Jean-Louis Merrien
email: jmerrien@insa-rennes.fr

Tomas Sauer
email: tomas.sauer@uni-passau.de

Extra Regularity of Hermite Subdivision Schemes

Keywords: Taylor operator, Hermite subdivision, Vector Subdivision, Smoothness

Hermite subdivision schemes act on vector valued data that is not only considered as functions values of a vector valued function from R to R r , but as evaluations of a function and its consecutive derivatives. Starting with data on r (Z), r = d + 1, interpreted as function value and d = r -1 consecutive derivatives, we compute successive iterations to define values on r (2 -n Z) and an r -vector valued limit function for whose first component C d -smoothness is generally expected.

In this paper, we construct Hermite subdivision schemes such that, beginning with the same data, it is possible to reach a limit function with smoothness d +p for any p > 0. The result is obtained with a generalized Taylor factorization and a smoothness condition for vector subdivision schemes.

Introduction

Subdivision schemes create curves or surfaces by applying stationary refinement rules on data defined on the integers. This refinement process extends the data to a discrete function defined on the half integers, quarter integers and so on, until eventually the values become so dense that one could speak of a limit function. Stationary subdivision [START_REF] Cavaretta | Stationary subdivision[END_REF] means that any step of the subdivision process is a stationary process which defines data on next level in a convolution like way as g n+1 = S a g n := β∈Z a(• -2β) g n (β); in this expression, a stands for the mask, a finitely supported sequence and the values g n on different iteration levels are normalized to be discrete functions g n : Z → R with the understanding that g n (α) stands for a value at 2 -n α, α ∈ Z. Such subdivision schemes with scalar coefficients can be trivially extended to the generation of curves by acting componentwise, on vector data, resulting in the iteration

g n+1 = S a g n := β∈Z a(• -2β) g n (β), g n : Z → R r .
Vector subdivision goes one step further by applying a matrix valued mask to the data, allowing for interaction between the components of the data vectors:

g n+1 = S A g n := β∈Z A(• -2β) g n (β), A : Z → R r ×r ,
again with the assumption that A is finitely supported. Finally, in Hermite subdivision the components of the vector f n (α) ∈ R r , r = d + 1, are considered as function value and d consecutive derivatives of a function at 2 -n α. Due to the chain rule, the refinement scheme now takes a level dependent form, that is, the operator depends on the iteration level n as

f n+1 = D -n-1 S A D n f n = β∈Z D -n-1 A(• -2β) D n f n (β), D =       1 1 2 . . . 2 -d      
.

All such types of subdivision schemes are covered extensively in the literature, see e.g. [START_REF] Conti | Hermite subdivision schemes, exponential polynomial generation, and annihilators[END_REF][START_REF]Convergence of level dependent Hermite subdivision schemes[END_REF][START_REF] Conti | Approximation order and approximate sum rules in subdivision[END_REF][START_REF] Cotronei | A note on Hermite multiwavelets with polynomial and exponential vanishing moments[END_REF][START_REF] Han | Noninterpolatory Hermite subdivision schemes[END_REF][START_REF] Moosmüller | C 1 analysis of Hermite subdivision schemes on manifolds[END_REF], just to name a few specific references on Hermite subdivision schemes. Standard questions to consider are the convergence of the iterative schemes and the regularity of the associated limit functions. This is well-known to be closely related to the way how the subdivision operators act on polynomial sequences, a property that can in turn be conveniently characterized by means of operator factorizations.

In the next section, we will review the basic definitions of vector and Hermite subdivision schemes and the appropriate notions of convergence. We will point out what vector subdivision schemes and Hermite subdivision schemes have in common and where they differ. Introducing Taylor operators, we will also present the transformation of a Hermite subdivision scheme into vector subdivision schemes via the Taylor factorizations. We illustrate the different schemes with an example where, in particular the limit functions are shown.

