B R Akula

V A Yastrebov

J Vignollet
email: julien.vignollet@safrangroup.com

G Cailletaud
email: georges.cailletaud@mines-paristech.fr

5-19 Mai 2017, Presqu'île de Giens (Var) Domain tying across virtual interfaces: coupling X-FEM with the Mortar method

Keywords: mortar method, extended finite element method, coupling

In this paper we propose a unified framework of the mortar domain decomposition method and extended finite element method (X-FEM). This framework allows to deal in an efficient manner with two cumbersome aspects of the finite element methods, namely incompatible interface discretizations and internal discontinuities. Features of mortar methods in the context of mesh tying, and of X-FEM in the context of void/inclusion treatment are exploited to formulate the weak coupling along an inclusion's surface and the virtual surface of the host mesh. It has a potential to address a multitude of problems from accurate substructuring to efficient wear simulation in contact problems.

Introduction

The finite element methods (FEM) is one of the widely used methods to solve mechanical engineering problems. The FEM is extremely flexible and can handle complex geometries, highly non-linear material models, large deformations and multi-physical problems. With the emergence of high performance computing its applicability to larger and more complex problems was only made easier. Among the wide spectrum of engineering applications, the class of problems dealing with interface mechanisms (composites, fracture, contact) is complex both with regard to its mathematical description and the difficulties arising in numerical treatment. Understanding and accurate modeling of these phenomena would contribute significantly to the progress across all fields of mechanical engineering scaling from material microstructure modeling to complex structural problems involving frictional contacts. However, the classical FEM is constrained in handling many problems encountered in interface mechanics. These limitations stem from its inability to independently represent the interface geometry from the underlying discretization. One such constraint imposed by conventional FEM methods is the requirement of conformal meshes along the interfaces. This lays significant restrictions on the possible complexities of geometries that can be handled.

The family of mortar methods has a potential to address these limitations. These are a subclass of domain decomposition methods (DDM), that are tailored for large class of non-conformal spatial interface discretizations [START_REF] Bernardi | A new nonconforming approach to domain decomposition : the mortar element method[END_REF][START_REF] Faker | The mortar finite element method with Lagrange multipliers[END_REF]. These were originally introduced as DDM for spectral elements [START_REF] Belgacem | A spectral element methodology tuned to parallel implementations[END_REF][START_REF] Bernardi | Coupling finite element and spectral methods : First results[END_REF]. The coupling of different physical models, discretization schemes, or non-matching meshes/discretizations along interior interfaces of the domain can be ensured by mortar methods. The mathematical optimality and applicability of the mortar methods in spectral and finite element frameworks were studied extensively for elliptic problems [START_REF] Bernardi | Coupling finite element and spectral methods : First results[END_REF][START_REF] Belgacem | A spectral element methodology tuned to parallel implementations[END_REF][START_REF] Wohlmuth | Discretization Methods and Iterative Solvers Based on Domain Decomposition[END_REF].

The other notable class of problems are those involving discontinuities. The fundamental requirement of the classical FEM is that the mesh has to conform to the geometry and with this it is possible to handle complex geometries comfortably. Since FEM strongly depends on the smoothness of the approximation polynomials, non-smooth behavior like high gradients or singularities in the stress and strain fields, or strong discontinuities in the displacement field as encountered with cracked bodies, pose a computational challenge to attain optimal convergence. However, having conformal geometries across different domains or contacting solids is not always possible or might demand computationally intensive re-meshing and field transfer procedures.

An attractive alternative is the partition of unity based enrichment method (PUM) [START_REF] Babuška | The partition of unity method[END_REF] for disconti-nuous fields referred to as the extended FEM (X-FEM) in [START_REF] Everett | An extended finite element method with discontinuous enrichment for applied mechanics[END_REF]. In X-FEM, special enrichment functions are added to the finite element approximation using the framework of PUM, to account for non-smooth behavior such as high gradients or singularities in the stress and strain fields, or strong discontinuities in the displacement field as in the case of cracked bodies without compromising on the optimal convergence [START_REF] Ferté | Interface problems with quadratic X-FEM : design of a stable multiplier space and error analysis[END_REF].

