Cássio F Dantas

Jérémy E Cohen

Rémi Gribonval

HYPERSPECTRAL IMAGE DENOISING USING DICTIONARY LEARNING

Keywords: Hyperspectral image, denoising, sparsity, low-rank, dictionary learning

Hyperspectral images are corrupted by noise during their acquisition. In this work, we propose to efficiently denoise hyperspectral images under two assumptions: (i) noiseless hyperspectral images in matrix form are low-rank, and (ii) image patches are sparse in a proper representation domain defined through a dictionary. These two assumptions have already led to state-of-the-art denoising methods using fixed Wavelet transforms. We propose to rather learn the dictionary from hyperspectral images, a task commonly known as dictionary learning. We show that the dictionary learning approach is more efficient to denoise hyperspectral images than state-ofthe-art methods with fixed dictionaries, at the cost of a larger computation time.

INTRODUCTION

Hyperspectral imaging has become a major image modality over the last years, largely thanks to the development of spectral sensors. Hyperspectral images collect reflectance spectra with significant spectral resolution (~200 wavelengths) for a large number of pixels (~1000×1000 pixels). These images contain a lot of information regarding the composition of the scene, and can therefore be used in remote sensing for monitoring forest or coastal regions, or in chemometrics to study the composition of chemical compounds [1].

However, Hyperspectral Images (HSI) are very often corrupted by various types of noise. In particular, for remotely acquired HSI, at least two kinds of noise are of importance: noise due to the spectral sensor sensitivity, and noise due to the swiping pattern of the sensors which yields stripes. Moreover, in the presence of clouds or other atmospheric perturbation, missing data may be present as well [2]. In this work, we will suppose that HSI are only corrupted with anisotropic Gaussian noise in order to simplify the analysis of our denoising method, but other types of noise as well as missing data can be tackled with similar tools.

The authors are with PANAMA research group. This work was supported by the Becose ANR project (ANR-15-CE23-0021).

Removing noise in HSI is an important pre-processing step for any learning task such as segmentation, detection or spectral unmixing, and has therefore been studied extensively in the literature. State-of-the-art techniques exploit two key properties [2]: HSI typically are low rank matrices once vectorized in the spatial dimensions; and HSI (or small patches of it) are sparse in some well chosen bases such as wavelets. In particular, the state-of-the-art method HyRes [START_REF] Rasti | Automatic hyperspectral image restoration using sparse and low-rank modeling[END_REF] makes use solely of these two assumptions to very efficiently denoise remotely acquired HSI with minimal computation time.

In this work, we propose to push further the boundaries of HSI denoising performance by switching from a fixed dictionary setting (wavelets) to a learned dictionary setting. Dictionary Learning (DL) is well established in HSI applications such as unmixing [START_REF] Castrodad | Learning discriminative sparse representations for modeling, source separation, and mapping of hyperspectral imagery[END_REF] and classification [START_REF] Chen | Hyperspectral image classification using dictionary-based sparse representation[END_REF]. It has also been previously applied to HSI denoising in several occasions (see [START_REF] Xing | Dictionary learning for noisy and incomplete hyperspectral images[END_REF] and references therein) but rarely coupled with a low-rank decomposition. Although the low-rankness of the HSI is enforced via nuclear norm in [START_REF] Zhao | Hyperspectral image denoising via sparse representation and low-rank constraint[END_REF], the dimensionality reduction promoted by an explicit low-rank factorization is not exploited at all as the full-size HSI is manipulated. Moreover, in [START_REF] Zhao | Hyperspectral image denoising via sparse representation and low-rank constraint[END_REF], like in most DL methods for traditional image denoising [START_REF] Elad | Image denoising via sparse and redundant representations over learned dictionaries[END_REF], the noise variance is supposed known, while here we use an efficient heuristic to estimate it.

In what follows, we first formalize the denoising task and the low-rank and image sparsity assumptions in Section 2. In Section 3 we present our generic DL-based algorithm for HSI, and instantiate it with two DL algorithms, K-SVD [START_REF] Aharon | K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation[END_REF] and SuKro [START_REF] Dantas | Learning dictionaries as a sum of kronecker products[END_REF]. Finally, in Section 4, we show on artificially noised HSIs that the proposed approach outperforms stateof-the-art methods for HSI denoising at the cost of increased computation time that is reasonable for offline applications.

SPARSE AND LOW-RANK MODELING

A hyperspetral image is composed of two spatial dimensions (image space) and one spectral dimension, and thus naturally represented as a 3 dimensional cube of data, say H ∈ R n1×n2×p , where n 1 and n 2 correspond to the spatial dimensions and p is the number of spectral bands.

Nevertheless, a hyperspectral image is also often represented in its matrix (2 dimensional) form by vectorizing its spatial dimensions at each spectral band. In an additive noise model, this leads to:

H = X + N (1)
where H ∈ R n×p (with n = n 1 n 2 ) containing as its j-th column the vectorized observed image at band j. X and N are respectively the noise-free unknown signal, and the noise matrix, both (n × p) matrices.

Low-rank assumption

An HSI X is commonly modeled as a low-rank matrix in the literature [START_REF] Rasti | Automatic hyperspectral image restoration using sparse and low-rank modeling[END_REF][START_REF] Golbabaee | Hyperspectral image compressed sensing via low-rank and joint-sparse matrix recovery[END_REF]. The classical linear mixture model [START_REF] Manolakis | Hyperspectral subpixel target detection using the linear mixing model[END_REF] is itself a low-rank model X=AS T , in which the image is decomposed into a few subregions called source images (as the columns of A) each containing a certain material with distinct spectral signature (in the columns of S). The HSI rank r in this case is the number of materials present in the image. The singular value decomposition (SVD) is a way of obtaining an analogous low-rank decomposition of X, even though it does not, in principle, promote spectral unmixing. It leads to the noisy low-rank model

H = UΣV T + N (2)
where the columns of U, V ∈ R n×r are respectively a set of vectorized eigen-images and the associated spectral components, and Σ ∈ R r×r is a diagonal singular value matrix. Figure 1 illustrates the described models and notations.

Sparsity assumption

The eigen-images U k = unvec(u k ) (with u k the k-th column of U) can, then, be sparsely represented in a well-suited base -or, more generally, representation domain. In the noisy case, sparse reconstruction thus works as a form of denoising. Indeed, sparse modeling for images has been used for decades now [START_REF] Elad | Image denoising via sparse and redundant representations over learned dictionaries[END_REF][START_REF] Bruckstein | From sparse solutions of systems of equations to sparse modeling of signals and images[END_REF], with very good performance in denoising tasks. State-of-the art techniques in Hyperspectral image restauration [START_REF] Rasti | Automatic hyperspectral image restoration using sparse and low-rank modeling[END_REF] use a 2-D (Kronecker) orthogonal wavelet basis B in which the vectorized eigen-images u k are represented as a sparse set of wavelet coefficients w k (k-th column of sparse coefficient matrix W): U = BW. The full model becomes:

H = BWΣV T + N (3)

A typical processing pipeline

In view of model ( 3), a usual denoising approach given the corrupted signal H -which is no longer low-rank due to the noise-consists in estimating Û, Σ and V via SVD truncation and futher denoising Û via 2D-Wavelet shrinkage. The spectral component of the low-rank model V can also be denoised (for instance, [START_REF] Chen | Denoising of hyperspectral imagery using principal component analysis and wavelet shrinkage[END_REF] uses a 1D-Wavelet shrinkage).

PROPOSED APPROACH

A first part of our contribution consists in replacing the classic orthogonal Wavelet basis in the described pipeline by a dictionary both 1) overcomplete and 2) learned from data.

Overcomplete learned dictionary

Dictionary learning is a consolidated research topic [START_REF] Elad | Image denoising via sparse and redundant representations over learned dictionaries[END_REF][START_REF] Aharon | K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation[END_REF][START_REF] Dantas | Learning dictionaries as a sum of kronecker products[END_REF].

Both the overcompleteness and the fact of learning a dictionary from a targeted type of data has been shown to improve the performance of natural image denoising tasks [START_REF] Elad | Image denoising via sparse and redundant representations over learned dictionaries[END_REF] when compared to the classical orthogonal Wavelet-based approach. Any existing dictionary learning approach could be used for our purpose here. We propose to use two methods:

1. K-SVD [START_REF] Aharon | K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation[END_REF] as a standard (baseline) algorithm for learning unstructured dictionaries; 2. SuKro [START_REF] Dantas | Learning dictionaries as a sum of kronecker products[END_REF] as an algorithm for learning Kroneckerstructured dictionaries. This type of structure arises naturally when dealing with vectorized 2D-signals (the eigen-images in the columns of Û). The usual Wavelet approaches (e.g. in the state-of-the-art HyRes [START_REF] Rasti | Automatic hyperspectral image restoration using sparse and low-rank modeling[END_REF]) are themselves Kronecker-structured.

An important drawback of the overcompleteness (and consequently, non-orthogonality) of the dictionary can be pointed out: it significantly complicates the estimation of the sparse coefficients when compared to the orthogonal case (for instance, in HyRes [START_REF] Rasti | Automatic hyperspectral image restoration using sparse and low-rank modeling[END_REF], this step comes down to a simple thresholding operation). Here, we will need to resort to the so-called sparse coding algorithms [START_REF] Bruckstein | From sparse solutions of systems of equations to sparse modeling of signals and images[END_REF] which are a sub-optimal way of solving this -now ill-posed-problem.

Consequently, computational complexity (and running time) is significantly bigger for the proposed method when compared to the literature. However, we will see in Section 4 that a performance gain is obtained. Therefore, except in very constrained cases, the proposed technique is worthy.

A patch-based denoising approach

In order to avoid having a prohibitively large dictionary D ∈ R n×m (with m > n), which would be very hard to train, we rely on a patch-based approach as generally used in the DL literature for denoising tasks [START_REF] Elad | Image denoising via sparse and redundant representations over learned dictionaries[END_REF]. The dictionary is learned over

{ D, Γ} = argmin D,Γ DΓ -Y 2 F + g(Γ) (4)
where g(•) is a sparsity-inducing penalty function (for instance an 0 or 1 norm regularization on the columns of Γ).

Figure 2 summarizes the data structure and adopted notation.

