Jinglei Shi
email: jinglei.shi@nankai.edu.cn

Christine Guillemot

LIGHT FIELD COMPRESSION VIA COMPACT NEURAL SCENE REPRESENTATION

Keywords: Light Field, Compression, Low Rank, Network Distillation

In this paper, we propose a novel light field compression method based on a low rank-constrained neural scene representation. While most existing methods directly compress the light field views, our method first learns a Multi-Layer Perceptron (MLP)-based Neural Radiance Field (NeRF) from the input views. To be able to efficiently compress the NeRF scene representation, the weights of the MLP are optimized under a low-rank constraint using the Alternating Direction Method of Multipliers (ADMM) optimization method. The weights of NeRF are then decomposed into Tensor Train (TT) components which allow us to distill original NeRF network into a slimmer one. The slim NeRF is then refined using a quantization-aware training procedure. Experimental results show that this low rank-constrained NeRF-based light field compression method can achieve better rate-distortion than reference methods, while keeping the free-viewpoint reconstruction capability.

INTRODUCTION

In recent decades, light fields have emerged as promising technology in the field of computational photography. They record the position and direction of light rays in the 3D space after only one single exposure, thus enabling applications such as digital refocusing [START_REF] Ng | Light field photography with a hand-held plenoptic camera[END_REF], depth estimation [START_REF] Shi | A framework for learning depth from a flexible subset of dense and sparse light field views[END_REF] and view synthesis [START_REF] Shi | Learning fused pixel and feature-based view reconstructions for light fields[END_REF] etc. However, the spatio-angular information captured by light fields also means large volumes of data and consequently huge information redundancy, with strong implications on storage, transmission and display systems. Many solutions have been proposed to address this problem which can be roughly categorized as Pseudo Video Sequence (PVS)-based methods, transform-based methods and reconstruction-based ones.

Light field views can be reordered as a pseudo video sequence (PVS), and encoded using video compression standards such as HEVC or its variants [START_REF] Conti | HEVC-based light field image coding with bi-predicted self-similarity compensation[END_REF]. The authors in [START_REF] Dai | Lenselet image compression scheme based on subaperture images streaming[END_REF] propose to follow raster or spiral-like scan orders to generate the PVS for compression. State-of-the-art learning-based video compression methods [START_REF] Yang | OpenDVC: An open source implementation of the DVC video compression method[END_REF][START_REF] Yang | Learning for video compression with hierarchical quality and recurrent enhancement[END_REF][START_REF] Yang | Learning for video compression with recurrent auto-encoder and recurrent probability model[END_REF] are also applicable to compress light fields. However, due to the different coding tools used for the keyframe coded in the I mode, and for the P and B frames, and due to the rate control used to have high quality reference frames (I to predict all the other frames, P to predict the B frames), the quality varies from one frame to the other. Transformations have also been applied to light fields to reduce their redundancy. A 4D-DCT transform-based solution, known as the Multidimensional Light field Encoder (MuLE) [START_REF] Carvalho | A 4D DCT-based lenslet light field codec[END_REF], has been adopted in the 4D transform mode of JPEG Pleno. The authors in [START_REF] Rizkallah | Geometryaware graph transforms for light field compact representation[END_REF] propose a geometryaware graph-transform based light field compression method. New models such as homography-based low rank models and mixture of experts have also been investigated in [START_REF] Jiang | Light field compression with homography-based low-rank approximation[END_REF] and [START_REF] Verhack | Steered mixture-of-experts for light field images and video: Representation and coding[END_REF] respectively. Some transform-based methods like [START_REF] Jiang | Light field compression with homography-based low-rank approximation[END_REF] work well on narrow-baseline data, but performance quickly drops when data has large baseline with less correlation. Predictive coding solutions have also been designed based on view synthesis methods. A sparse set of light field views is first encoded and used to synthesize (predict) the remaining views using e.g., Depth Image Based Rendering (DIBR), as in [START_REF] Astola | WaSP: Hierarchical warping, merging, and sparse prediction for light field image compression[END_REF], transform-assisted [START_REF] Ahmad | Shearlet transform based prediction scheme for light field compression[END_REF], or learning-based view synthesis [START_REF] Liu | View synthesis-based light field image compression using a generative adversarial network[END_REF][START_REF] Shi | Deep residual architecture using pixel and feature cues for view synthesis and temporal interpolation[END_REF] approaches. The performance of view synthesis-based predictive coding solutions depends on the performance of DIBR techniques used for view synthesis, and on the choice of the reference views used for synthesis. Though the aforementioned categories of methods follow different design philosophies, they all directly compress the light field views. NeRF [START_REF] Mildenhall | NeRF: Representing scenes as neural radiance fields for view synthesis[END_REF], as an implicit scene representation learned from a set of input views, is much less impacted by the camera baseline during training, and does not require prior disparity estimation (e.g. using DIBR techniques), it naturally gets rid of the limitations of the previous methods. Several variants of NeRF have been proposed to improve its performance, to accelerate its training or test efficiency [START_REF] Müller | Instant neural graphics primitives with a multiresolution hash encoding[END_REF], to reduce the number of input views [START_REF] Chen | MVSNeRF: Fast generalizable radiance field reconstruction from multi-view stereo[END_REF], or for deterring aliasing [START_REF] Barron | Mip-nerf: A multiscale representation for anti-aliasing neural radiance fields[END_REF].

In this paper, we adopt a simplified NeRF having only one MLP to represent a light field, whose weights are optimized under a low-rank constraint using the ADMM optimization method. These low-rank weights of LR-NeRF are then decomposed into TT components to initialize a slimmer network D(istilled)LR-NeRF, scene information is thus distilled and represented with fewer parameters. These parameters are finally quantized with an optimized codebook to obtain a compact neural scene representation of light field, namely Q(uantized)DLR-NeRF. The performance of our proposed QDLR-NeRF has been assessed against HEVC and JPEG Pleno compression standards, and also three recent learningbased video compression methods [START_REF] Yang | OpenDVC: An open source implementation of the DVC video compression method[END_REF][START_REF] Yang | Learning for video compression with hierarchical quality and recurrent enhancement[END_REF][START_REF] Yang | Learning for video compression with recurrent auto-encoder and recurrent probability model[END_REF]. Experimental results show that our method achieves a good rate-distortion performance on tested light fields. Moreover, our scene representation-based method, when compared with other methods, can generate coherent views and is not limited by the baseline and depth priors. It is also capable of retrieving views in any viewpoint at decoder side.