In an attempt to combine these two requirements of incompatibilities and discontinuities, we propose a unified solution to these problems. As illustrated in Figure 1, we propose to tie the virtual X-FEM boundary Γ v (discontinuity) introduced on the coarse mesh domain, with the outer boundary Γ g of the finely meshed inclusion which will be embedded into the X-FEM void of the coarse mesh. The enriched displacement field of the coarse mesh along Γ v is constrained to be equal to the displacement of the boundary (Γ g ) of the fine mesh via the mortar method. The planned approach for this unified method includes :

-using the level set method (LSM) [START_REF] Albert | Level set methods and fast marching methods : evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science[END_REF] to describe the contour of the discontinuity, -using X-FEM Heaviside enrichment (topological) for the nodes of the elements intersected by the discontinuity, [10] -using the mortar method [START_REF] Michael | A 3d mortar method for solid mechanics[END_REF] to tie both the discretized models along the boundary of the finermesh patch and the virtual boundary of this patch, which is represented in the parent (coarse) mesh by a level set.

FIGURE 1 -A finely meshed inclusion with border Γ g is embedded in a coarse mesh and is tied with the corresponding virtual surface Γ v using the mortar method : (a) coarse host mesh, (b) fine patch to be embedded, (c) the resulting combined mesh.

The X-FEM methods are extensively used in applications such as crack modeling, inclusion and void effects, shock wave front and oxidation front propagation, and other discontinuities (both strong and weak). Coming to the mortar methods they have been the topic of research in a multitude of applications like the non-overlapping domain decomposition methods (DDM), contact, wear etc. There are few works on similar lines where the advantages of these two methods are harnessed in a single application. To name a few we refer to the following works, an exhaustive pointers on the topic can be found in references therein. In [START_REF] Laborde | High-order extended finite element method for cracked domains[END_REF], the authors try to use the node based bonding between domains with crack. In [START_REF] Chahine | Etude mathématique et numérique de méthodes d'éléments finis étendues pour le calcul en domaines fissurés[END_REF][START_REF] Chahine | A non-conformal eXtended Finite Element approach : Integral matching Xfem[END_REF], the authors used the mortar type integration methods to glue a fine mesh surrounding the crack and the coarser mesh domain, where the crack is represented in the X-FEM formulation. But their coupling is limited to existing explicit interfaces, i.e. the inclusion does not cut the elements of the coarse domain but is aligned to the element edges. A dual mortar contact formulation integrated into X-FEM fluidstructure interaction approach is presented in [START_REF] Mayer | 3d fluid-structure-contact interaction based on a combined XFEM FSI and dual mortar contact approach[END_REF]. In [START_REF] Cuba Ramos | A new formulation for imposing Dirichlet boundary conditions on non-matching meshes[END_REF], the authors have emphasized on a way to impose boundary conditions on the internal boundaries represented by virtual X-FEM interfaces. The proposed method of coupling mortar and X-FEM competes with the volumetric coupling via the Arlequin method [START_REF] Hashmi | The Arlequin method as a flexible engineering design tool[END_REF] close to the overlapping DDM [START_REF] Tony | Domain decomposition algorithms[END_REF], where unlike the conventional non-overlapping DDM the artificial internal boundary condition is provided by its neighboring subdomains. The convergence of the solution on these internal boundaries ensures the convergence of the solution in the entire solution domain.