We will see that the definitions of the smoothness of the two schemes are significantly different. By construction, the limit function

φ =    φ 0 . . . φ d    ,
of a Hermite scheme satisfies φ j = φ ( j ) 0 for j = 0, . . . , d , so that φ 0 ∈ C d whenever all components of φ are continuous. The limit of a Hermite subdivision scheme always has to have a certain amount of regularity in the sense of differentiability. In this paper we investigate the question under which circumstances we can have extra regularity, that is, φ 0 ∈ C d +p for some integer p ≥ 0. We will relate this to a combined factorization, one due to the nature of Hermite subdivision schemes and one coming from a smoothness condition for vector subdivision schemes that is due to [START_REF]On vector subdivision[END_REF]. A similar approach has been used to characterize overreproduction of polynomials as an algebraic properties of the matrix symbols in [15]. Section 3 will be devoted to the B-spline case. Here the splines are obtained as the limit of, firstly, a scalar subdivision scheme, secondly, a Hermite subdivision scheme. The smoothness of such functions are well known and can be as large as wanted.

In the final Section 4, we will give a generic construction to obtain convergent Hermite subdivision schemes with any order of extra smoothness.

Vector and Hermite subdivision schemes

We begin by fixing some notation to describe subdivision schemes. Vectors in R r , r ∈ N, will generally be labeled by lowercase boldface letters: y = y j j =0,...,r -1 or y = y ( j ) j =0,...,r -1 , where the latter notation is used to highlight the aforementioned fact that in Hermite subdivision the components of the vectors correspond to consecutive derivatives. Moreover, in Hermite subdivision we denote the highest derivative by d , so that throughout the paper we will always have the relationship r = d + 1.

Matrices in R r ×r will be written as uppercase boldface letters such as A = a j k j ,k=0,...,r -1 . The space of polynomials in one variable of degree at most n will be written as Π n , with the usual convention Π -1 = {0}, while Π will denote the space of all polynomials. Vector sequences will be considered as functions from Z to R r and the vector space of all such functions will be denoted by r (Z). For a sequence y ∈ r (Z), the forward difference is defined as ∆y := y(• + 1)y, and iterated to

∆ j y := ∆ ∆ j -1 y = ∆ j -1 y(• + 1) -∆ j -1 y(•) = j k=0 (-1) k-j j k y(• + k), j ≥ 1.
We use 0 to indicate zero vectors and matrices. If we want to highlight the dimension of the object, we will use a subscript like 0 r , but to avoid too cluttered notation, we will often drop them if the size of the object is clear from the context. For a finitely supported sequence of matrices A ∈ r ×r 0 (Z), called the mask of the subdivision scheme, we define the associated stationary subdivision operator

S A : g → β∈Z A(• -2β) g (β), g ∈ r (Z).
Using potentially different masks A n ∈ r ×r 0 (Z), n ∈ N, these operators can be iterated into a subdivision scheme that creates sequences g n ∈ r 0 (Z), n ≥ 0,

g n+1 := S A n g n := β∈Z A n • -2β g n (β), n ≥ 0, (1)
from a given g 0 . An important algebraic tool for stationary subdivision operators is the symbol of the mask, the matrix valued Laurent polynomial

A * (z) := α∈Z A(α) z α , z ∈ C \ {0}. (2)
In a vector subdivision scheme as defined in [START_REF] Cavaretta | Stationary subdivision[END_REF], we simply set A n = A ∈ r ×r 0 (Z) and define convergence as follows.

Definition 1

The vector subdivision operator S A : r (Z) → r (Z) is called C p -convergent, p ≥ 0, if for any data g 0 := g ∈ r (Z) and the refinements from (1) there exists a function ψ g : R → R r with C p components such that for any compact K ⊂ R there exists a sequence ε n with limit 0 that satisfies max

α∈Z∩2 n K g n (α) -ψ g 2 -n α ∞ ≤ ε n . (3)
For a Hermite scheme, in (1), we set

A n (α) = D -n-1 A(α)D n , α ∈ Z, D :=       1 1 2 . . . 1 2 d       , (4)
so that r = d + 1 and for k = 0, . . . , d the k-th component of c n (α) corresponds to an approximation of the k-th derivative of some function ϕ n at α2 -n . Starting from an initial sequence f 0 ∈ r (Z), a Hermite scheme

f n+1 := H A n f n := D -n-1 S A D n f n , n ≥ 0,
can be rewritten as

g n+1 := D n+1 f n+1 = S A D n f n = S A g n , n ≥ 0, (5)
based on the relation

g n = D n f n , n ≥ 0. ( 6
)
To capture the intuition of vectors with consecutive derivatives, the convergence of Hermite schemes is a little bit more intricate and defined as follows.

