task completion across three distinct categories:
(1) interactive gameplay performing skill in Genshin Impact; (2) operate
within office application VooV Meeting; (3) recreational operation,
exemplified by navigating in Netflix.

###  4.1 Benchmark and setting

We evaluate the performance of MMAC-Copilot using two distinct benchmarks
designed to challenge and assess the capabilities of AI assistants in diverse
interaction environments. The first is the General AI Assistant Benchmark
(GAIA) [mialon2023gaia], which includes 466 rigorous question-answering tasks.
GAIA primarily tests AI assistants in a question-answering (QA) format,
focusing on their ability to utilize API calls for information retrieval and
task execution. However, GAIA restricts the evaluation to textual interactions
and API-based actions, which do not fully capture the broader range of GUI
interactions that are prevalent in real-world applications.

To address these limitations and to evaluate the agents’ capabilities in a
more complex and visually oriented environment, we introduced the Visual
Interaction Benchmark (VIBench). VIBench comprises cases from three categories
of applications: 3D gaming, recreation, and office environments. Unlike GAIA,
VIBench emphasizes the necessity for agents to interact with applications
through graphical user interfaces (GUIs) that cannot typically be controlled
via standard APIs such as Win32 APIs.

##### Mathematical Representation of VIBench

The core interaction process in VIBench can be mathematically described as
follows: Let $S_{0}$ represent the initial system state which includes the
current GUI appearance and available actions. Given a user request $R$, the
system must transition through a series of states $S_{1},S_{2},\ldots,S_{n}$
via actions $a_{1},a_{2},\ldots,a_{n}$, ultimately achieving the final state
$S_{f}$ that satisfies $R$. This process is represented by the sequence:

$S_{0}\xrightarrow{a_{1}}S_{1}\xrightarrow{a_{2}}S_{2}\xrightarrow{\cdots}S_{n}\xrightarrow{a_{n}}S_{f}$

The primary goal of VIBench is not merely to evaluate the agent’s ability to
execute each step efficiently but to assess whether the final state $S_{f}$
satisfies the initial user request $R$. This objective emphasizes the
effectiveness of the final outcome over the process, reflecting the practical
focus of real-world applications:

$\text{Objective}:\text{Verify that }S_{f}\text{ meets }R$

VIBench thus complements GAIA by adding a layer of complexity and realism,
challenging agents to not only respond correctly in a QA format but also to
effectively navigate and interact within software environments that mirror
everyday user challenges.Examples are shown in Figure 4

##### Setting

To ensure a fair comparison with previous work, we employed the same versions
of the ChatGPT API used in prior studies. Specifically, we utilized
GPT-4-turbo-1106 and gpt-4-vision-preview. For the Video Analyst, we used
gemini-1.5-pro-vision.

###  4.2 Evaluation Method

For GAIA, each question calls for a specific response: string, number, or
floats. Evaluation is conducted through a exact match between the model’s
answer and ground truth.

For our VIBench, we adopt human evaluation which follow the SIMA evaluation
method [sima2024scaling], recognizing that in numerous instances, task success
cannot be automatically inferred. We engage human judges who possess expert-
level familiarity with the specific applications, defined as having at least
20 hours of application usage. These experts are asked to assess recorded
videos to determine whether the user requests have been accurately executed.
During the assessment, human experts are instructed to disregard any
irrelevant actions performed by the agents. Their judgments are solely based
on whether the stated objectives of the tasks are achieved, without
considering the intermediate steps or unrelated actions. Meanwhile, to
minimize human subjectivity, we have imposed a maximum limit of 30 execution
rounds for task. If the task is not completed within 30 rounds, it is judged
as a failure.

###  4.3 Baseline

For GAIA, we present the performance of GPT-3.5 and GPT-4 with and without
manually set plugins, along with results from AutoGPT, Chamonmile and FRIDAY
which all use GPT-4 as their backend. The selection of Plugins for GPT-4 is
conducted manually, with individual choosing based on specific user query. For
comparative purposes, we also include data on human performance sourced from
GAIA [mialon2023gaia].

For VIBench, we report the performance of UFO and FRIDAY. Notably, since
FRIDAY does not natively support the Windows platform, we have modified
FRIDAY’s code to adapt it to Windows. Also we set UFO’s Vision model same as
our framework.

###  4.4 Results

The results of our experiments with MMAC-Copilot provide a comprehensive
assessment of its capabilities in handling complex, multi-modal tasks with in
digital environments. In this session, we present and discuss the findings
from our evaluations on both the GAIA benchmark and our newly developed Visual
Interaction Benchmark(VIBench).

Table 1: Performance comparison on GAIA. All performance are reported on the private test set. Model | Level 1 | Level 2 | Level 3 | Average
---|---|---|---|---
Human* | 93.90 | 91.80 | 87.30 | 91.00
GPT-3.5 [achiam2023gpt] | 4.3 | 1.89 | 2.08 | 2.67
GPT-4 [achiam2023gpt] | 9.68 | 1.89 | 0.00 | 4.00
AutoGPT4 [autogpt] | 15.05 | 0.63 | 0.00 | 5.00
GPT4 Turbo [achiam2023gpt] | 9.68 | 6.92 | 0.00 | 6.67
GPT-4 Plugins [achiam2023gpt] | 30.30 | 9.70 | 0.00 | 14.60
Chamomile [Camomile] | 16.13 | 17.61 | 2.08 | 14.67
FRIDAY [wu2024oscopilot] | 40.86 | 20.13 | 6.12 | 24.25
Ours | 45.16 | 20.75 | 6.12 | 25.91 (+6.8%)

MMAC-Copilot shows significant performance improvement across all levels of
GAIA. As illustrated in Table 1, MMAC-Copilot achieved the highest scores in
Level 1 (45.16) and Level 2 (20.75) tasks, matching the highest score in Level
3 tasks (6.12). Also on average, it outperformed the closest competing system,
FRIDAY, by 6.8% in overall task performance. Given that tasks in GAIA
encompass various complex AI assistant capabilities, from basic information
retrieval to interactive task requiring multiple steps and integration of
external data sources.

Table 2: Evaluation results on VIBench. Model | 3D Gaming | Recreation | Office | Average
---|---|---|---|---
UFO [Camomile] | 0.00 | 28.57 | 15.38 | 14.65
FRIDAY [wu2024oscopilot] | 31.58 | 42.86 | 30.77 | 35.07
Ours | 63.16 | 69.23 | 78.57 | 70.32 (+35.25)

In VIBench as shown in Table LABEL:tab:VIBench, MMAC-Copilot outperformed
other systems across all application categories (3D gaming, recreation, and
office scenarios). Our model achieved average scores fo 63.16% in 3D gaming,
69.23% in recreation, and 78.57% in office applications, with an overall
average of 70.32%. It indicates MMAC-Copilot’s robust capability in navigating
and interacting within complex GUI environments, where direct API interaction
is not possible. MMAC-Copilot’s superior performance can be attributed to its
effective utilization of image and video analysis, enabling precise navigation
and interaction through the GUIs of different software.

##  5 Conclusion

This paper presents MMAC-Copilot, a novel framework designed to enhance the
interaction capabilities of autonomous virtual agents with operating systems.
Our approach capitalizes of autonomous virtual agents with operating systems.
Our approach capitalizes on a team collaboration chain that harnesses the
collective expertise of each agents, including ”Planner”, ”Librarian”,
”Programmer”, ”Viewer”, ”Video Analyst”, and ”Mentor”. Each agent brings a
unique set of skills and perspectives, enabling an adaptive interaction with
complex digital environments. Also MMAC-Copilot innovatively mitigates the
issues with hallucination among the agents. By facilitating dynamic and team
collaboration among the agents, our framework allows for continuous adaptation
and refinement of plans based on environmental changes. The performance of
MMAC-Copilot has been tested on GAIA benchmark and our newly introduced
VIBench, showing significant improvements over existing state-of-the-art
systems.

