Rafaela C Brum

Luciana Arantes

Maria Clicia Castro

Pierre Sens

Lúcia M A Drummond

Evaluating Execution Times and Costs of a Federated Learning Application on different Cloud Providers

Keywords: Cloud Computing, Federated Learning, Time and Cost Evaluation

Federated Learning (FL) is a new area of distributed Machine Learning (ML) that emerged to deal with data privacy concerns. In FL, each client has access to a local private dataset. At every round, a client trains the model with its local dataset and sends the weights to a central server. The latter aggregates all client weights and then sends the final weights back to the clients. This approach is attractive in many domains as it allows multiple institutions to collaborate on an ML task without sharing their data. However, most ML models used in FL have millions of weights exchanged in each message. The messages sent between a client and the server can achieve gigabytes of size and are exchanged several times in the whole FL execution. This work presents a preliminary analysis of execution times and costs of a FL application in a multi-cloud scenario. Experiments were conducted considering executions on the Amazon Web Services, Google Cloud Provider, and also in both cloud providers at the same time.

Introduction

Federated Learning (FL) is a recent type of distributed Machine Learning (ML) in which the participating clients do not share their private data [START_REF] Shen | From distributed machine learning to federated learning : In the view of data privacy and security[END_REF]. The clients federation solves the learning task coordinated by a central server without sharing the data. Instead, each client computes and communicates only the model weights to update the current global model kept by the server. This server-clients architecture of FL is classified into Cross-Device or Cross-Silo Federated Learning, depending on the connected client's type. A Cross-Device Federated Learning has low-powered devices as clients [START_REF] Mcmahan | Communication-Compas'2022 : Parallélisme/ Architecture/ Système MIS/UPJV -Amiens France, 5-8 juillet 2022 Efficient Learning of Deep Networks from Decentralized Data[END_REF][START_REF] Nguyen | Efficient federated learning algorithm for resource allocation in wireless iot networks[END_REF] while a Cross-Silo Federated Learning has different companies or institutions as clients (e.g., hospitals [START_REF] Rajendran | Cloud-Based Federated Learning Implementation Across Medical Centers[END_REF]) with private datasets willing to collaborate to create a global model. In this work, we focus on the second type of Federated Learning, in which the central server can assume that all clients are available during the whole execution, and there are usually fewer clients (less than 10 [15]). McMahan et al. [START_REF] Mcmahan | Communication-Compas'2022 : Parallélisme/ Architecture/ Système MIS/UPJV -Amiens France, 5-8 juillet 2022 Efficient Learning of Deep Networks from Decentralized Data[END_REF] proposed the term Federated Learning in 2017 as a learning technique that allows users to collectively benefit from shared models trained from distributed data without centrally storing them. The authors presented a Cross-Device FL in which the clients are mobile phones. Since then, this area has received much attention from researchers due to the increasing concern with data privacy. In traditional distributed ML, participants usually exchange data to balance their execution times. However, with the rise of data protection laws (e.g., GPDR 1 in Europe and LGPD 2 in Brazil), application domains that handle sensitive data cannot share it anymore and depends on FL to create good prediction models. Moreover, the digital data created by an institution and used to train ML algorithms increases rapidly. Most institutions cannot upgrade their data centers due to high financial costs. One viable option is the use of cloud storage services, in which the user only pays for the amount of stored data. Many cloud providers offer these storage services with different privacy guarantees and data availability. For example, in the Google Cloud provider, the user can store the data in a multi-region scenario. The data is available in different regions of the same country, and the user can determine who can access it. In this scenario, each participating institution of the FL environment can choose the best storage option among the different cloud providers. According to Li et al. [START_REF] Li | CloudCmp : Comparing Public Cloud Providers[END_REF], the response time to access a file in one cloud provider can be twice the time to access it in another provider. Besides, they show that the maximum throughput of one cloud provider can vary up to 57%, and in another cloud provider varies less than 2%, depending on the cloud region used. Moreover, inside the same cloud provider, diverse storage options have different response times and throughput, as presented by Teylo et al. in [START_REF] Teylo | Developing Checkpointing and Recovery Procedures with the Storage Services of Amazon Web Services[END_REF]. Thus, each institution chooses the best cloud provider, cloud region, and storage service combination based on its respective needs, leading to a scenario where FL clients have their data in different cloud providers and, to preserve privacy, it is prohibitive to transfer all data to the same cloud provider. In previous works [START_REF] Brum | Towards optimizing computational costs of federated learning in clouds[END_REF][START_REF] Brum | Evaluating federated learning scenarios in a tumor classification application[END_REF], we present FL scenarios for a biomedical application that searches for tumor-infiltrated lymphocytes to estimate the patient survival rate using Convolutional Neural Networks (CNNs) as models. These models extract knowledge from images using millions of neurons, which are grouped into layers. Each neuron consists of an activation function and receives the results of the previous layer multiplied by weights, which are arbitrary float pointing values, as input. Thus, to train the model, it is necessary to multiply each image (represented by a matrix) by all layers of the network several times, which requires a huge computational power. One option to achieve this performance is using accelerators, such as Graphics Processing Units (GPUs). Besides the storage services, cloud providers offer Virtual Machines (VMs) with GPUs in a service generically called Infrastructure-as-a-Service (IaaS). The user only pays for what he/she uses (pay-as-you-go scheme), with no costs for hardware maintenance or energy consumption. Furthermore, the user has access to the newest computational resources as soon as the provider renders them available to hire. In Amazon Web Services (AWS), for example, there are many predefined types of VMs, also called instances, with different NVIDIA GPU architectures, from Kepler to Volta [START_REF] Services | Recommended GPU Instances[END_REF]. In Google Cloud Platform (GCP), the user can attach GPUs to a predefined or custom instance type [START_REF]GPUs on Compute Engine[END_REF]. To the best of our knowledge, few works tackle the problem of FL on clouds in the related literature. Liu et al. [START_REF] Liu | Client-edge-cloud hierarchical federated learning[END_REF] present a hierarchical FL architecture with mobile devices as clients while Fang et al. [START_REF] Fang | Highly efficient federated learning with strong privacy preservation in cloud computing[END_REF] propose an architecture for Federated Learning in the cloud focusing on privacy. However, both of them simulate the FL environment in a local machine and present only these results in their evaluation. Rajendran et al.present in [22] a FL approach whose results concern two Machine Learning models in a scenario with two clients in both a simulated environment and on Microsoft Azure Cloud Databricks [START_REF] Docs | What is Azure Databricks ?[END_REF], an open-source tool for data engineering and collaborative data science, to exchange the ML model between two institutions.

In our previous work, all the FL clients were in the same zone of Amazon Web Services (AWS), in us-east-1 (North Virginia). As explained before, this scenario does not properly reflect real choices, as institutions can store their datasets in different cloud providers and cloud regions. Thus, in this paper, we investigate how the location of both clients and datasets impacts the total FL execution time, and particularly the spent time in the synchronization steps in different cloud scenarios. Thus, we deploy the clients and the server across two providers, AWS and GCP. Our results show that the allocation of clients and servers has a key role in the execution times and financial costs of the FL application. The remaining of this paper is organized as follows. Section 2 presents concepts of cloud infrastructures. The used application in our experiments, a Federated Learning for solving a Tumor-Infiltrating Lymphocytes Classification Problem, is presented in Section 3. Section 4 shows our experimental results. Finally, Section 5 presents some conclusions and future work.

Cloud Infrastructures

Cloud providers divide their physical infrastructure into cloud regions, which are independent and isolated geographic areas [START_REF] Services | Region and Zones -Amazon Elastic Compute Cloud[END_REF][START_REF]Geography and regions -Documentation[END_REF]. For example, the first region in the eastern United States in AWS is in North Virginia ( us-east-1 region), while in GCP, it is in South Carolina (us-east1 region). We can see a map with all cloud regions in AWS' or on GCP's website [START_REF] Services | Global Infrastructure[END_REF][START_REF]Global Locations -Regions & Zones[END_REF]. The price of a VM type in each cloud provider varies among different cloud regions. For example, in April 2022, the GCP's VM type e2-standard-4, with 4 vCPUs and 16 GB of RAM, costed $0.13402 in Iowa (us-central1 region) and $0.16098 in Salt Lake City (us-west3 region) 3 . While the AWS' VM type t2.xlarge, with the same configuration, costed $0.1856 in North Virginia (us-east-1 region) and $0.2208 in North Carolina (us-west-1 region). Besides the VM allocation costs, there are two other costs to be considered : (i) the communication costs, since an FL application exchanges many messages during its execution, and (ii) the storage systems costs, necessary to store the whole dataset. In each FL round, the server sends all model weights to each client, receives them back after the training, and sends the aggregated ones so that the clients can evaluate the model. In terms of size, the VGG16 model, used in our previous work [START_REF] Brum | Evaluating federated learning scenarios in a tumor classification application[END_REF], has 132 million weights, which are float pointing variables using a total of 5.2 GB of memory. Thus, in each FL round, the server exchanges up to 15.6 GB in messages. In AWS, the price is $0.09 per GB sent from AWS to the Internet in the first 10 TB/month. In GCP, there are two different network tiers, called Standard and Premium Tier [START_REF]Network Service Tiers overview[END_REF]. While the Premium Tier uses GCP's high-speed internal network as much as possible, the Standard Tier uses the Internet to deliver the exchanged data. In the Premium Tier, we pay $0.12 per GB sent from GCP to any destination worldwide, except in China and Australia, in the first 1 TB/month, after this first TB, we pay $0.11 per GB. In the Standard Tier, we pay $0.085 per GB sent from GCP to any destination worldwide, except in China and Australia, in the first 10 TB/month. Regarding the storage services, AWS offers a total of 11 storage services [START_REF] Services | Cloud Storage on AWS[END_REF] and GCP offers nine different storage services [START_REF]Cloud Computing Services[END_REF]. Each storage service focuses on different needs in a company's workflow. In this work, we use the object storage service from each of the providers : Amazon Simple Storage Service (Amazon S3) [START_REF] Services | Amazon S3[END_REF] and Cloud storage [START_REF]Cloud Storage[END_REF]. These services similarly represent the objects, using a two-level organization [START_REF] Services | Amazon S3[END_REF][START_REF]Cloud Storage[END_REF]. At the higher level, they use buckets, structures similar to folders having a unique global name. These buckets help to organize the data of different users, identifying and billing them accordingly. S3 restricts each bucket to a single region, and each account can associate up to 100 buckets with it. In Cloud Storage, the user can configure the bucket availability to a single cloud region, in two close regions (dual-region), or several regions spread in a larger area (multi-region). There is no limit on the number of buckets per account in GCP, but there are limitations regarding the bucket's name and creation rate [START_REF]Quotas & limits -Cloud Storage[END_REF].

Objects are the lower level of these two storage services. They contain the user stored data represented by a name and a unique key used to access the object 4, 5 . Both services have an upper limit to a single object size of 5TB [START_REF] Services | Amazon S3[END_REF][START_REF]Quotas & limits -Cloud Storage[END_REF] and allow the user to create, change and, read objects from a bucket using a single operation. However, if the user wants to rename or move the object to another place, it takes at least two operations, downloading the object to a local system and uploading it with the new name or to the new location. The two cloud providers allow users to choose the privacy level for each object. By default, AWS makes all objects stored in S3 private, allowing only access from the resource owner and account administrator 6 . If a user wants to let others see his/her data, he/she needs to grant access to each object explicitly. On the other hand, GCP does not assume any privacy level but requests the user to set the requested level of external access when uploading new files to Cloud Storage 7 . AWS and GCP charge for storing the data and for each operation in it, and the price varies among different regions. In Amazon S3, users pay $0.023 per GB to store their data in N. Virginia (us-east-1 region) and $0.005 per 1000 operations in this region while in N. Carolina (uswest-1 region), users pay $0.026 per GB stored and $0.0055 per 1000 performed operations. In Google Cloud Storage, users also pay $0.023 per GB to store their data in N. Virginia (us-east4 region) and $0.020 per GB in Iowa (us-central1 region). The price per operation is the same in all regions of Cloud Storage, being $0.005 per 1000 operations.