Definition 2

The Hermite subdivision scheme with respect to the mask A ∈ r ×r (Z) as defined by ( 5) is called convergent if for any data f 0 ∈ r (Z) there exists a function Φ ∈ C (R, R r ) such that for any compact K ⊂ R there exists a sequence ε n with limit 0 which satisfies

max 0≤ j ≤d max α∈Z∩2 n K f ( j ) n (α) -φ i 2 -n α ≤ ε n . ( 7
)
Moreover, the scheme

H A n is said to be C p -convergent with p ≥ d if in addition φ 0 ∈ C p (R, R) and φ ( j ) 0 = φ j , 0 ≤ j ≤ d .

Remark 3 Since the intuition of Hermite subdivision schemes is to iterate on function values and derivatives, it usually only makes sense to consider C p -convergence for p ≥ d . Note, however, that the case p > d leads to additional requirements.

Remark 4 The two concepts of convergence based on S A and H

A n are significantly different as can be seen immediately from [START_REF] Dubuc | Hermite subdivision schemes and Taylor polynomials[END_REF]. Indeed, if the Hermite subdivision scheme is convergent it follows that

g n =       f (0) n 2 -n f (1) n . . . 2 -nd f (d ) n       →       φ 0 0 . . . 0       , hence Ψ g = φ 0 e 0 .
In particular, the components of g n have to converge to zero even with a prescribed rate. Therefore, in general it cannot be ensured that D -n g n converges or is bounded, even if g n converges to a multiple of e 0 . This is the reason why the factorization properties and the convergence analysis for Hermite subdivision cannot be obtained in a straightforward way from that of the vector subdivision operator, even if they are based on the same mask, see Fig. 1 for a particular example.

As a consequence of Remark 4 we observe that whenever a mask A defines a convergent Hermite subdivision scheme, the associated vector subdivision scheme based on S A is a socalled rank-1 subdivision scheme as defined in [START_REF] Micchelli | Regularity of multiwavelets[END_REF][START_REF]On vector subdivision[END_REF]. In concrete terms, this means that the mask as to satisfy

Q 0 e 0 = Q 1 e 0 = e 0 , Q := α∈Z A 2α+1 , ∈ {0, 1},
and that the two matrices Q 0 and Q 1 have no further common eigenvector with respect to the eigenvalue 1.

To give convergence criteria for vector and Hermite subdivision schemes, we need three different types of difference operators from [START_REF]From Hermite to stationary subdivision schemes in one and several variables[END_REF][START_REF]Generalized Taylor operators and polynomial chains for Hermite subdivision schemes[END_REF].

Definition 5

The simple difference operator (of rank-1 type) is defined as

D d := ∆ e d e T d =       1 . . . 1 ∆       (8)
A generalized incomplete Taylor operator is an operator of the form

T d :=          ∆ -1 * . . . * . . . . . . . . . . . . . . . . . . * ∆ -1 1          = ∆I 1 + t j k j ,k=0,...,d , (9)
where t j , j +1 = -1 and t j k = 0, k ≤ j.

In the same way, the generalized complete Taylor operator is of the form

T d := D d T d =          ∆ -1 * . . . * . . . . . . . . . . . . . . . . . . * ∆ -1 ∆          = ∆I + t j k j ,k=0,...,d . (10)
The name Taylor operator stems from the fact that, motivated by observations from [START_REF] Dubuc | Hermite subdivision schemes and Taylor polynomials[END_REF], the (incomplete) operator had originally be introduced in [START_REF]From Hermite to stationary subdivision schemes in one and several variables[END_REF] as

T d :=          ∆ -1 -1 2 . . . -1 d ! . . . . . . . . . . . . . . . . . . -1 2 ∆ -1 1         
as a comparison between the difference of function values and the derivative terms of the Taylor expansion. Since for any φ

∈ C d +1 (R) one has that Td       φ φ . . . φ (d )       (x) =       φ (d +1) (ξ 0 ) φ (d +1) (ξ 1 ) . . . φ (d +1) (ξ d )       , ξ j ∈ (x, x + 1), j = 0, . . . ,
T d S A = 2 -d S B T d and T d S A = 2 -d S B T d , (11)
respectively.