METHOD

Overall workflow

The workflow of our method is demonstrated in Fig. 1. Light field information is first learned and stored in NeRF, optimized with respect to the target rank r in ADMM method. The optimized low-rank weights are decomposed to transfer scene information from LR-NeRF to DLR-NeRF. The weights of DLR-NeRF are finally quantized to obtain compact QDLR-NeRF.

NeRF initialization from light field

NeRF network takes 5D coordinates (x, y, z, θ, ϕ) of light rays as inputs to predict a color and density (RGB, σ), like formulated as follows:

RGB, σ = F Θ (x, y, z, θ, ϕ), (1)
during this procedure, scene information is recorded by model parameters Θ = {W i , b i }, with W i and b i are weights and biases of the MLP.

Compared with the original NeRF presented in [START_REF] Mildenhall | NeRF: Representing scenes as neural radiance fields for view synthesis[END_REF], two modifications are made to adjust to the compression context: a) Instead of using two MLPs, we adopt only one MLP to sample rays, which halves the parameter number with little performance degradation (<0.2dB). b). Since COLMAP package [START_REF] Schonberger | Structure-from-motion revisited[END_REF] fails to estimated camera pose for narrow-baseline light fields, we simplify the camera pose of structured light fields into focal length f and offset ∆ (proportional to the baseline), and set f and ∆ as trainable parameters like in [START_REF] Wang | NeRF--: Neural radiance fields without known camera parameters[END_REF]. The learning process of NeRF can be described as:

{W i , b i , f, ∆} = argmin {Wi,bi,f,∆} ||c ′ -c gt || 2 2 , (2)
where c ′ and c gt are respectively rendered and ground-truth pixel values.

Rank optimization in ADMM

To reduce the bitrate needed to encode the NeRF MLP, we can apply TT decomposition on W i ∈ R n1×n2 , ∀i to further decrease the parameter number:

W i = Q 1 i • Q 2 i , (3)
where

Q 1 i ∈ R 1×n1×r and Q 2 i ∈ R r×n2×1
are the TT components, and r is the target rank. Let's note that other tensor or matrix decomposition methods are totally applicable as well. However, decomposing the full-rank W i into low-rank matrices Q 1 i and Q 2 i would lead to information loss. Similar to [START_REF] Yin | Towards efficient tensor decomposition-based dnn model compression with optimization framework[END_REF], we optimize NeRF under a rank constraint, and iteratively update its weights W i using the ADMM method. The problem is first formulated as:

W i = argmin Wi ||c ′ -c gt || 2 2 , (4) s.t. rank(W i ) < r, (5)
By introducing an indicator function g(•):

g(W i ) = 0, rank(W i ) < r +∞, otherwise. (6)
and an auxiliary variable Z, the above problem is reformulated as:

argmin {Wi,Zi} ||c ′ -c gt || 2 2 + g(Z i ) (7) s.t. W i = Z i (8)
The augmented Lagrangian in the scaled dual form of the above optimization problem is defined as:

L(W i , Z i , U i ) = l(W i )+g(Z i )+ ρ 2 ||W i -Z i +U i || 2 F + ρ 2 ||U i || 2 F , (9)
with ρ being a positive penalty parameter, U i being the dual multiplier, and l = ||c ′ -c gt || 2 2 being the rendering error. Such an optimization problem can be solved in an alternative manner as:

W t+1 i = argmin {Wi} L(W t i , Z t i , U t i ), (10)
Z t+1 i = argmin {Zi} L(W t+1 i , Z t i , U t i ), (11)
U t+1 i = U t i + W t+1 i -Z t+1 i ( 12
)
where the updates of W i in Eq. 10 can be achieved by gradient descent, and the updates of Z i , according to [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF], can be solved as:

Z t+1 i = Π r (W t+1 i + U t i ), (13)
with Π r being the operation that decomposes the matrix into the TT format truncated to the desired rank r. This step of rank optimization in ADMM forces the network weights to be low-rank without large performance degradation for TT decomposition in next stage.

Network Distillation

The optimization in terms of tensor rank allows decomposing W r i into TT components with a small approximation error. We can thus use TT components

Q 1 i ∈ R 1×n1×r and Q 2 i ∈ R r×n2×1
to initialize a smaller (distilled) network after neglecting the first dimension of Q 1 i and the last dimension of Q 2

i . Let us note that a layer having weight W i in LR-NeRF will be decomposed into two layers with weights Q 1 i and Q 2 i in DLR-NeRF only if the parameter number is reduced after decomposition.

Network distillation is crucial in our workflow. One may further reduce the model size by quantizing Q 1 i and Q 2 i , and retrieve Ŵ r i from quantized Q1 i and Q2 i when rendering views. However, the difference between Ŵ r i and W r i caused by quantization error makes the workflow unstable and leads to rendering artifacts. The distillation operation allows us to quantize Q 1 i and Q 2 i without reconstructing Ŵ r i during rendering, keeping the workflow stable and avoiding rendering artifacts. This step of distillation reduces the number of parameters via the TT decomposition and the initialization of DLR-NeRF. It will also facilitate the quantization in next stage.

Rate-constrained Quantization

As explained in Sec. 2.4, DLR-NeRF can be further compacted after being quantized. We found that only 0.5% of the parameters in {Q 1 i ,Q 2 i } are outside the interval [-1, 1], and these parameters are essential for good rendering quality and should not be coarsely quantized, we thus adopt different quantization strategies for values inside and outside [-1, 1].

The values outside [-1, 1] are quantized using a uniform 16-bit quantizer to keep enough precision. While the values within [-1, 1] that occupy most of the proportion are quantized using a non-uniform optimized quantizer. The codebook C of such a quantizer is obtained by applying k-means to cluster parameters into N centroids. Different from the work in [START_REF] Fan | Training with quantization noise for extreme model compression[END_REF], where the authors learn layer-wise codebooks, we instead optimize one single codebook for all layers, and adopt a quantization-aware training strategy to reduce errors caused by quantization. More precisely, after quantizing {Q 1 i , Q 2 i } of a certain layer i = n, we continue to train the network and update {Q 1 i , Q 2 i }∀i > n and {b i }∀i. Once all layers are quantized, we update the learned codebook and start a new iteration. Though more iterations can produce better global codebook, we found in the experiments that one iteration is good enough. During the training process, the biases b i can compensate in some degree the quantization error of {Q 1 i , Q 2 i }, we thus prefer to precisely quantize b i with a 16-bit uniform quantizer.