Federated Learning for solving a Tumor-Infiltrating Lymphocytes Classification Problem

In this work, we have executed the FL approach proposed in [START_REF] Brum | Evaluating federated learning scenarios in a tumor classification application[END_REF] to a biomedical application, which deals with a Tumor-Infiltrating Lymphocytes (TIL) classification problem, described in [START_REF] Saltz | Spatial Organization And Molecular Correlation Of Tumor-Infiltrating Lymphocytes Using Deep Learning On Pathology Images[END_REF], in different cloud scenarios. This application receives as input Whole-Slide Images (WSIs) and presents TIL maps as results. These maps show the spatial distribution and the density of TILs in each patient to help cancer treatment. There are two main phases in this application : training and production. In the training phase, the application divides each WSI into patches of smaller sizes, presents them to experts to classify as TIL-positive or TIL-negative, and uses them as a training dataset for the CNN. In the production phase, new WSIs are divided into patches and the CNN classifies these new patches into having or not having TILs. With the TIL-positive patches, the application creates the TIL maps. The FL approach focuses on the CNN training part and implements the VGG16 model [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF]. The CNN input size is 224×224, the patch dimension from the previous steps. We set the number of output classes to two, positive if the patch contains a TIL or negative otherwise. The centralized approach needs two separated datasets to train the CNN : one to train the model and another to test it. In our FL approach, we divided the two datasets among all clients homogeneously. Thus, each FL client accesses two separated and private datasets : one for training the local model and another to test it. The FL approach is composed of communication rounds which in turn are composed of 5 steps. In the first step, the server sends the current model weights to all participating clients. After receiving the weights, each client trains for several epochs on the local training dataset and sends the updates back to the server (step 2). Then, the server receives all updates, aggregates them, and sends the final weights to the clients (step 3). Next, each client updates their weights, tests the model with the testing dataset, and sends its evaluation metrics (e.g., accuracy and precision) to the server (step 4). Finally, the server aggregates the evaluation metrics of all clients to present global metrics (step 5).

Experimental Results

Our experiments were conducted on the framework Flower, a FL framework proposed by Beutel et al. [START_REF] Beutel | Flower : A friendly federated learning research framework[END_REF] that focuses on the FL execution in real scenarios. The authors developed Flower to support any ML framework underneath it (TensorFlow, PyTorch, or a custom one) and several client environments, with different operating systems or hardware settings. Moreover, they implemented some FL algorithms, like FedAvg [START_REF] Mcmahan | Communication-Compas'2022 : Parallélisme/ Architecture/ Système MIS/UPJV -Amiens France, 5-8 juillet 2022 Efficient Learning of Deep Networks from Decentralized Data[END_REF], FedProx [START_REF] Li | Federated optimization in heterogeneous networks[END_REF], and Q-FedAvg [START_REF] Li | Fair resource allocation in federated learning[END_REF], and the communication layer between server and clients using the gRPC protocol [START_REF]gRPC[END_REF], a highperformance implementation of the Remote Procedure Call (RPC) protocol that provides communication among several tasks with minimum overhead. Since Flower is open-sourced, it is possible to modify the source code to add specific log messages or create other FL algorithms within it. We used the Python 3.6 programming language and executed both server and clients into different VMs of AWS and GCP. We deployed each task into a different VM to avoid unintended data access from other clients and reinforce the privacy issues.

Regarding the application execution parameters, we executed the FL application with 4 clients and 10 communication rounds using 5 training local epochs in each, which are the best execution parameters, as presented in [START_REF] Brum | Evaluating federated learning scenarios in a tumor classification application[END_REF]. Each client works on a dataset containing 948 training patches and 522 test patches collected from the TCGA repository [START_REF]The Cancer Genome Atlas[END_REF]. We added log messages into the FL framework to get the time the server and clients send and receive each message. From these log messages, we computed the FL synchronization time and the computation time of each client. The FL synchronization time is the sum of three times per round : (1) the initial sync time, (2) the aggregation sync time, and (3) the test sync time. The initial sync time is the time between the server sending the first message with the current weights and the last client receiving it. The aggregation sync time is the time between the first client sending the updated weights back to the server and the last client receiving the final weights, which includes the server aggregation in step 3. The test sync time is the time between the first client sending the evaluation metrics and the server presenting the global metrics. The computation time of each client is the sum of the training and the testing steps in each client, including the time to access the dataset. We present here only the longest computation time.

Concerning the cloud environments, in both providers, each client executes in a VM with 8 vCPUs, 32GB of memory, and an Nvidia T4 Tensor Core GPU while the server in a VM with 4 vCPUs and 16GB of memory, as the server does not need a GPU. The cloud region, storage service, type of VMs (client and server), VM price (client and server), and network costs are summarized in Table 1 for the two cloud providers. We chose the Premium Network Tier of GCP to use the high-speed internal network as much as possible. unique cloud provider. As can be seen in S2, GCP presented better results than the AWS ones (seen in S1). Even if more expensive than the standard one, the choice of the Premium network showed to be a good option when compared with the AWS results. The second set of tests evaluated two cases : (i) the server, three clients, and their corresponding data sets on GCP, and one client and the corresponding data set on AWS, case S3 ; and (ii) all clients and the server on GCP, but one data set allocated on AWS, case S4. This last test showed that the allocation of the client in the same cloud provider of the dataset is the best option, in this case, even considering that GCP has presented better results than AWS in the other cases. Finally, we considered two scenarios where two datasets and two clients were allocated to each cloud provider. In the first case, the server was allocated on AWS, and in the second case on GCP. We observed similar execution times in the two cases, despite the use of the Premium Tier of GCP, with a 12% increase in the total costs using the server in GCP.

Execution times and Financial Costs

Conclusion

This paper presented a preliminary analysis of FL executions in several scenarios of clients and server allocation on GCP and AWS clouds. We could observe different costs and execution times when using those providers, even using similar infrastructures. These experiments showed that the allocation problem of clients and servers is not trivial and that the financial costs and execution times can vary in accordance with the used infrastructure. Moreover, it is not worth using the best VMs or network in one cloud provider if we have a dataset allocated in another one with worse performance, since the bigger access time of the slowest cloud provider will dominate the execution time. In future work, we intend to model that problem using optimization tools in order to obtain, given the characteristics and location of the datasets, and cloud providers' costs and infrastructure performance, the best clients and server allocation with the aim of reducing execution times and/or financial costs.

  4. https ://docs.aws.amazon.com/AmazonS3/latest/userguide/UsingObjects.html 5. https ://cloud.google.com/storage/docs/naming-objects 6. https ://aws.amazon.com/s3/security/ ?pg=ln&sec=be#Access_management_and_security 7. https ://cloud.google.com/storage/docs/access-control

	Compas'2022 : Parallélisme/ Architecture/ Système
	MIS/UPJV -Amiens France, 5-8 juillet 2022

  Table2presents the averages of three executions in different scenarios, where the costs are presented in $, and the time in hours, minutes and seconds. Our initial tests aimed at evaluating the execution times and financial costs when the complete FL application was executed on a

	Compas'2022 : Parallélisme/ Architecture/ Système
	MIS/UPJV -Amiens France, 5-8 juillet 2022

TABLE 1 -

 1 Cloud region, storage service, VM and cost for both client and server, and network transfer costs in each cloud providers used in our experiments

		Region	Storage Service	VM	Client ($ per hour) Cost	VM	Server ($ per hour) Cost	Network Costs ($ per GB)
	AWS	N. Virginia (us-east-1)	Amazon S3	g4dn.2xlarge	0.752	t2.xlarge	0.1856	0.090
	GCP	Iowa (us-central1)	Cloud Storage	n1-standard-8	0.730	e2-standard-4	0.13402	0.120

TABLE 2 -

 2 Average times and costs of 4 scenarios : S1-all FL application on AWS, S2-all FL application on GCP, S3 -3 clients and 3 data sets on GCP, 1 client and 1 data set on AWS, S4-4 clients and 3 data sets on GCP and one data set on AWS.

	Scenarios	Total time	Total Syncronization Message exchange Computing cost $ time cost $ time	VM cost $
	S1 (AWS) 1 :28 :32 10.51	0 :34 :47	5.80	1 :17 :55	4.71
	S2 (GCP) 0 :36 :54	9.61	0 :04 :35	7.73	0 :29 :44	1.88
	S3 (3 GCP,1 AWS) 1 :25 :22 12.04	0 :56 :58	7.57	1 :15 :42	4.47
	S4 (4 GCP,1 DS AWS) 1 :48 :21 13.25	1 :18 :26	7.73	1 :45 :20	5.52

Compas'2022 : Parallélisme/ Architecture/ Système MIS/UPJV -Amiens France, 5-8 juillet 2022

Acknowledgment

This research is supported by project CNPq/AWS 440014/2020-4, Brazil, by Programa Institucional de Internacionalização (PrInt) from CAPES (process number 88887.310261/2018-00) and CNPq (process number 145088/2019-7).


Paul Galanopoulo

Dr Noémie Gil

Dr Didier Gigmes

Dr Catherine Lefay

Yohann Guillaneuf

Dr Maëlle Lages

Julien Nicolas

Dr Muriel Lansalot
email: muriel.lansalot@univ-lyon1.fr

Franck D'agosto
email: franck.dagosto@univ-lyon1.fr

Franck D'agosto

Dr ] P Galanopoulo

F D'

One-Step Synthesis of Degradable Vinylic Polymer-Based Latexes via Aqueous Radical Emulsion Polymerization