In a certain sense, the factorization always exists. We simply have to write 6 as

T * (z)A * (z) = 2 -d B * (z)T * (z 2 ), T * (z) :=          z -1 -1 -1 * . . . * . . . . . . . . . . . . . . . . . . * z -1 -1 -1 1         
with an analogous identity for the complete factorization, to obtain that

B * (z) = 2 d T * (z)A * (z)T * (z 2 ) -1 and A * (z) = 2 -d T * (z) -1 B * (z)T * (z 2 ),
respectively. Given a finitely supported mask A, the resulting B * (z) is usually a nonpolynomial rational function, hence the factor B is an infinitely supported mask. Unfortunately, the same also holds true for A * which, for given B can only be guaranteed to be a rational function, even if

det A * (z) = 2 -d det T * (z) -1 det B * (z) det T * (z 2 ) = 2 -d (z -2 -1) d +1 (z -1 -1) d +1 det B * (z) = 2 z -1 + 1 2 d +1 det B * (z)
is a Laurent polynomial in z. This is in contrast to scalar subdivision schemes where raising the order of the zero at -1 is the standard way to increase the smoothness of the limit function. Nevertheless, the existence of a factorization is the key to the construction of convergent Hermite subdivision schemes.

Theorem 7 ([10], Corollary 4) If a given mask A has a complete Taylor factorization T d S

A = 2 -d S B T d where 1. B ∈ r ×r (Z) is finitely supported, 2. S B is a contraction on r ∞ (Z), 3. (B * (1)) 11 = 1, then H A n is C d -convergent.
Hence, in order to construct a C d -convergent subdivision scheme, we can start with a finitely supported B such that the associated subdivision satisfies the contractivity condition 2) and the normalization condition 3) at the same time. Note that the latter prohibits a simple rescaling of B , i.e., a multiplication with a small constant. This, however, is not enough as one also has to ensure that T * (z) -1 B * (z)T * (z 2 ) is a matrix valued Laurent polynomial which leads to additional conditions on B * . A generic construction for such a B has been given, for any generalized Taylor operator, in [START_REF]Generalized Taylor operators and polynomial chains for Hermite subdivision schemes[END_REF] which shows that for any generalized Taylor operator and any d there exists a C d convergent Hermite subdivision scheme that if factorizable with respect to this generalized Taylor operator. We will later extend this construction by means of a supercomplete Taylor factorization, but first we illustrate the concept by looking at a special case that actually motivated the development of generalized Taylor factorizations.

In Figure 1, with a given mask, {A(•)}, we plot the "limit" functions for the two vector schemes, S A , S B (after Taylor incomplete factorization), and the Hermite scheme H A n . We notice that the first functions for S A and H A n are identical, corresponding to φ 0 in Definition 2 and similarly the last ones of S B and H A n corresponding to φ (d ) 0 in the same definition. In this section, we rewrite the well known cardinal splines, [START_REF] Schoenberg | Cardinal spline interpolation[END_REF] in term of a scalar subdivision scheme and extend it into Hermite schemes of different orders. From the properties of cardinal splines, we have convergence of the schemes and regularity of the limit. Our presentation, already proposed in [START_REF] Merrien | A generalized Taylor factorization for Hermite subdivisions schemes[END_REF], is based on a construction detailed by Michelli in [START_REF] Micchelli | Mathematical aspects of geometric modeling[END_REF] and in summarized in the following. Let

ϕ 0 (x) = χ [0,1] = 1 if x ∈ [0, 1], 0 if x [0, 1].
For m = 1, 2, . . ., we build ϕ m by means of autoconvolution as