##  6 Discussion

We acknowledge that while MMAC-Copilot has achieved a level of generality in
application through the integration of GUI models for visual detection, it is
still limited by the GUI model’s ability to understand UI components.
Consequently, MMAC-Copilot struggles with interpreting complex UI interfaces
and performing multi-step operations that depend on such complexity.
Additionally, MMAC-Copilot’s capabilities in real-time 3D gaming environments
reveal gaps in the spatial understanding, which is cruical for tasks such as
navigation, aiming and shooting in 3D environments. The framework’s longer
inference time impair the ability to perform tasks with higher temporal
sensitivity.

To solve these, we suggest integrating external knowledge databases via online
search engines, providing additional context for decision-making processes.
This approach will not only help in understanding complex UI layouts but also
in recognizing user intentions and environmental cues in real-time games.

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page intended for references.


# Studies of Thick Gas Electron Multipliers fabricated in India

Promita Roy  Saha Institute of Nuclear Physics, Kolkata-700064, a CI institute
of Homi Bhabha National Institute, Mumbai-400094<EMAIL_ADDRESS>Purba Bhattacharya  School of Basic and Applied Sciences, Adamas University,
Kolkata-700126  Vishal Kumar  Supratik Mukhopadhyay  Nayana Majumdar  Sandip
Sarkar

###### Abstract

THick Gas Electron Multipliers (THGEMs) are robust and high gain Micro Pattern
Gaseous Detectors which are economically manufactured by standard drilling and
etching of thin printed circuit boards. In this paper, we present our recent
studies on thick GEMs which have been fabricated in India using local
expertise. Two types of THGEMs have been fabricated; one set has holes without
any external rim and another set has holes with rims. These detectors have
been characterized using argon-carbon dioxide and argon-isobutane gas
mixtures. Electron transmission, effective gain, energy resolution and
optimized working range studies have been presented for both the sets of
THGEMs.

###### keywords:

Micro Pattern Gaseous Detectors , Gas Electron Multipliers , Thick Gas
Electron Multipliers , electron transmission , rim , gain , energy resolution
, simulation

††journal: Nuclear Instrumentation Methods A

##  1 Introduction

The introduction of the Micro Pattern Gaseous Detectors opened a new era of
state-of-the-art detector technologies and they now represent the benchmarks
for the detector developments in future accelerator-based experiments.
Introduced by Fabio Sauli in 1997, the Gas Electron Multiplier (GEM) is a
powerful addition to the family of fast radiation detectors [1]; originally
designed for various particle physics experiments around the world. The THick
Gas Electron Multiplier (THGEM) is a (geometrically) ten-fifteen fold expanded
version of the standard Gas Electron Multipliers. It was first introduced at
the Weizmann Institute of Science [2]. This version of GEM is robust and
comparatively easier to fabricate. Secondary multiplication of primary
electrons on irradiating the detector with a radiation source occurs within
holes of few 100s of micron diameter upon drilling across 0.2-1mm thick
copper-cladded insulator plates. These detectors offer high readout gain of
103-105 depending on the gas mixture [3]. In recent years, they have been
widely used in large experiments such as dark matter measurement [4],
Cherenkov radiation imaging [5], and x-ray and neutron detection [6].
Performance of THGEMs depend on various factors including its geometrical and
electrical configurations. The size of the etched rim around the THGEM holes,
is essential for reducing discharge-occurrence probability significantly; this
permitted operation at higher THGEM voltages and hence yields higher detector
gains [7]. Drift, transfer and induction field values are also found to affect
gain and energy resolution apart from applied voltage across THGEMs [8]. With
hole size of few 100s of micron (200-600 $\mu m$), they are found to have
sufficiently good position resolution for muon tracking and imaging[9].

In the process of our search for a robust detector with decent spatial
resolution, two types of THGEMs have been planned and fabricated and their
experimental characterization have been carried out in different gas mixtures.
The results have been compared with a CERN made THGEM. Section 2 provides the
details of the THGEM geometries used, experimental setup and experimental
procudures. This is followed by experimental results in section 3 and
simulation results in section 4.

##  2 Geometry of THGEM and experimental setup

A 40$\times$48 mm FR4 PCB based thick GEM manufactured in CERN have been used
for the initial testing and characterization. This PCB based detector is
800$\mu$m thick with around 50$\mu$m of copper coating on each side. It has
straight cylindrical drills of diameter 500$\mu$m and rim of 100$\mu$m width
arranged in hexagonal pattern. Figure 1 shows the picture of the THGEM
procured from CERN and its microscpic view.

Figure 1: THGEM PCB from CERN

Another two sets of THGEMs of same size(40$\times$48 mm) with and without rims
have been designed in our lab using AutoCAD software. These have been
fabricated by drilling straight holes in FR4 based printed circuit boards. The
chosen FR4 is based on IS410 Lead-free Epoxy Laminate and Prepreg PCB
material. These FR4-PCBs are 370-380$\mu$m thick with nearly 50$\mu$m copper
coating on each side. Holes are arranged in hexagonal pattern similar to CERN
THGEM to increase the optical transparency of the foils. Both the sets of
copper coated FR4-PCBs have holes of diameter 500$\mu$m and pitch of 1mm.
However, just one set has 100 $\mu$m wide rims around the holes, while the
other set has no rim around the holes. Figure 2 shows the locally fabricated
THGEMs without and with rim respectively.

Figure 2: Locally fabricated THGEM without rim (left) and with rim (right) Table 1: Different geometries of THGEMs used THGEM no. | Type |  Thickness(d)($\mu m$) |  Hole radius(r)($\mu m$) |  Pitch(p)($\mu m$) |  Rim(a)($\mu m$)
---|---|---|---|---|---
TGA | CERN THGEM | 800 | 250 | 1000 | 100
TGB | SINP THGEM-1 | 380 | 250 | 1000 | none
TGC | SINP THGEM-2 | 370 | 270 | 1020 | 120

A stretched mesh secured in a rectangular frame has been used as the drift
cathode and and a copper plated PCB has been used as the readout anode. These
electrodes along with the THGEM PCB have been kept inside an aluminium box
which has all the electrical connections as well as gas inlet and outlets as
shown in the figure 3. Teflon holders and nylon screws have been used to
secure the drift plate, THGEM and readout anode plate inside the Al box. The
chosen drift gap is 10mm and induction gap is 3mm. The lid of the Al box has a
mylar window to place the source and irradiate the detector.

Figure 3: Aluminium box containing the THGEM

###  2.1 Measurement details and procedure

Voltages to drift electrode and THGEM PCB are applied using CAEN power supply
modules N1471 and N471A. Amptek pocket MCA has been used for getting the
${}^{55}Fe$ spectra and the readout current measurement has been done using
Keithley 6487 pico-ammeter. Temperature, pressure and humidity have been
constantly monitored using an Arduino based sensor. Ortec 142 IH preamplifier
and Ortec 672 spectroscopic amplifier have been used for shaping and
amplifying the readout signal. Figure 4 shows the typical experimental set up
containing the Al box, filter box, pre-amplifier, spectroscopic amplifier, MCA
and PC. Data from pre-amplifier is fed to spectroscopic amplifier, which is
then fed to the MCA interfaced with a desktop.

Figure 4: Experimental setup with THGEM

Data from readout anode is also fed to the pico-ammeter as shown in the figure
4.

###  2.2 Pre-characterization procedures

Prior to the characterization of the THGEMs, resistivity of all the detectors
(PCBs) have been checked and they have been found to have resistivity more
than 11G$\Omega$ on application of 1000V. Microscopic images have been saved
for all the three THGEMs as shown in figure 5. Copper coating of THGEM B and C
have 2-3$\mu$m thick gold plating to protect it from corrosion. Microscopic
images have been used to cross-check the geometrical parameters of these
locally fabricated THGEMs and they have been mentioned in table 1.

Figure 5: Microscopic images of THGEM A(top left), THGEM B (top right) and
THGEM C (bottom)

The next procedure was to clean the THGEM PCBs and that has been done using
the conventional cleaning measures which include soaking the detectors in iso-
propanol for hours and baking them in nitrogen gas with high flow-rate until
the test Al box is flushed entirely around 5 times. Initial high voltage
testing has been performed for all the three types of THGEMs using nitrogen
gas keeping in mind the breakdown limits from Paschen curve. Argon-carbon
dioxide gas mixture in the volumetric ratio 90:10 and argon-isobutane mixture
in the ratio of 95:5 have been used for various measurements. These
measurements have been carried out using a 55Fe extended source and a point
source whose rate of radiation have been controlled by collimating the source
using collimators having different apertures.