Keywords: Degradable, Emulsion, rROP, Thionolactone, Particles

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Introduction

The development of new synthetic routes to produce degradable and/or recyclable/reusable polymer materials has seen a surge of interest lately for their potential application as biomaterials or to address environmental concerns. [1,2] One possible strategy is the incorporation of specific chemical bonds into the polymer backbone. Upon external stimuli such as pH, chemical reagents, enzymes or light, these bonds break, resulting in polymer chains with lower molar masses. Such bonds can intrinsically be part of the chemical structure of the polymer chains like in polyesters that is a very well-known representative class of recyclable commodity polymers. [START_REF] Cornell | Modern Polyesters: Chemistry and Technology of Polyesters and Copolyesters[END_REF] However, for vinylic polymers produced by the formation of carbon-carbon bonds, such as those obtained by radical polymerization, the challenges may consist in finding appropriate comonomers that could not only allow the easy and controllable introduction of these cleavable chemical groups along the chains, but also be compatible with the main processes used to produce the corresponding polymers. The design of a specific class of monomers known as cyclic ketene acetals (CKAs) has made possible the synthesis of estercontaining vinylic polymers through radical polymerization. [START_REF][END_REF] CKAs are known to copolymerize with traditional vinylic monomers through radical ring-opening polymerization (rROP) (Scheme 1a). [5,6] The main advantage of such a process is to obtain a vinylic copolymer with properties similar to those of the vinylic homopolymer, but with the ability to be degraded into oligomers by cleavage of the ester functions distributed along the backbone. The possibility of forming reactive oligomers being reprocessable into valuable polymers is even more interesting for recyclability in the context of circular economy. Even though CKAs have been known since the pioneering work of Bailey in the 80's, [7] rROP has only recently attracted extensive interest and has mainly been used for the synthesis of biodegradable polymers for biomedical applications. [8] However, the use of CKAs for the synthesis of degradable vinylic polymers remains relatively restrained due to the inherent issues related to the chemical nature of CKAs. [9] Indeed, their low reactivity toward traditional vinylic monomers (notably toward more activated monomers such as (meth)acrylics and styrenics) lessens the range of accessible types of polymers. [START_REF][END_REF]9] In addition, CKAs exhibit competing radical mechanism between radical ring-opening and ring-retaining propagation; the latter leading to non-degradable ketal units in the chain. Last but not the least is the strong intolerance of CKAs to aqueous media as they are rapidly hydrolyzed. [10] This is indeed a major drawback for aqueous emulsion polymerization, one of the main production processes of polymers by radical polymerization, that leads to high solids content dispersions of polymer particles. The fact that, after 40 years exploring all types of radical (co)polymerizations of CKAs, only two examples of polymerization in dispersed aqueous media can be found, attests to the difficulty of the task. [11,12] As an alternative to CKAs, the groups of Roth [13] and Gutekunst [14] proposed to use the biphenylic cyclic thionolactone: dibenzo[c,e]oxepane-5-thione (DOT, Scheme 1b) capable of sustaining a ring-opening mechanism to afford in-chain thioester functions. Compared to the radical ring-opening of CKAs, DOT cannot lead to any type of ring-retention meaning that all the polymerized cyclic monomers will be converted to in-chain thioester functions. Furthermore, the authors reckon that, unlike CKAs, DOT could present favorable reactivity ratios with styrene or (meth)acrylates. Both teams identified acrylates, among other comonomers, to be particularly suitable for copolymerization with DOT. Bingham and coworkers successfully copolymerized DOT with methyl acrylate (MA), poly(ethylene glycol) methyl ether acrylate (PEGA), acrylonitrile and N,N-dimethyl acrylamide through organic solution radical polymerization reaching high conversions for each monomer. [13,15] Additionally, they successfully showed that reversible-deactivation radical polymerization via reversible addition-fragmentation chain transfer (RAFT) can be applied to DOT copolymerizations. In parallel, Gutekunst and coworkers successfully used DOT for RAFT copolymerization in dimethylformamide with both n-butyl acrylate (BA) and tert-butyl acrylate, and showed that DOT was consumed more rapidly than the acrylates leading to copolymers with a slight gradient of comonomer distribution. [14] Eventually, the copolymers all proved to be able to degrade through several techniques ranging from methanolysis in sodium methoxide [14] to hydrolysis in aqueous NaOH solution [15] and to aminolysis with isopropylamine. [13,15,16] All degradations were performed at room temperature. Later, Roth and coworkers extended the range of comonomers able to copolymerize with DOT to maleimides to afford alternating copolymers in dimethylsulfoxide, acrylonitrile or aniline, with tunable degradation. [16] However, the copolymerization with other vinylic monomers proved to be either difficult or impossible. Notably, Bingham et al. found that DOT inhibits radical polymerization of vinyl acetate and N-vinylcarbazole. [13] During copolymerization with methyl methacrylate, DOT remains a bystander, whereas it was originally found to retard the polymerization of styrene without being incorporated. [13] However, we recently showed that DOT and styrene could be efficiently copolymerized under appropriate experimental conditions. At high temperature (i.e., 150 °C), the reactivity ratios are close to 1, allowing a statistical incorporation of the two monomers regardless of the feed ratio. At lower temperature, only a feed ratio of 5 mol% of DOT could be used due to the lower reactivity of DOT. [17] The development of this new type of monomer opens a completely new field for the synthesis of degradable polymer materials. While its use remains yet to be fully investigated, the high stability of DOT in ambient conditions, and most of all its stability toward hydrolysis, is a significant advantage compared to CKAs when considering syntheses in aqueous media and particularly their transposition to emulsion polymerization. Besides its fundamental interest, the success of such transposition would put forward an original way of introducing degradability into latex technology and related applications. It could additionally give access to original degradable particles complementary to polyester counterparts, yet obtained by tricky nanoprecipitation methods of preformed polyester chains, which usually lead to low solids content dispersions. [18,19] Building on the very recent works with DOT mentioned above, we anticipated that (co)polymerizations involving DOT would provide a sustainable system for the formation of degradable particles obtained by aqueous emulsion polymerization. Since BA has shown good reactivity with DOT [14] and is one of the most widely used vinylic monomers for aqueous emulsion, the copolymerization of DOT and BA under emulsion conditions is first addressed in the present paper. Furthermore, while the copolymerization of DOT with styrene was shown to occur efficiently under harsh conditions in homogeneous media (T = 150 °C), [17] we also anticipated in a second step that emulsion copolymerization of DOT and styrene could benefit from a compartmentalization of the polymerization into nanoparticles in an emulsion process and allow successful copolymerization under milder conditions. Finally, terpolymerizations of DOT with styrene and BA were studied to demonstrate the ability of DOT to implement the synthesis of degradable polymers with tuneable thermal properties.

Results and Discussion

Emulsion copolymerization of DOT with n-butyl acrylate. Preliminary experiments. With the aim of identifying the simplest system possible, aqueous emulsion copolymerizations of DOT and BA were thus conducted using the most basic ab initio batch system, with sodium dodecyl sulfate (SDS) as surfactant and potassium persulfate (KPS) as the sole water-soluble initiator. In a first experiment, a mixture of BA containing 5 mol% of DOT was prepared prior to its addition to the aqueous medium to target a 18 wt% solids content (A1, Table 1). Despite stirring, the monomer mixture remained biphasic as BA seemed to be unable to fully solubilize DOT, the latter remaining visible as a fine solid. After addition of the aqueous phase, the polymerization carried out at 80 °C led to a white milky dispersion in about one hour (Figure 1). Coagulum was however formed, to the extent of 18 wt% of initial monomer mass preventing an accurate conversion of BA through gravimetric analysis (estimated to 66% for the stable latex fraction). The success of an emulsion polymerization is subjected to the use of a hydrophobic monomer with some solubility in water, even very low. This not only ensures an efficient initiation in water but also its diffusion from the monomer droplet reservoirs to the nucleated particles through the aqueous phase. While this is the case for BA, the very low solubility in water for DOT, combined with its low solubility in BA, may not allow for an efficient diffusion of both monomers through the water phase leading to some bulk copolymerization. Nonetheless, as shown by DLS analyses of the filtered recovered latex, the synthesis led to stable monodisperse particles of about 110 nm in diameter (see also the cryo-TEM image in Figure 1). 1 H NMR spectrum of the dry extract of this latex is presented in Figure 2. Compared to DOT monomer, the lower chemical shift and the broadening of the signals corresponding to the aromatic protons confirm the presence of ring-opened DOT units in the chains. This spectrum can be compared to that of PBA chains obtained from the dry extract of a latex synthesized under the exact same conditions but without DOT (A0 in Table 1, stable latex, particle diameter by cryo-TEM of 103 nm). A shoulder (e') broadens the signal of the methylene protons adjacent to the ester moiety in BA units at ca. 4 ppm. In accordance with previous work on DOT copolymerization with acrylates, [13,14] this signal is attributed to the proton corresponding to a BA unit next to a thioester group. The presence of additional aromatic resonances in the spectrum corresponding to latex A1 and the resonance assignments made in Figure 2 indeed confirm the incorporation of DOT into the PBA backbone during copolymerization. (kg mol -1 ) The dry extract from latex A1 was treated with isopropylamine, as thioester containing polymers are known to be sensitive to aminolysis. [13,15,16] The SEC trace of the recovered product (Figure 3) showed a much lower molar mass (from 160.8 kg mol -1 to 16.4 kg mol -1 after isopropylamine treatment), which accounts for a decrease in Mn of ~90%. 1) with the corresponding assignments, and from latex A0 obtained with BA as sole monomer (blue). Several experiments were then conducted to reduce the amount of coagulum and increase the yield (see Supporting Information, Table S1). Lowering the solids content to 10 wt% and varying the DOT content in the monomer mixture from 5 mol% in A1 down to 1 mol% in A5 (Table S1) allowed to recover a stable latex in absence of coagulum incorporating 0.9 mol% of DOT after conversion of 98% of BA in 75 min. Again, degradation was quite significant with a decrease in Mn of ~95%. This last experiment shows that the simplest emulsion polymerization system allows to yield, in less than an hour, stable nanoparticles made of a degradable polymer resulting from the copolymerization of DOT with BA.

Mn [h] (kg mol -1 ) Ɖ Mn [h] (kg mol -1 ) Ɖ Mn [h] (kg
Pseudo-miniemulsion / sonication-assisted emulsion copolymerization of DOT and n-butyl acrylate. Miniemulsion polymerization protocols were then investigated to decrease the impact of DOT solubility issues in BA and monomer diffusion issues through the aqueous phase. Indeed, in a miniemulsion polymerization, the nucleation takes place in pre-stabilized monomer droplets formed after applying high shear to the polymerization mixture. [20] Hexadecane was added in the monomers as a co-stabilizer, and after addition of the aqueous phase, the mixture was sonicated. However, this protocol did not provide well-defined miniemulsions as DLS analysis showed the presence of droplets with various sizes. Yet the dispersions remained visually rather stable and looked quite homogenous even for high DOT content (for fDOT = 5% for instance). Even though this process did not allow to provide a proper miniemulsion system, polymerization of the resulting dispersion can be regarded as sonication-assisted emulsion polymerizations. [21] These polymerizations indeed showed better reproducibility and much less or no coagulum, which allows to obtain representative samples and thus run kinetic measurements up to a content of 2 mol% of DOT in the initial feed. Syntheses with DOT feed ranging from 1 mol% to 4.8 mol% were thus conducted using those media as they stood (Table 1). The temperature was reduced to 70 °C to afford slower reaction and enable kinetics monitoring aiming for 10 wt% solid content. Those kinetic studies (Figure 4a) give an insight into the reactivity of both monomers. While almost complete conversion was reached in less than 20 min for a DOT-free formulation (A0* in Table 1), 40 and 60 min are required for DOT feeds of 1% and 2% (A1* and A2* in Table 1), respectively. This is nevertheless in strike contrast to the copolymerizations of DOT carried out in organic solution where reactions last at least several hours and for which conversions were not always quantitative. [14][15][16] This is due to the beneficial effect of compartmentalization of the emulsion polymerization process. DOT seems to have a retarding effect on the emulsion polymerization of BA. As mentioned previously, those results are in accordance with the already observed retardation in the cases of radical copolymerization of methyl acrylate with DOT by Bingham et al. [11] The authors explained this behavior by analogy to retardations due to slow fragmentation in the case of RAFT polymerization using dithiobenzoates. [22] In addition, the previously observed tendency of DOT to be inserted at a faster rate than acrylates [13,14] is confirmed here meaning that a drift in composition, i.e., in the content of degradable units in the chains over the course of the copolymerization, should be expected. Indeed, the observed decrease of the average molar fraction of DOT units in the copolymers over the course of the reaction (Figure 4b) corroborates this assumption. The formation of PBA chains might therefore take place towards the end of the reaction when DOT has been entirely consumed. It is worth mentioning that the size of the particles decreases as the molar fraction of DOT in the initial feed increases (from 71 nm (A0*) to 45 nm (A3*)). This phenomenon is not necessarily expected and remains difficult to explain in such a system were several factors influence the particle size. Moreover, the curves obtained from DSC analysis performed on the dried latex all showed a single glass transition temperature which increases with the average molar fraction of DOT in the copolymers (Figure 5 and Figure S6). Those results prove that the final material is made of random copolymers and tends to show that although a composition drift may take place, homopolymer chains of PBA did not form as it should have led to two distinct glass transition temperatures. However, given the low DOT content the Tgs of the different copolymers remain rather close to each other and a second glass transition would be difficult to identify in a such narrow range. Knowing that DOT homopolymerization is challenging and is yet to be fully investigated, the Tg value for PDOT homopolymer based on Fox equation can be estimated at around 110 °C. [14,16] In addition, in the case of homogeneous copolymerization of styrene and DOT, we found a similar Tg than the reference polystyrene, suggesting that the Tg value of the PDOT is close to 100 °C. [14,17] It is thus expected that the Tg of P(BA-co-DOT) would increase with the fraction of DOT units in the copolymer as the Tg of PBA is around -50 °C.