ϕ m = ϕ 0 * ϕ m-1 or ϕ m (x) = x x-1 ϕ m-1 (t )d t . Let us recall that ϕ m is a C m-1 piecewise polynomial of degree m with finite support [0, m + 1]. Moreover, ϕ m (x) = 1 2 m α∈Z m+1 α ϕ m (2x -α) where i j = i ! j !(i -j )! if 0 ≤ j ≤ i , 0 otherwise. Considering v(x) = α∈Z f (0) 0 (α)ϕ m (x -α), which is a finite sum for any x ∈ R since ϕ m has finite support, we deduce for n ∈ N 0 that v(x) = α∈Z f (0) n (α)ϕ m (2 n x -α)
where

f (0) n+1 (α) = 1 2 m β∈Z m + 1 α -2β f (0) n (β) =: β∈Z a m (α -2β) f (0) n (β), α ∈ Z, (12)
that is,

a m (α) = 1 2 m m + 1 α , α ∈ Z. (13)
This is a scalar subdivision scheme. Then, the well-known derivative formula for cardinal B-spline yields

d i v d x i (x) = α∈Z 2 ni ∆ i f (0) n (α -i ) ϕ m-i 2 n x -α , i = 0, . . . , m -1. ( 14
)
We have a particular case when i = m -1. Since the function ϕ 1 is piecewise linear with ϕ 1 (α) = δ 1α , we obtain

d m-1 v d x m-1 (β/2 n ) = 2 n(m-1) ∆ m-1 f (0) n (β -m + 1).
With this formula, we define a Hermite subdivision scheme of order d < m with mask {A(α)} and support [0, m + d + 1] by applying differences to the mask a m , yielding

A(α) =       a m (α) 0 . . . 0 ∆a m (α -1) 0 . . . 0 . . . ∆ d a m (α -d ) 0 . . . 0       , ( 15
)
thus

A * (z) = (1 + z) m+1 2m       1 0 . . . 0 (1 -z) 0 . . . 0 . . . (1 -z) d 0 . . . 0       . ( 16
)
Beginning with f 0 ∈ r , defined by ( 5) we notice that for n ≥ 1 and i = 1, . . . , d :

f (i ) n (α) = 2 i n ∆ i f (0) n (α -i ). ( 17
)
Now with ( 12) and ( 14), for n > 0,

d i v d x i (x) = α∈Z f (i ) n (α)ϕ m-i (2 n x -α), i = 0, . . . , d .
In [START_REF] Merrien | A generalized Taylor factorization for Hermite subdivisions schemes[END_REF], we had proved that the generalized Taylor operators are given by

T d :=       ∆ -1 . . . -1 . . . . . . . . . ∆ -1 1      
and

T d :=       ∆ -1 . . . -1 . . . . . . . . . ∆ -1 ∆       (18)
Thus the corresponding vector scheme in the factorization is given by

B * (z) = z(1 + z) m-d 2 m-d       1 1 . . . 1       (1 -z 2 ) d z 2 ((1 -z 2 ) d -1 . . . z 2 (1 -z 2 ) z 2 . ( 19
)
It is of rank 1. Let us also notice that

T * (z)A * (z) = 2 -d B * (z)T * (z 2 ) = 2 -m z -1 (1 + z) m+1 (1 -z) d +1       1 0 . . . 0 1 0 . . . 0 . . . . . . . . . 1 0 . . . 0       .
We did not plot the graphs of the B-splines which are well known and probably everyone has seen one already.

Hermite schemes with extra regularity: a generic construction

In this section we will show that for any generalized Taylor operator of any order d there exists a C d -convergent subdivision scheme with an a extra regularity of p for any given p ≥ 0.

Theorem 8 Given p ≥ 0 and a Taylor operator T d of order d , there exists a finitely mask A ∈

r ×r such that the subdivision scheme is C d -convergent with a limit function φ ∈ C d +p (R) and S A admits a Taylor factorization with respect to T d .