With the emphasis laid on the interface discretizations, many applications such as substructuring, arbitrary gluing, mesh construction of complex micro structures, localized mesh refinement near the crack tip, general static and dynamic mesh refinement, and potentially others can be of particular interest with this unified method of the X-FEM and the mortar. However, the applications are not limited only to mesh tying. In analogy to how the mortar method, initially used for tying was extended to contact problems by Belgacem et al. [START_REF] Belgacem | The mortar finite element method for contact problems[END_REF], Fischer and Wriggers [START_REF] Fischer | Frictionless 2d Contact formulations for finite deformations based on the mortar method[END_REF][START_REF] Fischer | Mortar based frictional contact formulation for higher order interpolations using the moving friction cone[END_REF], and Popp et al. [START_REF] Popp | Mortar Methods for Computational Contact Mechanics and General Interface Problems[END_REF] the method suggested here can be used to solve contact problems between a virtual surface (represented by the X-FEM) and an explicit surface of the homologue solid [START_REF] Vladislav | Numerical methods in contact mechanics[END_REF]. This extension would allow to treat efficiently wear problems without costly remeshing techniques. The document is structured as follows : In Section 2, we present the core philosophy of the mortar method with respect to the tying problem and the X-FEM with respect to the modeling of voids (closed discontinuity). The equations necessary for FEM framework are given for the unified method in Section 2.3. In Section 3 we show few initial results from the mortar mesh tying problem. In Section 4, we give the concluding remarks on our objective and state the prospective works.

Methodology

Mortar mesh tying

Let us consider a classical boundary value problem for a domain Ω. We assume that the domain is split into n subdomains Ω = ∪Ω i which are glued across their interfaces Γ i j . The resulting optimization problem should then include displacement-equality constraints imposed on glued boundaries, e.g. they take form g(u) = u i ju ji = 0, where u i j and u ji are the displacements of initially coinciding points of subdomains Ω i and Ω j , respectively. These constraints can be incorporated into the system energy functional (W (u)) using the Lagrange multiplier method forming a Lagrangian :

L(u,λ) = W (u) + ∀i j Γ i j λ • g(u) dΓ i j , (1)
where λ are the Lagrange multiplier functions representing interface tractions needed to ensure domain tying. Applying the principle of virtual work, equilibrium is ensured when the total virtual work on the solution path is zero : δW ext (u, δu) -δL(u, δu, λ, δλ) = 0, where δW ext corresponds to the virtual work of external forces, and δu, δλ are test functions chosen from respective functional spaces for primal and dual quantities. This weak form can be directly used in the discretized finite element framework formulated for an extended vector of unknowns including primal (displacements) and dual (Lagrange multipliers) degrees of freedom : isoparametric elements are used with similar interpolation functions for geometry and displacement on both tied subdomains, whose surfaces will be termed mortar and non-mortar ones ("m" and "nm" indices are used for them, respectively) :

u m (t, ξ, η)| Γ m i j = n m ∑ k=1 N m k (ξ, η) u m k (t), u nm (t, ζ, µ)| Γ nm i j = n nm ∑ l=1 N nm l (ζ, µ) u nm l (t), (2)
where n m is the total number of nodes on the discretized mortar side Γ m i j of the interface, and n nm is on the discretized non-mortar side Γ nm i j . The classical isoparametric shape functions N m k and N nm l (which are not necessary similar or of the same order) defined on the parametric space {ξ, η} and {ζ, µ} are used. The Lagrange multipliers (defined on the mortar surface) are also interpolated using classical shape functions

Φ : λ(ξ, η,t) = n lm ∑ i=1 Φ m i (ξ m , η m ) λ i (t). (3)
Here n lm refers to the number of nodes of the mortar side of the interface that carry additional dual dofs. The virtual work of the equality constrains embedded in the Lagrangian (1) takes the following form :

δW g = Γ m i j λ • δg(u) dΓ i j + Γ m i j δλ • g(u) dΓ i j (4)
Using Eq. 2, Eq. 3 in Eq. 4 results in the discretized interface virtual work (Eq. 5)

δW g = n lm ∑ i=1 n m ∑ k=1 λ i • δu m k Γ m i j Φ i N m k dA 0 D ik - n lm ∑ j=1 n nm ∑ l=1 λ j • δu nm l Γ m i j Φ j N nm l dA 0 M il + n lm ∑ i=1 n m ∑ k=1 δλ i • u m k D ik - n lm ∑ j=1 n nm ∑ l=1 δλ j • u nm l M jl (5)
Evaluation of the interface mortar integrals [D] and [M] involves first the determination of the integration domain by projecting the glued surface elements on an auxiliary plane and their clipping, and second, triangulation of the resulting polygon for Gauss integration (details can be found in [START_REF] Michael | A 3d mortar method for solid mechanics[END_REF][START_REF] Popp | Mortar Methods for Computational Contact Mechanics and General Interface Problems[END_REF]). The coupling achieved with mortar method is through weak satisfaction of the equality constraints, which is much more accurate than the conventional strong node-wise coupling like in the node-to-segment method [START_REF] Wriggers | Finite element algorithms for contact problems[END_REF].