The proposed denoising approach is described in Algorithm 1. Note that a new dictionary is learned for each eigenimage U k . To fully characterize the algorithm, some elements are still to be defined: the rank and noise estimation, sparse coding, dictionary update and denoising functions (respectively in lines 4, 5, 13, 14 and 16) are further detailed in sections 3.

Γ i = SparseCoding(Y, D i-1 , σ k ) 14: D i = DictionaryUpdate(Y, Γ i ) 15:
end for

Dictionary Learning Algorithms

The customary DL framework to tackle problem (4) consists in iteratively alternating two optimization steps:

1. Sparse coding: updating Γ with fixed D.

Dictionary update: optimizing for D with fixed Γ.

Sparse coding There is a extensive literature on how to address this problem [START_REF] Bruckstein | From sparse solutions of systems of equations to sparse modeling of signals and images[END_REF]. Although in theory any of the existing algorithms would serve our purposes, we use the traditional Othogonal Matching Pursuit (OMP) algorithm [START_REF] Pati | Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition[END_REF] which allows us to directly control the reconstruction precision of the input data. As a stopping criterion, we will use a threshold τ on the reconstruction error, i.e. the energy of the residual, proportional to the noise standard deviation σ (see Section 3.4 for more details on the noise level estimation) τ = aσ √ d, with a=1.16 an empirically-set gain following [START_REF] Elad | Image denoising via sparse and redundant representations over learned dictionaries[END_REF].

Dictionary update This is the step which distinguishes one DL algorithm from another. Two dictionary learning approaches are used:

1. K-SVD [START_REF] Aharon | K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation[END_REF]: the columns of D are updated one at a time via a rank-one SVD approximation of the error matrix with respect to the current atom.

2. SuKro [START_REF] Dantas | Learning dictionaries as a sum of kronecker products[END_REF]: it learns structured dictionaries which can be written as a sum of a few Kronecker products (denoted ) of smaller matrices

D j,q ∈ R √ d× √ m . D = nkron q=1 D 1,q D 2,q . ( 5
)
For the SuKro dictionary update step we use an Alternating Least-Squares (ALS) procedure where the blocks D j,q are optimized alternatively with a closedform optimal update in each step. This algorithm, which is more efficient than the one originally proposed in [START_REF] Dantas | Learning dictionaries as a sum of kronecker products[END_REF], was proposed in [START_REF] Dantas | Learning Tensorstructured Dictionaries with Application to Hyperspectral Image Denoising[END_REF] for a higher-order generalization of SuKro, i.e. for sums of Kronecker products of any number of blocks, and we apply it for the particular case of two blocks described in [START_REF] Chen | Hyperspectral image classification using dictionary-based sparse representation[END_REF].

Computation of the denoised image

Once the learning process is finished, the denoised version of the eigen-image Ûk is computed as follows. All possible patches from the noisy image -i.e. not only the N training patches, but all

(n 1 - √ d + 1)(n 2 - √ d + 1)
overlapping patches with a onepixel translation from its neighbors-are reconstructed with the learned dictionary using OMP with the same threshold τ . Each pixel in the final denoised image is then calculated as the weighted average of all reconstructed patches covering that pixel (each with weight one) and its value in the noisy image (with weight λ = (σ √ n) -1 ). See [START_REF] Elad | Image denoising via sparse and redundant representations over learned dictionaries[END_REF] for details on why, in some sense, this is an optimal recovering procedure.

Noise and Rank Estimation

In this subsection we show how the noise level can be efficiently estimated in practice under the assumption that the original image is low-rank and for a Gaussian distribution on the entries of the noise matrix N. First of all, as n p 1, the singular values of the noise matrix N are very close to a constant. Indeed, using the limit case of the Marčenko-Pastur distribution [START_REF] Marčenko | Distribution of eigenvalues for some sets of random matrices[END_REF] when the column dimension is large in front of the row dimension, we get that all singular values of N are close to σ √ n. Moreover, as both the noiseless matrix X and the noise N are tall matrices generated by a priori independent processes, they are likely to be almost orthogonal to each other. Then,

H T H ≈ X T X + N T N ≈ X T X + σ 2 nI (6)
and we get that the eigenvalues of H T H are the sum of those of X T X and a constant term σ 2 n. Finally, since X is (approximately) a rank r matrix, any singular value of H after the r-th index is (approximately) equal to σ √ n. Therefore, the noise level can be estimated by looking at the tail value ς tail of the singular values of H. Figure 3 shows that we indeed observe such a singular value profile in practice when adding Gaussian noise to a real HSI.

While σ = ς tail / √ n estimates the noise level in H, we are rather interested in the noise level σ k at an eigen-image U k . Considering that each column u k of U is scaled to unit-norm regardless of the associated singular value (denoted ς k for the k-th largest singular value), we observed that σ k is inversely proportional to ς k and can be estimated as follows:

σk = σ/ς k = ς tail /(ς k √ n) (7)
The rank estimation is based on this same observation: the singular values saturate as the noise dominates. A suitable approximate rank r is obtained by detecting this saturation. We set r equals to the first singular value ς k such that

(ς k -ς tail )/ς tail < . EstimateRank(Σ) = min{k | (ς k -ς tail )/ς tail < } (8)
where is a threshold empirically calibrated to = 3×10 -2 .

EXPERIMENTAL RESULTS

Four hyperspectral images were used in the experiments with (n 1 × n 2 × p) pixels: San Diego (400 × 400 × 158), Houston (349 × 1905 × 144), Washington DC (1280 × 256 × 191), Urban (307 × 307 × 162). The images were corrupted with additive white Gaussian noise with standard deviation σ uniform over all spectral modes and adjusted to give the desired input signal-to-noise ratio (SNR).

For each eigen-image (columns of U k ), we collect N = 20000 patches with (6 × 6) pixels from the noisy image to form the training data. As a pre-processing, each patch is centralized to zero mean in its pixel values. The patches are taken uniformly-spaced and partially overlapping. A dictionary is learned from this data with n it = 20 iterations of alternating optimization. Dictionaries are initialized with an Overcomplete (cropped) 2D-DCT (ODCT). In the case of SuKro, the remaining terms are initialized with unit-norm Gaussian random matrices, with no relevant impact on its performance.

The results reported in Table 1 are averaged over 10 runs with independent noise realizations. Standard deviations are usually around 0.005dB (and never bigger than 0.02dB).

Three variations of the proposed method are tested, with three different types of dictionary: K-SVD unstructured learned dictionary, SuKro learned structured dictionary with n kron = 3 terms and an overcomplete but fixed ODCT dictionary (i.e. the learning process is skipped). The proposed methods are compared to: a Wavelet 3D approach [START_REF] Abdullah | Wavelet-based noise reduction in multispectral imagery[END_REF] which uses only the sparsity prior by applying a 3D-wavelet basis directly in the data cube H; and the HyRes [START_REF] Rasti | Automatic hyperspectral image restoration using sparse and low-rank modeling[END_REF] method, which combines both sparse and low-rank assumptions. The latter is a state-of-the-art approach (see [START_REF] Rasti | Automatic hyperspectral image restoration using sparse and low-rank modeling[END_REF] for a detailed comparison of this method with other methods in the literature).

The proposed methods with K-SVD and SuKro dictionaries consistently outperform the state-of-the-art HyRes approach by about 1.5dB. A performance gain of about 1.3dB is already obtained by replacing the orthogonal Wavelet basis in HyRes by a fixed overcomplete DCT dictionary allied to the patch-based procedure. The learning process then brings an additional 0.2dB in denoising performance. A visual comparison is provided in Figure 4.

Execution times are reported in Table 2. The literature methods HyRes and Wavelet 3D take about one and two minutes respectively, independently of the noise level. The running times for the proposed method, on the other hand, increase as the noise level decreases. That's because the OMP reconstruction threshold gets smaller, requiring more iterations. The execution times range from 120 to 760s in the ODCT case and from 260 to 1600s with a learned dictionary (actually, SuKro is slightly faster than K-SVD). Therefore, the proposed technique with dictionary learning takes around 5 to 30 times longer than HyRes.

Fig. 1 :

 1 Fig. 1: Hyperspectral image modeling: 3D data cube (left), 2D representation with vectorized spatial dimensions (center) and low-rank factorization model (right).

Fig. 2 :

 2 Fig. 2: N patches ( √ d × √ d) are extracted from a eigenimage U k to form the training data matrix Y, which is used to learn a dictionary D and a sparse reconstruction matrix Γ.

3 and 3. 4 . 8 :

 48 Algorithm 1 Dictionary-based HSI denoising approach1: INPUT: Noisy hyperspectral image H 2: Low-rank step 3: [U, Σ, V] = SVD(H) 4: r = EstimateRank(Σ) 5: [σ 1 , . . . , σ r ] = EstimateNoise(Σ) 6: Û = U(:, 1 : r) , V = V(:, 1 : r)Σ(1 : r, 1 : r) for k = [1, . . . , r] do 9:Extract N patches from Ûk to form Y for i = [1, . . . , n it ] do 13:

16 :

 16 Ûk = Denoise( Ûk , D nit , σ k ) 17: end for 18: Û = [ Û1 , . . . , Ûr ] 19: OUTPUT: Recovered image X = Û VT

Fig. 3 :

 3 Fig. 3: Singular values of the San Diego HSI before (dotted green line) and after the noise addition (full blue line), as well as the noise singular values (dashed red) in logarithmic scale. Note that the noise singular values are very close to a constant σ √ n (= 912 × 400 = 3.648 • 10 5 ) as predicted by the theory.

Method

Input SNR [

CONCLUSION

In this paper, we proposed to replace fixed sparsifying transforms by overcomplete learned dictionaries to improve stateof-the-art performance in the HSI denoising task.

Although the proposed patch-based approach is markedly slower than other previous methods, running times remain reasonable for an offline application, even for large images. A potential improvement would be to also denoise the right term of the low-rank decomposition (spectral component V).