##  3 Experimental results

CERN THGEM (THGEM A) as well as SINP THGEMs (B and C) have been characterized
using two 5.9keV 55Fe sources of different rates in argon based gas mixtures.
Extensive gain measurements have been performed for different applied voltages
within the discharge limit. Table 2 lists the various experimental
measurements and some of the important parameters used.

Table 2: Experimental configuration Section | THGEM | Measurement | Gas mixture | Source (kHz)
---|---|---|---|---
3.1.1 | A | Characterization |  $Ar:CO_{2}$=90:10, $Ar:iC_{4}H_{10}$=95:5 | 1.2, 4.8
3.1.2 | B and C | Characterization | -do- | 1.8, 4.8
3.2 | A, B and C | Gain measurement | -do- | 1.65
3.3 | A, B and C | Energy resolution | -do- | 1.7
3.4 | B and C | Simulation |  $Ar:iC_{4}H_{10}$=95:5 |  1cm ${}^{55}Fe$ track

###  3.1 Characterization

Characterization of THGEM includes getting 55Fe spectra for different rates of
the source, studying the optimum working voltage range and optimizing the
electron transparency to maximize the gain.

####  3.1.1 THGEM A (CERN)

Figure 6 shows the output from the spectroscopic amplifier in the oscilloscope
on irradiating the detector with a source as well as the fitted photopeak of
the 55Fe spectrum from the MCA. Both the photopeak and Ar-escape peak look
well separated.

Figure 6: Oscilloscope signal(left) and Fe-55 spectrum(right) obtained using
THGEM A Figure 7: Operating range in argon based gas mixtures for THGEM A

To study the optimum working range, two rates of 55Fe source have been used in
two different argon based gas mixtures and corresponding MCA spectra have been
recorded while varying the THGEM voltage. Drift and induction fields have been
kept constant throughout the measurement. The photopeak of the obtained MCA
spectra have been fitted with a gaussian function to get the mean photopeak
channel number corresponding to each appled voltage. Figure 7 shows the
operating voltage range for THGEM A for two different rates (1.2kHz and
4.8kHz) in two different gas mixtures; Ar:$CO_{2}$ = 90:10 and
Ar:$iC_{4}H_{10}$ = 95:5. The maximum attainable gain or MCA channel number
was defined as the one at which micro-discharges were not observed for at
least 1 minute. It has been noted that for a low rate source, operating
voltage range is higher for both the gases. Furthermore, the voltage range is
higher in argon-isobutane mixture than in argon-carbon dioxide mixture due to
the former having good discharge quenching property.

Figure 8: Variation of gain with drift field for THGEM A

To optimize the drift ($E_{D}$) and induction field ($E_{I}$) values or to
maximize the electron collection within the hole and electron transmission to
the anode at a given voltage, MCA spectra have been saved by varying the drift
and induction fields keeping the THGEM voltage constant. Figure 8 shows the
variation of mean MCA photopeak channel number with different values of
$E_{D}$ keeping THGEM voltage and $E_{I}$ constant. It is observed that the
mean channel number increases with the drift field, forms a plataeu-like
region and finally decreases for both the gases.

Similarly, figure 9 shows the variation of mean photopeak MCA channel number
for both the gases with different values of $E_{I}$ keeping $\Delta$VGEM and
$E_{D}$ constant. It is observed that MCA channel number increases with
increasing $E_{I}$.

Figure 9: Variation of gain with induction field for THGEM A

####  3.1.2 THGEM B and C

The same procedure has been adopted to characterize the locally fabricated
SINP THGEMs (B and C). Details of the experimental cofiguration are given in
the table 2.

Figure 10: Fe-55 spectrum obtained using THGEMs B and C in gases $Ar:CO_{2}$ =
90:10 (top) and $Ar:iC_{4}H_{10}$ = 95:5 (bottom)

Figure 10 shows the 55Fe spectra for both the types of THGEMs (B and C) with
and without rim in the two argon based gas mixtures. The next step has been to
measure the optimum working voltage range for both the THGEMs. Figure 11 shows
the operating range for THGEMs B and C in argon-carbon dioxide gas mixture. It
is observed that working voltage range is different for both THGEM B and C.
THGEM C has a working voltage range on the higher side. The reason is the
presence of rims in THGEM C which decreases the ampliplication field inside
the holes and hence, shifts the working voltage range to the higher values.
Another important point is that the voltage range for THGEM C is slightly
larger than that for THGEM B. This is again due to the presence of rims in
THGEM C which reduces the discharges at higher voltages.

Figure 11: Operating range in argon carbon dioxide based mixture

Furthermore, using two different rates of source also makes a difference in
the allowed voltage range. Voltage range is less for both the THGEMs (B and C)
when a high rate of source(4.8kHz) is used in argon-carbon dioxide mixture. On
testing the operating voltages for THGEM B and C in argon-isobutane mixture,
it is observed that THGEM C(with rim) is able to withstand gain higher than
THGEM B as shown in figure 12. In addition, on increasing the rate of the
source, there is no discharge obtained for THGEM C at the highest operating
voltage for the low source, whereas THGEM B could not withstand the highest
operating voltage obtained with the low source on irradiation with a high
source. As a result, the working voltage range for THGEM B decreases on
increasing the rate of the source. This is due to the absence of the rim in
THGEM B which makes it more prone to discharges and thus, decreases the
working voltage range.

Figure 12: Operating range in argon isobutane based mixture Figure 13:
Variation of gain with drift field for argon carbon dioxide based gas mixture

To optimize the drift field and induction field values for gain measurement,
mean of the fitted MCA photo peak have been recorded by varying the drift
field and induction field values while keeping the $\Delta$VGEM constant. In
both the cases, drift and induction field values have not been increased
beyond a value which can initiate an avalanche in the concerned gas.

Figure 14: Variation of gain with drift field for argon isobutane based
mixture

Figure 13 shows the variation of MCA channel number with $E_{D}$ at constant
values of $\Delta$VGEM and $E_{I}$ for argon carbon dioxide based gas mixture.
It has been observed that channel number increases with $E_{D}$, reaches a
maximum(plateau-like region) and sligtly decreases on further increasing
$E_{D}$. The value of $E_{D}$ corresponding to the maximum gain is different
for both THGEMs B and C. This difference is due to the different ratio of
field values at the entrance of holes of both the THGEMs. Similarly, for argon
isobutane based gas mixture(14), variation of MCA photopeak channel number has
been recorded for different $E_{D}$ keeping the other two parameters
$\Delta$VGEM and $E_{I}$ fixed. Gain is found to increase, reach a maximum and
then decrease finally. Once again the value of $E_{D}$ corresponding to the
maximum gain has been found to be different for both the THGEMs B and C.

Figure 15: Variation of gain with induction field for argon carbon
dioxide(left) and argon isobutane based(right) mixtures

Variation of gain with induction field ($E_{I}$) have been studied keeping the
$\Delta$VGEM and $E_{D}$ fixed. Figure 15 shows that the gain increases with
increasing induction field for both the THGEMs B and C in both the argon based
gas mixtures.

###  3.2 Gain measurement

For measuring the gain of the detectors, drift and induction fields have been
kept fixed at the values around which the effective gain is close to maximum.
The desired voltage at which the gain is to be measured, has been applied to
the detector and kept undisturbed till the time the THGEM PCBs are fully
polarized. This dielectric polarization time is found to be different for
different THGEMs depending on their geometry. Once the polarization is
complete at a given THGEM voltage, the detector is irradiated with a source
and large number of MCA spectra have been saved till the time the photo peak
position in the MCA saturated. These 5.9 keV x-ray spectra have been fitted
with a Gaussian function. The area under such a gaussian distribution has been
calculated which is equal to the total number of counts for that measurement.
Finally, the equivalent rate of irradiation for each measurement has been
estimated by dividing the area of the spectra by the duration of each
measurement. Upon the saturation of photo peak position, current from readout
is measured using a pico-ammeter. Current data for 5 minutes have been saved
and fitted with a gaussian to obtain the average value of saturation current.
Gain is calculated using the formula

$\begin{split}G=I/(n_{p}\times R\times e)\,,\end{split}$ (1)

where ’G’ is the effective gain, ’np’ is the mean number of primaries obtained
from HEED simulation, R is the rate of the source used and ’e’ is the
electronic charge.