The quantized matrices {Q 1 i , Q 2 i }, are then encoded using Huffman coding to further reduce the bitrate. Considering that separately coding the two sets of {Q 1 i , Q 2 i } (inside and outside [-1, 1]) will require one extra bit as indication flag, we directly code the union of the two sets at the cost of a small bitrate increase to obtain Q1 i and Q2 i . The quantized version of DLR-NeRF, referred to as QDLR-NeRF, is a compact model that represents the input light field. One can transmit the quantized matrices Q1 i , Q2

i , bi (16 bits), the camera parameters f (32 bits), ∆ (32 bits) and the codebook (32 bits) to reconstruct the light field. The quantization finally reduces the number of bits needed for each parameter.

EXPERIMENTS

Training details

We adopt a 8-layer MLP with 256 channels as NeRF architecture. The sampling rate of light ray is set to 128, camera pose parameters f and ∆ are initialized to 0.01W (W is the image width) and 10, the penalty factor ρ is set to 10. During the training and rank optimization steps of NeRF, learning rate is set to 5 × 10 -4 with an exponential decay rate 0.1, each step is trained for 3 × 10 5 iterations. After initialization of DLR-NeRF, we further finetune it for 2 × 10 5 iterations for better performance. In the final quantization step, the codebook C is obtained after 20 iterations of clustering in the kmeans algorithm. After quantizing each layer, we train the subsequent layers for 3 × 10 4 iterations. The entire workflow is implemented using the Pytorch learning framework and has been trained using one GPU GeForce RTX 2080Ti with 11GB memory.

Settings

Both the target rank r and the centroids number N can control the model size. To test the performance of our method at different bitrates, we first optimized the model using different rank values r = {150, 90, 70, 40} with a fixed N = 256. However, we found that the reconstructed light fields are prone to artifacts when using a rank r inferior to 40. We therefore decrease the centroid number N = {256, 64, 32} after setting r = 40 to further compress the model. The method has been tested using both real-world and synthetic light fields. We took four Illum-captured scenes Bikes, Danger, StonePillarsOutside(SPO), FountainVincent2(FV2) from the EPFL dataset [START_REF] Rerabek | New light field image dataset[END_REF] as real-world test data, where each light field has a resolution of 432 × 624 × 9 × 9. For synthetic data, we took four scenes boxes, sideboard, cotton, dino from the HCI dataset [START_REF] Honauer | A dataset and evaluation methodology for depth estimation on 4D light fields[END_REF], where each light field has a resolution of 512 × 512 × 9 × 9. We use estimated camera poses in our test in spite of the availability of ground truth camera pose for synthetic data.

Performance

We compare our method with two typical non-learning-based compression standards HEVC [START_REF]High Efficiency Coding and Media Delivery in Heterogeneous Environments -Part 2: High Efficiency Video Coding[END_REF] and JPEG Pleno [START_REF]Jpeg pleno[END_REF], and with three learning-based methods RLVC [START_REF] Yang | Learning for video compression with recurrent auto-encoder and recurrent probability model[END_REF], HLVC [START_REF] Yang | Learning for video compression with hierarchical quality and recurrent enhancement[END_REF] and OpenDVC [START_REF] Yang | OpenDVC: An open source implementation of the DVC video compression method[END_REF]. We assess the BD-PSNR gains using the Bjontegaard measure, and taking the results with HEVC as a reference. Table 1 shows that our method outperforms other compared methods with a large margin. Although Lytro-captured light fields contain artifacts such as vignetting and blurriness caused by hardware limitations, which prevents our NeRFbased method from retrieving a precise scene, our method can still yield good results.

Let us note that some compared methods may select certain views as anchors to compress the other views, leading to a high variation of quality across views (lower view coherence), as shown in Fig. 3. Therefore we show in Fig. 2 the error maps averaged across ALL views under similar bitrates. And by observing Fig. 2 and Fig. 3, we found that our method can produce better decompressed views with both higher inter-view coherence and lower errors. More detailed explanations and further results can be found in [START_REF] Shi | Distilled low rank neural radiance field with quantization for light field compression[END_REF].

Contributions of each step

To give insights on the contributions of each functional block, we show in Table 2 the contributions of each step in terms of PSNR and model size. By applying each operation, our method can yield a very compact neural scene representation which has a model size divided by 33, compared with the original NeRF, and with an acceptable quality loss.

CONCLUSION

In this paper, we have proposed a novel light field compression method based on compact neural scene representation.

Results show that our method can achieve very high compression ratio while keeping better view coherence and reconstruction quality when compared with other methods.

Fig. 1 .

 1 Fig. 1. Overall workflow of our proposed QDLR-NeRF method.

2 .

 2 (0.12bpp) 37.63dB (0.11bpp) 38.55dB (0.11bpp) 41.90dB (0.13bpp) 42.81dB (0.13bpp) Danger 31.50dB (0.10bpp) 31.86dB (0.14bpp) 31.39dB (0.11bpp) 32.79dB (0.10bpp) 33.04dB (0.11bpp) Fig. Averaged error maps of decompressed light fields using different methods, along with the PSNR and bitrate values.

Fig. 3 .

 3 Fig. 3. PSNR variation across views for the scene 'boxes'.

Table 1 .

 1 BD-PSNR gains with respect to HEVC baseline (covering bpp range 0-0.2). The best results are in bold.

	LFs	JPEG Pleno RLVC HLVC OpenDVC Ours
	boxes	0.65	-0.41	-0.79	-1.46	2.24
	sideboard	1.77	-0.52	-1.75	-2.12	3.88
	cotton	0.98	-2.95	-3.44	-3.18	0.34
	dino	2.56	-0.18	-1.07	-1.28	3.33
	Bikes	0.54	0.19	-0.22	-0.70	0.67
	Danger	1.17	0.25	-0.05	-0.78	1.58
	SPO	0.40	0.15	-0.24	-0.60	0.49
	FV2	0.36	0.25	-0.15	-0.57	0.49
	Average	1.05	-0.40	-0.96	-1.34	1.63

Table 2 .