[1] P. Ghamisi, N. Yokoya, J. Li, W. Liao, S. Liu, J. Plaza, B. Rasti


Olivier Cuisinier
email: olivier.cuisinier@univ-lorraine.fr

Abdelwadoud Mehenni

Farimah Masrouri

Emmanuel Lavallée

au ciment ou à base d'argile sur l'érodabilité d'un limon compacté

Keywords: Érosion interne, traitement des sols, chaux, ciment, argiles internal erosion, soil treatment, lime, cement, clays

archive ouverte pluridisciplinaire HAL, est destinée au dépôt

Impact d'un traitement à la chaux,

INTRODUCTION

Les sols constituant les ouvrages en terre peuvent être sujets à divers phénomènes d'érosion qui peuvent provoquer des dégâts importants sur la structure de l'ouvrage, allant même jusqu'à sa ruine. Un moyen pour prévenir le risque d'érosion est d'utiliser des matériaux présentant une érodabilité qui répond aux critères d'érosion requis, et aussi aux autres critères portant sur le comportement hydrique et mécanique des sols. Lorsque les sols disponibles pour la construction ne permettent pas de satisfaire aux critères nécessaires, le traitement constitue une alternative intéressante. Il existe ainsi plusieurs techniques de traitement qui permettent d'améliorer les sols. Plusieurs auteurs ont montré que le traitement à la chaux améliore principalement la maniabilité du sol ainsi que les performances mécaniques du sol [START_REF] Bell | Lime stabilization of clay minerals and soils[END_REF][START_REF] Brandl | Alteration of soil parameters by stabilization with lime[END_REF][START_REF] Eades | Reaction of hydrated lime with pure clay minerals in soil stabilization[END_REF][START_REF] Le Runigo | Performance of limetreated silty soil under long-term hydraulic conditions[END_REF][START_REF] Little | Handbook for stabilization of pavement subgrades and base courses with lime[END_REF][START_REF] Robin | Chemo-mechanical modelling of lime treated soils[END_REF][START_REF] Stoltz | Weathering of a lime-treated clayey soil by drying and wetting cycles[END_REF][START_REF] Stoltz | Multi-scale analysis of the swelling and shrinkage of a lime-treated expansive clayey soil[END_REF]. Les effets de la chaux sur la conductivité hydraulique des sols dépendent en revanche des conditions de compactage [START_REF] Cuisinier | Microstructure and hydraulic conductivity of a compacted lime-treated soil[END_REF]Herrier et al., 2012b;[START_REF] Le Runigo | Impact of initial state on the fabric and permeability of a lime-treated silt under long-term leaching[END_REF][START_REF] Mccallister | Physical property changes in a lime-treated expansive clay caused by leaching[END_REF]. Plusieurs études ont montré l'amélioration des caractéristiques mécaniques des sols après traitement au ciment (par exemple [START_REF] Al-Amoudi | Characterization and chemical stabilization of Al-Qurayyah sabkha soil[END_REF][START_REF] Sariosseiri | Effect of cement treatment on geotechnical properties of some Washington State soils[END_REF]. Selon [START_REF] Bellezza | Effectiveness of cement on hydraulic conductivity of compacted soil-cement mixtures[END_REF] l'évolution de la conductivité hydraulique est principalement liée à la nature du sol et peut augmenter ou diminuer après le traitement au ciment. Les sols argileux sont souvent utilisés comme produit de traitement spécifiquement pour réduire la conductivité hydraulique des sols afin d'atteindre une valeur cible. L'ampleur de la réduction de la conductivité hydraulique dépend du dosage et de la nature du matériau argileux utilisé (par exemple [START_REF] Chapuis | The 2000 R.M. Hardy Lecture: Full-scale hydraulic performance of soilbentonite and compacted clay liners[END_REF][START_REF] Sivapullaiah | Hydraulic conductivity of bentonite-sand mixtures[END_REF]. L'objectif principal de l'article est de présenter les résultats d'une campagne d'études concernant l'impact de différents traitements sur l'érodabilité d'un limon compacté. Dans ce qui suit, après la description du processus d'érosion et sa caractérisation expérimentale, une revue des données disponibles sur les effets des traitements sur les caractéristiques de l'érosion interne est présentée. Les phénomènes d'érosion qui se produisent sur les structures hydrauliques en terre peuvent être classés en deux catégories : l'érosion externe et l'érosion interne. L'érosion externe concerne les surfaces extérieures des structures, alors que l'érosion interne se produit à l'intérieur du corps de la structure. Cette étude est axée sur l'érosion de conduit, qui est un mécanisme d'érosion interne. Selon Fell et al. (2003), ce processus d'érosion interne peut être divisé en quatre phases : l'initiation de l'érosion par une concentration de la circulation d'eau au sein de la structure, la poursuite de l'érosion, la progression de l'érosion qui aboutit à former une conduite, et la formation d'une brèche conduisant à la destruction de la structure. Pour étudier l'érosion interne, le « hole erosion test » (HET) a été développé (Wan andFell, 2004a, 2004b). Cet essai dérive d'autres dispositifs expérimentaux d'érosion interne (ASTM-D4647, 1998;[START_REF] Lefebvre | Erosivity of natural intact structured clay: evaluation[END_REF][START_REF] Pham | Comparison of internal and surface erosion using flow pump tests on a sand-kaolinte mixture[END_REF][START_REF] Rohan | A new technique to evaluate erosivity of cohesive material[END_REF][START_REF] Sherard | Pinhole test for identifying dispersive soils[END_REF]. Le HET ne caractérise le comportement des sols qu'une fois qu'un conduit est complètement formé. L'essai ne permet pas d'étudier la phase initiale du processus d'érosion interne (c'est-à-dire la formation de la conduite dans le sol). En fait, dans le HET, un conduit préformé dans une éprouvette de sol est créé pour simuler ce conduit. L'érodabilité des parois du conduit est ensuite étudiée en le soumettant à un écoulement d'eau. La croissance du diamètre du conduit est surveillée par la mesure des paramètres hydrauliques. Les résultats permettent de déterminer la loi empirique d'érosion du matériau, qui peut être exprimée comme suit :

0 = ε si 𝜏𝜏 < 𝜏𝜏 𝑐𝑐 ) ( c er k τ τ ε - =  si 𝜏𝜏 > 𝜏𝜏 𝑐𝑐
Eq. 1 où :

𝜀𝜀̇ : taux d'érosion par unité de surface par unité de temps [kg/s/m 2 ] ;

𝜏𝜏 : contrainte de cisaillement hydraulique [Pa] ;

𝜏𝜏 𝑐𝑐 : contrainte critique [Pa] ; 𝑘𝑘 𝑒𝑒𝑒𝑒 : coefficient d'érosion [s/m].
La contrainte τc exprime le seuil de déclenchement d'érosion du sol. Plusieurs modifications ont été apportées sur les dispositifs HET pour permettre un meilleur contrôle des paramètres hydrauliques, comme la chute de pression hydraulique dans l'échantillon ou le débit d'eau [START_REF] Benahmed | Investigating conc entrated leak erosion behaviour of cohesive soils by performing Hole Erosion Test[END_REF][START_REF] Haghighi | Improvement of hole erosion test and results on reference soils[END_REF][START_REF] Indraratna | Modelling the erosion rate of chemically stabilized soil incorporating tensile force-deformation charcacteristics[END_REF][START_REF] Lüthi | A Modified Hole Erosion Test (HET-P) Device[END_REF][START_REF] Pham | Comparison of internal and surface erosion using flow pump tests on a sand-kaolinte mixture[END_REF] Le HET a été utilisé pour évaluer l'impact des paramètres de compactage, c'est-à-dire la teneur initiale en eau, la densité sèche et l'énergie de compactage sur l'érodabilité des sols compactés.

Certains auteurs [START_REF] Attom | The effect of compaction and initial water content on soil erosion[END_REF][START_REF] Lim | Experimental investigation of erosion in variably saturated clay soils[END_REF]Wan and Fell, 2004a) ont montré que pour une même densité sèche, les sols présentent une érodabilité plus faible s'ils sont compactés à une teneur en eau plus élevée que la teneur en eau de de l'optimum (côté humide de la courbe de compactage). L'augmentation de la densité sèche du sol induit une diminution de l'érodabilité et une contrainte critique plus élevée. Pour les sols compactés, les effets de la densité sèche, sur la baisse de l'érodabilité, sont également liés à la teneur en eau initiale de compactage. Le long de la courbe de compactage, l'érodabilité la plus faible est atteinte pour les valeurs couplées de la teneur en eau initiale optimale et de la densité sèche maximale [START_REF] Attom | The effect of compaction and initial water content on soil erosion[END_REF][START_REF] Wahl | A comparison of the hole erosion test and jet erosion test[END_REF].

Quelques [START_REF] Benahmed | Investigating conc entrated leak erosion behaviour of cohesive soils by performing Hole Erosion Test[END_REF][START_REF] Bennabi | Some elements of comparison between two laboratory devices for soil erosion testing[END_REF][START_REF] Haghighi | Improvement of hole erosion test and results on reference soils[END_REF].

Il apparaît ainsi que le traitement à la chaux ou au ciment ou avec une argile, pourrait réduire l'érodabilité d'un sol compacté. La plupart des résultats disponibles restent cependant qualitatifs. L'objectif principal de cette étude est d'étudier l'impact des traitements du sol sur les caractéristiques de l'érosion. Un nouveau dispositif HET amélioré a été mis en place pour quantifier les effets de différents traitements. Dans les sections suivantes, les procédures d'essais et d'interprétation sont présentés, et les modifications produites par les traitements sont discutées.

MATÉRIAUX ET MÉTHODES

Matériaux

Le sol support de cette étude est un limon fin (wL = 28,5 % ; wP = 20,5 % ; VBS = 1,56 ;

pourcentage < 80 µm = 99,2 %). Quatre produits de traitement ont été utilisés avec différents dosages : une chaux vive (1 et 3 %), un ciment CEM II (3 et 6 %), une bentonite calcique activée (2, 5 et 9 %), et une kaolinite (2, 5 et 9 %). Les éprouvettes du limon traité à la chaux/ciment ont subi différents temps de cure (0, 7, 30, 90 jours) avant les essais.

Il est bien établi que le compactage du côté humide de l'optimum de la courbe de compactage permet d'atteindre la perméabilité la plus faible (par exemple [START_REF] Mitchell | Permeability of compacted clays[END_REF]. Les valeurs de wOPN et de γdmax ont été déterminées pour chaque traitement (Table 1). Pour l'ensemble des modalités, les éprouvettes ont été compactées du côté humide de l'optimum (w = wOPN+3% et γd = 0,96 γdmax), avec des valeurs de w variant entre 17,5 et 20,0 % et des valeurs de γd variant entre 1,68 et 1,75 Mg/m 3 , selon la modalité de traitement.