Figure 5. Evolution of the Tg depending on the average molar fraction of DOT units in the P(BA-co-DOT) copolymers for sonication-assisted emulsion process (Table 1).

Eventually, molar mass distributions and degradation experiments provide very valuable data. First, it can be noticed that overall, the molar masses of the copolymers tend to decrease with the increase of the molar fraction of DOT (Table 1). Interestingly, the molar mass distributions present a shoulder toward the higher molar mass values that seems to increase as the content of DOT decreases (Figure S3). BA, and more generally acrylic monomer polymerization, is prone to irreversible chain transfer to polymer leading to branching, especially at higher temperature. [23] The presence of this shoulder may thus rather be linked to chain transfer reactions. Anyhow, the increase in DOT contents seems to reduce the phenomenon as the shoulder disappears for latex A4* (Figure S3) for which fDOT,0 = 4.8%. It is possible that DOT exerts some sort of control over branching during BA polymerization as styrene is known to. [24] Indeed, for the control experiments with only BA (A0 and A0*, Table 1) it was not possible to perform any SEC analysis; neither on the original copolymers nor after the degradation procedure (see below), as the samples could not be filtered before injection, while no issues were found with any of the samples obtained from synthesis involving DOT.

The degradation of the copolymer chains incorporating ringopened monomer units is convenient and already well documented. Several routes are possible for acrylic polymers containing in-chain thioester link as shown by the experiments conducted by the groups of Roth [13] and Gutekunst. [14] The most convenient way is to use isopropylamine. It was notably shown to be particularly efficient in the case of BA-based copolymers. It is also possible to use concentrated solutions of KOH. [15] Hence, alternative routes were studied, and a very fast and complete degradation was achieved through use of 1,5,7-triazabicyclo [4.4.0]dec-5-ene (TBD), a strong organo-soluble base. [17] In the frame of the present paper, the chain degradation can be considered on the dry extract itself or after an appropriate treatment of the latex. As such, particle degradation is more challenging as the degradation process will also depend on the ability of the nucleophile to diffuse through the water phase and reach the (inside of) the particles. To evaluate the efficiency of the degradation, we compared the SEC traces of dry extracts before and after treatments with isopropylamine and TBD (Scheme 2), first for latex A2* (Table 1). Scheme 2. Degradation of P(BA-co-DOT) with isopropylamine and TBD. The polymer chains are shown as having acyl-TBD chain-ends. Please note that these could readily hydrolyze into carboxylic acids in water.

Figure 6 shows in both cases the formation of much lower molar masses than observed in the starting samples confirming once again the incorporation of the DOT units. As expected, the molar masses of the degradation products decrease with the increase of the molar fraction of DOT in the copolymer (Table 1, Figure 7). It is possible to estimate the molar mass after degradation by assuming a perfect distribution of DOT units along the PBA chain and no consecutive DOT units, leading to a statistical copolymer (see Supporting Information). The comparison between the experimental molar mass of the degraded products and the theoretical one (Table 1) shows that, even if isopropylamine is convenient and most of the time sufficient to degrade in-chain thioester functions, TBD is more suitable to ensure complete degradation. Control experiments A0 and A0* (Table 1) also underwent the degradation treatment to attest that the observed shift in molar masses was specific of the in-chain thioester linkages. Neither the original dried latex (as mentioned above) nor the treated one were soluble enough for SEC, indirectly showing that treatment with TBD or isopropylamine was not efficient. 1, plain line) and its degradation product after treatment of the dry extract with isopropylamine (dotted line) and TBD (dashed line), and direct treatment of the latex with TBD (short dashed line). 1).

For every experiment, degradation products also present a broadened molar mass distribution as shown by the chromatograms and the dispersity of the degradation product (Table 1, Figure S3). Relation between the molar mass of the degraded product and the reactivity ratios of both the vinylic and the monomer bringing the degradable units have already been simulated in the case of copolymerization of 2-methylene-1,3dioxepane (MDO). [25] It was notably shown that the copolymerization of comonomers with disparate reactivity ratios would result in a drift of composition leading to intermolecular heterogeneity in the case of radical polymerization (or intramolecular heterogeneity with gradient copolymer in the case of controlled radical polymerization), which would result on degradation products presenting broad molar mass distribution in both cases. As for radical emulsion copolymerization of DOT and BA, the increased dispersity of degradation products thus settles the kinetic results, as this phenomenon is symptomatic of an uneven distribution of DOT units in the different polymer chains. Eventually straightforward approaches to degrade the latex were probed. First, dilution of the latex in KOH solutions was tried but without success. SEC analyses of the resulting chains did not show any sign of decrease of molar mass compared to the starting latex. In addition, the aspect of the latex remained visually unchanged apart when high KOH concentration was used for which the latex ended up precipitating/coagulating probably through a salting out effect. Isopropylamine led to partial degradation of the copolymers and larger particles. Ultimately, by diluting the latex in a concentrated aqueous TBD solution, complete degradation of the copolymers was observed as shown by the SEC results (A1*, A2*, A4*, Table 1, Figure S4). It was expected that TBD could readily degrade polymer chains inside dispersed particles as it is both water-soluble and organosoluble and may thus able to diffuse from the aqueous phase to the core of the particles. Interestingly, the aspect of the latex did not change substantially after treatment with TBD although SEC of the resulting product did show that the copolymers were degraded (Table 1). Nonetheless, DLS analyses performed on the resulting medium attest that it was still composed of particles even though they were larger, and the particle size distribution was much broader. For instance, degradation on latex A2* led to particles almost twice as big as the initial ones (Zave = 75 nm, PDI = 0.21). This feature is indeed very interesting on an applicative point of view, as it allows to employ the target latex and manipulate the degraded chains as stable dispersions if required.

Emulsion copolymerization of DOT with styrene. Following the promising results obtained for emulsion copolymerization of DOT and BA, we suspected DOT to be able to readily copolymerize with styrene in emulsion under appropriate conditions. While copolymerization of styrene and DOT in homogeneous media was found to be only accessible at high temperature or in presence of a low amount of DOT, [17] we reckon that the mechanism of compartmentalization associated with emulsion polymerization may be a trigger to the copolymerization. [26] The temperature was raised to 80 °C as a high-energy activation is required to achieve the copolymerization between DOT and styrene, varying the DOT content between 0 and 5 mol% (Table 2). [17] The use of styrene instead of BA very slightly improved the solubility of DOT in the monomer (2 mol% (3.5 wt%) in BA against 3 mol% (6 wt%) in styrene). Syntheses led to the formation of coagulum at DOT content above 3 mol% although the impact on the stability of the latex seems to be less prominent as shown by the lower coagulum content for S4 (Table 2) versus A4* (Table 1) . Nonetheless, from 1 to 5 mol% of DOT in the initial monomer feed, syntheses led to stable latex and high conversion for styrene (> 90 %) (Table 2).

Table 2. Ab initio aqueous emulsion copolymerization of styrene and DOT. Experimental conditions, results and molar mass characteristics of the copolymers and their degradation products. The incorporation of DOT into the polystyrene backbone was confirmed by 1 H NMR spectroscopy of the dried latexes, all of which showing a broad signal at 4.1 ppm (as illustrated in Figure 8 for latex S3, Table 2), which is absent from the spectrum of pure polystyrene. This signal is attributed to the proton b' characteristic of a styrene unit next to the thioester of a DOT unit. The multiplicity of the signal is presumed to be linked with polymer tacticity. [13] The integration of this signal was compared to that of signal a to estimate the molar fraction of DOT units in the copolymer, giving very consistent results. For every synthesis, the molar fraction of DOT in the obtained copolymer remains very close to the molar fraction of DOT in the initial monomer mixture (Table 2). FTIR spectra recorded for the powders obtained after drying the latex confirm further the incorporation of DOT units in the copolymers (Figure 9). In accordance with Spick et al. works, [16] the presence of an absorbance band between 1650 and 1680 cm - 1 , which intensity increases with the amount of DOT, attests to the presence of -SC=O bond in the copolymer. Both styrene conversion and the composition of the copolymer were followed over the course of the synthesis for fDOT,0 = 3% and compared to the DOT-free synthesis (respectively S3 and S0, Figure 10, Table 1). Unlike BA, styrene copolymerizations did not seem to be impacted by the presence of DOT as both experiments reached quantitative conversion at similar rates within about one hour. The DOT content in the polymer increased slightly over the course of the reaction as the lowest average content of DOT units observed is about 2.5 mol% for 50% styrene conversion and reaches 3 mol% at higher conversion. In homogeneous solution, the incorporation of DOT is halved compared to that of styrene at 80 °C and reached a similar conversion at 150 °C. [17] In emulsion, the compartmentalization seems to even lower the reactivity difference between styrene and DOT and thus it is not necessary to increase the temperature to 150 °C as for solution copolymerization to have a statistical incorporation of thioester moieties into the PS backbone. [17] 2).

Unlike the copolymers obtained by the copolymerization of DOT and BA, molar mass distributions do not show sign of tailing and present rather narrow traces. Degradation experiments yielded oligomers that seem to keep the same molar mass distribution as their original copolymer with no enlargement of the signal or increase in the dispersity after treatment of the dried latex by TBD (Figure 11). In addition, the molar mass of the degraded copolymers decreases with the increase of their DOT fraction following the trend of the theoretical values. Combining those results with the kinetic data shows that the insertion of DOT units in the case of copolymerization with styrene seems to be much more regular thus yielding truly random copolymers. Unlike what was achieved with P(BA-co-DOT), it was however not possible to degrade the P(S-co-DOT) copolymers as waterborne particles with TBD. Several factors may hinder TBD diffusion inside the particle core, among which the ability of the degrading agent to swell the entangled polymer chains, which may itself be linked with the Tg of the copolymers. Whereas PBA-based copolymers present Tg way below ambient temperature (-40/-50 °C, Table 1), PS-based copolymers present Tg way above (about 100 °C, Figure 12). In the case of PS, the "glassiness" of the particles may thus limit the access to the core.

Emulsion copolymerization of DOT, n-butyl acrylate and styrene.