 2 PSNR and model size in percentage after each step. PSNR is averaged on 8 tested light fields.

	PSNR	41.98dB	38.83dB	38.72dB	38.18dB
	Size	100%	50%	16.7%	3.3%

Metrics NeRF-org LR-NeRF DLR-NeRF QDLR-NeRF

* Most of work has been done when the first author was in INRIA Rennes, funded by the DeepCIM project in the context of the French ANR programme on Artificial Intelligence.


Douriya Z Khedaioui

Camille Tribout

Julie Bratasanu

Franck D ' Agosto

Christophe Boisson

Damien Montarnal

Deciphering Siloxane Bond Exchanges: from a Molecular Study to vitrimerization and recycling of Silicone Elastomers

Keywords: siloxane exchange, dynamic networks, vitrimers, rheological model

The activity of various additives promoting siloxane equilibration reactions is examined and quantified on model compounds. We found in particular that the "superbase" phosphazene derivative P4tBu can promote very fast exchanges (a few seconds at 90 °C) even at low concentration (<0.1 wt%). We demonstrate that permanent silicone networks can be transformed into reprocessable and recyclable dynamic networks by mere introduction of such additives. Annealing at high temperature degrades the additives and deactivates the dynamic features of the silicone networks, reverting them back into permanent networks. A simple rheological experiment and the corresponding model allow to extract the critical kinetic parameters to predict and control such deactivations.

INTRODUCTION

Silicone elastomers feature high flexibility, high thermal stability, and both hydrophobic and oleophobic surface properties. They are used in a vast range of applications such as heat-resistant sealants or encapsulants, moulds, cosmetics, drug delivery or bio-implants [START_REF] Shit | A Review on Silicone Rubber[END_REF][START_REF] Mazurek | How to Tailor Flexible Silicone Elastomers with Mechanical Integrity: A Tutorial Review[END_REF] While recent academic and industrial developments have led to silicone thermoplastic elastomers (SiTPEs), such as segmented block copolymers [START_REF] Yilgör | Siloxane-Urea Segmented Copolymers -1. Synthesis and Characterization of Model Polymers from MDI and α, ω-Bis(Aminopropyl)Polydimethylsiloxane[END_REF][START_REF] Falco | Linear and Nonlinear Viscoelastic Properties of Segmented Silicone-Urea Copolymers: Influence of the Hard Segment Structure[END_REF][START_REF] Zhu | Silicone Urethane Urea Copolymer and Preparation and Use Thereof[END_REF] or thermoplastic vulcanizates (SiT-PVs), [START_REF] Mani | Morphology Development in Novel Composition of Thermoplastic Vulcanizates Based on PA12/PDMS Reactive Blends[END_REF][START_REF] Gornowicz | THERMOPLASTIC SILICONE VULCANIZATES PREPARED BY CONDENSATION CURE[END_REF] the vast majority of silicone elastomers are still obtained through conventional cross-linking methods such as hydrosilylation, hydrolysis-condensation or radical reactions (essentially peroxides-or UV-induced) [START_REF] Stark | [END_REF] . Unlike conventional cross-linked elastomers, recycling routes for silicone elastomers is practically and economically feasible through chemical depolymerization with acids, bases or fluorides [START_REF] Rupasinghe | Full Circle Recycling of Polysiloxanes via Room-Temperature Fluoride-Catalyzed Depolymerization to Repolymerizable Cyclics[END_REF] , which is particularly favored in the case of poly(dimethyl siloxane) by the equilibration between (macro)cyclic and linear forms and the high volatility of the smaller cyclic species (D4 to D6) that can be recovered by distillation and repolymerized. [START_REF] Okamoto | Polysiloxane Depolymerization with Dimethyl Carbonate Using Alkali Metal Halide Catalysts[END_REF][START_REF] Chang | Nucleophilic Cleavage of Crosslinked Polysiloxanes to Cyclic Siloxane Monomers: Mild Catalysis by a Designed Polar Solvent System[END_REF][START_REF] Julian | THE USE OF ORGANIC POLYMER AND DEPOLYMERIZATION CATALYST[END_REF][START_REF] Fouquet | Characterization of Ethanolysis Products of Poly(Dimethylsiloxane) Species by Electrospray Ionization Tandem Mass Spectrometry[END_REF] Alternatively, covalent adaptable network (CAN) strategies [START_REF] Chakma | Dynamic Covalent Bonds in Polymeric Materials[END_REF][START_REF] Elling | Reprocessable Cross-Linked Polymer Networks: Are Associative Exchange Mechanisms Desirable?[END_REF][START_REF] Zheng | Dynamic Covalent Polymer Networks: A Molecular Platform for Designing Functions beyond Chemical Recycling and Self-Healing[END_REF][START_REF] Scheutz | Adaptable Crosslinks in Polymeric Materials: Resolving the Intersection of Thermoplastics and Thermosets[END_REF][START_REF] Cuminet | Neighboring Group Participation and Internal Catalysis Effects on Exchangeable Covalent Bonds: Application to the Thriving Field of Vitrimer Chemistry[END_REF][START_REF] Van Zee | Permanently Crosslinked Polymers with Dynamic Network Topology[END_REF][START_REF] Guerre | Directing Chemical Reactivity to Control Material Properties[END_REF][START_REF] Winne | Dynamic Covalent Chemistry in Polymer Networks: A Mechanistic Perspective[END_REF] have been recently applied to silicone materials with the prospect of directly healing cracks or reprocessing the materials by grinding and molding steps. Three approaches can be distinguished: i) the use of dissociative crosslinks such as Diels-Alder adducts [START_REF] Gou | Thermally Self-Healing Silicone-Based Networks with Potential Application in Recycling Adhesives[END_REF] or borate esters (the original "silly putty" formulation [START_REF] Mcgregor | Treating Dimethyl Silicone Polymer with Boric Oxide[END_REF] ) that provide viscoelastic or TPE materials, ii) the use of associative exchanges: UV-induced disulfide exchanges, [START_REF] Xiang | A Facile Method for Imparting Sunlight Driven Catalyst-Free Self-Healability and Recyclability to Commercial Silicone Elastomer[END_REF] heat-activated transesterifications, [START_REF] Zhang | Recyclable Polydimethylsiloxane Network Crosslinked by Dynamic Transesterification Reaction[END_REF][START_REF] Tran | Formation of Cross-Linked Films from Immiscible Precursors through Sintering of Vitrimer Nanoparticles[END_REF] imine exchange, [START_REF] Feng | Multifunctional Vitrimer-Like Polydimethylsiloxane (PDMS): Recyclable, Self-Healable, and Water-Driven Malleable Covalent Networks Based on Dynamic Imine Bond[END_REF] vinylogous urethanes [START_REF] Stukenbroeker | Polydimethylsiloxane Quenchable Vitrimers[END_REF] or derivatives of Meldrum's acids [START_REF] Ishibashi | Vitrimeric Silicone Elastomers Enabled by Dynamic Meldrum's Acid-Derived Cross-Links[END_REF] and iii) the direct and controlled catalysis of siloxane exchanges.

