Mathieu Léonardon

Vincent Gripon

Using Deep Neural Networks to Predict and Improve the Performance of Polar Codes

Polar codes can theoretically achieve very competitive Frame Error Rates. In practice, their performance may depend on the chosen decoding procedure, as well as other parameters of the communication system they are deployed upon. As a consequence, designing efficient polar codes for a specific context can quickly become challenging. In this paper, we introduce a methodology that consists in training deep neural networks to predict the frame error rate of polar codes based on their frozen bit construction sequence. We introduce an algorithm based on Projected Gradient Descent that leverages the gradient of the neural network function to generate promising frozen bit sequences. We showcase on generated datasets the ability of the proposed methodology to produce codes more efficient than those used to train the neural networks, even when the latter are selected among the most efficient ones.

I. INTRODUCTION

Polar Codes are a family of Error Correcting Codes that are used in the 5G NR standard. In order to construct these codes, traditional methods consist in estimating the bit error probabilities with mathematical models. Usually these approaches consider the Successive Cancellation (SC) decoding algorithm as it allows such mathematical formulations. In practice the Successive Cancellation List (SCL) decoder allows to achieve much better error rates. However predicting the bit error probabilities for SCL is not as easy as for SC. Finding good code constructions is therefore challenging, and many works have been published with that aim [START_REF] Elkelesh | Decoder-tailored polar code design using the genetic algorithm[END_REF]- [START_REF] Liao | Construction of polar codes with reinforcement learning[END_REF]. Most of these methods require estimating bit/frame error rates (BER/FER) using Monte Carlo simulations, and training models to lower them. Ideally, a well trained machine learning algorithm could lead to finding error rates lower than those encountered on its training data.

In this paper, we aim at using neural networks to predict the FER of a polar code from its construction parameters. Once such a neural network is trained, it is possible to use it to generate competitive codes. To this end we propose to use an adaptation of a classical algorithm typically meant to generate adversarial inputs, i.e. inputs that are specifically designed to fool the neural network decision. In our case, these frozen bit sequences inputs will be generated with the aim at lowering the error rate. We propose two datasets made of bit frozen sequences and associated FERs, obtained using Gaussian Approximation or Density Evolution. For each one, we show that it is possible to train neural networks able to predict the FER with high confidence, resulting in an error inflation of about 5% in average on previously unseen inputs. Using these neural networks, we propose new polar codes achieving lower FERs than the ones used during training. We release both the datasets and the code used in this paper at the following address: https://github.com/brain-bzh/PolarCodesDNN.

II. CONSTRUCTING POLAR CODES

A. Conventions

In this paper we consider (N, K) polar codes, where N is the codeblock length and K the length of the information sequence. We are more precisely interested in polar codes with Arıkan kernels, as defined in [START_REF] Arikan | Channel polarization: A method for constructing capacityachieving codes for symmetric binary-input memoryless channels[END_REF]. Apart from both N and K, such a code is defined thanks to the frozen bit sequence f ∈ {0, 1} N . Namely, the 1s in f = {f i } 0≤i<N correspond to the indices of the frozen bits.

B. Mathematical models

Constructing polar codes refers to the way to generate a frozen bits sequence f , given N , K, and a channel model. In [START_REF] Arikan | Channel polarization: A method for constructing capacityachieving codes for symmetric binary-input memoryless channels[END_REF], construction was only defined for Binary Erasure Channels. Methods to construct polar codes for Additive White Gaussian Noise (AWGN) channels were later given in [START_REF] Tal | List decoding of polar codes[END_REF], [START_REF] Trifonov | Efficient design and decoding of polar codes[END_REF]. All three methods sort the positions of frozen bits, from the most to the least reliable, considering SC decoding. However, freezing the least reliable bits is not guaranteed to offer the best performance, expressed as the Frame Error Rate (FER) for a targeted SNR in more complex situations. As a matter of fact, finding the best frozen bit sequence (i.e. that yielding the lower Frame Error Rate (FER)), for a given channel, target SNR and decoder, is a difficult challenge that has to be addressed specifically, using complex models or empirical approaches.

C. Learning Approaches

Some learning-based methods have been recently proposed to further improve the error correcting performance of polar codes. They are based on the mathematical constructions mentioned above and particularize this construction to the specific channel and decoding conditions. In [START_REF] Elkelesh | Decoder-tailored polar code design using the genetic algorithm[END_REF], a genetic algorithm is used to modify the frozen bits set in order to improve the performance of polar codes with different channel models (AWGN, Rayleigh) and different decoding algorithms e.g. Belief Propagation (BP), SCL, etc. Reinforcement Learning (RL) has also been used to address the same problematic [START_REF] Liao | Construction of polar codes with reinforcement learning[END_REF]. The polar code construction is formulated as a maze-traversing game, which is solved using RL methods. This allows to improve the performance of polar codes constructed with [START_REF] Tal | List decoding of polar codes[END_REF] in some of the studied cases.