Figure 16: Current measurement using pico-ammeter

A typical readout current data measured using pico-ammeter is shown in the
figure 16. Mean value of the gaussian fitted photopeak is used for the gain
measurement. Effective saturated gain once the radiation charging up process
gets over for all the three THGEMs(A, B and C) have been measured using the
formula given in equation 1. Figure 17 shows the gain curve for THGEM A in
argon based mixtures.

Figure 17: Effective gain values for THGEM A in argon-carbon dioxide and
argon-isobutane mixture with a source of rate 1.2kHz Figure 18: Variation of
gain for THGEMs B and C in argon carbon dioxide based mixture Figure 19:
Variation of gain for THGEMs B and C in argon isobutane based mixture

Similarly, effective gain values for locally fabricated THGEMs B and C in
argon carbon dioxide and argon isobutane gas mixtures are shown in figure 18
and figure 19 respectively. It is observed that the working voltage range is
higher in THGEM C. In addition to a larger working voltage range, THGEM C also
yields higher effective gain value. The reason could be higher collection and
extraction efficiency for THGEM C as compared to B. In other words, higher
gain value in THGEM C can be attributed to the absence of rim in THGEM B which
results in more loss of electrons on the top and bottom copper electrodes of
THGEM B, thus, resulting in lower effective gain. Similar results have been
observed while measuring effective gain for THGEMs B and C in argon isobutane
gas mixture as plotted in figure 19. Gain for THGEM C has been found to be
higher as compared to THGEM B. This could be again due to higher values of
collection and extraction efficiencies for THGEM C as compared to THGEM B.

###  3.3 Energy resolution

${}^{55}Fe$ spectrum obtained during gain measurement for each applied THGEM
voltage is fitted with a gaussian function. The mean and sigma values of the
fitted gaussian photopeak have been used to calculate the energy resolution at
that voltage. Energy resolution is defined to be

$\begin{split}E_{res}=FWHM/E=2.355\times\sigma_{E}/E\end{split}$ (2)

where E and $\sigma_{E}$ are the mean and sigma of the gaussian fitted MCA
spectrum.

Figure 20: Variation of energy resolution with the applied THGEM voltage for
CERN THGEM

These energy resolution values have been calculated once the gain is found to
saturate for a partcular applied THGEM voltage. $E_{res}$ value is found to
decrease on increasing the THGEM voltage, reach a minimum value and then again
it increases on further increasing the voltage across THGEM. The same
variation has been observed for all the three THGEMs A, B and C. Figure 20
shows the variation of energy resolution with applied voltage for THGEM A in
both argon carbon dioxide and argon isobutane based gas mixtures. The best
resoltion obtained THGEM A in both the gases is around 32-33%. This value of
energy resolution is found to be affected by the applied drift and induction
fields.

Figure 21: Variation of energy resolution with the applied THGEM voltage for
argon based mixtures

Similarly, the energy resolution values for THGEMs B and C in both the argon
based gas mixtures are shown in figure 21.There are two points to be discussed
in the context of energy resolution. First is the presence and absence of rim
affecting the energy resolution, second is the gas mixture. In argon-carbon
dioxide gas mixture, THGEM C is observed to have a better resolution for
higher voltage points than THGEM B. THGEM B fails to yield better resolution
than 52% due to it having discharges at higher voltage points. However in
argon-isobutane mixture, THGEM B seems to have slightly better energy
resolution than THGEM C. This is due to the quenching property of isobutane
which reduces discharges at higher voltages. Furthermore, THGEM C is observed
to yield a decent resolution fluctuating around 45% for a larger voltage
range. This can be attributed to the presence of rim in THGEM C and a quencher
gas (isobutane) in the studied gas mixture. This combination rim and isobutane
increases the voltage range for THGEM C and a saturation valley is observed
around 45%. Finally, the overall energy resolution for both the THGEMs is
expected to improve by tuning the drift and induction field values.

##  4 Simulation results

Figure 22: Simulated results showing variation of collection, extraction
efficiency(top) and transmission efficiency(bottom) with THGEM voltages in
Argon-isobutane mixture

To verify the reason behind the difference in effective gain for THGEMs with
and without rim, GARFIELD simulation has been done for THGEMs B and C,
electric field values and different efficiencies have been calculated. One cm
long 55Fe track has been simulated and collection, extraction and transmission
efficiencies have been calculated for both the THGEMs B and C in the argon
isobutane gas mixture.

Transmission efficiency determines the effective gain and is defined as the
product of collection and extraction efficiencies at a given applied THGEM
voltage. Figure 22 shows the simulated collection, extraction and transmission
efficiency for THGEMs B and C in argon isobutane mixture. It has been observed
that collection and extraction efficiency for THGEM C is around 90% throughout
the working voltage range, whereas these efficiencies are quite less for THGEM
B. Extraction efficiency for THGEM B is almost half of that of THGEM C. Due to
the absence of rim in B, there are significant number of electron losses on
top and bottom copper electrodes. This makes transmission efficiency for THGEM
B significantly lower, less than half of that of THGEM B and hence, results in
lower gain in case of B. Similar results are expected in argon-carbon dioxide
mixture too. This satisfactorily justifies the results obtained in section 3.2
.

## Conclusion

Two THGEMs with and without rim fabricated locally in SINP have been tested
and characterized using 55Fe source in argon based gas mixtures. Their
performance have been compared with the performance of THGEM A from CERN.
Operating voltage range for THGEMs B and C do not overlap. Voltage range for
THGEM C is found to shift towards the higher magnitudes, which can be
attributed to the presence of rim in THGEM C. Electric field inside the holes
decreases due to the presence of rim, as a result of which voltage range for
THGEM C shifts towards higher magnitudes of THGEM B in both the gases.
Moreover, operating voltage range is found to depend on the rate of
irradiation of source. Presence of rim is found to reduce discharges at higher
voltages for high rate sources, thus allowing higher working voltage range.
Furthermore, these THGEMs specially THGEM B is found to perform better in
argon isobutane gas mixture due to its quenching property.

Gain of more than $3.2\times 10^{3}$ have been achieved with THGEM C. The
presence of the etched rim around the THGEM holes, is found to be essential
for reducing discharge-occurrence probability significantly. This permitted
operation at higher voltages and at higher detector gains. THGEM B is found to
have a maximum gain of $2.2\times 10^{3}$ in argon-isobutane gas mixture which
is better than the maximum gain obtained in argon carbon dioxide gas mixture.
Simulation results show that the electron transparency of THGEM C is almost
twice of that of THGEM B. In terms of energy resolution, THGEM B is found to
yield slightly better resolution THGEM C in argon isobutane mixture, but the
latter yields a satisfactorily good resolution for a larger voltage range.

To conclude, THGEM C performs better close to the discharge limit by
significatly reducing the discharges and allowing operation at high gains.
Furthermore, it has excellent electron transmission close to 90%. Presence of
a quencher gas improves the operation of THGEM B. Finally both the THGEMs B
and C yield reasonably rood energy resolution. In future, we want to use
THGEMs in double THGEM configuration and study the position resolution for
muon tracking purposes.

## Acknowledgement

Authors would like to acknowledge their respective Institutes and Universities
for the funding, lab and computational facilities to carry out the experiment
and simulation. They would like to thank Mr. Biswajit Majumdar and Mr.
Samarjit Majumdar for helping them in the fabrication process. Authors would
also like to thank Mr. Subhendu Das, Mrs. Lipi Das and Mr. Arindam Das for
their help in the THGEM gerber file design. Finally they would like to thank
Prof. Chinmay Basu for his support. This work has partially been performed in
the framework of the RD51 Collaboration and they would like to acknowledge the
members of the RD51 Collaboration for their help and suggestions.