The latter has been known for over 70 years, with the pioneering work on siloxane bond equilibration of Osthoff in the 1950s. [START_REF] Hurd | The Mechanism of the Base-Catalyzed Rearrangement of Organopolysiloxanes[END_REF][START_REF] Osthoff | Chemical Stress-Relaxation of Polydimethylsiloxane Elastomers[END_REF][START_REF] Kantor | The Mechanism of the Acid-and Base-Catalyzed Equilibration of Siloxanes[END_REF] The underlying admitted mechanism for bases or fluorides is the cleavage of the Si-O-Si bond by nucleophilic displacement, which forms a silanolate anion. This silanolate moiety is further involved and regenerated through nucleophilic attacks on other Si-O-Si bonds. This equilibration reaction is also at the core of the equilibrium between rings and linear chains in PDMS polymers that occurs to a certain extent in anionic ring-opening polymerization (ROP) of silicones, and that started to be studied by Wright and Semlyen in the 70's. [START_REF] Wright | Cyclization Equilibrium Constants and the Conformations of Polydimethylsiloxane Chains[END_REF][START_REF] Wright | Equilibrium Ring Concentrations and the Statistical Conformations of Polymer Chains: Part 3. Substituent Effects in Polysiloxane Systems[END_REF] Exploitation of such equilibration reactions in dynamic crosslinked silicone networks was however only brought up by McCarthy and Zheng in 2012. [START_REF] Zheng | A Surprise from 1954: Siloxane Equilibration Is a Simple, Robust, and Obvious Polymer Self-Healing Mechanism[END_REF] They conducted an anionic ROP of octamethyl tetrasiloxane (D4) in presence of a bis-D4 crosslinker initiated by an oligosiloxane functionalized with tetramethylammonium silanolate endgroups (TMA2-PDMS). In absence of quenching, the living silanolate group enables dynamic chain exchange and welding of fractured parts at 90°C for 24h. Although precise measurement of exchange dynamics or network relaxation times were not carried out, these materials were further studied by Seiffert and coworkers, using this time as catalyst tetramethylammonium hydroxide (TMAOH) at higher concentrations than McCarthy. [START_REF] Schmolke | Dynamically Cross-Linked Polydimethylsiloxane Networks with Ambienterature Self-Healing[END_REF] Using home-made piercing experiments that monitor the flow of dynamic networks, they found very fast relaxations, typically in the 1-2 s range, with only weak dependencies to catalyst concentrations or to temperature in the 5-60°C range. Exciting benefits of malleable silicone networks have also been explored by Xie and coworkers, who combined plastic deformation of micropatterned silicones with pyrolysis to obtain well defined silicon oxycarbide ceramics. [START_REF] Zheng | Mechano-Plastic Pyrolysis of Dynamic Covalent Polymer Network toward Hierarchical 3D Ceramics[END_REF] The malleable silicone network was obtained this time directly by adding sodium octanoate (NaOct) from a commercial two-component formulation undergoing platinum-catalyzed hydrosilylation (Sylgard 184 from Dow Corning). Much higher temperatures were however required for malleability, typically between 180 and 200°C. More recently, the group of Terentjev has also investigated the use of siloxane crosslinks for the reprocessing of liquid crystalline elastomers obtained after thiol-acrylate Michael addition between mesogenic diacrylates and isotropic dithiols. [START_REF] Saed | Catalytic Control of Plastic Flow in Siloxane-Based Liquid Crystalline Elastomer Networks[END_REF] Several additives were found to induce relaxations in these networks, such as TMA2-PDMS, triphenylphosphine (TPP), triazobicyclodecene (TBD) and NaOct. TBD, also known to ring-open and polymerize effectively cyclic siloxane, [START_REF] Fuchise | Organocatalytic Controlled/Living Ring-Opening Polymerization of Cyclotrisiloxanes Initiated by Water with Strong Organic Base Catalysts[END_REF] was found very effective. In this case also, very high temperatures from 190°C to 210°C were required, even in the case of tetramethylammonium-based catalysts that are yet known to degrade above 150 °C. [START_REF] Jehanno | Organocatalysed Depolymerisation of PET in a Fully Sustainable Cycle Using Thermally Stable Protic Ionic Salt[END_REF][START_REF] Van Herck | Covalent Adaptable Networks with Tunable Exchange Rates Based on Reversible Thiol-Yne Cross-Linking[END_REF] Last, Du Prez and coworkers recently investigated epoxy resins containing bisamine disiloxane hardeners and various catalysts (strong organic bases and silanolates). While studies of siloxane exchanges on disiloxane model molecules barely showed any effect of TBD (5 mol%) over 16 h at 120 °C, the addition of alcohols slightly accelerated the exchange. Mechanical relaxation of the epoxy networks containing 10 mol% TBD/Si and a large amount of hydroxyls issued from the epoxy-amine reaction showed however dramatically faster relaxations, down to about 5 s at 220°C which eventually enabled reprocessing and thermoforming of composites. [START_REF] Debsharma | Fast Dynamic Siloxane Exchange Mechanism for Reshapable Vitrimer Composites[END_REF] Given the historical and technological relevance of dynamic siloxane exchanges, the apparent discrepancy within all these studies of the effect of network composition and the nature or concentration of catalysts on the exchange dynamics of siloxanes prompted us for a more thorough comparison of various catalysts using model compounds. Recognizing the importance of the effective solubility of catalysts in highly apolar environments and the economical necessity to use catalytic amounts of additives, we ensured in particular to study purely siloxane environments, either as small model molecules or as target dynamic networks.