X-FEM for void modeling

The model presented in Figure . 1(a) involves a strong discontinuity (Γ v ) encompassing a square void. Across this virtual interface, the displacement field has a jump. To take this jump into account, the nodes of blending elements (elements intersected by the discontinuity) are enriched by multiplying the classical shape function of the node by a Heaviside function. The effective support size of these elements for integration is reduced to its volume fraction [START_REF] Sukumar | Arbitrary branched and intersecting cracks with the extended finite element method[END_REF]. For the elements which lie entirely within the void, their dofs are removed from the global system. Here we use a Heaviside function defined for a level set function φ as

H(X) = 1, φ(X) > 0 0, φ(X) < 0 . (6)
Within every blending element, the interface Γ v , defined by the zero level-set function φ(ξ, η) = 0, can be redefined as : ξ = ξ v (η). The shape function of the enriched element then become N i (X) = H(X) N i (X). These enriched blending elements bring in the change in the volume on which the total system potential is integrated. The blending elements are remeshed to integrate the virtual energy integrals only in the effective support zones (see Figure . 2). The displacement on the surface Γ v within every blending element is given by classical interpolation function :

ũ(ξ, η,t)| Γv = ∑ i N i (ξ v (η), η)u i (t).

Mortar coupling along level set

The mortar and the X-FEM methods can be combined to ensure weak mortar-type coupling along virtual interface Γ v and the inclusion surface Γ g . The enriched blending elements, the standard elements of the coarsely discretized domain Ω o the elements from discretized inclusion domain Ω i along with the new mortar-interface elements are to be incorporated into the virtual work formulation of the global system [see Figure . 1(c)] :

Ω o \Ω i σ : δε dΩ o δW int o + Ω i σ : δε dΩ i δW int i + Γ g δ [ λ • g( ũ, u) ] dΓ δW tie oi = Γ f o t 0 • δũ dΓ + Γ f i t i • δu dΓ δW ext , (7)
where the tilde notation is used for the coarser host mesh quantities. Note that the volume forces are omitted for simplicity. The displacement test functions for both domains are chosen from appropriate functional spaces and have to satisfy Dirichlet boundary conditions. The contribution δW int o is evaluated only outside the inclusion domain by performing additional triangulation [START_REF] Sukumar | Modeling holes and inclusions by level sets in the extended finite-element method[END_REF] as shown in Figure . 2. Of special interest is the virtual work contribution from the tied interface boundaries Γ v and Γ g . Typical to the mortar terminology we classify the virtual boundary Γ v as the non-mortar side and the inclusion boundary Γ g as the mortar side, where the mortar integrals are evaluated as in Eq. ( 5), where the number of nonmortar nodes n nm are typically the intersection points of the element edges of the blending elements and Γ v . The intersections are not conventional FEM nodes, and so the interpolated values of the element displacement field ũ of Ω o is constrained to be integrally equal to the displacement on Γ g . The mortar algorithms of projections, clipping and triangulation are all done with these intersections and the faces of Γ g .

Examples

In Figure . 3, the mortar methods are used to tie the two domains along the non-matching curved interface. The equality constraints are enforced to a high degree precision which is reflected in a smooth normal stress profiles σ yy and σ xx across the curved (Figure.

Conclusions and perspectives

The proposed method makes the FEM flexible in its ability to treat efficiently two distinct and fully non-conformal domains, i.e. incompatibilities both at the interface and within the bulk of elements. It encompasses the intricacies of a diverse and well established fields of the non-overlapping domain decomposition methods and X-FEM, working together to simplify and provide a good alternative solution compared to the conventional ones (node-wise coupling and/or remeshing). The spectrum of applications is wide and includes mesh tying and substructuring and a generic framework for frictional contact including the wear model handled by X-FEM.