## References

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# Limiting propulsion of ionic microswimmers

Evgeny S. Asmolov (Евгений С. Асмолов)  Frumkin Institute of Physical
Chemistry and Electrochemistry, Russian Academy of Science, 31 Leninsky
Prospect, 119071 Moscow, Russia  Institute of Mechanics, Lomonosov Moscow
State University, 1 Michurinskiy Prospect, 119991 Moscow, Russia
<EMAIL_ADDRESS>Olga I. Vinogradova (Ольга И. Виноградова)  Frumkin Institute
of Physical Chemistry and Electrochemistry, Russian Academy of Science, 31
Leninsky Prospect, 119071 Moscow, Russia

###### Abstract

Self-propulsion of catalytic Janus swimmers in electrolyte solutions is
induced by inhomogeneous ion release from their surface. Here, we consider the
experimentally relevant cases of particles which emit only one type of ions
(type I) or equal fluxes of cations and anions (type II). In the limit of a
thin electrostatic diffuse layer we derive a nonlinear outer solution for the
electric field and concentrations of active (i.e. released from the surface)
and passive ionic species. We show that for swimmers of type I both the
maximum ion flux and propulsion velocity are constrained. This suggests that
the propulsion of Janus swimmers can be optimized by tuning the concentration
of active ions.

##  I Introduction

Catalytic Janus swimmers have drawn a considerable interest during last
decadesmoran2017 ; dey2017 ; asmolov2022COCIS  because of their ability to
move autonomously. The potential applications include drug delivery, lab-on-a-
chip devices and nano-roboticshu2020micro . The self-propulsion mechanism of
ionic swimmers is based on an inhomogeneous ion release at the particle
surface. Ion diffusion induces concentration gradient and electric
fieldgolestanian2005propulsion .

Despite the variety of catalytic reactions, two main models for ionic flux are
acceptedasmolov2022COCIS ; Wang2020practical ; peng2022generic  (see Fig. 1):
only one type of ions is released (Type I) and equal fluxes of cations and
anions (Type II). The classical examples of the first type include
bimetallicpaxton2004  and metallic-insulator howse2007  Janus particles. As a
result of hydrogen peroxide decomposition on the catalytic particle side,
protons are released into solution and then are adsorbed on the non-catalytic
side. The ionic flux for the second type can be generated by the salt
dissolution in watermcdermott2012 , enzyme-enhanced decomposition of organic
compoundsdey2015micromotors ; patino2018fundamental  or by photochemical
reactionszhou2018 .

Charged particle immersed in an electrolyte solution is surrounded by an
electric diffuse layer (EDL), where charge imbalance is large and the electric
field is strong. Concentration of ionic species, electric potential and fluid
flow near catalytic swimmers are governed by a system of the Nernst-Planck,
Poisson and Stokes equations. The EDL is thin for swimmers of micron size, and
the method of matched asymptotic expansions is usually applied for the
theoretical description of self-propulsionprieve.dc:1984 ; anderson.jl:1989 ;
golestanian2007 . The EDL effect appears macroscopically as a slip velocity at
the particle surface. The electric field in the outer region (at distances of
the order of the particle size) is not zero due to ion diffusion and small
charge imbalance and can effect on the slip velocitynourhani2015 ;
asmolov2022self . The analytical solutions in the outer region has been
obtained for the swimmer of Type Inourhani2015 ; ibrahim2017  under the
assumption that field disturbances are small and the Nernst-Planck equations
can be linearized.

Recently, we have proposedasmolov2022self ; asmolov2022MDPI  non-linear
analytical solutions to Nernst-Planck and Poisson equations for the cases when
bulk electrolyte contains active (released from the swimmer surface) ions
only. However, the self-propulsion in experiments is carried out in media
containing both active and passive ions. The addition of passive salt reduces
the swimmer velocitymoran2017 ; ibrahim2017 ; arque2020ionic . In the present
paper, we extend our previous solutions to include the effect of passive ions
in the bulk. We obtain the analytical solutions in the outer region for both
swimmer types and all ionic species. The depletion of active ions arise for
the swimmer of Type I near an adsorbing particle side. This imposes a
constraint on the maximum swimmer velocity.

Our paper is organized as follows. The governing equations are formulated in
Sec. II. Their solution at distances of the order of particle size (outer
region) is given in Sec. III. Our numerical results for concentrations and for
the limiting particle velocity are presented in Sec. IV and are summarized in
Sec. V.

##  II Governing equations

Figure 1: Two types of ionic swimmers. (a) Type I releases cations only at the
active side and adsorb them at the other side. (b) Type II releases both
cations and anions at the active side.

We consider an 1:1 aqueous electrolyte solution with the bulk ion
concentration $c_{\infty}.$ It includes passive salt ions with dimensionless
concentrations $C^{s\pm}\left(\mathbf{r}\right)\ $(scaled by $c_{\infty})$ and
ions which are released from the particle surface with concentrations
$C^{r\pm}\left(\mathbf{r}\right)$. Here the upper (lower) sign corresponds to
the positive (negative) ions. The Peclet number is assumed to be small, so
that the convective fluxes are ignored, and the ion fluxes are governed by
dimensionless Nernst-Planck equations

$\nabla\cdot\mathbf{J}^{i}=0,$ (1) $\mathbf{J}^{i}=-\nabla C^{i}\mp
C^{i}\nabla\Phi.$ (2)

Here $i=r\pm,$ $s\pm,$ the electric potential $\Phi$ is scaled by $k_{B}T/e,$
where $e$ is the elementary positive charge, $k_{B}$ is the Boltzmann
constant, $T$ is the temperature of the system, and coordinates are scaled by
the particle size $a.$

Unequal distributions of oppositely charged ions generate an electric field,
governed by the Poisson equation for the electric potential,

$\Delta\Phi=\lambda^{-2}\frac{C^{s+}+C^{r+}-C^{s-}-C^{r-}}{2},$ (3)

where the dimensionless parameter $\lambda=\left(\kappa a\right)^{-1}$ is the
ratio of the Debye length $\kappa^{-1}=\left(8\pi e^{2}c_{\infty}/\epsilon
k_{B}T\right)^{-1/2}$ to the particle size, $\epsilon$ is the permittivity of
the fluid. We consider large particles with a thin electric double layer
(EDL), i.e. $\lambda\ll 1,$ and construct the asymptotic solution using the
method of matched expansions in two regions with different lengthscales. The
lengthscale of the outer region is the particle size $a$ and that for the
inner region is the Debye length $\kappa^{-1}$.

The boundary conditions at the particle surface $S$ (defined by
$\mathbf{r}=\mathbf{r}_{s}$) read,

$\displaystyle\mathbf{r}\in S:\
\mathbf{J}^{r+}\cdot\mathbf{n}=-\mathrm{Da}j^{+}\left(\mathbf{r}_{s}\right),$
(4)
$\displaystyle\mathbf{J}^{r-}\cdot\mathbf{n}=-\mathrm{Da}j^{-}\left(\mathbf{r}_{s}\right),$
(5) $\displaystyle\mathbf{J}^{s\pm}\cdot\mathbf{n}=0,$ (6) $\displaystyle\
\Phi=\phi_{s}\left(\mathbf{r}_{s}\right),$ (7)
$\displaystyle\mathrm{Da}=\frac{Ja}{c_{\infty}D^{r+}},$ (8)

where $\mathrm{Da}$ is the Damköhler number and $J$ is the average module of
dimensional flux of released cations. The first two conditions define the
surface fluxes of active ions, and the third one the zero fluxes for passive
ions. The functions $j^{\pm}\left(\mathbf{r}_{s}\right)$ characterize
dimensionless inhomogeneous ion productions at the particle surface. Because
of definition (8) the cation flux satisfies the condition

$\frac{1}{S}\int_{S_{p}}\left|j^{+}\right|dS=1.$ (9)

The net flux for the swimmers of Type I should vanish in a steady state:

$\int_{S_{p}}j^{+}dS=0.$ (10)

The boundary condition (7) sets the surface potential $\phi_{s}$ which are
assumed to be constant.