Deep Learning Techniques have also been used to improve existing codes [START_REF] Huang | Ai coding: Learning to construct error correction codes[END_REF]. It is mentioned that neural networks are used to predict performance of a polar code under certain channel and decoding conditions, and some results are provided. However, the methods used to construct and train the network are not described. In this paper, we propose to further investigate the capabilities of neural networks to model and construct polar codes. We give hints about how to construct these neural networks, with detailed experiments and results. Finally, we propose methods that allow to use neural networks to improve the construction of polar codes for a given channel and decoding algorithm.

III. POLAR CODES DESIGN WITH NEURAL NETWORKS

A. Dataset Generation

We propose to train a neural network to predict the FER performance of a polar code. The neural network is trained on a dataset consisting of pairs of frozen bits sets f associated with their corresponding FERs. The FERs are obtained by Monte Carlo simulations, using the AFF3CT toolbox as a library [START_REF] Cassagne | Aff3ct: A fast forward error correction toolbox![END_REF]. The frozen bits set is the only parameter that change in the communication chain accross the simulations. All other parameters are constant, e.g. N , K, E b /N 0 . The frozen bits sets space is a high dimensional space N K in which most of its elements yield bad FERs. Directly training on poorly performing codes would inevitably lead the neural network to focus predicting high FERs.

In order to generate a relevant subset, we first use the Gaussian Approximation (GA) method to generate a list of the frozen bits positions p ∈ {0, 1, . . . N -1} N -1 , that are sorted according to the reliability that GA associates with each of them. Usually, this sorted positions are directly used to generate the frozen bits set f , where

f pi = 1 if i < K 0 if i ≥ K , ∀0 ≤ i < N (1)
To generate our dataset, a subset of the values in p are randomly shuffled to get a new vector of frozen bits positions p d :

p d i = π d (p i ), ∀K -r ≤ i < K + r (2)
where {π d } 0≤d<D is a set of uniformly distributed random permutations. The shuffling range r is chosen empirically. According to our experiments, the best results are obtained when r is chosen so that there is a one to ten ratio between the worst and the best measured FER in the generated dataset.

In order to get a dataset of size D, the shuffling operation is performed D times, to get D variants of the frozen bits positions p d and the corresponding frozen bits sets f d .

B. Neural network

Let us first recall that a neural network can be modelled as a mathematical function that is obtained by assembling elementary subfunctions called layers. A typical layer is a function of the form F : x → σ(W x + b ), where W and b , respectively a matrix and a vector, are made of values that are trainable parameters, and σ is a nonlinear predetermined function which acts on each component of its input independently. The number of dimensions in the input x is denoted d input and the number of dimensions in the output F (x) is denoted d output . Assembling layers can be performed by performing composition, additions, concatenations, etc.

In our work, we consider very simple architectures that are defined using three hyperparameters:

• The number of layers L. We denote each associated layer function (F ) 1≤ ≤L . We use Rectified Linear Units (ReLU) as nonlinear functions in all layer functions, except the last for which we do not use a nonlinear function.

• The number of neurons in hidden layers, denoted H. We then fix the dimensions:

d input 1 = N , (d input = H, ∀ > 1), (d output = H, ∀ < L), d output L = 1.
• The shortcut gap G defined in analogy to celebrated

ResNet architectures [START_REF] He | Deep residual learning for image recognition[END_REF] (see details below). Once all parameters L, H and G are fixed, we can define the neural network function F . Denote f an input sequence of bits. We compute the following sequence:

       f 0 = f f +1 =    F +1 (f ) + f +1-G if ∃k ∈ N * , = kG and + 1 < L F +1 (f ) otherwise (3) Then, we define F (f ) = f L , ∀f .
In particular, when G is no lesser than L -1, this sequence boils down to composing

F = F L • F L-1 • • • • • F 1 .
A toy depiction of this architecture is shown in Figure 2.

To train F to associate an input bit sequence f with the corresponding FER denoted FER f , we use a variant of the Stochastic Gradient Descent algorithm named Adam [START_REF] Kingma | Adam: A method for stochastic optimization[END_REF], which is meant to minimize the Mean Square Error between F (f ) and FER f in logscale. In other words, we aim at approximately solving:

arg min {W ,b } 1≤ ≤L E[(F (f ) -log(FER f )) 2 ]. (4)
We use a logscale to better encompass for the range of FERs we expect to observe in applications.

C. Inflation Of Error (IOE)

To train a neural network, we make use of a training set T and a validation set V , both made of pairs (f , FER f ). The idea is to define these sets so that V ∩ T = ∅. Once a neural network has been trained using T as a proxy to the expectation of Equation ( 4), we evaluate on V its ability to generalize predictions to previously unseen inputs.