As DOT proved to readily copolymerize with both styrene and BA, we sought to implement the emulsion terpolymerization of DOT with both BA and styrene in order to obtain degradable copolymer latexes with a wide range of Tg. A series of experiments was thus conducted with 2 mol% of DOT in a mixture of styrene and BA with varying molar ratios. Syntheses all led to stable and coagulum-free latexes (Table 3), with conversion of BA and styrene above 90% in about 60 min. DSC results show a single glass transition temperature that increases with the amount of styrene, suggesting the formation of a statistical copolymer (Figure S7). As expected, it was possible to vary the Tg of the copolymers in a range between -45 °C and 100 °C by adjusting the molar ratio BA/styrene in the initial monomer mixture (Figure 12). 3).

Table 3. Ab initio aqueous emulsion copolymerization of DOT, BA and styrene. Experimental conditions, results and molar mass characteristics of the copolymers and their degradation products. SEC analysis on dry extracts and their corresponding degradation products gives an insight on the impact of styrene in this copolymerization system (Figure 13). Indeed, molar mass distribution of the copolymers and their degradation products differ significantly depending on the amount of styrene in the initial batch of monomers. The shoulder toward high molar masses observed for the styrene-free synthesis disappears as the amount of styrene increases. In addition, molar mass distributions of degradation products show the disappearance of the tailing toward the higher molar masses and overall narrower molar mass distributions for the syntheses using styrene and BA compared to the ones using only BA (Figure 13). Once again, this is shown by the decrease of the dispersity but also in lower Mn for degradations products. Combining the fact that the shape of the distribution and the dispersity are left almost unchanged after the degradation process, it is reasonable to assume that the incorporation of DOT units is more homogeneous in these systems than in the ones involving BA and DOT. The average fraction of DOT units in the copolymers could not be determined by NMR. The spectra of those copolymers did not present any isolated-enough signals that can be specifically related to a DOT unit. Nonetheless, incorporation of DOT could be indirectly confirmed by degradation experiments. Indeed, while degradation occurred for the copolymers obtained using DOT in the initial feed, no degradation was observed in the case of DOTfree synthesis (AS0, Table 3). 3).

Run

Conclusion

The aqueous emulsion copolymerization of dibenzo[c,e]oxepane-5-thione with either n-butyl acrylate, styrene or both was conducted under standard emulsion polymerization conditions to afford stable latexes in less than two hours. Syntheses all led to the quantitative incorporation of thioester units into the backbone of the vinylic copolymers. Relatively low content of cleavable thioester functions is required to obtain degradable copolymers. Indeed, successful degradation of the copolymers through the use of isopropylamine or TBD led to oligomers with molar masses in agreement with the molar fraction of integrated DOT units. The terpolymerization of DOT with both styrene and n-butyl acrylate allowed to afford degradable copolymers with tunable glass transition temperature. The emulsion copolymerization of DOT thus provides a straightfoward one-step process to obtain waterborne particles made of degradable vinylic polymers. The development of more specific systems may open new perspectives in latex production for a broad range of applications in various fields, such as coatings and drug delivery, by producing degradable copolymers additionnaly addressing environmental issues.

Entry for the Table of Contents

Emulsion copolymerization of dibenzo[c,e]oxepane-5-thione (DOT) with either styrene, n-butyl acrylate or a mixture of both were carried out in water using SDS as surfactant and KPS as initiator. Stable waterborne particles of vinylic polymers (red) bearing cleavable thioester units (blue) along the chains were obtained. The polymer chains were degraded to oligomers using a strong base (1,5,7triazabicyclo [4.4.0]dec-5-ene) or isopropylamine.

Institute and/or researcher Twitter usernames: @cp2m_lab ; @DAgostoFranck ; @LansalotMuriel

Water, SDS, KPS Degradation

Base/Amine Emulsion copolymerization

Scheme 1 .

 1 Scheme 1. Radical ring-opening mechanism in the case of (a) CKAs and (b) DOT.

Table 1 .

 1 Ab initio and sonicated-assisted emulsion copolymerization of DOT and BA. Experimental conditions, results and molar mass characteristics of the copolymers and their degradation products.

  [a] Molar fraction of DOT in the initial monomer mixture. [b] Theoretical solids content (th) and experimental final solids content (s) measured by gravimetric analysis. [c] Weight fraction of coagulum in the final medium compared to the initial monomer mass. [d] Yield of polymers under the for m of particles calculated from the solids content after filtration. Equivalent to conversion when no coagulum is formed. [e] Average molar fraction of inserted DOT in the resulting copolymers estimated by 1 H NMR. [f] Glass transition temperature estimated from DSC analysis. [g] Intensity-average diameter and polydispersity of the final particles from DLS. [h] Experimental number-average molar mass, Mn, and dispersity, Mw/Mn, of the copolymer or degradation product determined by SEC-THF using a calibration based on PMMA standards. [i] Calculation of the theoretical Mn for degradation products (equation S1). IPA stands for isopropylamine and TBD for 1,5,7-triazabicyclo [4.4.0]dec-5-ene.

Figure 1 .

 1 Figure 1. Cryo-TEM image and photo of latex A1 (Table1) obtained by copolymerizing DOT and BA in emulsion (FDOT = 3.2 %).

Figure 2 . 1 H

 21 Figure2.1 H NMR spectrum (CDCl3, 256 scans) of the dried extract obtained from latex A1 (red, Table1) with the corresponding assignments, and from latex A0 obtained with BA as sole monomer (blue).

Figure 3 .

 3 Figure 3. Molar mass distribution for latex A1 (plain line) and its degradation product after treatment of the dry extract with isopropylamine (dotted line).

Figure 4 .

 4 Figure 4. (a) Evolution of BA conversion versus time during sonication-assisted emulsion copolymerizations of DOT and BA (Table 1) for experiments A0* (black circles), A1* (blue squares) and A2* (red triangles). (b) Evolution of BA conversion (squares), DOT conversion (dash) and average molar fraction of DOT in the copolymers (FDOT, circles) versus time for experiment A1*.

Figure 6 .

 6 Figure 6. Molar mass distribution of the dry extract for latex A2* (Table1, plain line) and its degradation product after treatment of the dry extract with isopropylamine (dotted line) and TBD (dashed line), and direct treatment of the latex with TBD (short dashed line).

Figure 7 .

 7 Figure 7. Evolution of experimental (dotted line) and theoretical (dashed line) number-average molar masses of the degradation products as a function of the average molar fraction of inserted DOT units in the copolymers (from A1*, A2*, A3* and A4*, Table1).

  [a] Molar fraction of DOT in the initial monomer mixture. [b] Theoretical solids content (th) and experimental final solids content (s) measured by gravimetric analysis. [c] Weight fraction of coagulum in the final medium compared to the initial monomer mass. [d] Yield of polymers under the form of particles calculated from t he solid content after filtration. Equivalent to conversion when no coagulum was formed. [e] Average molar fraction in the resulting copolymers estimated by 1 H NMR. [f] Intensity-average diameter and polydispersity of the final latex from DLS. [g] Experimental number-average molar mass, Mn, and dispersity, Mw/Mn, of the copolymer or degradation product determined by SEC-THF using a calibration based on polystyrene standards. [h] Calculated theoretical Mn for degradation products (equation S1).

Figure 8 . 1 H

 81 Figure 8. 1 H NMR spectrum (CDCl3, 256 scans) of the dried extract from latex S3 (Table2) with the corresponding assignments.

Figure 9 .

 9 Figure 9. FTIR spectra of dried extract from latex S0 (black), S1 (blue), S2 (red), S3 (orange) and S4 (green). All the curves were normalized based on the absorbance band at 1440 cm -1 that is linked with methylene group both present in styrene and DOT unit (C-H bending).

Figure 10 .

 10 Figure 10. Evolution of styrene conversion versus time for experiments S0 (blue squares) and S3 (plain red circles) and the molar fraction of DOT units in the copolymer for S3 (open red circles). See Table2for details on S0 and S3.

Figure 11 .

 11 Figure 11. a) Molar mass distribution of dry extracts (plain lines) and degradation products (dashed lines) obtained through degradation of dry extracts using TBD obtained from latexes S1 (blue), S2 (red), S3 (orange) and S4 (green). (b) Evolution of experimental (squares) and theoretical molar masses (dashed line) of the degradation products as a function of the average molar fraction of inserted DOT units in the copolymers (from S1, S2, S3 and S4, Table2).

Figure 12 .

 12 Figure 12. Evolution of the Tg depending on the average molar fraction BA/styrene in the monomer mixture for emulsion polymerization with 2 mol% of DOT (Table3).

  Molar fraction of DOT in the initial monomer mixture. [b] Theoretical solids content (th) and experimental final solids content (s) measured by gravimetric analysis. [c] Total conversion of BA and styrene determined by gravimetric analysis. [d] Glass transition temperature estimated from DSC analysis. [e] Intensity-average diameter and polydispersity of the final latex from DLS. [f] Experimental number-average molar mass and dispersity, Mw/Mn, of the copolymer or degradation product determined by SEC-THF using a calibration based on polystyrene standards.

Figure 13 .

 13 Figure 13. Molar mass distribution of dry extracts (plain lines) and degradation products (dashed lines) obtained through degradation of dry extracts using TBD for latexes synthesized with varying BA/S molar ratio (blue: A3; purple: AS1; red: AS2; yellow: AS3; green: S2, Table3).

mol -1 ) Ɖ

																Zave
																(nm)
																PDI
	Conventional emulsion polymerizations:
	A1 4.8 17.5 120 12.5 18.0	66	3.2	n.d. 110	161	3.1	n.d.	16.4	n.d.	n.d.
										0.01	4.0			3.9
	A0	-17.5 65 17.7	-	99	-	-49	61	insoluble Insoluble	n.d.	insoluble	n.d.	n.d.
										0.04
	Sonicated-assisted experiments:
	A1* 1.0 10.7 60 10.5	-	98	0.8	-47	60	360	15.5	15.7	30.1	18.8	108
										0.02	3.0		6.9	7.7	8.2	0.29
	A2* 2.0 10.0 70	9.7	-	97	1.8	-45	52	330	7.3	8.8	10.1	8.9	75
										0.02	3.0		6.4	7.5	7.8	0.21
	A3* 3.0 9.8	90	9.3	4.5	93	2.2	-45	45	307	5.9	7.0	12.4	6.6	n.d
										0.03	3.1		3.8	3.5	2.8
	A4* 4.8 8.9 140 7.7	13	84	3.9	-43	44	143	3.4	4.2	16.8	4.6	n.d
										0.04	3.5		4.0	2.6	4.1
	A0*	-	9.9	20	9.7	-	96	-	-49	71	insoluble	n.d.	insoluble	insoluble	n.d.
										0.04

n.d.

Acknowledgements

The authors acknowledge the financial support from the Agence Nationale de la Recherche (grant number ANR-18-CE06-0014 CKAPART). Pierre-Yves Dugas (CP2M) is acknowledged for the cryoTEM observations.


Angela Felisia

Paola Chiarello

Goatin

Chiarello

Paola Goatin
email: paola.goatin@inria.fr

A non-local system modeling bi-directional traffic flows

We present a non-local model describing the dynamics of two groups of agents moving in opposite directions.

The model consists of a 2 × 2 system of conservation laws with non-local fluxes, coupled in the speed functions. We prove local in time existence of weak solutions and present some numerical tests illustrating their behaviour.

Introduction

Conservation laws with non-local flux are suitable to describe several phenomena arising in many fields of application.