Boundary conditions at infinity are,

$r\rightarrow\infty:\ C^{s\pm}=C_{\infty}^{s\pm},\
C^{r\pm}=C_{\infty}^{r\pm},\ \Phi=0.$ (11)

The bulk electrolyte solution is electroneutral, so that

$C_{\infty}^{s+}+C_{\infty}^{r+}=C_{\infty}^{s-}+C_{\infty}^{r-}=1.$

The fluid flow satisfies the Stokes equations,

$\mathbf{\nabla\cdot
v}=0\mathbf{,\quad}\Delta\mathbf{v}-\mathbf{\nabla}p=\mathbf{f}.$ (12)

Here $\mathbf{v}$ and $p$ are the dimensionless fluid velocity and pressure
(scaled by $\dfrac{\epsilon k_{B}^{2}T^{2}}{4\pi\eta e^{2}a}$ and
$\dfrac{\epsilon k_{B}^{2}T^{2}}{4\pi e^{2}a^{2}}$), $\eta$ the viscosity and
$\mathbf{f}=-\Delta\Phi\mathbf{\nabla}\Phi$ is the electrostatic body force.

##  III Theory

In this section we present the solution of the system (1)-(8) in the outer
region with the lengthscale $a.$ Since $\lambda\ll 1,$ the leading-order
solution of (3) in the outer region is

$C^{s+}+C^{r+}=C^{s-}+C^{r-}=C\left(\mathbf{r}\right),$ (13)

i.e. the electroneutrality holds to $O\left(\lambda^{2}\right).$ However, one
should take into account the small charge
$C^{s+}+C^{r+}-C^{s-}-C^{r-}=O\left(\lambda^{2}\right)$, since it induces a
finite potential difference in the outer region. We sum up the equations (1)
for positive and negative ions, and the system then becomes

$\displaystyle\Delta C+\nabla\cdot\left(C\nabla\Phi\right)$ $\displaystyle=$
$\displaystyle 0,$ (14) $\displaystyle\Delta
C-\nabla\cdot\left(C\nabla\Phi\right)$ $\displaystyle=$ $\displaystyle 0.$
(15)

Summing up and subtracting equations (14) and (15) we obtainnourhani2015 ;
ibrahim2017 ; asmolov2022MDPI :

$\displaystyle\Delta C$ $\displaystyle=$ $\displaystyle 0,$ (16)
$\displaystyle\nabla\cdot\left(C\nabla\Phi\right)$ $\displaystyle=$
$\displaystyle 0.$ (17)

Likewise, by calculating the sum and difference of Eqs.(4), (5) and (6) we
derive the boundary conditions to (16) and (17),

$\displaystyle\mathbf{r}\in S:\
\mathbf{\nabla}C\cdot\mathbf{n}=-\frac{\mathrm{Da}}{2}\left(j^{+}+j^{-}\right),$
(18) $\displaystyle
C\mathbf{\nabla}\Phi\cdot\mathbf{n}=-\frac{\mathrm{Da}}{2}\left(j^{+}-j^{-}\right).$
(19)

Solution of Laplace equation (16) makes it possible to find the net ion
concentrations $C$. However, the solution of a non-linear equation for the
electric potential (17) is generally not straightforward. We consider here the
case of the constant ratio of the surface fluxes,

$j^{-}\left(\mathbf{r}_{s}\right)/j^{+}\left(\mathbf{r}_{s}\right)=\alpha=\text{const}.$
(20)

The condition is valid for the two swimmer type: $\alpha=0$ for the Type I
(cations are released only) and $\alpha=D^{r+}/D^{r-}$ for the Type II. The
solution of (17) satisfying the boundary conditions (11) and (19)
isasmolov2022MDPI

$\Phi=-\beta\ln\left(C\right),$ (21) $\beta=\dfrac{\alpha-1}{\alpha+1}.$ (22)

The above dimensionless parameter is limited, namely $\left|\beta\right|\leq
1.$

To find the concentrations $C^{i}$ we do not need to solve the system (1)-(6).
The boundary fluxes for passive ions $C^{s\pm}$ are zero, so their fields obey
the Boltzmann distribution,

$C^{s\pm}=C_{\infty}^{s\pm}\exp\left(\mp\Phi\right)=C_{\infty}^{s\pm}C^{\pm\beta}.$
(23)

Once $C$ and $C^{s\pm}$ are known we can also determine the distributions of
active ions from Eqs. (13), (23):

$\displaystyle C^{r+}$ $\displaystyle=$ $\displaystyle
C-C^{s+}=C-C_{\infty}^{s+}C^{\beta},$ (24) $\displaystyle C^{r-}$
$\displaystyle=$ $\displaystyle C-C^{s-}=C-C_{\infty}^{s-}C^{-\beta}.$ (25)

Thus, we obtain concentrations of all ionic species in the outer region for
arbitrary $\mathrm{Da}$ and $j^{\pm}\left(\mathbf{r}_{s}\right).$ This is one
of the main results of our work. However, physically reasonable results may
not be found for any boundary conditions. The restriction comes from the
condition that the concentrations may not be negative for all $\mathbf{r}$ and
$i,$

$C^{i}\left(\mathbf{r}\right)\geq 0.$ (26)

This implies physical constraints on $\mathrm{Da}$ for given $C_{\infty}^{i}$
and $j^{\pm}\left(\mathbf{r}_{s}\right)$. When the normal gradient
$\mathbf{\nabla}C\cdot\mathbf{n}$ is negative over the entire particle
surface, i.e. the net flux of released ions
$j^{+}\left(\mathbf{r}_{s}\right)+j^{-}\left(\mathbf{r}_{s}\right)$ is always
positive as for the Type II, the solution of (16) satisfies the condition
$C\geq 1$ for any $\mathbf{r,}$ i.e. the net concentration exceeds the bulk
value. This means that the concentration of released ions (Eqs. (24), (25)) is
always positive since $C_{\infty}^{s\pm}\leq 1.$ The opposite situation takes
place for the Type I when ions are adsorbed at some surface portion, where the
surface flux is negative and $\mathbf{\nabla}C\cdot\mathbf{n}$ is positive.
Then we may obtain physically impossible solution $C\leq 0,$ when
$\mathrm{Da}$ is large enough. Moreover, negative solutions for active ions
can follow from Eqs. (24), (25) even at smaller $\mathrm{Da}$ (see Sec. IV).
Therefore, there is a limiting value $\mathrm{Da}_{\mathrm{max}}$ for which
the minimum concentration $C_{\infty}^{r\pm}=0.$

Solution in the inner region expresses the slip velocity $\mathbf{v}_{s}$ at
the outer edge of the EDLprieve.dc:1984 ; anderson.jl:1989

$\mathbf{v}_{s}=\psi\nabla_{s}\Phi_{s}+4\nabla_{s}(\ln
C_{s})\ln\left[\cosh\left(\frac{\psi}{4}\right)\right],$ (27)
$\psi=\phi_{s}-\Phi_{s},\quad\Phi_{s}=\Phi\left(\mathbf{r}_{s}\right),\quad
C_{s}=C\left(\mathbf{r}_{s}\right).$

Here $\nabla_{s}$ is the gradient operator along the particle surface, the
first and the second terms in Eq.(27) are associated with electro- and
diffusio-osmotic flows, respectively, $\psi$ is the potential drop in the
inner region, $\Phi_{s}$ and $C_{s}$ are the outer potential and the
concentration at the dividing surface between the inner and outer regions.