To properly reflect the exponential scale of FER, we define the Inflation Of Error (IOE) of (f , FER f ) ∈ V for the neural network function F as: f

IOE(f , F ) = max FER f exp(F (f )) , exp(F (f )) FER f -1. (5
F (f ) F 1 F 2 F G+1
Fig. 2. Depiction of the architectures used to predict the FER based on input frozen bit sequences. The direct flow of data from input to output is periodically combined with shortcuts of length G.

As an example, an IOE of 100% means that the FER predicted using the neural network is half or twice the actual FER of the considered validation sample.

In our experiments, we are typically interested in measuring the average IOE or the worst (i.e. maximal) IOE on the validation set.

D. Constructing codes using neural networks

Once neural networks have been trained to predict FER based on input frozen bit sequences, we can leverage them to propose new efficient codes.

We adapt the methodology described in [START_REF] Madry | Towards deep learning models resistant to adversarial attacks[END_REF], called Projected Gradient Descent (PGD). PGD has been widely used in the context of adversarial attacks, where the aim is to generate inputs meant to fool the neural network predictions. The idea is to fix the neural network parameters, and use gradient descent to update inputs so that it translates the corresponding output towards new decisions. In our case, the goal is to use the prediction of the neural network as the function to minimize, since it means lowering the FER.

In more details, we implement a straight-through [START_REF] Bengio | Estimating or propagating gradients through stochastic neurons for conditional computation[END_REF] procedure, in which we manipulate inputs that are not necessarily binary: such inputs can be seen as a relaxation of bit sequences into real-valued ones. When we estimate the FER using the neural network, we manipulate a quantized version of the input, where all values below the median are set to one bit and all others the other one. We compute the gradient with respect to this input, but apply it to its real-valued version. More details are available in Algorithm 1. It comes with two parameters: a number of iterations I and a gradient step µ.

y ← F ( f ) 5 for 0 ≤ i < N do 6 f i ← f i -µ ∂y ∂ fi 7 Return quantized version of f IV. EXPERIMENTS
Throughout this section, we consider two types of polar codes, namely (256, 128), termed small and (1024, 512), termed large. In both cases, an SCL decoder is used, with a list depth of respectively L = 4 and L = 32. The SNR values used for any construction method are respectively E b /N 0 = 3.2dB and E b /N 0 = 2.7dB. For the small codes, we generated a total of 77 466 frozen bit sequences f , split arbitrarily in 80% used for training and 20% for validation. For the large codes, we generated a total of 15 862 frozen bit sequences, with the same proportions for training and validation.

We use classical data standardization techniques, where we center and reduce both input and output components. On the input, this has the effect of replacing 0s with -1s. We also remove coordinates in the inputs that are constant on our datasets. We end up using 36 out of the 256 input dimensions for small codes and 112 out of the 1024 input dimensions for large codes. The minimal FER in the training and validation sets are 1.67e-4 for small codes and 5.75e-5 for large ones. As a reference for upcoming experiments, we also computed the average IOE and worst IOE obtained when using a constant predictor (predicting the average of outputs). We obtained: average IOE: 86.03% and worst IOE: 1541.32% for small codes and average IOE: 47.90% and worst IOE: 4716.20% for large ones. These quantities can be thought of as "chance levels". For all reported results, we compute at least an average over 10 runs, where each run is obtained with different randomly drawn initial weights for the layer parameters, and different order of presenting data during training.

A. Effect of hyperparameters on IOE

As a first series of experiments, we wanted to empirically study the impact of hyperparameters in our architectures on the average and worst IOEs. We first randomly searched for an efficient starting set of hyperparameters, and then looked at the influence of varying a single parameter, while leaving the others constant. We ended up using 20 epochs for training for small codes and 100 epochs for training for large ones. In both cases, we used 320 neurons in the hidden layers, a network depth of 6 and a shortcut gap of 3.

Prediction performance of neural networks: On small codes, we obtain after 10 runs the following performance: average IOE: 2.65% ± 0.06%, worst IOE: 29.36% ± 5.63% for small codes (confidence intervals obtained at 95%). For the large codes, we obtain: average IOE: 5.46% ± 0.04%, worst IOE: 31.28% ± 0.04%. In other words, the ratio between predicted FER and actual one is no more than 6% in average for our datasets, which is way smaller than the chance level we estimated. Also, in the worst case, the actual FER is about 30% larger or smaller than the actual one, which demonstrates the outstanding generalization ability of the trained neural networks. It is also worth pointing out that the estimation of FER using Monte Carlo simulations is subject to a small error, and that this error could be significant with respect to the average IOE measured in our experiments.

Effect of the number of epochs in training:

In Figure III-A, we report the average and worst IOEs obtained depending on the number of epochs used to train the neural network architectures. As expected the small codes require less epochs to reach convergence, as the dataset is way larger than in the case of large codes. Interestingly, we observe a small increase of IOEs when training large codes for too many epochs, which can indicate a small risk of overfitting.