Pour la préparation des éprouvettes, la teneur en eau du sol a été ajustée à la teneur en eau souhaitée pour le compactage. Après une période de stockage de 24 h, le sol et le produit de traitement ont été soigneusement mélangés dans un malaxeur à couteaux. Le mélange sol + chaux a été laissé au repos pendant 1 h dans un récipient hermétique avant de passer au compactage. Lors de l'utilisation du traitement au ciment, le compactage a été effectué quelques minutes (maximum 30 minutes) après le traitement pour tenir compte du temps de prise du ciment. Lorsque des traitements à l'argile ont été utilisés, aucun temps spécifique n'a été nécessaire entre le mélange et le compactage ; le mélange a été compacté statiquement jusqu'à la densité sèche cible. Le compactage a été réalisé en plusieurs couches de l'ordre de 2 cm afin limiter l'hétérogénéité des éprouvettes. Lorsqu'une période de cure avant l'essai était nécessaire, les éprouvettes compactées étaient enveloppées dans des feuilles de plastique et maintenues à 20,0 ± 1,5 °C afin d'éviter toute perte d'eau.

Protocole expérimental de l'essai HET

Un dispositif HET amélioré a été mis au point pour mesurer l'érosion interne des sols compactés et traités (Figure 1). Ce dispositif comprend plusieurs optimisations qui permettent l'application d'une pression d'entrée élevée, permettant une grande chute de pression à travers l'éprouvette et donc une forte contrainte de cisaillement hydraulique le long du conduit. Des améliorations majeures ont été apportées au système de pression : la pression de l'eau à l'entrée de l'échantillon a été appliquée grâce à un réservoir spécial alimenté en air comprimé. Une électrovanne permet effluents. À la fin de l'essai, le conduit érodé est rempli avec de la paraffine liquide, ce qui permet de déterminer sa géométrie et son diamètre moyen.

IMPACT DES TRAITEMENTS SUR L'ÉRODABILITÉ DES SOLS

Sol non traité

La répétabilité des paramètres d'érosion obtenus avec le nouveau dispositif HET a été vérifiée avec le sol non traité (Figure 2). Les résultats montrent que ker varie entre 1,13 x 10 -04 et 3,04

x 10 -04 s/m, et τc varie entre 291 et 615 Pa (Table 2). Cette dispersion est du même ordre de grandeur que dans les études disponibles (par exemple [START_REF] Haghighi | Improvement of hole erosion test and results on reference soils[END_REF]Wan and Fell, 2004a). Les valeurs moyennes pour le limon non traité sont ker= 2,07 x 10 -04 s/m et τc= 429 Pa.

Le limon étudié présente une contrainte de cisaillement critique élevée par rapport aux valeurs généralement observées dans les sols non traités (Wan and Fell, 2004a). Ceci peut être lié au pourcentage élevé de la fraction de particules fines (< 80 µm), 99,2%, et aux conditions de compactage. En fait, le compactage du côté humide de l'optimum permet un meilleur réarrangement des particules du sol que le côté sec et réduit la porosité (par ex. [START_REF] Benson | Influence of clods on hydraulic conductivity of compacted clay[END_REF]. Le compactage du côté humide (w = 17,5%) combiné à la densité sèche élevée (ρd = 1,73 Mg/m 3 ) induit une microstructure particulière qui limite le détachement des particules du sol et nécessite un plus grand niveau de contrainte de cisaillement hydraulique pour initier l'érosion. Cependant, une fois que cette valeur de contrainte de cisaillement hydraulique est atteinte, le taux d'érosion progresse rapidement, comme le montre la valeur du coefficient d'érosion du sol (ker= 2,07 x 10 -04 s/m).

Impact du traitement à la chaux

Pour un traitement à 1% de chaux, ker varie entre 1,09 x 10 -04 et 2,87 x 10 -04 s/m quel que soit le temps de cure (Table 3 et Figure 5). Pour un traitement à 3% de chaux, ker se situe entre 1,54

x 10 -04 et 4,52 x 10 -04 s/m. Le traitement à la chaux augmente également la contrainte de cisaillement critique du sol non traité, qui peut atteindre des valeurs allant jusqu'à 1 770 Pa pour 1 % de chaux et 1 889 Pa pour 3 % de chaux.

Le taux d'érosion au cours de l'essai était plus élevé pour les éprouvettes traitées avec 3 % de chaux que pour le limon non traité. Cette observation pourrait s'expliquer par la taille maximale des agrégats dans le sol traité, qui est supérieure à celle du sol non traité. Il a également été observé au cours des essais que l'érosion des particules de sol traitées à la chaux se produit souvent par le détachement d'agrégats de sol, et pas seulement de particules de sol individuelles.

L'amélioration des caractéristiques d'érosion interne du limon induite par le traitement à la chaux est donc caractérisée principalement par l'augmentation de la contrainte de cisaillement critique.

Apport d'un traitement au ciment

Dans le cas du ciment à 3 %, il a été nécessaire d'appliquer une pression d'entrée allant jusqu'à 450 kPa pour amorcer l'érosion du sol (Figure 6 et Table 4). Ceci se traduit par la forte contrainte de cisaillement hydraulique appliquée relativement aux autres traitements. Pour la plupart des éprouvettes testées, l'érosion a commencé à une contrainte de cisaillement hydraulique de 2 000 Pa.

Dans le cas d'un traitement avec 6 % de ciment, la pression d'entrée appliquée était supérieure à 500 kPa, et la longueur des éprouvettes a été réduite à 70 mm pour appliquer une contrainte de cisaillement hydraulique plus élevée. Le détachement des particules de sol a commencé après 4 000 Pa pour la plupart des éprouvettes. Cependant, la loi sur l'érosion présentait des caractéristiques non linéaires pour la plupart des spécimens, surtout au-delà de 7 jours de cure.

Ainsi, il était difficile de déterminer la valeur de ker pour certaines éprouvettes. Dans de tels cas, la contrainte de cisaillement critique correspond à la valeur de la contrainte de cisaillement hydraulique à la fin du processus d'érosion (Table 4).

Le traitement au ciment apporte une augmentation de la contrainte de cisaillement critique, associée à une diminution du coefficient d'érosion du sol.

Impact de l'addition d'argile

L'utilisation de 2 % de kaolinite n'a pas changé de façon significative ker, avec cependant une légère tendance à l'augmentation de τc ( Table 5 et Figure 3). L'addition de 9% de kaolinite a diminué ker d'un ordre de grandeur et a induit une faible augmentation de τc avec une valeur moyenne de 513 Pa. Pour le traitement avec 2% de bentonite, ker reste du même ordre de grandeur pour toutes les éprouvettes et est d'environ 5,00 x 10 -05 s/m (Figure 4 et Table 5) ce qui est inférieur d'un ordre de grandeur à celui du sol non traité. τc varie entre 0.123 et 0.483 kPa, ces valeurs sont en accord avec l'intervalle de variation de τc pour les limons non traités.

Le traitement de l'argile induit principalement une réduction du coefficient d'érosion du sol en fonction de la nature et du pourcentage du produit ajouté.

DISCUSSION

Les deux paramètres qui caractérisent la résistance à l'érosion des sols, le coefficient d'érosion Pour des dosages supérieurs à 6 %, l'ajout de différents produits de traitement (la chaux qui n'a pas été testée) réduit le coefficient d'érosion. Dans cette gamme de dosage, le temps de cure semble avoir un effet favorable pour diminuer le coefficient d'érosion du limon traité au ciment.

ker
L'évolution de la contrainte critique (valeurs moyennes) en fonction du dosage utilisé pour différents produits de traitement et différents temps de cure montre que les produits argileux n'apportent pas de modifications significatives sur la contrainte critique (Figure 8). Le traitement par des liants induit une augmentation de la contrainte critique. L'ampleur de l'augmentation de la contrainte critique est par ailleurs liée aux trois paramètres suivants : nature de traitement, dosage utilisé et temps de cure. Les effets de ces trois paramètres sont interdépendants. Ainsi, l'augmentation de la contrainte critique est plus marquée dans le cas du traitement au ciment. L'augmentation du dosage utilisé favorise l'augmentation de la contrainte critique mais avec un effet plus marqué dans le cas du ciment que dans le cas de la chaux. Le temps de cure semble favoriser l'augmentation de la contrainte critique, mais avec un effet plus ou moins important selon le dosage et la nature du produit de traitement.

L'impact des traitements sur les caractéristiques d'érosion du limon (𝜏𝜏 𝑐𝑐 ,𝑘𝑘 𝑒𝑒𝑒𝑒 ) est aussi illustré par la Figure 9. Ce mode de présentation permet de s'affranchir de la barrière de classification qui se limite à un seul paramètre (ker). Les effets de chaque produit de traitement peuvent donc être visualisés avec la prise en compte de la dispersion des résultats. Ainsi, l'influence est notable uniquement pour un dosage de 9 % de kaolinite, et elle est marquée par la diminution du coefficient d'érosion, d'un ordre de grandeur, sans grande modification de la contrainte critique.

L'action principale de la bentonite est la diminution du coefficient d'érosion, d'un ordre de grandeur, sans modification significative de la contrainte critique. Ces effets sont obtenus même avec un faible dosage de bentonite. L'ajout de la chaux ne modifie pas de manière significative le coefficient d'érosion du sol. L'effet principal est l'augmentation de la contrainte critique. Par ailleurs, l'augmentation de la contrainte critique semble être plus importante avec l'augmentation du temps de cure. L'impact de l'ajout du ciment est lié au dosage utilisé.