In this paper, we are interested in describing two groups of agents moving in opposite directions using this class of equations.

, the authors consider a multi-class traffic model expressed by a system of conservation laws with non-local fluxes obtained generalizing the n-populations model for traffic flow introduced in [1], where each equation of the system describes the evolution of the density ρ i of the vehicles belonging to the i-th class. In particular, the non-local multi-class model takes into account the distribution of heterogeneous agents characterized by their maximal speeds and look-ahead visibility in a traffic stream.

, where the authors study a mixed type system of conservation laws describing two populations moving in opposite directions. The latter model is not hyperbolic for certain density values, because the Jacobian matrix of the flux exhibits complex eigenvalues in a subset of the phase space, and oscillations arise in the elliptic region. In particular, existence and uniqueness of solutions are still open problems. On the contrary, we will show that introducing a non-local dependence in the speed function allows to prove existence of solutions through the convergence of a suitable finite volume scheme, at least for sufficiently small times.

The paper is organized as follows. In Section 2 we present a non-local version of the mixed system studied in [START_REF] Goatin | A mixed system modeling two-directional pedestrian flows[END_REF] and we discretize it with an upwind scheme; in Section 3 we recover uniform L ∞ and BV bounds on the sequence of approximate solutions, which allow to prove the existence of weak solutions locally in time. Finally, in Section 4, we exhibit some numerical simulations showing the behaviour of weak solutions of our system considering different kernel and velocity functions. In particular, we compare non-local solutions to their local counterpart presented in [START_REF] Goatin | A mixed system modeling two-directional pedestrian flows[END_REF].

A non-local bi-directional traffic flow model

We consider the following system

       ∂ t ρ 1 + ∂ x (ρ 1 v 1 (r * ω 1 )) = 0, ∂ t ρ 2 -∂ x (ρ 2 v 2 (r * ω 2 )) = 0, (1)
where

r = ρ 1 + ρ 2 , r * ω 1 (t, x) = ∫ x+η 1
x ω 1 (yx)r(t, y)dy, r * ω 2 (t, x) = ∫ x x-η 2 ω 2 (yx)r(t, y)dy.

In [START_REF] Benzoni-Gavage | An n-populations model for traffic flow[END_REF], ρ 1 = ρ 1 (t, x) is the density of the population moving in the direction of increasing space coordinate with speed v 1 = v 1 (r * ω 1 ) depending on the downstream weighed mean of the total density r, while ρ 2 = ρ 2 (t, x) is the density of the population moving in the opposite direction, with non-local speed v 2 .

In the following, we will assume (H1) ω 1 ∈ C 1 ([0, η 1 ]; R + ), ω 2 ∈ C 1 ([-η 2 , 0]; R + ), η i > 0 for i ∈ {1, 2}, ω 1 ≤ 0 and ω 2 ≥ 0, ∫ η 1 0 ω 1 (x)dx = ∫ 0 -η 2 ω 2 (x)dx = 1; we set ω max := max{ω 1 (0), ω 2 (0)};

(H2) v i : R + → R + , i ∈ {1, 2}, are smooth non-increasing functions such that v i (0) = V max i > 0 and v i (r) = 0 for r ≥ 1.

We couple [START_REF] Benzoni-Gavage | An n-populations model for traffic flow[END_REF] with an initial datum

ρ i (0, x) = ρ 0 i (x), i = 1, 2, (2)
and we construct a sequence of finite volume approximate solutions as follows. We fix a space step ∆x and a time step ∆t subject to a CFL condition that will be specified later. Let x j+1/2 = j∆x for j ∈ Z and n ∈ N be the cells interfaces,

x j = ( j -1/2)∆x the cells centers and t n = n∆t the time mesh. We aim at constructing finite volume approximate solutions of (1) of the form

ρ ∆x = (ρ ∆x 1 , ρ ∆x 2 ) with ρ ∆x i (t, x) = ρ n i, j for i = 1, 2, (t, x) ∈ C n j = [t n , t n+1 [ ×[x j-1/2 , x j+1/2
[. To this end, we approximate the initial data with piece-wise constant functions

ρ 0 i, j = 1 ∆x ∫ x j+1/2 x j-1/2 ρ 0 i (x)dx, ∀j ∈ Z, i = 1, 2. (3)
Similarly, for the kernels, we set

ω i,k := 1 ∆x ∫ (k+1)∆x k∆x ω i (x)dx, ∀k ∈ Z, i = 1, 2, (4)
so that ∆x +∞ k=-∞ ω i,k = ∫ +∞ -∞ ω i (x)dx = 1 (extending ω i (x)
= 0 outside their domain). We use the following upwind scheme:

ρ n+1 1, j = ρ n 1, j - ∆t ∆x ρ n 1, j v 1 (R n 1, j+1 ) -ρ n 1, j-1 v 1 (R n 1, j ) , (5a)
ρ n+1 2, j = ρ n 2, j + ∆t ∆x ρ n 2, j+1 v 2 (R n 2, j ) -ρ n 2, j v 2 (R n 2, j-1 ) , (5b)
where

R n 1, j = ∆x +∞ k=0 ω 1,k r n j+k , R n 2, j = ∆x 0 k=-∞ ω 2,k r n j+k , (6)
where the sums are indeed finite due to the compact supports of ω 1 and ω 2 .

Existence of weak solutions

In this section, we prove existence of weak solutions for sufficiently small times. The proof relies on the following estimates, which ensure the convergence of the numerical scheme [START_REF] Chiarello | Lagrangian-antidiffusive remap schemes for non-local multi-class traffic flow models[END_REF].

Theorem 1 Let ρ 0 i (x) ∈ (BV ∩ L ∞ ) (R; R + ), for i = 1,

Lemma 1 (Positivity)

For any T > 0, under the CFL stability condition

∆t ∆x ≤ 1 max{ v 1 ∞ , v 2 ∞ } , (1)
the scheme (5) is positivity preserving on [0,T] × R.

Proof Let us assume that ρ n i, j ≥ 0 for all j ∈ Z and i ∈ {1, 2}. Let us show that the same holds for ρ n+1 i, j . Let us compute

ρ n+1 1, j = 1 - ∆t ∆x v 1 (R n 1, j+1 ) ρ n 1, j + ρ n 1, j-1 ∆t ∆x v 1 (R n 1, j ), ρ n+1 2, j = 1 - ∆t ∆x v 2 (R n 2, j-1 ) ρ n 2, j + ρ n 2, j+1 ∆t ∆x v 2 (R n 2, j ).
By assumption (1), we ensure ρ n+1 i, j ≥ 0 for i ∈ {1, 2} and for all j ∈ Z.

Corollary 1 (L 1 -bound)

If ρ 0 i, j ≥ 0 for all j ∈ Z and i ∈ {1, 2} and the CFL condition (1) holds, the approximate solutions constructed via the scheme (5) satisfy

ρ n i 1 = ρ 0 i 1 , i = 1, . . . , M, (2)
for any n ∈ N, where ρ n i 1 := ∆x j ρ n i, j denotes the L 1 norm of the i-th component of ρ ∆x .

Proof Thanks to Lemma 1, we have

ρ n+1 1 1 = ∆x j ρ n+1 1, j = ∆x j ρ n 1, j - ∆t ∆x ρ n 1, j v 1 (R n 1, j+1 ) -ρ n 1, j-1 v 1 (R n 1, j ) = ∆x j ρ n 1, j , ρ n+1 2 1 = ∆x j ρ n+1 2, j = ∆x j ρ n 2, j + ∆t ∆x ρ n 2, j+1 v 2 (R n 2, j ) -ρ n 2, j v 2 (R n 2, j-1 ) = ∆x j ρ n 2, j , proving (2)
.

Lemma 2 (L ∞ -bound)

If ρ 0 i, j ≥ 0 for all j ∈ Z and i ∈ {1, 2} and (1) holds, then the approximate solution ρ ∆x constructed by the algorithm (5) is uniformly bounded on [0,T] × R for any T such that

T ≤ 1 4 max{ v 1 ∞ , v 2 ∞ }w max ρ 0 ∞ . Proof Let us define ρ = max{ρ n 1, j-1 , ρ n 1, j , ρ n 2, j , ρ n 2, j+1 }. Then we get ρ n+1 1, j = 1 - ∆t ∆x v 1 (R n 1, j+1 ) ρ n 1, j + ∆t ∆x v 1 (R n 1, j )ρ n 1, j-1 ≤ 1 - ∆t ∆x v 1 (R n 1, j+1 ) -v 1 (R n 1, j ) ρ ρ n+1 2, j = 1 - ∆t ∆x v 2 (R n 2, j-1 ) ρ n 2, j + ∆t ∆x v 2 (R n 2, j )ρ n 2, j+1 ≤ 1 - ∆t ∆x v 2 (R n 2, j-1 ) -v 2 (R n 2, j ) ρ,
and

v 1 (R n 1, j ) -v 1 (R n 1, j+1 ) = v 1 (ξ j+1/2 ) ∆x +∞ k=0 ω 1,k r n j+k+1 -∆x +∞ k=0 ω 1,k r n j+k = ∆x v 1 (ξ j+1/2 ) +∞ k=1 (ω 1,k-1 -ω 1,k )r n j+k -ω 1,0 r n j ≤ 4 ∆x ω 1 (0) v 1 ∞ ρ n ∞ , (3)
v 2 (R n 2, j-1 ) -v 2 (R n 2, j ) = v 2 (ξ j-1/2 ) ∆x 0 k=-∞ ω 2,k r n j+k-1 -∆x 0 k=-∞ ω 2,k r n j+k = ∆x v 2 (ξ j-1/2 ) -1 k=-∞ (ω 2,k+1 -ω 2,k )r n j+k -ω 2,0 r n j ≤ 4 ∆x ω 2 (0) v 2 ∞ ρ n ∞ , (4)
where

ξ j+1/2 ∈ i(R n 1, j , R n 1, j+1 ), with I(a, b) = [min{a, b}, max{a, b}], and ρ n ∞ = (ρ n 1 , ρ n 2 ) ∞ = max i, j ρ n i, j . So, until ρ n ∞ ≤ K for some K ≥ ρ 0 ∞ , we get ρ n+1 ∞ ≤ ρ n ∞ 1 + 4K max v 1 ∞ , v 2 ∞ ∆t w max with w max = max{ω 1 (0), ω 2 (0)}. This implies ρ n ∞ ≤ ρ 0 ∞ e Cn∆t ,
where

C = 4K max{ v 1 ∞ , v 2 ∞ }w max . Therefore we get that ρ(t, •) ∞ ≤ K for t ≤ 1 4K max{ v 1 ∞ , v 2 ∞ }w max ln K ρ 0 ∞ ≤ 1 4e max{ v 1 ∞ , v 2 ∞ }w max ρ 0 ∞ ,
where the maximum is attained for K = e ρ 0 ∞ . Let us iterate the procedure: at time t m , m ≥ 1, we set K = e m ρ 0 ∞ and we get that the solution is bounded by K until t m+1 such that

t m+1 ≤ t m + m 4e m max{ v 1 ∞ , v 2 ∞ }w max ρ 0 ∞ .
Therefore, the approximate solution remains bounded, uniformly in ∆x, at least for t ≤ T with

T ≤ 1 4 max{ v 1 ∞ , v 2 ∞ }w max ρ 0 ∞ +∞ m=1 m e m ≤ 1 4 max{ v 1 ∞ , v 2 ∞ }w max ρ 0 ∞ . Lemma 3 (Spatial BV-bound) Let ρ 0 i ∈ (BV ∩ L ∞ ) (R; R + ) for i ∈ {1, 2}. If (1)
holds, then the approximate solution ρ ∆x (t, •) constructed by the algorithm (5) has uniformly bounded total variation for t ∈ [0,T] for any T such that

T ≤ min i=1,2 1 H TV(ρ 0 i ) + 1 , ( 5
)
where

H = 2 ρ n ∞ w max 12 ρ n ∞ max{ v 1 ∞ , v 2 ∞ } + max{ v 1 ∞ , v 2 ∞ } .
Proof Let us consider the component ρ 1 and subtract the identities

ρ n+1 1, j+1 = ρ n 1, j+1 - ∆t ∆x ρ n 1, j+1 v 1 (R n 1, j+2 ) -ρ n 1, j v 1 (R n 1, j+1 ) ρ n+1 1, j = ρ n 1, j - ∆t ∆x ρ n 1, j v 1 (R n 1, j+1 ) -ρ n 1, j-1 v 1 (R n 1, j ) ,
analogously for the second component ρ 2 ,

ρ n+1 2, j+1 = ρ n 2, j+1 + ∆t ∆x ρ n 2, j+2 v 2 (R n 2, j+1 ) -ρ n 2, j+1 v 2 (R n 2, j ) ρ n+1 2, j = ρ n 2, j + ∆t ∆x ρ n 2, j+1 v 2 (R n 2, j ) -ρ n 2, j v 2 (R n 2, j-1 )
.