Effect of the number of neurons in hidden layers:

Next, we wanted to study the impact of the number of neurons in hidden layers H in the architecture. In Figure IV-A, we depict the evolution of IOEs as a function of H. Except for the small outlier obtained with the blue curves -which is probably due to the limited number of runs -, we observe an expected global decrease of IOE as the number of neurons grow. A plateau is soon reached around H = 300. Interestingly, the worst IOE for small codes seems to increase with the largest values of H, making one suspect about potential overfitting.

Effect of the depth of the architecture:

In Figure IV-A, we vary the depth of trained architectures and report the corresponding IOEs. The effect of the depth seems to play a role analogous to that of the number of neurons in hidden layers, where too small or too large values can cost higher IOEs. The best obtained average IOEs are for depth 3 and 9 respectively, with a pretty insignificant effect between these values. On the contrary, increasing the depth seems to cause way larger worst IOEs for small codes.

Effect of the shortcut gap length: Next, we tested the effect of the shortcut gap G on the IOE of trained architectures. Obtained results are presented in Figure IV-A. Of all tested parameters, the shortcut gap length seems to be the one having the smallest influence on obtained IOEs. It is worth noting that a depth of 6 is equivalent to not using shortcuts in our architectures. When zooming, we still observe a significant improvement when using small gaps in both the case of small and large codes.

Using standard methods to increase predicted IOEs: Finally, we implemented standard techniques from the field of Deep Learning meant to either prevent overfitting or improve prediction performance. We implemented DropOut [START_REF] Srivastava | Dropout: a simple way to prevent neural networks from overfitting[END_REF], including on the input [START_REF] Devries | Improved regularization of convolutional neural networks with cutout[END_REF], where we randomly mask coordi- E b /N 0 (dB) Fig. 6. Frame Error Rate performance of small (left) and large (right) codes, with the GA construction (blue circles), the generated datasets (blue areas), and our proposed construction (red circles). Green vertical lines correspond to the target SNR.

nates of input or output vectors computed throughout the architecture during training, BatchNorms [START_REF] Ioffe | Batch normalization: Accelerating deep network training by reducing internal covariate shift[END_REF], which center and reduce each dimension of computed vectors during training, and learn corresponding coefficients to be used on validation data, and Mixup [START_REF] Zhang | mixup: Beyond empirical risk minimization[END_REF], a simple data-augmentation procedure in which training inputs are linearly interpolated and trained to associate the corresponding linear interpolation of the outputs.

Results are presented in Table I. Interestingly, none of the tested methods resulted in improved IOEs on our tests. We suspect that it is because these techniques are mostly meant to be applied when dealing with raw input signals, such as in the case of vision or audio signals. In our case, their discrete and combinatorial nature might make these techniques unsuitable.

B. Constructing Codes

After having explored the impact of hyperparameters on the IOEs, we fixed parameters for constructing interesting codes. We chose: (L, H, G) = (3, 640, 3) trained for 100 epochs for large codes (i.e. not using shortcuts) and (L, H, G) = (5, 320, 3) trained for 40 epochs for small codes. When using Algorithm 1, we used a maximum of 5000 iterations and a gradient step µ = 0.1.

With large codes, we obtained a frozen bit sequence reaching as low as 1.01e-5 as FER, validated using Monte Carlo simulations, which is way lower than the minimum observed on the training set (5.75e-5). With small codes, we obtained several interesting candidates that reached competitive FERs, but not surpassing by a large margin the examples of the training set. This was expected due to the relative small size of these codes and the abundance of already very optimized bit sequences in the training set. The corresponding FERs of best candidates are depicted in Figure 6.

Fig. 1 .

 1 Fig. 1. Evolution of validation Inflation Of Errors depending on the number of epochs used for training, in both the case of large codes (left) and small codes (right). Standard deviation obtained on 10 runs is also shown.

Algorithm 1 :

 1 Algorithm used to generate low FER polar codes. 1 f ← random binary initialization 2 for a fixed number of iterations I do 3 f ← quantized version of f

4

 4

Fig. 3 .

 3 Fig.3. Evolution of validation Inflation Of Errors depending on the size of the hidden layers in the trained architectures, for large codes (blue and cyan) and small codes (red and orange). Standard deviation obtained on 10 runs is also shown.

Fig. 4 .

 4 Fig.[START_REF] Arikan | Channel polarization: A method for constructing capacityachieving codes for symmetric binary-input memoryless channels[END_REF]. Evolution of validation Inflation Of Errors depending on the depth of the trained architectures, for large codes (blue and cyan) and small codes (red and orange). Standard deviation obtained on 10 runs is also shown.

Fig. 5 .

 5 Fig.5. Evolution of validation Inflation Of Errors depending on the shortcut gaps in the trained architectures, for large codes (blue and cyan) and small codes (red and orange). Standard deviation obtained on 10 runs is also shown.

TABLE I INFLUENCE

 I OF COMMON TECHNIQUES IN DEEP LEARNING ON THE AVERAGE IOE (CONFIDENCE INTERVALS AT 95% ARE INDICATED):DROPOUT, BATCH-NORMS (BNS) AND MIXUP.