The next step is the extension of the implementation to three-dimensional problems. In perspective the method will be taken forward into the realm of parallel computing, thus enabling to solve large and complex problems in a flexible and accurate way.

Acknowledgement

FIGURE 2 -

 2 FIGURE 2 -Triangulation of the blending element area which lies out the void defined by the level set function φ. The boundary Γ v intersects the element edges (black points). The dofs of nodes marked x are removed from the global system.

uFIGURE 3 -

 3 FIGURE 3 -Mortar method is used for mesh tying along the non-matching (a) curved and (b) flat interfaces.

The authors acknowledge financial support of the ANRT (grant CIFRE no 2015/0799).

Références


Carlos Antonio

Albuquerque Pinheiro

Claire Roiland

Philippe Jehan

Gilles Alcaraz

Solventless and Metal-Free Synthesis of High-Molecular-Mass Polyaminoboranes from Diisopropylaminoborane and Primary Amines

Keywords: polyaminoboranes, inorganic polymers, aminoboranes, diisopropylaminoborane, BN compounds

Solventless and Metal-Free Synthesis of High Molecular Mass

Polyaminoboranes from Diisopropylaminoborane and Primary

Amines

Carlos Antonio De Albuquerque Pinheiro, [a] Claire Roiland, [a] Philippe Jehan [b] and Gilles Alcaraz* [a] Abstract: The solventless reaction of diisopropylaminoborane with n-butylamine, at room temperature, leads to a mixture of well identified B(sp2)H, B(sp3)H2 and B(sp3)H3 boron containing species. At low temperature, the reaction outcome is completely modified leading selectively to the formation of high mass polybutylaminoborane. Extended to a variety of primary amines under solventless conditions and at low temperature, this reaction provides a new, efficient and direct metal-free access to high molecular mass polyaminoboranes in good to high yield, in very mild conditions.

The field of inorganic polymers incorporating main group elements in their backbone is a very promising research area that remains relatively underdeveloped due to the lack of efficient synthetic methods available to connect inorganic repeating unit together successively. [1] Among them and over the last decade, polyaminoboranes that incorporate a BN unit in their backbone, (RHN-BH2)n (I-R), have represented an area of considerable interest for novel BN-containing polymers and materials. [2] They strongly benefited from the development of new systems involved in dihydrogen release derived from catalyzed dehydrocoupling strategies. [3] To date, the preparation of linear polyaminoboranes is largely dominated by the catalyzed dehydropolymerization of a restricted number of primary amine-boranes or ammoniaborane [2a, 4] involving expensive metal catalysts and going hand-in-hand with the production of dihydrogen. The lack of simple and efficient synthetic approaches still hampers their promising development, the only metal catalyst-free syntheses resulting in the deprotonation of sophisticated amine-boronium cations and related polar-covalent analogs, [5] leading only to low molecular mass polyaminoboranes along with salts.

As a part of our program on the aminoboranes chemistry, we report for the first time the use of diisopropylaminoborane ( i Pr2NBH2, DIAB) [6] as a convenient "BH2" transfer reagent in the presence of a variety of primary amines for an original and efficient metal-free and solventless synthesis of very high molecular mass polyaminoboranes (RNH-BH2)n (I-R) with neither dihydrogen production nor salt formation, in very mild conditions.