While we found unexpectedly that several additives mentioned above do not promote sizeable siloxane exchanges, we also discovered that a phosphazene derivative, P4tBu (please see full name in the Materials section in SI), is extremely effective. Incorporation of this additive in both model and commercial silicone networks proved a highly efficient way to transform them into malleable dynamic networks. Last, we also discovered that at temperatures higher than 130°C thermal degradation of P4tBu effectively deactivates the dynamic siloxane exchanges and the networks are reverted to non-dynamic networks. We empha- size that proper control of such deactivation and corresponding predictive mathematical modelling opens exciting perspectives for dynamic networks in general, and may solve the difficult compromise that must be made between fast dynamics required for (re)processing the networks at high temperatures and very slow dynamics required for effective creep-resistance at service temperatures. [START_REF] Jourdain | Rheological Properties of Covalent Adaptable Networks with 1,2,3-Triazolium Cross-Links: The Missing Link between Vitrimers and Dissociative Networks[END_REF]

RESULTS AND DISCUSSION

Molecular study of catalyzed siloxane exchanges. While the equilibration of siloxanes has been studied as early as the 1950s the kinetics of these reactions was never, to the best of our knowledge, studied explicitly with model compounds. We chose short disiloxane compounds bearing different substituents (hexamethyldisiloxane -HMDS and divinyltetramethyldisiloxane -DVTMS). The expected cross-exchange product (vinylpentamethyldisiloxane -VPMDS) is also commercially available and can be easily quantified by gas chromatography (GC, Figure S1 in SI). The established exchange mechanism involves the formation of a silanolate ion pair as the active catalytic center (See Figure 1d). In practice, the various molecules added to accelerate the siloxane exchanges may only be precursors (e.g. organic bases, hydroxides) to the silanolates species. For clarity throughout the manuscript we will refer to them as additives. HMDS and DVTMS were mixed in equimolar ratios in presence of various additives (Figure 1b) at low concentration (0.1 wt%) and in absence of any additional solvents. Samples withdrawn at different times from the reaction medium were analyzed by GC to quantify the evolution of the fractions of HDMS, DVTMS and VPDMS during the reaction. The resulting equilibration kinetics is displayed in Figure 1c. Surprisingly, no formation of VPMDS was observed after 24 hours of reaction when TPP, NaOct and TBD were used as additives. Widely used tetramethyl ammonium basic salts, TMAOH and TMA2-PDMS display characteristic equilibration times of ca. 2 h and 1 h, respectively, while CsOH is much slower (ca. 150 h). Such differences are significant given that TMA2-PDMS and TMAOH involve the same counter ions (tetramethylammonium) and should lead to the same silanolate active species after the first exchange with disiloxanes (see Figure 1d). We attribute thus the significant induction time with TMAOH to a poor solvatation of this salt in the apolar disiloxanes and correspondingly to a slow conversion in tetramethyl ammonium silanolate. The same reasoning might also explain the lower activity of CsOH.

Further investigation about the nature of the silanolate active center was obtained by directly using lithium, sodium, or potassium silanolates. In the three cases no VPDMS was observed, thus demonstrating that highly dissociated ion pairing is critical for the reactivity of silanolates. This prompted us to test bases with highly delocalized charge and excellent solubility in apolar media. In this regard, the P4tBu "superbase" is known to form phosphazenium hydroxide [HP4tBu] + [HO] -or alcoxides in presence of adventitious water or alcohols (See Scheme 1), [START_REF] Boileau | Activation in Anionic Polymerization: Why Phosphazene Bases Are Very Exciting Promoters[END_REF][START_REF] Köhler | Industrial Synthesis of Reactive Silicones: Reaction Mechanisms and Processes[END_REF][START_REF] Hupfield | Ring-Opening Polymerization of Siloxanes Using Phosphazene Base Catalysts[END_REF] and is also amongst the most efficient known initiators for ROP of D4 cyclic siloxanes. [START_REF] Gornowicz | THERMOPLASTIC SILICONE VULCANIZATES PREPARED BY CONDENSATION CURE[END_REF][START_REF] Van Herck | Covalent Adaptable Networks with Tunable Exchange Rates Based on Reversible Thiol-Yne Cross-Linking[END_REF][START_REF] Debsharma | Fast Dynamic Siloxane Exchange Mechanism for Reshapable Vitrimer Composites[END_REF][START_REF] Jourdain | Rheological Properties of Covalent Adaptable Networks with 1,2,3-Triazolium Cross-Links: The Missing Link between Vitrimers and Dissociative Networks[END_REF][START_REF] Boileau | Activation in Anionic Polymerization: Why Phosphazene Bases Are Very Exciting Promoters[END_REF][START_REF] Köhler | Industrial Synthesis of Reactive Silicones: Reaction Mechanisms and Processes[END_REF] P4tBu was indeed found extremely efficient for these siloxane equilibrations, with ca. 3 min of exchange time at 0.1 wt% and 90°C (Figure 1c). We assume that the traces of water enables the in situ formation of [HP4-t Bu] + [HO] -initiator.

As expected, increasing the amount of additives and heating accelerates the exchanges kinetics. The corresponding mastercurve of exchange kinetics at different temperatures for TMA2-PDMS (Figure S2) demonstrates the absence of latent effects in these reactions (e.g. sudden increase of reactivity above a temperature threshold) and allows to determine for 0.1 wt% additive an exchange time about 900 s at 90°C and an activation energy Ea, of 92 kJ mol -1 . The exchange kinetics also depend on the concentration, with a 10-fold acceleration from 0.05 wt% to 5 wt% of additive (Figure S3). Above 110°C, however, the reaction becomes slower, which might be related to the thermal decomposition of the tetramethylammonium silanolate in trimethylamine and methoxysilane, or its subsequent condensation products as reported in the literature. [START_REF] Zheng | A Surprise from 1954: Siloxane Equilibration Is a Simple, Robust, and Obvious Polymer Self-Healing Mechanism[END_REF] No clear evidence of such decomposition products were however identified on the gas chromatograms as trimethylamine is highly volatile. TMSOMe and DVSOMe might also condense readily to reform HMDS, DVTMS and VPMDS. [START_REF] Köhler | Industrial Synthesis of Reactive Silicones: Reaction Mechanisms and Processes[END_REF] The deactivation of TMA2-PDMS at 120°C was further verified by pre-equilibrating the additive with HMDS for 2h before adding 1 eq. of VDTMS. In this case, no cross-exchanged VPDMS could be detected after 2h.