The velocity of a freely moving in the $x-$direction particle can be
determined by using the reciprocal theoremteubner1982

$v_{p}=-\frac{1}{6\pi}\int_{V_{f}}\mathbf{f}\cdot\left(\mathbf{v}_{1}-\mathbf{e}_{x}\right)dV.$
(28)

The integral is evaluated over the whole fluid volume $V_{f}$ and
$\mathbf{v}_{1}\left(\mathbf{r}\right)$ represents the velocity field for the
particle of the same shape that translates with the velocity $\mathbf{e}_{x}$
in a stagnant fluid. The velocity can be presented as a superimposition of the
contributions of the outer and inner regionsasmolov2022self ,

$v_{p}=v_{po}+v_{pi},$ (29)
$v_{po}=-\frac{\beta^{2}}{6\pi}\int_{V_{f}}\frac{\left(\mathbf{\nabla}C\right)^{2}\mathbf{\nabla}C\cdot\left(\mathbf{v}_{1}-\mathbf{e}_{x}\right)}{C^{3}}dV,$
(30)
$v_{pi}=-\frac{1}{S}\int_{S_{p}}\left(\mathbf{v}_{s}\cdot\mathbf{e}_{x}\right)dS.$
(31)

Equations (27)-(31) do not include the concentrations of each ionic species,
but only the net value $C.$ Hence, the velocity is independent of
$C_{\infty}^{s+}$ provided
$\mathrm{Da}<\mathrm{Da}_{\mathrm{max}}\left(C_{\infty}^{s+}\right).$

When the Damköhler number is small, the concentration and potential fields are
slightly disturbed in the outer region,
$C=1+O\left(\mathrm{Da}\right)\mathrm{,\ }\Phi=O\left(\mathrm{Da}\right),$
$\psi\simeq\phi_{s}$ as $\mathrm{Da}\ll 1.$ Contribution of the outer region
to the velocity is $O\left(\mathrm{Da}^{3}\right)$ and can be neglected, since
the integrand in Eq. (30) is proportional to
$\left(\mathbf{\nabla}C\right)^{3}$. Then the dominant contribution of the
order of $\mathrm{Da}$ comes from the inner region (31) with the linearized
slip velocity:

$\mathbf{v}_{s}=\phi_{s}\nabla_{s}\Phi_{s}+4\nabla_{s}C_{s}\ln\left[\cosh\left(\frac{\phi_{s}}{4}\right)\right]=O\left(\mathrm{Da}\right).$
(32)

Below we show that this linear solution approximates well the exact results up
to $\mathrm{Da}\sim 1.$

##  IV Results and discussion

We consider spherical particle with symmetric ion release with respect to $x-$
axis, so that $j^{\pm}$ depends on azimuthal angle $\theta$ only. The boundary
condition (18) is rewritten as

$r=1:\ \partial_{r}C=-\frac{\mathrm{Da}}{2}\left(j^{+}+j^{-}\right).$ (33)

We use the spherical coordinate system to solve Eq. (16) with the boundary
conditions (33) and present the solution in terms of Legendre polynomials
$P_{n}\left(\cos\theta\right)$,

$C=1+\frac{\mathrm{Da}}{2}\xi\left(r,\theta\right),$ (34) $\displaystyle\xi$
$\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\frac{j_{n}}{n+1}P_{n}\left(\cos\theta\right)r^{-n-1},$
(35) $\displaystyle j_{n}$ $\displaystyle=$
$\displaystyle\left(n+\frac{1}{2}\right)\int_{0}^{\pi}\left(j^{+}+j^{-}\right)P_{n}\sin\theta
d\theta.$ (36)

We model inhomogeneous distributions
$j^{+}\left(\theta\right),j^{-}\left(\theta\right)$ by piecewise constant
functions, i.e. the ions are released from the surface for
$\theta\leq\theta_{r}.$ At the other sphere side, $\theta_{r}<\theta\leq\pi,$
cations are adsorbed for the swimmer Type I (see Fig. 1(a)), and the fluxes
are zero for the Type II (see Fig. 1(b)). Then the surface flux distributions
satisfying conditions (9) and (10) are

$\begin{array}[]{cc}0\leq\theta\leq\theta_{r}:&j^{+}=\frac{1}{2\left(1+\cos\theta_{r}\right)}<0,\
j^{-}=0,\\\
\theta_{r}<\theta\leq\pi:&j^{+}=-\frac{1}{2\left(1-\cos\theta_{r}\right)}>0,\
j^{-}=0,\end{array}$ (37)

for the first type, and

$\begin{array}[]{cc}0\leq\theta\leq\theta_{r}:&j^{+}=j^{-}/\alpha=\frac{1}{\left(1+\cos\theta_{r}\right)}>0,\\\
\theta_{r}<\theta\leq\pi:&j^{+}=j^{-}=0,\end{array}$ (38)

for the second one.

We evaluate numerically the integrals in Eq. (36) to obtain a reduced ion
concentration $\xi\left(r,\theta\right)$ with the surface fluxes (37) and (38)
on a uniform grid in $\theta$ with $N_{\theta}=200.$ To evaluate the particle
velocity we calculate the surface integral in Eq. (31), and the volume
integral in Eq. (30) using the same grid in $\theta$ and a non-uniform grid in
$r$ (with a grid step varying as $r^{2}$) with $N_{r}=100$ nodes and a cut-off
radius $R_{out}=100$. Figure 2$\left(a\right)$ shows the functions
$\xi_{s}\left(\theta\right)=\xi\left(1,\theta\right)$ for the swimmer of the
Type I with $\beta=-1$ (solid and dashed-dotted lines) and Type II with
$\beta=-0.5,\ \alpha=D^{r+}/D^{r-}=1/3$ (dashed and dotted lines). The
functions decay with $\theta$, i.e. the ion concentration is always greater at
the catalytic section where ions are released. For the Type II, $\xi_{s}>0$
for any angle $\theta_{r}$ over the entire surface, so that $C\geq 1$ (see Eq.
(34)), i.e. the net concentration is always greater than the bulk value. In
contrast, for the Type I, $\xi_{s}\left(\theta\right)$ changes the sign so
that $C_{s}<1$ at the rear sphere side where the ions are adsorbed. We obtain
from Eq. (34) non-physical values $C_{s}<0$ when
$\mathrm{Da>Da}_{\mathrm{max}}=-2/\xi_{\mathrm{min}}$ where
$\xi_{\mathrm{min}}=\xi\left(1,0\right)<0$ is the absolute minimum of the
function $\xi$ attained at the rear sphere pole. The minimum
$\xi_{\mathrm{min}}$ grows and the limiting Damköhler number decreases with
$\theta_{r}$ (smaller area of cation adsorption).

The limiting value of $\mathrm{Da}$ becomes even smaller when we take into
account the condition that the concentrations of all ionic species should also
be positive. We evaluate $C^{i}$ by using Eqs. (23)-(25) and present the
results in Fig. 2$\left(b,c\right).$ For the Type II, Eqs. (23)-(25) predict
positive concentrations for all species (see Fig. 2$\left(c\right))$ as $C\geq
1,$ $C_{\infty}^{s+}\leq 1,\ \left|\beta\right|\leq 1,$ so that there is no
limitation on $\mathrm{Da.}$ The concentrations of passive ions (dashed-dotted
and dotted lines) are rather small, while those for the released ions (solid
and dashed lines) are large and surpass the bulk value $C_{\infty}^{r+}=0.7.$
Another behavior can be viewed in Fig. 2$\left(b\right)$ for the Type I. The
concentration of released cations (solid line) is close to zero at
$\theta=\pi.$ Therefore, the Figure illustrates the limiting regime which is
attained at $\mathrm{Da}_{\mathrm{max}}\simeq 6$ for given $C_{\infty}^{r+}$
and $j^{+}\left(\theta\right).$ Greater $\mathrm{Da}$ is impossible since Eq.
(25) gives $C^{r+}<0.$ We note that $C_{s}^{r+}\left(\theta\right)$ (solid
line) decreases significantly along the sphere surface while the concentration
of passive cations $C_{s}^{s+}\left(\theta\right)$ (dash-dotted line) grows,
so that the variation of the net concentration remains not too large. Thus the
presence of passive salt reduces the limiting $\mathrm{Da.}$

Figure 2: (a) Reduced ion concentration at the particle surface as a function
of $\theta$. Solid and dashed-dotted lines correspond to the swimmer Type I
with $\theta_{r}=\pi/2$ and $\theta_{r}=\pi/3$, respectively, dashed and
dotted lines to the Type II, $\theta_{r}=\pi/2$ and $\theta_{r}=\pi/3$.
Concentrations of ion species for swimmers of (b) the Type I and (c) Type II
calculated using $\theta_{r}=\pi/2$, $\mathrm{Da}=3$ and
$C_{\infty}^{s+}=0.3$. Solid and dashed lines are concentrations of released
cations and anions, respectively, dashed-dotted and dotted lines are
concentrations of salt cations and anions.