Setting ∆ n i, j+1/2 = ρ n i, j+1 -ρ n i, j for i ∈ {1, 2}, we get ∆ n+1 1, j+1/2 = ∆ n 1, j+1/2 - ∆t ∆x ρ n 1, j+1 v 1 (R n 1, j+2 ) -2ρ n 1, j v 1 (R n 1, j+1 ) + ρ n 1, j-1 v 1 (R n 1, j ) , ∆ n+1 2, j+1/2 = ∆ n 2, j+1/2 + ∆t ∆x ρ n 2, j+2 v 2 (R n 2, j+1 ) -2ρ n 2, j+1 v 2 (R n 2, j ) + ρ n 2, j v 2 (R n 2, j-1 ) .
We can write

∆ n+1 1, j+1/2 = 1 - ∆t ∆x v 1 (R n 1, j+2 ) ∆ n 1, j+1/2 (6)
- ∆t ∆x ρ n 1, j v 1 (R n 1, j+2 ) -2v 1 (R n 1, j+1 ) + v 1 (R n 1, j ) (7)
+ ∆t ∆x ∆ n 1, j-1/2 v 1 (R n 1, j ), and
∆ n+1 2, j+1/2 = 1 - ∆t ∆x v 2 (R n 2, j-1 ) ∆ n 2, j+1/2 (8)
+ ∆t ∆x ρ n 2, j+1 v 2 (R n 2, j+1 ) -2v 2 (R n 2, j ) + v 2 (R n 2, j-1 ) (9)
+ ∆t ∆x ∆ n 2, j+3/2 v 2 (R n 2, j+1 ).
Observe that assumption (1) guarantees the positivity of ( 6) and [START_REF] Goatin | Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity[END_REF]. The term (7) can be estimated as

v 1 (R n 1, j+2 ) -2v 1 (R n 1, j+1 ) + v 1 (R n 1, j ) = v 1 ∆x +∞ k=0 ω 1,k r n j+k+2 -2v 1 ∆x +∞ k=0 ω 1,k r n j+k+1 + v 1 ∆x +∞ k=0 ω 1,k r n j+k = v 1 (ξ n 1, j+3/2 )∆x +∞ k=0 ω 1,k r n j+k+2 - +∞ k=0 ω 1,k r n j+k+1 -v 1 (ξ n 1, j+1/2 )∆x +∞ k=0 ω 1,k r n j+k+1 - +∞ k=0 ω 1,k r n j+k = v 1 (ξ n 1, j+3/2 )∆x +∞ k=1 (ω 1,k-1 -ω 1,k )r n j+k+1 -ω 1,0 r n j+1 -v 1 (ξ n 1, j+1/2 )∆x +∞ k=1 (ω 1,k-1 -ω 1,k )r n j+k -ω 1,0 r n j = v 1 ( ξn 1, j+1 )(ξ n 1, j+3/2 -ξ n 1, j+1/2 )∆x +∞ k=0 ω 1,k 2 i=1 ∆ n i, j+k+3/2 + v 1 (ξ n 1, j+1/2 )∆x +∞ k=1 2 i=1 (ω 1,k-1 -ω 1,k )∆ n i, j+k+1/2 -ω 1,0 ∆ n i, j+1/2 , with ξ n 1, j+1/2 ∈ I(R n 1, j , R n 1, j+1 ) and ξn 1, j+1 ∈ I(ξ n 1, j+1/2 , ξ n 1, j+3/2
). For some θ, µ ∈ [0, 1], we compute

ξ n 1, j+3/2 -ξ n 1, j+1/2 = θ∆x +∞ k=0 ω 1,k 2 i=1 ρ n i, j+k+2 + (1 -θ)∆x +∞ k=0 ω 1,k 2 i=1 ρ n i, j+k+1 -µ∆x +∞ k=0 ω 1,k 2 i=1 ρ n i, j+k+1 -(1 -µ)∆x +∞ k=0 ω 1,k 2 i=1 ρ n i, j+k = θ∆x +∞ k=1 ω 1,k-1 2 i=1 ρ n i, j+k+1 + (1 -θ)∆x +∞ k=0 ω 1,k 2 i=1 ρ n i, j+k+1 -µ∆x +∞ k=0 ω 1,k 2 i=1 ρ n i, j+k+1 -(1 -µ)∆x +∞ k=-1 ω 1,k+1 2
i=1 ρ n i, j+k+1 = ∆x +∞ k=1 θω 1,k-1 + (1 -θ)ω 1,k -µω 1,k -(1 -µ)ω 1,k+1 2 i=1 ρ n i, j+k+1 + ∆x(1 -θ)ω 1,0 2 i=1 ρ n i, j+1 -∆x µ ω 1,0 2 i=1 ρ n i, j+1 -∆x(1 -µ) ω 1,0 2 i=1 ρ n i, j + ω 1,1 2 i=1 ρ n i, j+1
.

By monotonicity we have

θω 1,k-1 + (1 -θ)ω 1,k -µω 1,k -(1 -µ)ω 1,k+1 ≥ 0.
Taking the absolute values, we get

ξ n 1, j+3/2 -ξ n 1, j+1/2 ≤ ∆x +∞ k=1 θω 1,k-1 + (1 -θ)ω 1,k -µω 1,k -(1 -µ)ω 1,k+1 + 4 ω 1,0 2 ρ n ∞ ≤ 12 ∆x ω 1 (0) ρ n ∞ .
Applying the same argument to the second component, we compute

v 2 (R n 2, j+1 ) -2v 2 (R n 2, j ) + v 2 (R n 2, j-1 ) = v 2 ( ξn 2, j )(ξ n 2, j+1/2 -ξ n 2, j-1/2 )∆x 0 k=-∞ ω 2,k 2 i=1 ∆ n i, j+k+1/2 + v 2 (ξ n 2, j-1/2 )∆x 0 k=-∞ 2 i=1 (ω 2,k-1 -ω 2,k )∆ n i, j+k-1/2 -ω 2,0 ∆ n i, j-1/2 , with ξ n 2, j+1/2 ∈ I(R n 2, j , R n 2, j+1
) and ξn 2, j ∈ I(ξ n 2, j-1/2 , ξ n 2, j+1/2 ), and

ξ n 2, j+1/2 -ξ n 2, j-1/2 ≤ 12 ∆x ω 2 (0) ρ n ∞ .
Let now K 1 > 0 be such that j ∆ i, j ≤ K 1 for i ∈ {1, 2}, = 0, . . . , n. Taking the absolute values and rearranging the indexes, we have

j ∆ n+1 1, j+1/2 ≤ j ∆ n 1, j+1/2 1 -λ v 1 (R n 1, j+2 ) -v 1 (R n 1, j+1 ) + ∆t H K 1 , j ∆ n+1 2, j+1/2 ≤ j ∆ n 2, j+1/2 1 -λ v 2 (R n 2, j-1 ) -v 2 (R n 2, j ) + ∆t H K 1 ,
where

H = 2 ρ n ∞ w max 12 ρ n ∞ max{ v 1 ∞ , v 2 ∞ } + max{ v 1 ∞ , v 2 ∞ } . Therefore, by (3)-(4) we get j ∆ n+1 i, j+1/2 ≤ j ∆ n i, j+1/2 (1 + ∆t G) + ∆t H K 1 , with G = 4 max{ v 1 ∞ , v 2 ∞ }w max ρ n ∞ . We thus obtain j ∆ n i, j+1/2 ≤ e Gn∆t j ∆ 0 i, j+1/2 + e HK 1 n∆t -1,
that we can rewrite as

TV(ρ ∆x i )(n∆t, •) ≤ e Gn∆t TV(ρ 0 i ) + e HK 1 n∆t -1 ≤ e HK 1 n∆t TV(ρ 0 i ) + 1 -1 ,
since H ≥ G and it is not restrictive to assume K 1 ≥ 1.

Therefore, we have that TV(

ρ ∆x i ) ≤ K 1 for t ≤ 1 H K 1 ln K 1 + 1 TV(ρ 0 i ) + 1
, where the maximum is attained for some

K 1 < e TV(ρ 0 i ) + 1 -1 such that ln K 1 + 1 TV(ρ 0 i ) + 1 = K 1 K 1 + 1 .
Therefore the total variation is uniformly bounded for

t ≤ 1 H e TV(ρ 0 i ) + 1 .
Iterating the procedure, at time t m , m ≥ 1 we set K 1 = e m TV(ρ 0 i ) + 1 -1 and we get that the solution is bounded by K 1 until t m+1 such that

t m+1 ≤ t m + m H e m TV(ρ 0 i ) + 1 . (10)
Therefore, the approximate solution has bounded total variation for t ≤ T with

T ≤ 1 H TV(ρ 0 i ) + 1 . Corollary 2 Let ρ 0 i ∈ (BV ∩ L ∞ ) (R; R + ). If (1)
holds, then the approximate solution ρ ∆x constructed by the algorithm (5) has uniformly bounded total variation on [0,T] × R, for any T satisfying [START_REF] Chiarello | Lagrangian-antidiffusive remap schemes for non-local multi-class traffic flow models[END_REF].