In comparison, the P4tBu additive at 0.1 wt% shows much faster exchanges, ca. 75 s at 90°C (Figure S4) with comparable activation energy about 100 kJ mol -1 .

Silicone vitrimers.

Model polysiloxane networks were prepared to further study the effect of these siloxane exchanges on relaxations. Liquid silicone resins (LSR)-like formulations have been implemented using divinyl-terminated poly(dimethylsiloxane) (Mw = 25 000 g mol -1 ), polymethylhydrosiloxane crosslinker (Mw = 1900 g mol -1 , SiH/Vinyls=2) and a Pt-based NHC derivative (300 ppm/vinyls) [START_REF] Markó | Highly Active and Selective Platinum(0)-Carbene Complexes. Efficient, Catalytic Hydrosilylation of Functionalised Olefins[END_REF][START_REF] Khedaioui | Polyethylene Aerogels with Combined Physical and Chemical Crosslinking: Improved Mechanical Resilience and Shape-Memory Properties[END_REF] that allows a complete curing within 15h at 75°C (please see complete description in the SI). The resulting networks do not display any detectable extractible fraction ; they are permanent networks (see stress relaxations below) and relatively brittle (εbreak about 100%) given the relatively low molar mass of the divinyl PDMS precursors.

All additives accelerating siloxane exchanges are bases and thus incompatible with the crosslinking reaction by Pt-catalyzed hydrosilylation. The additives were thus post-added in the PDMS networks by solvent-assisted diffusion in toluene. To ensure homogeneous dispersion, 8 mm x ca. 1.5 mm disks of the PDMS networks were impregnated with

Scheme 1 Activation of P4tBu by the adventitious presence of water in HMDS and DVTMS

the exact amount of additive solution in toluene required for swelling to the equilibrium (e.g. a 1:1 weight ratio of solution to PDMS sample), and left to dry at 40°C overnight.

As P4tBu shows promising results in terms of solubility and equilibration times in the above study involving molecular compounds, we focused our study on its use to induce siloxane bond exchanges in PDMS networks. A control experiment on the pristine PDMS rubber (Figure 2) demonstrates the absence of any relaxation at 90 °C. Adding P4tBu to the networks in amounts from 16 to 100 ppm to Si atoms (e.g. from 0.01 wt% to 0.09 wt%) induces stress relaxation (Figure 2 -The corresponding frequency sweeps are shown in Figure S5 in SI), with corresponding characteristic times from 1700 to 3 s. Notably, these dynamic exchanges are complemented by a decrease in the plateau moduli (from about 350 kPa for the initial network to about 100 kPa for the highest P4tBu loading) that substantiates the expected mechanism of this additive. First, Si-O-Si chains are cleaved by the phosphazenium hydroxides to yield phosphazenium silanolates acting as persistent active centers. Further exchanges within the network allow strand exchanges and stress relaxation. It should be noted that such reorganizations also trigger the formation of volatile cyclic siloxanes by back-biting reactions. Consequently, the crosslinked samples gradually depolymerize and evaporation takes place over time at 90°C. Figure S6 shows the weight loss of ca. 70 mg disks after annealing in open air at 90°C for 6h: significant losses > 25wt% are observed for P4tBu concentrations above 100 ppm /Si. This limited stability illustrates a major drawback of fast dynamics in silicone networks and prompted us to characterize the time-dependent evolution of exchange dynamics.

Deactivation of exchange reactions. We carried out repeated stress relaxations (1% strain) to measure the evolution of characteristic relaxation times (Figure 3). The network containing 100 ppm P4tBu displays an initial increase of relaxation times that levels off to about 30 s after 1h. We hypothesized that this change is due to slow and partial deactivation of the silanolate groups due to impurities in the network, probably water. [START_REF] Cazacu | Silicone Rubbers. Ix. Contributions to Polydimethylsiloxane-α,ω-Diols Synthesis by Heterogeneous Catalysis[END_REF] We verified this by adding controlled amounts of benzyl alcohol (BzOH, 0.5 and 1 eq. to P4 t Bu); the corresponding deactivations proceed to much larger extents, effectively yielding kinetics comparable to networks initially loaded with ca. 33 ppm of P4tBu/Si. Extended uses of such exchange deactivations can be found at elevated temperatures, when thermal degradation of the silanolate salts occurs. A rapid screening of the timeand temperature-evolution of deactivation was found by monitoring the creep behavior while applying a heating ramp to the material (Figure 4). While the viscosity is expected to continuously decrease upon heating, temporary idleness around 80-100°C corroborates a partial deactivation. A subsequent dramatic increase of viscosity above 140°C indicates a second and total deactivation, which would effectively transform the highly dynamic networks into quasi-permanent networks. Deactivation dynamics and modelling. Further analysis of thermal deactivation in the 130-170°C range was carried out on networks containing 33 ppm of P4tBu/Si using repeated stress relaxation cycles (Figure 5). While relaxations at 130°C are slow (ca. 4000 s) but relatively reproducible over 5h, relaxations at 150 and 170°C are initially considerably faster, but slow down significantly over time. Delayed relaxations at 170°C are clearly not representative of classical mono-or stretched exponential decays encountered in dynamic networks, [START_REF] Jourdain | Rheological Properties of Covalent Adaptable Networks with 1,2,3-Triazolium Cross-Links: The Missing Link between Vitrimers and Dissociative Networks[END_REF] but are likely indicative that the deactivation occurs over similar timescales as the stress relaxation.

Figure 6. Evolution of relaxation times at different temperatures in model silicone networks containing 33 ppm of additives. Dots correspond to individual fits while continuous lines correspond to first-order deactivation models (eq 1).