Solventless reaction of n-butylamine with diisopropyl-aminoborane at room temperature, under stoichiometric conditions, results in an exothermic reaction leading to the complete disappearance of the starting borane reagent with formation of diisopropylamine along with a mixture of B(sp2)H, B(sp3)H2 and B(sp3)H3 boron containing species in a ca 1:2.5:1.4 integration ratio in the 11 B{ 1 H} NMR, respectively (Figure 1 and Scheme 1). Among them, diisopropylamine-borane (IV: δ( 11 Β) -21.4 ppm) and n-butylamine-borane (VI-Bu: δ( 11 Β) -19.4 ppm) for the B(sp3)H3 compounds and N,N',N"-tributylcycloborazane as a mixture of two isomers (II-Bu: δ( 11 Β) -5.9, -6.9 ppm) [7] for the B(sp3)H2 species could be unambiguously identified. In the presence of a twofold excess of n-butylamine, the distribution of the final compounds is modified with a decrease of II-Bu and the almost complete disappearance of IV, as seen by 11 B NMR (see Figure S1 in the Supporting Information). [8] Careful examination of the data revealed the doublet at δ( 11 Β) 27.7 ( 1 JBH obs = 113 Hz) that collapsed upon proton decoupling results from a mixture of (diisopropylamino)(butylamino)borane (III-Bu), and bis(butylamino)borane (V-Bu) that were both prepared separately. [9] Although suspected, N,N',N"-tributylborazine could not be unambiguously detected (see Figure S1 and S4 in the Supporting Information). In the 1 H{ 11 B} NMR spectrum of the reaction mixture, compounds III-Bu and V-Bu exhibit for the BH group a deshielded doublet at δ 3.92 ( 3 JBH-NH = 8.5 Hz) and a broad signal at δ 3.76, respectively. The use of a twofold excess of n-butylamine results in a signal of greater intensity at δ 3.76 whereas the one at δ 3.92 has almost disappeared, which in agreement with the 11 B NMR data recorded both in stoichiometric and super-stoichiometric conditions (see Figure S5 in the Supporting Information).

Surprisingly, performing the same reaction at low temperature led to a different reaction outcome (Scheme 2). With a stoichiometric amount of n-butylamine at -40 °C, the formation of a white precipitate was initially observed to end up with a gel-like solution. Here again, formation of diisopropylamine was clearly detected along with the expected side products identified as II-Bu, III-Bu, IV, V-Bu and VI-Bu (See Figure S2-S3 in the supplementary information). [10] After evaporation of the volatiles under vacuum and workup, a white solid of poor solubility in most common organic solvents was obtained. Solution NMR could however be recorded in deuterated chloroform and the 11 B{ 1 H} NMR spectrum exhibits a broad signal at δ -7.6 in agreement with previously reported data [2a] (see Figure S48-S49 in the Supporting Information). Further analyses of the isolated material revealed its polymeric nature identified as polybutylaminoborane I-Bu. Scheme 2. Reaction of DIAB with primary amines under stoichiometric conditions, at low temperature.

The reaction results in the transfer of the BH2 group from DIAB to the primary amine, probably via the transient formation of the amine-DIAB adduct, [11] followed by the elimination of diisopropylamine and formation of the corresponding alkylaminoborane that further polymerizes to afford polybutylaminoborane I-Bu in 78% isolated yield. [12] In the infrared spectrum, the presence of characteristic NH (ν = 3250 cm -1 ), CH (ν = 2960cm -1 ) and BH (ν = 2392, 2297 cm - 1 ) stretching vibrations could be clearly observed in agreement with the infrared data reported for polybutylaminoborane obtained by catalytic dehydrocoupling. [2a] The size-exclusion chromatography (SEC) analysis of the polymer in THF presented a bimodal distribution consisting of a minor narrow peak of low molecular mass oligomers and a major broad peak with a shoulder corresponding to high molecular mass polybutylaminoborane (I-Bu: Mw = 5450000 g.mol -1 -PDI = 6.9). I-Bu showed a good stability at room temperature as monitored by SEC analysis showing no evidence of polymer degradation after one week stirring in THF (see Figure S23 in the S.I.). Electrospray-ionization mass spectroscopy (ESI-MS) analyses indicated that the material displays a linear polymeric distribution with a n-butylaminoborane (n-BuNH-BH2) repeating unit (see Figure S33-S34 in the Supporting Information). The solid state 13 C{ 1 H} NMR spectroscopy reveals the presence of four characteristic broad carbon signals at δ 13.5, 20.2, 29.8 and 49.6 (Figure 2), ascertaining the presence of the N-butyl group of the isolated polybutylaminoborane (I-Bu).