	DropOut	BNs	Mixup	IOE (Large)	IOE (Small)
				5.46% (± 0.04%)	2.65% (± 0.06%)
				7.69% (± 0.06%)	3.08% (± 0.17%)
				8.95% (± 0.37%)	6.42% (± 0.80%)
				9.14% (± 0.50%)	8.05% (± 0.11%)
				36.78% (± 0.28%)	58.92% (± 0.37%)
				37.43% (± 0.44%)	60.78% (± 0.69%)
				32.77% (± 0.23%)	52.96% (± 0.21%)
				32.89% (± 0.33%)	53.31% (± 0.18%)

V. CONCLUSION

In this paper, we investigated how to build neural networks capable of predicting the error correction performance of polar codes. We proposed an algorithm to extract competitive codes from trained neural networks. Two datasets of frozen bits sets associated with their corresponding FER values, for a fixed SNR, were generated for two different code lengths, under SCL decoding. These datasets were used to train neural networks to predict the FER of any frozen bits set. We obtained in average a ratio of errors of less than 1.06 for the (1024,512) codes and 1.02 for (256,128) codes. The codes we generated using trained neural networks were shown to perform better than those of the training datasets. Finally, the source code used in this article as well as the generated datasets are published for reproducibility and reuse.


Jean-Christophe Maréchal

Bernard Ladouche

Claudine Lamotte

Benoît Dewandel

Vivien Hakoun

Pierre Perrochet

Saltwater intrusion and flow reversal at the submarine spring of a Mediterranean karst aquifer : explanation and modelling

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Saltwater intrusion and flow reversal at the submarine spring of a Mediterranean karst aquifer : explanation and modelling

Maréchal JC, Ladouche B, Lamotte C, Dewandel, B, Hakoun V, Perrochet P The Vise spring is the main outlet of a Jurassic karst aquifer located close to Montpellier city, South of France. The Vise spring is submarine, occurring at the bottom of the Thau lagoon at a depht of 30 m. The lagoon , made up of brachish water is connected to the Mediterranean sea. The fresh water from the karst aquifer as a whole, and especially from the spring, contributes to the qualitative state of the Thau lagoon which is well known for shellfish farming activities.

During the last fifty years (from 1967 to 2014), six occasional saltwater intrusions (called "inversac" in French) occurred, inversing the water flow at the submarine spring during a period varying from a few weeks to a few months. This backflooding process at the spring induces a very large saltwater intrusion into the karst aquifer. Given that this aquifer provides several highly important ecosystem services (drinking water supply for the coastal villages, fresh water to the Lagoon, thermal water to Balaruc spa and health resort), a large program of groundwater monitoring has been recently launched particularly at the submarine spring (discharge, temperature, quality) and several neighbouring borewells.

In november 2021, a seventh backflooding event started and is observed with the new monitoring system. From an initial flow rate of about 60 l/s from the aquifer to the lagoon through the spring, the flow inverted to about 350 l/s from the lagoon to the aquifer in a few minutes on the 28 th November 2020 at 9:40 AM. This sudden backflooding created a sudden water level rise of about 2.5 meters into the karst confined aquifer. A few months later, the saltwater intrusion into the aquifer is still high, equal to 150 l/s.

A physical mechanism is proposed to explain the sudden inversion of flow and its long duration after that the event has started. The propagation of a piezometric wave through the aquifer is simulated using simple analytical solutions. A preliminary meshed model of the aquifer and its interactions with the lagoon has been developed for identifying the main processes. First attempts for proposing alert indicators are also discussed.


François Stoffelbach

David M Haddleton
email: d.m.haddleton@warwick.ac.uk

Rinaldo Poli
email: poli@u-bourgogne.fr

Prof Rinaldo Poli

Controlled Radical Polymerization of Alkyl Acrylates and Styrene Using a Half-Sandwich Molybdenum(III) complex containing Diazadiene Ligands

Keywords: Atom Transfer Radical Polymerization, Molybdenum, polyacrylate, polystyrene, block copolymer

The half-sandwich molybdenum(III) complex CpMoCl2( i Pr2-dad) ( i Pr2-dad = i Pr-N=CH-CH=N-i Pr) proved to be an effective metal catalyst for the controlled radical polymerization of methyl acrylate, butyl acrylate, and styrene. In conjunction with an alkyl iodide [R-I : CH3CH(COOEt)I] as an initiator and in the presence or absence of Al(O-i-Pr)3 as a co-catalyst, the molybdenum-based system gave polymers with narrow molecular weight distributions. The in-situ addition of styrene to a macroinitiator of poly(methylacrylate) afforded an AB-type block copolymer.