  2000). Des essais d'étalonnage préliminaires ont permis de déterminer la relation entre la concentration et la turbidité pour chaque traitement et chaque dosage. Cette relation est définie sur deux domaines de concentrations : faibles [0 à ~1,5 kg/m 3 ] et élevées [~1,5 à 16 kg/m 3 ]. L'utilisation du domaine de faibles concentrations permet d'apporter plus de précision sur la mesure de concentration à partir du signal de turbidité. De plus amples détails sur la conduite des essais d'étalonnage sont disponibles dans Mehenni (2015) ainsi que dans Mehenni et al. (2016).

  et la contrainte critique 𝜏𝜏 𝑐𝑐 , peuvent être modifiés, de manière indépendante, par l'ajout d'un produit de traitement. La Figure 7 illustre l'évolution du coefficient d'érosion du limon étudié (valeurs moyennes) en fonction de l'augmentation du dosage pour les différents produits de traitement et les différents temps de cure utilisés. Pour des dosages inférieurs à 3 % les différents traitements ne modifient pas le coefficient d'érosion de manière significative. Seul le traitement à la bentonite (2 %) permet de réduire le coefficient d'érosion d'un ordre de grandeur. Pour cette gamme de dosage aussi, le temps de cure ne semble pas avoir un impact significatif sur le coefficient d'érosion.

Figure 2 .

 2 Figure 1. Détail de la nouvelle cellule HET développée.

Figure 3 .Figure 4 .

 34 Figure 3. Résultats d'érosion du limon traité à la kaolinite.

Figure 5 .

 5 Figure 5. Impact du traitement à la chaux sur l'érodabilité du limon.

Figure 6 .

 6 Figure 6. Impact du traitement au ciment sur l'érodabilité du limon.

Figure 7 .Figure 8 .

 78 Figure 7. Impact des traitements sur le coefficient d'érosion du limon (valeur moyenne).

  Pour l'ensemble des essais présentés dans l'article, l'observation de la forme finale du conduit montre que la section transversale finale peut, dans certains cas, présenter un écart par rapport à la forme circulaire. Cet écart pourrait être lié aux hétérogénéités dans l'éprouvette de sol, au départ non uniforme de matériau, ou encore à des réparations de pression non uniforme le long du conduit.Des investigations ont donc été menées pour vérifier la sensibilité des paramètres d'érosion calculés (𝜀𝜀̇ et 𝜏𝜏 et par conséquent ker et τc) vis-à-vis des petites variations de la forme de la section transversale entre la forme réelle et la forme théorique. Il a été alors nécessaire de modifier les équations Où : c(t) est la concentration du sol dans les effluents, q(t) est le débit d'eau à l'instant t.Dans un deuxième temps, la perte de masse est reliée à la géométrie du conduit érodée. Deux hypothèses sont alors supposées : i) l'érosion est uniforme le long du conduit ; ii) la relation entre les rayons de l'ellipse est unique [a(t)=α.b(t)], le coefficient α est mesuré à la fin de l'essai.Ainsi, la perte de masse est donnée par : 𝑑𝑑𝑑𝑑(𝑡𝑡) = 2. 𝛼𝛼. 𝜋𝜋. 𝜌𝜌 𝑑𝑑 . 𝐿𝐿. 𝑏𝑏 𝑡𝑡 . 𝑑𝑑𝑏𝑏

					𝑑𝑑𝑑𝑑(𝑡𝑡) = 𝑐𝑐(𝑡𝑡). 𝑞𝑞(𝑡𝑡). 𝑑𝑑𝑡𝑡	Eq. 4
								Eq. 5
	L'évolution du rayon peut être déduite en égalisant les deux équations (Eq. 4et Eq. 5) de la perte
	de masse :
					𝑑𝑑𝑏𝑏 =	𝑐𝑐(𝑡𝑡). 𝑞𝑞(𝑡𝑡) 2. 𝜋𝜋. 𝜌𝜌 𝑑𝑑 . 𝐿𝐿	.	𝑏𝑏 𝑡𝑡 𝑑𝑑𝑡𝑡	Eq. 6
	𝜀𝜀̇=	𝜌𝜌 𝑑𝑑 2	�	2 𝑎𝑎(𝑡𝑡) 2 + 𝑏𝑏(𝑡𝑡) 2 [𝑎𝑎(𝑡𝑡)𝑑𝑑𝑏𝑏 + 𝑏𝑏(𝑡𝑡)𝑑𝑑𝑎𝑎]	𝑑𝑑𝑡𝑡 1	Eq. 2
		𝜏𝜏 =	1 √2	.	∆𝑃𝑃(𝑡𝑡) 𝐿𝐿	.	�𝑎𝑎(𝑡𝑡) 2 + 𝑏𝑏(𝑡𝑡) 2 𝑎𝑎(𝑡𝑡). 𝑏𝑏(𝑡𝑡)	Eq. 3
	Afin de déterminer les paramètres d'érosion il est nécessaire de mesurer la variation des rayons
	du conduit durant l'érosion du sol. La perte de sol est utilisée pour mesurer l'évolution des
	rayons du conduit, a et b, durant l'essai. Dans un premier temps la perte de masse dm(t) est
	mesurée par rapport à la concentration du sol dans les effluents :

d'analyse, afin d'intégrer la forme réelle dans le calcul des paramètres d'érosion. La géométrie du conduit a été intégrée au modèle de calcul. Avec comme objectif de quantifier la sensibilité des paramètres d'érosion en fonction de la forme réelle de la section du conduit. Le modèle proposé se base sur une section ellipsoïdale (grand rayon : a ; petit rayon b ; rapport entres les rayons : α = a/b). Ce modèle a l'avantage de pouvoir se ramener à une section circulaire si a=b.. Le taux d'érosion et la contrainte de cisaillement hydraulique sont alors définis comme suit :

Où : ρd est la densité sèche, db et da sont les variations du petit et du grand rayon de l'ellipse, L est la hauteur de l'éprouvette du sol, ∆P(t) est le différentiel de pression entre l'amont et l'aval de l'éprouvette à l'instant t.

La quantité de sol érodée est calculée à partir des mesures du signal de turbidité

(Reddi et al.,

Table 1 : Caractéristiques de compactage.

 1 Toutefois, ces améliorations initiales sont susceptibles d'évoluer au cours du temps lorsque le sol traité est soumis à des sollicitations hydriques de séchage-humidification. L'enjeux majeur est de suivre cette évolution et de déterminer donc la durabilité des effets de traitements.

	Avec

Table 2 : Caractéristiques d'érosion du limon sans traitement.

 2

			Valeurs moyenne
	Essai	𝒌𝒌 𝒆𝒆𝒆𝒆 (s/m)	𝝉𝝉 𝒄𝒄 (Pa)
			𝒌𝒌 𝒆𝒆𝒆𝒆 (s/m)	𝝉𝝉 𝒄𝒄 (Pa)
	1	2.58 x 10 -04	391
	2	2.16 x 10 -04	389
	3	2.22 x 10 -04	398
			2.07 x 10 -04	429
	4	1.13 x 10 -04	491
	5	3.04 x 10 -04	615
	6	1.30 x 10 -04	291

Table 3 : Impact du traitement à la chaux sur les caractéristiques d'érosion du limon. 474 Test Dosage en chaux et temps de cure

 3

			Caractéristiques	Valeurs moyennes
			𝒌𝒌 𝒆𝒆𝒆𝒆 (s/m)	𝝉𝝉 𝒄𝒄 (Pa)	𝒌𝒌 𝒆𝒆𝒆𝒆 (s/m)	𝝉𝝉 𝒄𝒄 (Pa)
	1		1.25 x 10 -04	780
	2	+ 1% chaux	1.20 x 10 -04	618	1.25 x 10 -04	745
	3	(1 jour)	2.62 x 10 -04	836
	1		2.87 x 10 -04	1725
		+ 1% chaux
	2		2.35 x 10 -04	1770	2.63 x 10 -04	1660
		(7 jours)
	3		2.65 x 10 -04	1487
	1		2.71 x 10 -04	1336
		+ 1% chaux
	2		2.53 x 10 -04	1538	2.11 x 10 -04	1279
		(30 jours)
	3		1.09 x 10 -04	963
	1	+ 3% chaux	2.01 x 10 -04	990
					2.24 x 10 -04	1200
	2	(1 jour)	2.47 x 10 -04	1409
	1		1.54 x 10 -04	1857
	2		1.75 x 10 -04	1863
	3		4.52 x 10 -04	1889
		+ 3% chaux
	4		2.15 x 10 -04	1567	2.46 x 10 -04	1707
		(7 jours)
	5		2.28 x 10 -04	1522
	6		3.07 x 10 -04	1648
	7		1.81 x 10 -04	1608
	1		2.16 x 10 -04	1329
	2	+ 3% chaux	2.27 x 10 -04	1035
					1.71 x 10 -04	1362
	3	(30 jours)	1.78 x 10 -04	1197
	4		1.68 x 10 -04	1 845

Table 4 : Apport du traitement au ciment sur les caractéristiques d'érosion du limon.

 4

	478
	479

Table 5 : Caractéristiques d'érosion du limon traité à la kaolinite ou à la bentonite.

 5

	483

d'ajuster la pression d'air dans le réservoir air-eau pendant l'expérience. La pression d'eau appliquée sur la partie amont de l'éprouvette a été régulée en utilisant un capteur situé à proximité de l'entrée de l'éprouvette. La pression d'eau était également mesurée à la sortie de l'éprouvette. La pression d'entrée maximale dépassait 650 kPa, produisant la chute de pression nécessaire de 650 kPa entre les débits en amont et en aval des éprouvettes de sol. D'autres optimisations ont été apportées à la configuration de la cellule d'essai (Figure1). Les éprouvettes de sol étaient maintenues à l'intérieur du dispositif d'essai des deux côtés, ceci afin d'éviter toute perte de pression et toute fuite d'eau, notamment lors d'essais sous fortes pressions. Le débit d'eau a été mesuré à l'aide d'un débitmètre à ultrasons. La plage de mesure de vitesse est de 0,1 à 20 m/s, sur une canalisation de 19 mm cela se traduit par une gamme de débit de 1,7 à 340 L/min avec une précision de l'ordre de 3 %. Un turbidimètre a été raccordé à la sortie de la cellule d'essai pour mesurer la turbidité de l'effluent, et ainsi évaluer la quantité de matière érodée en cours d'essai.Pour éviter un endommagement de la paroi du conduit lors de sa réalisation, celui-ci a été percé juste après le compactage, avant la prise.[START_REF] Mehenni | Impact of Lime, Cement, and Clay Treatments on the Internal Erosion of Compacted Soils[END_REF]. L'éprouvette était ensuite placée dans la cellule d'essai. L'essai d'érosion était effectué en contrôlant et en surveillant les paramètres hydrauliques, la pression appliquée à l'éprouvette, la chute de pression entre l'amont et l'aval de l'éprouvette et le débit à travers le conduit.Des essais préliminaires ont été réalisés pour chaque traitement afin de définir la plage de pression nécessaire pour amorcer l'érosion en détachant les particules du sol[START_REF] Mehenni | Comportement hydromécanique et érosion des sols fins traités[END_REF].Cette pression est liée à la nature du sol et au produit de traitement utilisé. L'initiation du processus d'érosion a été déterminée par le signal de turbidité. L'essai d'érosion est réalisé par application d'une pression constante à l'entrée de la cellule. Le but étant de maintenir cette pression constante pendant l'essai et de mesurer pour chaque pas de temps le débit d'eau traversant le conduit, le différentiel de pression entre l'amont et l'aval et aussi la turbidité des