Accepted Manuscript

To go one step further and take advantage of the simplicity of the method involving diisopropylaminoborane as a convenient "BH2" transfer reagent, a variety of polymers (Figure 3) and random copolymers were prepared, at the gram scale, in good to high yields, from primary amines (table 1, entry 1-6) and mixtures of primary amines (table 1, entry 7-8), respectively.

All the prepared polymers and copolymers have been characterized by SEC when soluble, solution state 11 B{ 1 H} NMR in deuterated chloroform, 13 C{ 1 H} solid state NMR, infrared spectroscopy (Figure 4), thermogravimetric and elemental analyses, ESI-MS and solid state 11 B{ 1 H} NMR in the case of I-H and I-Me (Figure S43 and S44) (See the Supporting Information). Except for polyaminoborane I-H that was found insoluble in our conditions, the SEC analysis reveals the obtained polyaminoboranes display higher molecular mass compared to that of analogous polyaminoboranes prepared from their corresponding amine-boranes via metal catalyzed dehydropolymerization [2a, 4b] or by deprotonation of amineboronium cations and related compounds. [5] The relatively high PDI values measured suggest an uncontrolled polymerization process. [14] In the case of I-H and I-Me, the infrared spectra (Figure 4) are similar to those reported by Manners, Staubitz et al. under iridium catalysis, [2a, 4b] by Schneider et al. with ruthenium in the case of I-H, [15] and by Beweries et al. for the iron hydride catalyzed dehydrocoupling of methylamine-borane for I-Me. [4f] It is worth noting that this synthesis enables for the first time the preparation of polyaminoboranes incorporating unsaturated substituents at the nitrogen as illustrated with polyallylaminoborane (I-allyl) obtained in 48% isolated yield.

In summary, we have shown that the synthesis of very high molecular mass polyaminoboranes can be performed at the gram scale, in good to high yields, by simply reacting diisopropylaminoborane with a wide range of primary amines or mixture of primary amines under solventless and metal-free conditions, at low temperature, without production of dihydrogen.

The role of diisopropylaminoborane as an efficient "BH2" transfer reagent has been clearly demonstrated and efforts are currently underway to rationalize the reaction mechanism and the role of the temperature on the reaction outcome.

With the isolation of polyallylaminoborane (I-allyl), this study provides, for the first time, new insights into the design of sidechain unsaturated amine containing polyaminoboranes. Detailed studies of the synthetic potential of this new family of polyaminoboranes are also in progress.

Table 1. Synthesis and molecular mass data for polyaminoboranes.

Entry

Polymeric material I-R [a,b] (RNH-BH2)n (%) [c] Mw (g.mol 78 5450000 [e] 6.9 6 R = allyl [a] Homopolymer. [b] Copolymer. [c] Isolated yield. [d] Data given from SEC analysis in THF (with 0.1 w/w% nBu4NBr) and using standard calibration against polystyrene. [13] [e] Extrapolated data. [f] Not determined; insoluble product.

Accepted Manuscript

  Figure 1. 11 B{ 1 H} NMR spectrum of the crude reaction of n-butylamine with diisopropylaminoborane under stoichiometric conditions at room temperature.

Figure

  Figure 2. Solid state 13 C{ 1 H} NMR of polybutylaminoborane (I-Bu).

Figure 3 .

 3 Figure 3. Polyaminoboranes I-H, I-Et, I-Bu, I-allyl and random copolymer I-Et/Bu.

Figure 4 .

 4 Figure 4. Infrared spectra of polymers I-H, I-Me, I-Et, I-Pr, I-Bu and random copolymers I-Et/Bu and I-Et/2Pr.

avenue du Général Leclerc, 35042 Rennes cedex (France) E-mail: gilles.alcaraz@univ-rennes1.fr Accepted Manuscript

Acknowledgements

We thank the C.N.R.S.and the University of Rennes 1. C.A.D.A.P. thanks the Brazilian Ministry of Education and the CAPES for his fellowship. We thank S. Cammas-Marion for SEC measurements. A.G. thanks S. Guillaume for fruitful discussions.

The structures of these species have been ascertained by direct comparison with authentic samples prepared separately and/or by comparison with NMR data from available literature (See the Supporting Information).

[9] 11 B NMR chemical shift and