Introduction

Controlled radical polymerization has become a heavily investigated area of research due to its high tolerance of polar functionalities and the ability to develop well defined macromolecular architectures. [1][2][3][START_REF] Matyjaszewski | Controlled Radical Polymerization[END_REF][START_REF] Patten | [END_REF][6][7] Transition metal mediated controlled radical polymerisation has attracted considerable attention with many different catalyst systems based on Cu(I), [8][9][10][11][12][13][14][15] Fe(II), [16][17][18][19][20] Ni (II, 0), [21][22][23] Ru(II), [24][25][26][27] Re(V), 28 Mo(V and III), [29][30][31][32] Pd(0), 33 Co(II) 34 and Rh(I) 35,36 being studied. Our recent discovery that complexes CpMoCl2(PMe3)2 and CpMoCl2(dppe) are able to control the radical polymerization of styrene by two different mechanisms simultaneously, i.e., ATRP and SFRP, 30 prompted us to extend our study to other half-sandwich Mo complexes with different ligands. Recently, we have reported the controlled radical polymerization of styrene with new CpMoCl2(R2-dad) complexes (R2-dad = RN=CH-CH=NR with R = Ph, p-Tol, 2,6-i Pr2C6H3 and i Pr) under typical SFRP (Stable Free Radical Polymerization) and ATRP (Atom Transfer Radical Polymerization) conditions. The SFRP polymerization occurs by using the Mo(III) complex in conjunction with 2,2'azobis(isobutyronitrile) (AIBN) as a conventional radical initiator (the dormant species being an alkylmolybdenum(IV)-terminated polymer chain), whereas the ATRP polymerization occurs by using the Mo(III) complex and 1-bromoethylbenzene as an initiator. 32 In this paper, we present results demonstrating that CpMoCl2( i Pr2-diazadiene) is an efficient catalyst for the controlled free radical polymerization of methyl acrylate, butyl acrylate, and styrene, in the presence or absence of a catalytic promoter. We demonstrated that an iodide initiator works better for acrylates like in copper-mediated ATRP 37 and works well for styrene like in ruthenium-and rhenium-based ARTP. 28,38 As an extension of this investigation, we present the polymerization of methyl acrylate followed by styrene to afford the respective block copolymer, representing the first report of a block co-polymer prepared with a molybdenum-based catalyst.

Results and Discussion

Homopolymerization of methyl acrylate. The polymerization of acrylate monomers under ATRP conditions with a molybdenum based catalyst complex has not been reported to date.

Styrene has been reported to be polymerised by ATRP with Mo based systems but previous attempts to polymerise the acrylates via ATRP with complexes CpMoCl2L2 (L = PMe3 or L2 =  4 -C4H6) was unsuccessful. 30 A first attempt to carry out the ATRP of methyl acrylate using CpMoCl2( i Pr2-diazadiene)/BrEA (BrEA = CH3CH(Br)COOEt) at 80 °C in toluene solution (30% v/v) yielded no polymer after 24 hours. A small conversion (about 2%) was observed for the same mixture at 100 °C after 24 hours. This prompted us to use an iodide based initiator, since faster ATRP processes have been observed for the Cp*RuCl(PPh3)2/Al(O-i-Pr)3/RX initiating system when using iodide initiators instead of the bromide analogues due to the relatively weak C-I bond. 39 Using CH3CH(I)COOEt) (IEA) as an initiator under the same conditions as described above yielded the results shown in Figure 1. The polymerization exhibited excellent linear correlation of the Mn as a function of conversion and a decrease in PDI throughout the reaction, although the initiator efficiency (f) was low. 40 The f factor is calculated as the ratio between the theoretical slope and the slope of the least-squares fit of the experimental data for the Mn vs. conversion plot with the imposition of a zero intercept. The polymerisation follows first order kinetics as would be expected for a living polymerisation (see Figure 2). However, the apparent first order rate constant is low (1.2x10 -6 s -1 ) with six days being required to obtain 50% conversion. In an attempt to increase the polymerisation rate, we utilised the Lewis acid activator Al(O-i-Pr)3, which is known to accelerate the ATRP for other systems. [41][42][43] Indeed, the presence of Al(O-i-Pr)3 had a dramatic effect on the polymerisation rate (see Figure 3), with kapp increasing to 1.7x10 -5 s -1

for Al/Mo = 1.

(O-i-Pr)3. [Ma]/[IEA]/[Mo]/[ Al(O-i-Pr)3] = 151/1/1/1 () ; [Ma]/[IEA]/[Mo]/[Al(O-i-Pr)3] = 171/1/1/0.5 () ; [Ma]/[IEA]/[Mo] = 140/1/1 ().
The polymerisation rate increased with the concentration of aluminum compound, as expected. Even under these reaction conditions, the Mn increased linearly with conversion and the PDIs remain relatively low, indicating that the polymerisation remained controlled although still characterized by low f values. 40 Results consistent with living-type polymerisation with fast initiation and little or no termination were obtained at a lower temperature (80 °C, see Figure 4). The PDIs obtained were narrower (1.16< Mw/Mn <1.24) with a higher initiator efficiency. 50% conversion was reached at this temperature after only 24 hours.