L'


Mary-Ambre Carvalho

Hervédekkiche

Lydia Karmazin

Fabien Sanchez

Bruno Vincent

Masatoshi Kanesato

Yoshihiro Kikkawa

Romain Ruppert

Synthesis and Study at a Solid/Liquid Interface of Porphyrin Dimers Linked by Metal Ions

Keywords:

Several porphyrin dimers linked by metal ions were prepared. One trimeric compound was also isolated and one porphyrin dimer linked by palladium(II) could be structurally characterized. In solution, the size of the new compounds was estimated by DOSY NMR techniques. These compounds all contained long aliphatic chains (O-C 12 H 25 ), which were used to assemble them at a highly oriented pyrolytic graphite (HOPG)/liquid interface. The highly ordered arrays were visualized by scanning tunneling microscopy (STM). porphyrins bearing two external coordination sites. 9 Electronic 46 communication through the coordination linkage between the 47 individual macrocycles was demonstrated. More recently, we 48 have shown that this electronic delocalization was responsible 49 for ultrafast energy transfer from a zinc(II) porphyrin to a free 50 base porphyrin linked together by a palladium(II) or 51 platinum(II) ion. 10 In this study, we have modified our initial 52 porphyrins by adding alkyl chains at the periphery of these 53 molecules and attempted to organize and assemble them on a 54 HOPG surface. 55 ■ RESULTS AND DISCUSSION 56 Monomeric compounds bearing long aliphatic chains in 57 addition to external coordination sites were the first synthetic

■ INTRODUCTION

To build molecular scale devices on surfaces, it is essential to control the organization of the molecules at the surfaces or at the solid/liquid interface. 1 Self-assembly is often used as an organizational tool, resulting in nice structures on various substrates. [START_REF]•cm -1 ). 337 Porphyrin 4. A solution of nickel porphyrin 3 (200 mg, 0.158 338 mmol) and Lawesson's reagent (100 mg, 1.75 mmol) in benzene (100 339 mL) was refluxed under argon for 2 h. After evaporation of the solvent, 340 the solid residue is purified by column chromatography (silica gel, 341 dichloromethane/cyclohexane, 1/1). The desired compound 4 is 342 isolated in 95% yield after recrystallization from dichloromethane and 343 methanol (192[END_REF] Molecules like porphyrins, chlorins, or corroles are ubiquitous molecules and are in the center of numerous essential processes found in living systems. Tetrapyrrolic macrocycles are found in many light-harvesting complexes and in photosynthetic reaction centers because of their exceptional optical properties. These properties were recognized quite early by synthetic chemists and therefore these molecules were modified and designed to offer new interesting optical, electrochemical, photochemical, and chemical properties. 3 Because of these properties, it was obvious that these molecules should be studied at the solid/liquid interfaces. Until now, several authors have shown that porphyrins could be assembled on HOPG or metal surfaces. 4 The STM technique was used to visualize single molecules on the surface, thus acting as a well-known characterization tool for large molecules. 5 In addition, STM demonstrated how these molecules form highly ordered assemblies on these substrates. 6 We have shown earlier that metallo-porphyrins could be connected by coordination bonds if the aromatic core of the macrocycle was functionalized with an external chelating c1 coordination site (see Chart 1). [START_REF]A solution of porphyrin 4 (50 mg, 39 μmol) in 1,2dichloroethane (100 mL) was added dropwise to a solution of Pd(acac) 2 (238 mg, 720 μmol, 20 equiv) at 60 °C in 1,2dichloroethane (200 mL) under argon over a period of 3 h. After the mixture was cooled and the solvent was evaporated, two compounds were isolated after column chromatography (silica gel, cyclohexane/dichloromethane, 7/3). Dimer 6 (18 mg, 34%) and porphyrin 7 (23 mg, 40%) were obtained pure after crystallization from hexane, dichloromethane and methanol. 1 H NMR (CDCl 3 , 50 °C, 400 MHz): δ H 9.17 (d, 1H, J = 8 Hz[END_REF] The external chelate was coplanar and conjugated with the aromatic core, allowing electronic delocalization between two porphyrins linked by metal ions. 8 This electronic delocalization was extended to oligoporphyrins by using monomeric the other porphyrin in the dimer (chemical shift close to 0 ppm)(see Figure 27). Both isomers were observed previously, but only for porphyrins bearing external enaminoaldehydes. 8c

Here, the steric hindrance of two Ar 1 meso groups was not large enough to prevent the formation of the cis isomer 5, in contrast to the 3,5-t-Bu 2 -phenyl groups used previously. 8b However, by refluxing overnight a dichloroethane solution of the isolated mixture of the two isomers 5 and 6, the initial cis/ trans mixture was cleanly converted to the single trans isomer 6 f1 as shown by 1 H NMR (see Figure 1).

Reacting porphyrin 4 with an excess of Pd(acac) 2 gave, although in rather moderate yield (40%), porphyrin 7 bearing a palladium acetylacetonate group on the external coordination site. During this reaction, the porphyrin dimers 5 and 6 was s3 also isolated as side product in 34% yield (see Scheme 3). Even with a large excess of Pd(II) and by adding the porphyrin solution dropwise to the Pd(II) solution, the external soft chelate of porphyrin 4 reacted rapidly with the soft Pd(II) ion to afford also dimers 5 and 6.

Until now, we were unable to prepare a nickel porphyrin bearing two external coordination sites and two Ar 1 meso groups. Therefore, we prepared dimers and trimers by using an unsymmetrical bis-functionalized nickel porphyrin monomer 8 previously described. 9b During the course of this work, we f2 obtained single crystals of 8 and the structure was solved (see These rather short metal-nitrogen bonds and the presence of two additional fused aryl groups cause the distortion of the porphyrin ring, which is highly ruffled. For the two external coordination sites (N-O and N-S), intramolecular and intermolecular hydrogen bonds were found between N-H and CO (or CS).

By reacting 4, 8 and palladium(II), a complex mixture of products was obtained, but most of them were isolated cleanly s4 (see Scheme 4). As expected, the mixture of dimers 5 and 6 was isolated once again, along with the heterodimer 9 bearing one external coordination site and the symmetrical dimer 10, which bears two external coordination sites.

Thionation of the external coordination site of 9 with Lawesson's reagent gave dimer 11, which bears now an external s5 N-S coordination site (see Scheme 5).

By reacting 11 with 0.5 equiv of palladium(II), we expected to obtain a porphyrin tetramer. The formation of oligomers was indeed observed, but we were not able to separate the trimer 14 (vide infra) from the expected tetramer. The presence of both compounds was easily demonstrated by MALDI mass spectrometry, but was difficult to deduce from the 1 H NMR longer reaction times, indicating that 14 was probably one of to the presence of additional aromatic rings fused to the porphyrin core (see Figure 3). The coordination geometry around the linking palladium(II) is trans square planar and the Pd-S and Pd-N lengths are close to 2.3 and 2 Å, respectively. The three fused rings bearing the external coordination sites are almost planar and the geometry around the palladium(II) ion renders the two planes belonging to two different porphyrins almost coplanar. However, within a given porphyrin, the angle between the planes of the three fused rings are approximately 65°. In monomeric porphyrins, these angles are similar (see Figure 2 for compound 8 or ref 9b for a related structure). These angles being additive in dimer 12, the overall observed distortion between the two external sites is close to 130°.

The presence of trimer 14 in all reactions incited us to try to prepare it deliberately, because trimer 14 was always present in all attempts. The reaction of porphyrin 7 with the previously s7 described bis-functionalized porphyrin 13 gave cleanly trimer These compounds were characterized by standard spectroscopic techniques (NMR, mass, electronic spectra). To obtain additional information about the size and to show that dimers or the trimer were pure compounds and not a mixture of oligomeric species in solution, we used DOSY NMR techniques to determine the diffusion coefficients of the different isolated compounds. 14 The molecular volume, closely related to the translational self-diffusion coefficient, was assumed to be similar to a sphere. Indeed, the values obtained for some compounds t1 (listed in Table 1) were in good agreement with the size estimated from previous X-ray data. [START_REF]A solution of porphyrin 4 (50 mg, 39 μmol) in 1,2dichloroethane (100 mL) was added dropwise to a solution of Pd(acac) 2 (238 mg, 720 μmol, 20 equiv) at 60 °C in 1,2dichloroethane (200 mL) under argon over a period of 3 h. After the mixture was cooled and the solvent was evaporated, two compounds were isolated after column chromatography (silica gel, cyclohexane/dichloromethane, 7/3). Dimer 6 (18 mg, 34%) and porphyrin 7 (23 mg, 40%) were obtained pure after crystallization from hexane, dichloromethane and methanol. 1 H NMR (CDCl 3 , 50 °C, 400 MHz): δ H 9.17 (d, 1H, J = 8 Hz[END_REF][8][9] Our initial goal was to test the possibility of assembling these molecules on HOPG surfaces. Several compounds were tested at a solid/liquid interface consisting of HOPG and a 1phenyloctane solution of the compound. Unfortunately, the trimeric compound 14 did not assemble neatly on the surface. The crystallographic data from the STM images are listed in t2 Table 2. The nickel porphyrin monomer 4 and trans dimer 6 were found in a centro-symmetric unit cell, as was the case for the trans dimer 11. For compounds 4 and 6, only two of the three tails per porphyrin were in van der Waals contact with the HOPG surface, and followed one of the HOPG lattice direction, implying that the third tail near the external coordination site is dangling in the solvent. The lattice constants of 6 were almost identical to those of 4 despite the presence of coordinating metal between the ligands in 6. In the case of 11, the porphyrin core was aligned in a manner similar to that of 4 and 6. All the alkyl chains are attached along the HOPG lattice direction. Thus, porphyrin building blocks with an external coordination site could be assembled into wellordered two-dimensional structures on a solid substrate. Synthesis. The starting benzaldehyde derivatives (4-dodecyloxybenzaldehyde and 2-carbomethoxybenzaldehyde) were prepared using published procedures. 16,17 The nickel meso-tetraaryl porphyrin 1 with three dodecyloxy-phenyl and one phenyl-ester was obtained in 5 to 6% yield (overall yield for porphyrin synthesis followed by metalation with Ni(acac) 2 ) by using a procedure described previously for other benzaldehydes. 8b The nickel porphyrins 8 and 12 with two external coordination sites were available. 9b