Condition $C^{r+}\geq 0$ can be reformulated as the limitations of the minimum
bulk value or of the maximum Damköhler number. Equation (24) with $\beta=-1$
reads

$C^{r+}=C-C_{\infty}^{s+}/C\geq 0.$ (39)

Then by using (34) we obtain

$C_{\infty}^{s+}\leq
C_{\min}^{2}=\left(1+\frac{\mathrm{Da}}{2}\xi_{\mathrm{min}}\right)^{2},\
C_{\infty}^{r+}\geq 1-C_{\min}^{2}>0,$ (40)

or equivalently,

$\mathrm{Da}<\mathrm{Da}_{\mathrm{max}}=2\frac{\sqrt{C_{\infty}^{s+}}-1}{\xi_{\mathrm{min}}}.$
(41)

The maximum Damköhler number depend only on the salt concentration
$C_{\infty}^{s+}$ and the angle $\theta_{r}$. Figure 3$\left(a,b\right)$ shows
$\mathrm{Da}_{\mathrm{max}}$ as the functions of the two parameters. It
decreases with $\theta_{r}$ and with $C_{\infty}^{s+}.$ The active cations
cannot released on the catalytic surface
($\mathrm{Da}_{\mathrm{max}}\rightarrow 0$ and $v_{p}\rightarrow 0)$ when they
are absent in the bulk, $C_{\infty}^{r+}\rightarrow 0,$ or when the area of
cation adsorption tends to zero $\left(\theta_{r}\rightarrow\pi\right).$

Figure 3: Limiting Damköhler number calculated as functions (a) of
$\theta_{r}$ for $C_{\infty}^{s+}=0.3$ and (b) of $C_{\infty}^{s+}$ for
$\theta_{r}=\pi/2$.  Figure 4: Particle velocities calculated using
$\phi_{s}=-1$, $0$, $1$ (from top to bottom), $\theta_{r}=\pi/2$,
$C_{\infty}^{s+}=0.3$ as functions of $\mathrm{Da}$. Solid curves are the
total velocities $v_{pi}+v_{po}$, dashed lines correspond to the linear
solution (32), dotted line is the limiting Damköhler number.

The particle velocity $v_{p}$ depends on three dimensionless parameter,
$\mathrm{Da,}$ $\phi_{s},$ $\theta_{r}.$ Equations (27)-(29) do not include
the concentrations of each ionic species, but only the net value $C,$ so the
velocity remains the same for any $C_{\infty}^{s+}$ when
$\mathrm{Da}<\mathrm{Da}_{\mathrm{max}}\left(C_{\infty}^{s+},\theta_{r}\right).$
We evaluated $v_{p}$ earlier for the swimmer Type IIasmolov2022self  (see also
Fig. 4 in asmolov2022COCIS ). $\mathrm{Da}$ can be arbitrary in this case, and
$v_{p}$ grows logarithmically as $\mathrm{Da}\rightarrow\infty.$

We show the particle velocity for the swimmer Type I in Fig. 4 as function of
the Damköhler number for different $\phi_{s}.$ The velocity of an uncharged
particle $\left(\phi_{s}=0\right)$ is very small, and at finite $\phi_{s}$ and
$\mathrm{Da,}$ it is close to the predictions of the linear solution (32). The
maximum velocity, that is reached at
$\mathrm{Da}=\mathrm{Da}_{\mathrm{max}}\left(C_{\infty}^{s+},\theta_{r}\right),$
strongly depends on $C_{\infty}^{s+}$.

Figure 5: The contribution of the outer region to the particle velocity
$v_{po}$ calculated for $\mathrm{Da}=\mathrm{Da}_{\mathrm{max}}$ and
$\theta_{r}=\pi/2$ (solid line). Dashed line is the power function with the
exponent $-0.355$.

We further study the dependences of the maximum particle velocity
$v_{p}\left(\mathrm{Da}_{\mathrm{max}}\right)$ on $C_{\infty}^{s+},$
$\theta_{r}$ and $\phi_{s},$ and evaluate separately the contributions of the
outer and the inner regions, $v_{po}$ and $v_{pi}$. The velocity $v_{po}$ is
shown in Fig. 5 as the function of the salt concentration. It is negative for
any $C_{\infty}^{s+},$ and is extremely small at finite salt concentration,
but its magnitude grows infinitely like
$\left(C_{\infty}^{s+}\right)^{-0.355}$ (dashed line) as
$C_{\infty}^{s+}\rightarrow 0$. The reason is that the net ion concentration
$C\left(\mathbf{r}\right)=C^{r+}+C^{s+}$ is small near the rear sphere pole in
this case since both $C^{r+}$ and $C^{s+}\ $tend to zero, while the gradient
$\mathbf{\nabla}C$ is finite. As a result the integrand in (30) tends to
infinity.

Figure 6: Limiting velocities (solid lines) as functions of (a)
$C_{\infty}^{r+}$ for $\theta_{r}=\pi/2$ and (b) of $\theta_{r}$ for
$C_{\infty}^{s+}=0.3$. Dashed-dotted lines correspond to the contributions of
the inner region $v_{pi}$.

We show the contribution of the inner region $v_{pi}$ evaluated by using Eqs.
(31), (27) and the total particle velocity $v_{p}=v_{po}+v_{pi}$ in Fig. 6.
The velocity $v_{pi}$ is dominant while $v_{po}$ is negligible at finite salt
concentration $C_{\infty}^{s+}$ and any $\phi_{s}.$ Up to
$C_{\infty}^{s+}\simeq 0.4,$ the velocity $v_{pi}$ is nearly linear in
$C_{\infty}^{r+}$ (see Fig. 6(a)), but at small $C_{\infty}^{s+}$ it grows
infinitely, similar to $v_{po}$. The origin of the singularities for the two
contributions is the same: $C\rightarrow 0$ near the rear pole in the limit
$C_{\infty}^{s+}\rightarrow 0,$
$\mathrm{Da}\rightarrow\mathrm{Da}_{\mathrm{max}}.$ However, the singularity
for $v_{pi}$ is weaker than that for $v_{po}.$ The reason is that the
singularity of the slip velocity (Eq. (27)) in the integrand of (31) is weaker
(proportional to $C^{-1})$ than that of the integrand in Eq. (30) for $v_{po}$
(proportional to $C^{-3}).$ The contribution of the outer region becomes
dominant at very small $C_{\infty}^{s+}$, and the total velocity is negative
for any $\phi_{s}.$ The maximum particle velocity decays with the angle
$\theta_{r}$ i.e. with the area of the catalytic zone (see Fig. 6(b)).

##  V Conclusion

We have developed a theory of a self-propulsion of catalytic ionic swimmers in
a media containing both active (released from the swimmer surface) and passive
salt ions. The theory have involved the two types of swimmers, which release
only one type of ions (Type I) or equal fluxes of cations and anions (Type
II). In the limit of a thin EDL, we have derived the analytic solution for the
concentrations of all ionic species at distances comparable to the particle
size (outer region). For the swimmer of Type I, the constraint on the maximum
Damköhler number $\mathrm{Da}_{\mathrm{max}}$ follows from the depletion of
active ions near the adsorbing particle side (their concentration should be
positive). $\mathrm{Da}_{\mathrm{max}}$ decreases with the salt concentration
$C_{\infty}^{s+}$ but increases with the area of adsorbing part. The particle
propulsion velocity grows infinitely when
$\mathrm{Da}\rightarrow\mathrm{Da}_{\mathrm{max}},\ C_{\infty}^{s+}\rightarrow
0.$

###### Acknowledgements.

This work was supported by the Ministry of Science and Higher Education of the
Russian Federation.

## DATA AVAILABILITY

The data that support the findings of this study are available within the
article.

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