The intervention of degenerative transfer processes has been recognized as a possibility during ATRP when using iodide initiators. 44,45 In order to test for this possibility, we have carried out the following experiments. A polymerization of MA carried out in the presence of AIBN as a source or radicals, IEA and the aluminum co-catalyst but in the absence of the Mo Given the dramatic increase in the polymerisation rate on the addition of Al(O i Pr)3, we also examined the effect of adding the Al(O i Pr)3 in a polymerisation initiated by a bromide initiator. The rate of polymerisation was substantially increased, even at 60 °C (51 % conversion after 21 h). Conversely, the molecular weight control was poor, with observed PDI's > 2. This may result from the absence of degenerative exchange processes as compared with the iodide-initiated polymerization.

Homopolymerization of n-butyl acrylate.

Butyl acrylate (nBA) was polymerised using the CpMoCl2( i Pr2-dad)/IEA system in xylene (120 °C), toluene (100 °C), and in the presence and absence of Al(O-i-Pr)3 (see Figure 5). The rate of polymerisation increased with temperature and with catalytic promoter, as expected. The lowest PDIs (1.30< Mw/Mn <1.46) were obtained at 100 °C in the presence of Al(O-i-Pr)3 (Figure 6). It is interesting to observe that the initiator efficiency is greater in the absence of the Al(O-i-Pr)3 for this reaction, whereas the opposite trend has been observed in another case. 43 The collective results shown so far demonstrate that the CpMoCl2( i Pr2-diazadiene)/IEA/Al(Oi-Pr)3 initiating system is effective for the controlled/living radical polymerisation of acrylates with different alkyl groups.

Homopolymerization of styrene.

We have recently shown that styrene can be polymerised by ATRP in bulk at 90 °C using CpMoCl2( i Pr2-diazadiene) and 1-bromoethylbenzene (BEB) as an initiator. 32 In order to know if a macroinitiator of poly(methyl-acrylate)-CH2CH(COOMe)I would be able to initiate the styrene polymerisation and yield a block copolymer, we carried out a polymerisation of styrene with the CpMoCl2( i Pr2-dad)/IEA/Al(O-i-Pr)3 initiating system in solution at 90 °C.

The results, shown in Figure 7, demonstrate that the polymerisation seems to be better controlled under these conditions relative to those previously reported (greater polymerisation rate and lower PDIs). As for the MA polymerisation described above, we considered it interesting to examine the effect of Al(O-i-Pr)3 on a polymerisation initiated by a bromide initiator. As for that case, the Mn increased linearly with the conversion with f close to unity, but the PDIs were very high (> 2).

Block Copolymerization

On the basis of the living homopolymerisations of both methyl acrylate and styrene with an acrylate initiator, a sequential block co-polymerisation of MA and styrene was examined with the molybdenum-based catalyst. MA was first polymerised with CpMoCl2( i Pr2-dad)/IEA coupled with Al(O-i-Pr)3 at 80°C in toluene. After 31% consumption of MA, the residual monomer and the toluene solvent were removed by evaporation (a 1 H NMR spectrum of this mixture showed the absence of residual MA) and a fresh solution of styrene in toluene was added to the poly(MA). As shown in Figure 8, the molecular weight distribution shifted to higher molecular weights while retaining a relatively narrow molecular weight distribution.

These results show that each step of the sequential polymerisation is well controlled and virtually free from chain transfers and terminations to give homopolymers. Figure 9 shows 1 H NMR spectra of the PMA-b-PS obtained, showing signals of the styrene repeat units and those of the MA units. Furthermore, the two small absorptions a and b at  4.16 and 5.00, respectively, correspond to the ethyl CH2 group 22 contained in the initiator at the MA chain, and to the terminal proton adjacent to the iodine atom at -end. 28,46 No vinyl groups are visible on this spectrum. The Mn calculated from the peak intensity ratios c/a and d/a were 8280 and 9260, respectively, in close agreement with those obtained by sizeexclusion chromatography [Mn (SEC) = 7980 and 9940].

Conclusion

The CpMoCl2( i Pr2-dad)/IEA/Al(O-i-Pr)3 initiating system is capable of controlling the radical polymerisation of methyl and butyl acrylates and styrene. AB-type block copolymers of MA and styrene have been obtained by the ternary initiating systems. Furthermore, the Al(O-i-Pr)3 compound plays a role as co-catalyst also in combination with this family of organometallic Mo complexes.

Experimental Section

General remarks. All reactions were carried out under an atmosphere of dry and oxygen-free argon using standard Schlenk techniques. Styrene was washed with an aqueous NaOH solution (10%), followed by water, then dried over MgSO4 and finally distilled at 25°C under reduced pressure. Methyl and butyl acrylate were passed through an alumina column and then distilled under argon. Toluene and xylene were purified by distillation under argon after drying over sodium benzophenone ketyl. (1-Bromoethyl)benzene and 2-bromopropionate were purchased from Aldrich Chemical Co and degassed before use. Al(Oi-Pr)3 (Aldrich) was used as received and handled in a glovebox under dry and oxygen-free argon. Compounds CpMoCl2( i Pr2-dad) 32 and CH3CH(I)COOEt (IEA) 47 were obtained according to previously described synthetic procedures.