■ CONCLUSION

Porphyrin 2. A solution of lithium hydroxide (150 mg, 3.6 mmol)

and nickel porphyrin 1 (150 mg, 0.12 mmol) in dioxane/water (100 mL/5 mL) was refluxed under argon for 2 days. After evaporation of the solvents, the residue was filtered through a short silica gel column (eluent dichloromethane with 0.5% acetic acid). After evaporation and drying, the solid residue was taken up in toluene (100 mL), an excess of oxalyl chloride (5 mL) added, and after 1 h of stirring, the solution was heated up and the unreacted excess oxalyl chloride eliminated by distillation. After the mixture was cooled to RT, tin(IV) tetrachloride (1 mL) was added. After 0.5 h of stirring, dichloromethane (150 mL)

was added and the reaction was quenched with an aqueous sodium hydroxide solution (5%). The organic phase was washed several times with water and dried over sodium sulfate. After evaporation of the solvents, the desired compound 2 was obtained in 87% yield by precipitation from dichloromethane/methanol (127 mg, 0.102 mmol). 6, 159.4, 159.2, 159.1, 145.6, 145.1, 144.1, 143.6, 143.2, 141.1, 140.7, 139.6, 139.4, 135.5, 135.3, 134.6, 134.4, 134.3, 134.2, 133.7, 133.6, 132.9, 132.6, 132.5, 131.8, 131.7, 131.1, 127.4, 125.4, 121.5, 118.6, 113.3, 113.2, 108.6, 68.34, 68.31, 68.28, 31.97, 31.96, 29.75, 29.74, 29.73, 29.71, 29.70, 29.69, 29.68, 29.67, 29.53, 29.51, 29.44, 29.43, 29.42, 29.40, 26.21, 26.20, 22.74, 14.18. UV-vis (CH 2 Cl 2 ): λ max (ε) = 466 (108000), 650 (15000), 690 nm (9500sh from reflections taken from tree sets of 20 frames, each at 10 s exposure. The structure was solved by Direct methods using the program SHELXS-2013. 19 The refinement and all further calculations were carried out using SHELXL-2013. 20 The H atoms were included in calculated positions and treated as riding atoms using SHELXL default parameters. The non-H atoms were refined anisotropically, using weighted full-matrix least-squares on F2. A semiempirical absorption correction was applied using SADABS in APEX2 ;

M -1 •cm -1

Scheme 1 .

 1 Scheme 1. Preparation of the Starting Porphyrins 2-4 a

  140 the thermodynamic sinks of the reaction. Another synthetic 141 strategy was then envisaged. The porphyrin dimer 10 was first 142 treated with the Lawesson's reagent to afford the new 143 porphyrin dimer 12, now bearing two external enaminothiodimer 12 with two equivalents of the 146 functionalized porphyrin 7 gave again a mixture of products, 147 namely trimer 14 and a tetramer together with recovered 148 starting material. Single crystals of dimer 12 were obtained and 149 its X-ray structure represents the first structure of a porphyrin 150 dimer of this type. The porphyrins were extremely distorted 151 due to the very short Ni-N bond lengths (1.90 Å or less) and 152 f3

Figure 1 .

 1 Figure 1. 1 H NMR of the mixture of porphyrin dimers 5 and 6 (top spectrum) and of dimer 6 after heating in dichloroethane (pyrrolic region).

Figure 2 .

 2 Figure 2. X-ray structure of porphyrin 8. All hydrogen atoms (except those involved in H-bonds) and the tert-butyl groups are omitted for clarity.

  169 s714 in 77% yield (seeScheme 7). No tetramer was detected in 170 the MALDI mass spectrum and the very clean 1 H NMR 171 spectrum recorded at 50 °C confirmed this (see Figures 85 and

  172

  Figure 4 are shown the spectra of monomer 4, dimer 6, and

  However, monomer 4, and dimers 6 or 11 gave very nice f5 patterns, as shown in Figure 5. Bright spots represent the porphyrin core, whereas dark troughs are composed of dodecyloxy chains. Plausible molecular models are superimposed on each STM image. The vacant spaces between the alkyl chains (Figure 5c, 5d and 5e) may be filled with the coadsorbed solvent molecules. 15

  In conclusion, these preliminary experiments have shown that the assembly of metallo-porphyrins bearing external coordination sites on surfaces like HOPG is indeed possible. Work is in progress to prepare porphyrins bearing two external coordination sites and two aliphatic tails with different lengths. With these porphyrins it should then be possible to self-assemble them around metal ions directly on the surface to obtain finite

Figure 3 .

 3 Figure 3. (a) Top: Two views of the X-ray structure of porphyrin dimer 12. Five chlorobenzene solvent molecules, all hydrogen atoms and the tert-butyl groups are omitted for clarity. (b) Bottom: View showing the infinite dimer chains maintained together by intermolecular hydrogen bonds between the external enaminothioketone sites (chlorobenzene solvent molecules, meso-aryl groups and all hydrogen atoms, except the NH, are omitted for clarity).

Scheme 7 .

 7 Scheme 7. Preparation of Porphyrin Trimer 14

  × 3.0 × 0.9 = 18.9 nm 3 each compound is compared with the simulated spectrum. Measurement of self-diffusion coefficients were performed on a BRUKER 600 MHz spectrometerAvance III, equipped with a Bruker BBI probe developing a pulsed field gradient of 5.1 G/cm/A. Diffusion NMR data were acquired at 298 K using a Stimulated Echo pulse sequence with smoothed square bipolar z gradients in 34 increments. The duration of half gradients were 1 ms and the diffusion time was optimized for each sample between 120 and 140 ms. A recycle delay of at least 4 s was respected between each scans. DOSY spectra were generated by the dedicated module of NMRNotebook software, using Inverse Laplace Transform (ILT) driven by maximum entropy. STM Measurements. Two-dimensional structures of porphyrins with an external coordination site were visualized by using low-current STM (Nanoscope IIIa: Digital Instruments, CA) at HOPG/1phenyloctane interface. Pt/Ir wire (90/10) was mechanically cut for a STM tip. The compounds were dissolved in 1-phenyloctane at a concentration ranging from 0.75 to 1 mM, and the solution was deposited on HOPG (ZYB grade, NT-MDT, Russia). The reproducibility of the STM images was checked by using different tips and samples. All STM images were obtained in the constant current mode, and analyzed with the SPIP software (Image Metrology, Denmark). The STM images were corrected by a HOPG lattice under the physisorbed monolayer.

Figure 5 .

 5 Figure 5. STM images of compounds 4 (a, b), dimer 6 (c, d), and dimer 11 (e, f) at the HOPG/1-phenyloctane interface. One of the three dodecyloxy chains was omitted in the molecular models in (b) and (d) because it was not attached on the HOPG surface. A set of arrows indicates the underlying HOPG lattice direction. Tunneling conditions: (a) I = 2.0 pA, V = -1118 mV; (b) I = 1.0 pA, V = -1210 mV; (c) I = 1.1 pA, V = -1027 mV; (d) I = 1.5 pA, V = -1000 mV; (e) I = 1.0 pA, V = -1088 mV; (f) I = 1.0 pA, V = -992 mV.

  transmission factors: T min/ T max = 0.6231/0.7528. The SQUEEZE instruction in PLATON 21 was applied. The residual electron density was assigned to half a molecule of chlorobenzene. C 62 H 58 N 6 NiOS, M = 993.91 g•mol -1 , black prism, monoclinic, space group C2/c, a = 23.3114(6) Å, b = 14.1663(4) Å, c = 34.3124(10) Å, α = 90°, β = 90.3370(10)°, γ = 90°, V = 11331.0(5) Å 3 , ρ calc. = 1.165 g•cm -3 , Z = 8, T = 173 K, 3.792 < θ < 66.6, transmission factors T min /T max = 0.6231/ 0.7528, 54991 reflections measured, 9895 unique reflections, R 1 = 0.064, R 2 = 0.1693, GOF = 1.062. CCDC number: 1573335. Crystal Data 12. Transmission factors: T min/ T max = 0.4866/0.7528. The SQUEEZE instruction in PLATON was applied. The residual electron density was assigned to a molecule of chlorobenzene. The atoms C104, C105, and C106 are disordered over two positions with an occupancy ratio of 0.7/0.3. C 154 H 139 Cl 5 N 12 Ni 2 PdS 4 , M = 2687.07 g• mol -1 , black plate, monoclinic, space group P21/c, a = 20.6340(8) Å, b = 26.9516(16) Å, c = 28.2022(13) Å, α = 90°, β = 112.533(3)°, γ = 90°, V = 14486.5(13) Å 3 , ρ calcd = 1.232 g•cm -3 , Z = 4, T = 130(2) K, 4.0 < θ < 66.622. Transmission factors: T min /T max = 0.4866/0.7528, 109226 reflections measured, 25034 unique reflections, R 1 = 0.1637, R 2 = 0.4055, GOF = 1.200. CCDC number: 1573332.

Table 1 .

 1 NMR DOSY Data for Compounds 6,7, 11, and 14

Table 2 .