Proof If T ≤ ∆t, then TV(ρ ∆x i ; [0,T] × R) ≤ TTV(ρ 0 i ). Let us assume now that T > ∆t. Let n T ∈ N\{0} such that n T ∆t < T ≤ (n T + 1)∆t. Then, for i ∈ {1, 2}, TV(ρ ∆x i ; [0,T] × R) = n T -1 n=0 j ∈Z ∆t ρ n i, j+1 -ρ n i, j + (T -n T ∆t) j ∈Z ρ n T i, j+1 -ρ n T i, j ≤ T sup t ∈[0,T ] TV (ρ ∆x i )(t, •) + n T -1 n=0 j ∈Z ∆x ρ n+1 i, j -ρ n i, j .
We then need to bound the term

n T -1 n=0 j ∈Z ∆x ρ n+1 i, j -ρ n i, j .
From the definition of the numerical scheme (5a)-(5b), we obtain

ρ n+1 1, j -ρ n 1, j = - ∆t ∆x ρ n 1, j v 1 (R n 1, j+1 ) -ρ n 1, j-1 v 1 (R n 1, j ) , = ∆t ∆x ρ n 1, j-1 v 1 (R n 1, j ) -v 1 (R n 1, j+1 ) + v 1 (R n 1, j+1 ) ρ n 1, j-1 -ρ n 1, j , ρ n+1 2, j -ρ n 2, j = ∆t ∆x ρ n 2, j+1 v 2 (R n 2, j ) -ρ n 2, j v 2 (R n 2, j-1 ) = ∆t ∆x ρ n 2, j+1 v 2 (R n 2, j ) -v 2 (R n 2, j-1 ) + v 2 (R n 2, j-1 ) ρ n 2, j+1 -ρ n 2, j
.

Taking the absolute values and using ( 3)-( 4) we get

ρ n+1 1, j -ρ n 1, j ≤ ∆t ∆x 4 ∆x ω 1 (0) v 1 ∞ ρ n ∞ ρ n 1, j-1 + v 1 ∞ ρ n 1, j-1 -ρ n 1, j ρ n+1 2, j -ρ n 2, j ≤ ∆t ∆x 4 ∆x ω 2 (0) v 2 ∞ ρ n ∞ ρ n 2, j+1 + v 2 ∞ ρ n 2, j+1 -ρ n 2, j .
Summing on j, we get

j ∈Z ∆x ρ n+1 1, j -ρ n 1, j ≤ 4 ω 1 (0) v 1 ∞ ρ n ∞ ∆t j ∈Z ∆x ρ n 1, j-1 + v 1 ∞ ∆t j ∈Z ρ n 1, j-1 -ρ n 1, j j ∈Z ∆x ρ n+1 2, j -ρ n 2, j ≤ 4 ω 2 (0) v 2 ∞ ρ n ∞ ∆t j ∈Z ∆x ρ n 2, j+1 + v 2 ∞ ∆t j ∈Z ρ n 2, j+1 -ρ n 2, j .
which yields, for i ∈ {1, 2},

n T -1 n=0 j ∈Z ∆x ρ n+1 i, j -ρ n i, j ≤ max{ v 1 ∞ , v 2 ∞ }T sup t ∈[0,T ]
TV(ρ ∆x i )(t, •)

+ 4 max {ω 1 (0), ω 2 (0)} max { v 1 ∞ , v 2 ∞ } ρ n ∞ T sup t ∈[0,T ] ρ ∆x i (t, •) 1 ,
that is bounded by Corollary 1, Lemma 2 and Lemma 3.

Proof (of Theorem 1)

To complete the proof of the existence of solutions to the Cauchy problem ( 1)-( 2), we apply Helly's theorem and follow a Lax-Wendroff type argument as in [START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF][START_REF] Chiarello | Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel[END_REF][START_REF] Chiarello | Non-local multi-class traffic flow models[END_REF], see also [START_REF] Leveque | Finite volume methods for hyperbolic problems[END_REF], to show that the approximate solutions constructed by scheme ( 5) converge to a weak solution of ( 1)-( 2).

Numerical tests 4.1 Kernel support tending to zero

In this subsection, we observe the solution behaviour as the length of kernel support diminishes. We consider the space domain given by the interval [-1, 1] and the space discretization mesh size is set to ∆x = 0.001. We impose absorbing conditions at the boundaries, adding N = η i /∆x ghost cells at the right boundary for the first population ρ 1 and at the left boundary for the second population ρ 2 , where we extend the solution constantly equal to the last value inside the domain.

We take the following linear decreasing kernel and velocity functions:

ω 1 (x) = 2 η 1 1 - x η 1 , ω 2 (x) = 2 η 2 1 + x η 2 ; (1a) v 1 (ρ) = 1 -ρ , v 2 (ρ) = 1 -ρ . (1b)
In Figures 1 and2, we consider the same initial data of [7, Figures 5 and6], observing that the solutions have a similar behaviour, especially when the kernel supports η 1 = η 2 are taken sufficiently small. Moreover, comparing Figures 3 and4 with the numerical tests in [7, Figures 14 and15], we note that when we consider initial data in the elliptic region of the local system, oscillations increase as the kernel support diminish.

Asymptotic behaviour in a periodic setting

We consider now periodic initial data with periodic boundary conditions, see Figure 5. In the following we set ∆x = 0.001, the CFL condition is as in in [START_REF] Benzoni-Gavage | An n-populations model for traffic flow[END_REF]. We are interested in the behaviour of solutions with different types of kernel functions and velocities.

For Test 5, let us consider the linear kernels and velocity functions

ω 1 (x) = 2 η 1 1 - x η 1 , ω 2 (x) = 2 η 2 1 + x η 2 ; (2a) v 1 (ρ) = 1 -ρ , v 2 (ρ) = 1 -ρ , (2b)
see Figure 6. From Figure 7 we can observe that the solutions tend to a steady-state as the time increases.

1 (x) = 3 2η 3 1 η 2 1 -x 2 , ω 2 (x) = 3 2η 3 2 η 2 2 -x 2 ; (3a) v 1 (ρ) = V max 1 (1 -ρ) , v 2 (ρ) = V max 2 (1 -ρ) ; (3b)
In Fig. 8 we can see the profile of the solutions where the agents of the second class are faster than the agents belonging to the first class. Also in this case, reducing the supports of the kernels, oscillations increase. Figure 9 is very similar to Figure 7; here we observe that solutions tend to a steady-state after time 2.

For Test 7, let us consider the concave kernel functions (3a) with different supports and same maximum velocities

V max 1 = V max 2 = 1
in (3b). In Figure 10, we can observe the behaviour of the solutions when the agents of the two classes have different look-ahead distances, i.e. kernel supports. We can notice that, when one group of agents has a sufficiently large look-ahead distance, oscillations are not evident. From Figure 11, we see that the solutions reach the steady-state as the time increases.

Maximum principle

The simplex S := {(ρ 1 , ρ 2 ) ∈ R 2 : ρ 1 + ρ 2 ≤ 1, ρ i ≥ 0 for i = 1, 2}, is not an invariant domain, as we can see in Figure 12, as we already noticed for the multi-class system presented in [START_REF] Chiarello | Non-local multi-class traffic flow models[END_REF].

In particular, we observe that the sum r(t, x) can exceed 1 even if the initial condition is such that ρ 1 (0, x) + ρ 2 (0, x) ≤ 1.

2 ,

 2 and assumptions (H1) -(H2) hold. Then the Cauchy problem (1), (2) admits a weak solution on [0,T[ ×R, for some T > 0 sufficiently small.

Fig. 1 :

 1 Fig. 1: Test 1. Solution to (1) with (1) and ρ 0 1 = 0.2 * (x < 0) + 0.1 * (x >= 0) and ρ 0 2 = 0.1 * (x < 0) + 0.2 * (x >= 0) with η 1 = η 2 = 0.1 on the left and η 1 = η 2 = 0.01 on the right.

Fig. 2 :

 2 Fig. 2: Test 2. Solution to (1) with (1) and ρ 1 = 0.2 * (x < 0) + 0.1 * (x >= 0) and ρ 2 = 0.1 * (x < 0) + 0.3 * (x >= 0) with η 1 = η 2 = 0.1 on the left and η 1 = η 2 = 0.01 on the right.

Fig. 3 :

 3 Fig. 3: Test 3. Solution to (1) with (1) and ρ 0 1 = 0.1 * (x < 0) + 0.4 * (x >= 0) and ρ 0 2 = 0.2 * (x < 0) + 0.5 * (x >= 0). With η 1 = η 2 = 0.1 at left, η 1 = η 2 = 0.01 at right.

Fig. 4 :

 4 Fig. 4: Test 4. Solution to (1) with (1) and ρ 0 1 = 0.4 * (x < 0) + 0.1 * (x >= 0) and ρ 0 2 = 0.5 * (x < 0) + 0.2 * (x >= 0). With η 1 = η 2 = 0.1 at left, η 1 = η 2 = 0.01 at right.

Fig. 5 :

 5 Fig. 5: Initial condition for tests 5, 6 and 7 with periodic boundary conditions. ρ 0 1 = 0.3 + 0.2 sin(2πx) and ρ 0 2 = 0.1 + 0.1 sin(2π x).

For Test 6 ,max 1 = 0.8 and V max 2 =

 612 let us consider concave kernel functions and linear decreasing function velocities with different maximum velocities V 1.3:

Fig. 6 :

 6 Fig. 6: Test 5. Solution to (1) with (2) and ρ 0 1 = 0.3 + 0.2 sin(2π x) and ρ 0 2 = 0.1 + 0.1 sin(2πx) with η 1 = η 2 = 0.1 at left, η 1 = η 2 = 0.01 at right.

Fig. 7 :

 7 Fig. 7: (t, x)-plot solutions of (1) with (2) and initial conditions ρ 0 1 = 0.3 + 0.2 sin(2πx) and 0 2 = 0.1 + 0.1 sin(2π x), see Figure 5; from top to bottom: η 1 = η 2 = 0.1, 0.01; from left to right: ρ 1 , ρ 2 and ρ 1 + ρ 2 .

  ω

Fig. 8 : 1 = 0.8 and V max 2 = 1 . 3 ,

 81213 Fig. 8: Test 6. Solution to (1) with (3), V max 1 = 0.8 and V max 2 = 1.3, and ρ 0 1 = 0.3 + 0.2 sin(2πx) and ρ 0 2 = 0.1 + 0.1 sin(2πx); η 1 = η 2 = 0.1 on the left, η 1 = η 2 = 0.01 on right.

Fig. 9 :

 9 Fig. 9: (t, x)-plot solutions of (1) with (3) and initial conditions ρ 0 1 = 0.3 + 0.2 sin(2π x) and ρ 0 2 = 0.1 + 0.1 sin(2π x), see Figure 5; from top to bottom: η 1 = η 2 = 0.1, 0.01; from left to right: ρ 1 , ρ 2 and ρ 1 + ρ 2 .

Fig. 10 : 1 = V max 2 = 1 ,

 10121 Fig. 10: Test 7. Solution to (1) with (3), V max 1 = V max 2 = 1, and ρ 0 1 = 0.3 + 0.2 sin(2π x) and ρ 0 2 = 0.1 + 0.1 sin(2π x); η 1 = 0.1, η 2 = 0.01 on the left, η 1 = 0.01, η 2 = 0.1 on the right.

Fig. 11 :

 11 Fig. 11: (t, x)-plot solutions of (1) with (2) and initial conditions ρ 0 1 = 0.3 + 0.2 sin(2π x) and ρ 0 2 = 0.1 + 0.1 sin(2πx), see Figure 5; from top to bottom: η 1 = 0.1 η 2 = 0.01 and η 1 = 0.01 η 2 = 0.1; from left to right: ρ 1 , ρ 2 and ρ 1 + ρ 2 .

Fig. 12 : 1 = 1 .5 and V max 2 =

 12112 Fig. 12: Solution to (1) obtained with initial data ρ 0 1 = 0.9 * (x < 0) + 0.1 * (x >= 0) and ρ 0 2 = 0.1 * (x < 0) + 0.75 * (x >= 0), linear kernels, η 1 = 0.01 and η 2 = 0.1 , V max 1 = 1.5 and V max 2 = 0.8 and absorbing boundary conditions.