Polymerization Procedures. All ATRP polymerisation reactions were conducted following the same experimental procedure. A typical procedure is described as a representative example. CpMoCl2( i Pr2-dad) and Al(O-i-Pr)3 were added to a 25 mL Schlenk tube equipped with a stirring bar. Styrene, toluene and 2-iodopropionate were added to the reaction flask by a syringe after a 20 min Ar purge. The Schlenk tube was immersed in an oil bath heated at 90°C. Aliquots were withdrawn periodically for monitoring.

Measurements. The MWD, Mn and Mw/Mn of the polymers were measured by sizeexclusion chromatography (SEC) using THF as eluent (1mL/min) at room temperature on two polystyrene gel columns (B Jordi) equipped with a refractive index detector. The columns were calibrated against standard polystyrene and PMMA (for PMA and PBA) (Polymer Laboratories). The Mn of PMA and PBA were corrected by using the Mark-Houwink coefficients of ref. [48,49]. 1 H NMR spectra were recorded in CDCl3 at 25°C on a Brucker 500 MHz spectrometer. The peaks positions are reported with positive shifts in ppm downfield of TMS, as calculated from residual solvent peaks.

Figure 1 .

 1 Figure 1. Mn (black dots) and PDI (open triangles) vs. conversion for the polymerization of methyl acrylate using CpMoCl2( i Pr2-diazadiene)/IEA (MA/Mo/IEA = 140/1/1) in toluene (30% v/v) at 100 °C. The straight line is the least squares fit of the experimental data (f = 0.6).

Figure 2 .Figure 3 .

 23 Figure 2. First order kinetic plot for the polymerisation of MA obtained in the presence of CpMoCl2( i Pr2-diazadiene)/IEA (MA/Mo/IEA = 140/1/1) in toluene solution (30% v/v) at 100 °C.

Figure 4 .

 4 Figure 4. (a) Mn (black dots) and PDI (open triangles) vs. conversion for PMA obtained in the presence of CpMoCl2( i Pr2-diazadiene)/Al(O-i-Pr)3/IEA (MA/Mo/Al/IEA = 171/1/1/1) in toluene solution (30% v/v) at 80 °C. The straight line is the least squares fit of the experimental data (f = 0.85) (b) Size exclusion chromatograms of the polymer samples reported in part (a).

Figure 5 .Figure 6 .

 56 Figure 5. Polymerisation conversion vs. time for PBA in the presence of CpMoCl2( i Pr2diazadiene) in solution (30% v/v). [Ba]/[IEA]/[Mo]/[Al(O-i-Pr)3] = 155/1/1/0.5, T = 100°C () ; [Ba]/[IEA]/[Mo]/[Al(O-i-Pr)3] = 150/1/1/0, T = 120°C () ; [Ba]/[IEA]/[Mo]/[Al(O-i-Pr)3] = 155/1/1/0, T = 100°C ().

Figure 7 .

 7 Figure 7. (a) Mn (black circles) and PDI (open triangles) vs. conversion for PS obtained in the presence of CpMoCl2( i Pr2-diazadiene)/Al(O-i-Pr)3/IEA (S/Mo/Al/IEA = 141/1/1/1) in toluene solution (30% v/v) at 90 °C. The straight line is the least squares fit of the experimental data (f = 0.55). (b) Size exclusion chromatograms of the polymer samples reported in part (a).

Figure 8 .

 8 Figure 8. Size-exclusion chromatograms of the polymers obtained during the block polymerisation of methyl acrylate and styrene in the presence of CpMoCl2( i Pr2diazadiene)/Al(O-i-Pr)3/IEA (Ma/Mo/Al/IEA = 174/1/1/1) in toluene solution (30% v/v) at 80 °C and 128 equivalent of styrene (addition).

Figure 9 .

 9 Figure 9. 1 H-NMR spectrum of the PMA-b-PS sample with Mn = 7980-9940 shown in Figure 8.

Acknowledgements. We are grateful to the CNRS for support of this work through the programme "Catalyse et Catalyseurs pour l'Industrie et l'Environnement" and for a Ph.D. fellowship (BDI) to FS. We thank the Conseil Régional de Bourgogne for co-sponsoring FS's fellowship and for the funds used to purchase the SEC instrument. Part of this work was carried out by FS in the laboratory of DMH with support from COST D17 (short mission) and the Marie Curie Supramolecular and Macromolecular Training Site programme (MCFH-2001-00397).

Supporting Information

Tables of experimental and theoretical Mn and PDI as a function of conversion for all ATRP processes (3 pages).