 2 Unit Cell Parameters Measured from STM Images

	compound	a (nm)	b (nm)	γ (deg)
	4	1.73 ± 0.13	3.57 ± 0.16	81 ± 4
	6	1.79 ± 0.22	3.65 ± 0.20	83 ± 3
	11	1.85 ± 0.18	3.72 ± 0.18	84 ± 5

DOI: 10.1021/acs.inorgchem.7b02422 Inorg. Chem. XXXX, XXX, XXX-XXX B

DOI: 10.1021/acs.inorgchem.7b02422 Inorg. Chem. XXXX, XXX, XXX-XXX I


Mary-Ambre Carvalho

Hervédekkiche

Mayumi Nagasaki

Yoshihiro Kikkawa

Romain Ruppert

Coordination-Driven Construction of Porphyrin Nanoribbons at a Highly Oriented Pyrolytic Graphite (HOPG)/Liquid Interface

Nanostructures were built at the solid/ liquid interface by self-assembly and/or coordination bonds. Metalloporphyrins bearing two external coordination sites and long alkyl chains allowed the self-assembly of the compounds on highly oriented pyrolitic graphite. After addition of a metal ion, long transition-metal linked porphyrin nanoribbons were obtained and visualized by scanning tunneling microscopy. In these porphyrin ribbons electronic delocalization is possible through the d orbitals of the connecting metal ions.

 Well-organized structures can be generated this way,

It was possible to assemble porphyrins bearing external coordination sites with metal ions in solution. [START_REF] Richeter | Syntheses and Optical and Electrochemical Properties of Porphyrin Dimers Linked by Metal Ions[END_REF] Strong electronic interactions were demonstrated between the c1 individual subunits in porphyrin dimers (see 1 in Chart 1, top) by using enamino(thio)ketones as ligands for the connecting ions. These external coordination sites can also be described as noninnocent ligands, thus allowing electronic delocalization over the entire molecule through the d orbitals of the linking metal ion, and this was corroborated by DFT studies. [START_REF] Dekkiche | Ultrafast Singlet Energy Transfer in Porphyrin Dyads[END_REF] Larger molecules were obtained by using porphyrins bearing two external coordination sites (examples of such but interactions between the individual units are generally rather small. [START_REF] Marschall | Random Two-Dimensional String Networks Based on Divergent Coordination Assembly[END_REF][START_REF] Heim | Self-Assembly of Flexible One-Dimensional Coordination Polymers on Metal Surfaces[END_REF][START_REF] Koepf | Surface-Tuned Assembly of Porphyrin Coordination Oligomers[END_REF][START_REF] El Garah | Nanopatterning of Surfaces with Monometallic and Heterobimetallic 1D Coordination Polymers: A Molecular Tectonics Approach at the Solid/Liquid Interface[END_REF][START_REF] Velpula | Hydrogen-Bonded Donor-Acceptor Arrays at the Solution-Graphite Interface[END_REF][START_REF] Sun | Surface Confined Synthesis of Porphyrin Containing Two-Dimensional Polymers: The Effect of Rigidity and Preferential Adsorption of Building Blocks[END_REF][START_REF] Sakamoto | Coordination Nanosheets (CONASHs): Strategies, Structures and Functions[END_REF][START_REF] Cai | Competition between Hudrogen Bonds and Coordination Bonds Steered by the Surface Molecular Coverage[END_REF][START_REF] Sosa-Vargas | Decorative" 2D Supramolecular Self-Assembly: Strategies towards Functional Surfaces for Nanotechnology[END_REF][START_REF] Amabilino | Supramolecular Chemistry at Surfaces[END_REF] Porphyrin monomers or presynthesized dimers such as 1 bearing long alkoxy chains self-assembled at the liquid/highly oriented pyrolytic graphite (HOPG) surface. [START_REF] Carvalho | Synthesis and Study at a Solid/Liquid Interface of Porphyrin Dimers Linked by Metal Ions[END_REF] To prepare higher oligoporphyrins linked by metal ions, porphyrins bearing two external coordination sites in addition to long meso-alkylphenyl groups are needed. The Ar meso substituents used in the present studies were phenyl groups bearing C 18 H 37 alkyl chains in the para positions (Ph-C 18 H 37 ). To obtain linear porphyrin ribbons, two series of symmetrical porphyrin 2 and 3 (with a center or a plane of symmetry) were used for the self-assembly process. Porphyrins such as 2b and 3b were obtained by following procedures used earlier for the preparation of similar porphyrins (see Supporting Information (SI) for experimental details).

Before starting the on-surface studies, we had noticed in the X-ray structure of compound 3c that linear tapes with molecules held together by hydrogen bonds were present f1 (see Figure 1). In the solid state, the molecules of 3c (bearing two Ph-C 12 H 25 meso groups) are arranged in such a way that the two external coordination sites were already preorganized to build square planar metal complexes if divalent metal ions are added. 35 Similar arrangements of porphyrin 3b at the solid/liquid interface were therefore expected, and the construction of linear ribbons with porphyrins linked by metal ions could be envisaged directly at the interface (according to the schematic representation shown in Figure 1 at the right).

Drop casting a phenyloctane solution of porphyrin 3b onto a HOPG surface gave a well-organized arrangement of the porphyrins at the solid/liquid interface. However, the additional interactions of the alkyl chains with the hexagonal HOPG surface led to a different arrangement of the molecules than in the solid-state structure. Two porphyrins were assembled in centro-symmetric pairs held together by hydrogen bonds, and all pairs were separated by a space (see f2 Figure 2a and2b).

Apart from this observation, the expected alternating arrangement of molecules was present at the interface and the porphyrins were aligned in rows separated by spaces filled by the alkyl chains that interact with HOPG. The distance (L 1 in Figure 2b) between the porphyrin rows was 2.8 ± 0.1 nm. 146 gave images with more sparsely packed areas (see Figure 3). At 147 a concentration of 0.1 mM, well-separated ribbons were 148 formed almost exclusively. The distance L 3 between them was 149 2.7 ± 0.1 nm, which was almost identical to the value of L 1 150 found for 3b without Ni(II). This result suggests that the 151 nanoribbons are adsorbed on the HOPG with the assistance of 152 alkyl chains, which are packed between the porphyrin rows.

153 It is not possible to compare studies in solution with studies 154 at the HOPG/liquid interface. However, in order to have an 155 idea of the electronic structure of the linear arrays at the 156 surface, solution experiments were performed at concen-157 trations close to those used earlier to generate the nanoribbons 158 at the interface. An equimolar amount of Ni(acac) 2 was added 159 to a toluene solution of porphyrin 3b in a thin quartz cell. The 160 electronic spectrum was recorded as a function of time. The To demonstrate that this approach was not limited to porphyrin 3b, centro-symmetric porphyrin monomer 2b (Ar = PhC 18 H 37 ) was also prepared. Once more, the self-assembly of the individual molecules was studied at the HOPG/liquid interface. The STM images obtained for millimolar phenyloctane solutions of 2b were not as well resolved as for f5 porphyrin 3b (see Figure 5a). However, to see if the oligoporphyrins could be constructed and visualized at the interface, the same procedure described earlier for 3b was followed. The STM images in Figure 5b clearly showed that, again, long linear nanoribbons of porphyrins were present at the interface. As observed before for 3b, large areas of the surface were occupied by alkyl chains for monomer 2b, but upon addition of Ni(acac) 2 , the porphyrin nanoribbons were again densely packed on the surface.

In conclusion, we have demonstrated that long linear porphyrin nanoribbons can be constructed at the HOPG/ liquid interface by using a combination of weak interactions (hydrogen bonds, van der Waals interactions with the surface and/or between alkyl chains) and also stronger coordination linkages. By combining these two factors, the best of both

115

  Figure 2d. Note that similar observations of linear ribbons were

Figure 1 .

 1 Figure 1. X-ray structure of compound 3c (Ar = PhC 12 H 25 ). Hydrogen bonds between the individual porphyrins are schematically drawn in light blue, d(Ni-Ni) = 14.339 Å (left). Right: expected formation of nickel(II)-connected porphyrin nanoribbons.

Figure 2 .

 2 Figure 2. STM height images of compound 3b (Ar = PhC 18 H 37 ) before (a, b) and after the addition of nickel(II) (c, d) at the HOPG/ phenyloctane interface (c = 1 mM). Pairs are highlighted with brackets in (b). Proposed molecular models were superimposed on the STM images in (b) and (d). In (d), the C18 alkyl chains were replaced by CH 3 groups. Arrows indicate the HOPG lattice directions. Tunneling conditions: (a) I = 1.1 pA, V = -103 mV, (b) I = 1.2 pA, V = -164 mV, (c) I = 1.6 pA, V = -989 mV, (d) I = 1.4 pA, V = -818 mV.

  f4 161 results of the first measurements are depicted in Figure 4. As 162 the complexation reaction progressed in solution, a bath-163 ochromic shift of all absorption bands was observed. The reaction could not be monitored to completion because the metal ion linked oligoporphyrins formed after addition of Ni(acac) 2 were also adsorbed on the quartz faces of the cuvette, giving rise to light-diffusion phenomena. However, the evolution of the first four spectra clearly indicated that strong interactions between porphyrin subunits linked by nickel(II) ions were present in the oligoporphyrins. A flattening and broadening of the bands in the Soret area (from 450 to 600 nm) and a bathochromic shift of the Q bands to wavelengths higher than 900 nm are the signature of strong electronic interactions between the aromatic π-electrons of each individual porphyrin induced by the participation of the d orbitals of the linking metal ions.

Figure 3 .

 3 Figure 3. STM height images of porphyrin 3b at different concentrations after addition of 1 equiv of nickel(II). Assuming a similar orientation of the alkyl chains (as in Figure 2b), a molecular model of linear ribbons with sparse packing was superimposed on the STM image in (f). Concentrations of the solution are indicated on each STM image. Tunneling conditions: (a) I = 1.0 pA, V = -752 mV, (b) I = 2.0 pA, V = -948 mV, (c) I = 1.9 pA, V = -740 mV, (d) I = 3.8 pA, V = -823 mV, (e) I = 2.4 pA, V = -942 mV, (f) I = 3.1 pA, V = -744 mV, (g) I = 1.0 pA, V = -1000 mV, (h) I = 5.0 pA, V = -1000 mV.

Figure 4 .

 4 Figure 4. Electronic spectra of monomer 3b as a function of time after addition of 1 equiv of Ni(acac) 2 in toluene (t = 0, 5, 20, and 60 min).