Thus, we derived a low-parameter model for stress relaxations that takes into consideration a first-order deactivation, i.e. that effective relaxation times are increasing exponentially with time:

𝑮(𝒕) = 𝑮 𝟎 . 𝒆 -( 𝝉 𝒅 𝝉 𝟎 [𝒆 - 𝜟𝒕 𝝉 𝒅 -𝒆 - 𝒕+𝜟𝒕 𝝉 𝒅 ]) 𝜷 (eq. 1)
G0 is the network modulus, τ0 and τd are the initial relaxation time and the characteristic deactivation time of the relaxation, respectively, Δt is the time-delay between the initial heating of the sample (start of the deactivation) and each successive stress relaxation experiment, β is the stretch parameter characteristic of the breadth of distribution of relaxation times. Details of derivation of eq 1 and simulations for different sets of values are represented in SI. For each temperature, fitting of all successive stress relaxation data is carried using a single set of τ0, τd and β values and fixed Δt parameters using the "Global fit" feature of Origin software (Figure 5). The G0 parameters are individualized during the fitting to account for small variations of initial moduli.

Given the low amount of fitting parameters (τ0, τd, β) for such multiple datasets, the fit quality is remarquable and substantiates the validity of the model. As expected, the results indicate that initially, the relaxation times are significantly faster at 170°C than at 130°C. Yet, degradation occurs more significantly at higher temperatures and the situation becomes reversed after about 2h of heating (Figure 6).

In line with initial studies on model compounds, the deactivation was also studied by rheology for PDMS networks containing 33 ppm/Si of TMA2PDMS and CsOH additives.

(See Figure 6 and corresponding sets of stress relaxation in Figure S8) It was found that the TMA additive was initially moderately efficient, but degraded very quickly at 150°C (deactivation time τd about 20 min). In accordance with the above studies on model compounds, CsOH shows initially a much lower activity but also deactivates much more slowly (τd about 1.5 h at 160°C).

Reprocessability of PDMS elastomers. The maximal stress relaxation expected during a single step before the deactivation completely transforms the dynamic network into a permanent network can be determined from eq 1 using Δt = 0 s and is strongly driven by the ratio τd/τ0. Figure 7 gives in this way an estimation of the processability window after 30 min of deformation. The values obtained for the different additives at 33 ppm are reported, and indicate clearly that with such concentrations, proper reprocessing within 30 min could only be expected with P4tBu at 170°C. P4tBu recovers most of the elongation at break at the expense of a 70% decrease in modulus (Figure 8). In contrast, the sample containing 30 ppm P4tBu suffers from a significant decrease of elongation at break and a lower decrease in modulus (-55%).

In view of the very low amount of additives used, such important changes are strongly driven by the topology of the model PDMS networks chosen that feature mainly long linear chains (99 wt%) anchored at extremities by polyfunctional crosslinkers. It would be interesting in future developments of siloxane dynamic networks to either promote recoupling of cleaved chains in such topologies, or to focus on different topologies featuring permanent polymer backbones equipped with dynamic siloxane crosslinks.

CONCLUSIONS

We showed using model compounds that the dynamics of siloxane exchanges are extremely dependent on the environment of the silanolate living chain-ends, and that highly delocalized and hydrophobic cations such as the P4tBu phosphazene derivative are extremely active even in low amounts (below 0.1 wt%). When implemented in crosslinked PDMS networks, such siloxane exchanges promote dynamic strand exchanges and stress relaxation. We also show that the silanolate active centers can be lost by two distinct phenomena: i) a partial deactivation around 90°C that we attribute to reactions with impurities and ii) a complete thermal deactivation that occurs above 120°C depending on the additive.

The understanding at the molecular level of such thermal deactivations remains unclear and challenging given the low concentrations of additives involved. We show however that simple rheological experiments (e.g. repeated stress relaxations) and a low-parameter model enables to quantify very easily both the initial dynamics of exchange and the corresponding deactivation time.

In the scope of CANs and more specifically vitrimers, such deactivating relaxations could constitute an interesting alternative against the problematic tradeoff between creep resistance at low temperatures and processability at high temperatures. Activation of the relaxations could for instance be triggered only when needed (e.g. during the recycling or reshaping of the materials by locally adding the additives) and further deactivated by annealing at high temperatures to yield permanent networks.

ASSOCIATED CONTENT

Supporting Information. Experimental details on materials, instrumentation and synthetic procedures of model networks, Additional experimental results and derivation of deactivating stress relaxation models. This material is available free of charge via the Internet at http://pubs.acs.org.

Figure 1 .

 1 Figure 1. a) Molecular study of the siloxane bond exchange. b) Library of additives used to induce the dynamic exchange. c) Kinetic study at 90 °C in presence of 100 ppm of additive. d) Mechanism involved for the siloxane bond exchange.

Figure 2 .

 2 Figure 2. Stress relaxation at 90°C (1% strain) of model silicone networks containing 0-100 ppm P4tBu/Si.

Figure 3 .

 3 Figure 3. Evolution of relaxation times at 90°C in model silicone networks containing various amounts of P4tBu/Si and BzOH.

Figure 4 .

 4 Figure 4. Evolution of apparent creep viscosity in model network containing 100 ppm P4tBu during a 1°C min -1 heating ramp.

Figure 5 .

 5 Figure 5. Repeated stress relaxations (normalized) of model networks containing 33 ppm P4tBu/Si at 130, 150 and 170°C. The total heating time for each series is represented by the color scale. Black lines are fits corresponding to a) individual stretched exponential decays, and b,c) general model (eq. 1) accounting for repeated deactivating relaxations.

Figure 7 .

 7 Figure 7. Diagram of processability window, displaying the expected amount of relaxed stress after deforming a sample for 30 min, depending on the corresponding initial relaxation time τ0 and the deactivating time τd. The calculation was made with β = 0.66. Experimental values obtained for PDMS samples loaded with 33 ppm of additives are reported.

Figure 8 .

 8 Figure 8. Tensile testing of PDMS elastomers after reprocessing for 30 min at 170°C using 30 & 50 ppm P4tBu /Si. Young moduli are reported next to each curve. The effectiveness of the reprocessing was evaluated by impregnating shredded PDMS elastomer with 30 and 50 ppm P4tBu/Si, and mold-pressing the fragments under 40 MPa, for 30 min at 170°C. In absence of additive, only non-cohesive powder is obtained. The sample containing 50 ppm

ACKNOWLEDGMENT

We are indebted to Vincent Monteil, Jean Raynaud and Nam Vu for stimulating discussions.

Funding Sources

DM gratefully acknowledges funding from ANR (grant AEROLEFIN, ANR-17-CE07-0006) for financial support.

AUTHOR INFORMATION

Corresponding Author * damien.montarnal@univ-lyon1.fr

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.