The idea behind the supercomplete construction is simple and to some extent even follows the same concept as usual scalar subdivision: starting with the Taylor factor B such that T d S A = 2 -d S B T d , we create an additional the order of smoothness of the limit function of S B by constructing a scheme whose symbol has extra (matrix) factors. This means that S B from the (incomplete) Taylor factorization should be further factorizable into

I d ∆ p S B = 2 -p S B I d ∆ p , ( 20
)
where B is also a finitely supported mask. From [START_REF] Micchelli | Regularity of multiwavelets[END_REF][START_REF]On vector subdivision[END_REF] we recall the following result on smoothing limit functions.

Theorem 9

The vector subdivision scheme S B has C p limit function if

I d ∆ p+1 S B = 1 2 p S B I d ∆ p+1 (21)
and S B is contractive. The converse does not hold.

For the construction of an appropriate B , we partition it as

B * (z) = B * 11 (z) B * 12 (z) B * 21 (z) B * 22 (z) , B 11 ∈ d ×d (R), B 12 , B T 21 ∈ 1×d (R), B 11 ∈ 1×1 (R)
Since ( 21) can be rewritten as

B * (z) = 1 2 p I d z -1 -1 -p-1 B * 11 (z) B * 12 (z) B * 21 (z) B * 22 (z) I d z -2 -1 p+1 = 1 2 p B * 11 (z) (z -2 -1) p+1 B * 12 (z) (z -1 -1) -p-1 B * 21 (z) (z + 1) p+1 B * 22 (z) , (22)
we can record the following immediate consequence of Theorem 9.

Corollary 10 S B converges to a C p limit function of the form

f c = f c e d if 1. S B is contractive, 2. B *
21 has a zero of order p + 1 at 1,

3. B is normalized as B * 22 (1) = 1.
The corollary tells us that contractive schemes are at the heart of the construction of a convergent Hermite subdivision scheme. Note that contractivity of a scheme C means that the spectral radius

ρ(S C ) := lim sup n→∞ S n C 1/n , S C := sup c ∞ =1 S C c ∞ ,
based on the operator norm of the subdivision operator is less than one, where

c ∞ = sup α∈Z max 0≤k<r |c k (α)|.
The following simple sufficient condition for contractivity of a vector subdivision scheme is most likely known in the folklore, but state it and give a quick proof for the sake of completeness and the reader's convenience.

Lemma 11

If C is a lower triangular mask, i.e., all components of C (α) are lower triangular matrices and the diagonal elements c 00 , . . . , c r -1,r -1 ∈ (Z) are scalar contractive schemes, then C defines a contractive vector subdivision scheme.

Proof: Write C = D + N where D ∈ r ×r (Z) is a diagonal scheme and N ∈ r ×r (Z) is strictly lower diagonal one, then S n C = S n D + S N n for some strictly lower diagonal N n ∈ r ×r (Z), n ∈ N. Since the diagonal elements are contractions, there exist some n ∈ N such that S n D < 1. With

E ε :=       1 ε . . . ε r -1      
we have that

E ε S n C E -1 ε = S n D + S E ε N n E -1 ε , E ε N n E -1 ε =       0 ε * 0 . . . . . . . . . ε r -1 * . . . ε * 0      
and hence there exists ε > 0 such that ρ := E ε S n C E -1 ε < 1. Hence, for any m ∈ N,

S mn C ≤ E ε E -1 ε E ε S mn C E -1 ε = ε 1-r E ε S n C E -1 ε m ≤ ρ r
* ∆ -1 ∆ p+1          = I d ∆ p T d = I d ∆ p+1 T d . ( 23
)
The special cases are T d = T d ,-1 and T d = T d ,0 .

A factorization with respect to a supercomplete operator, T d ,p S A = 2 -p-d S B T d ,p is equivalent to

A * (z) = 1 2 p+d T * d ,p (z) -1 B * (z) T * d ,p (z 2 ) = 1 2 p+d T * (z) -1 I d (z -1 -1) -p B * (z) I d (z -2 -1) p T * (z 2 ) = 1 2 p+d T * (z) -1 I d z -1 -1 -p-1 B * (z) I d z -2 -1 p+1 T * (z 2 ) =: 1 2 d T * (z) -1 C * (z) T * (z 2 ).
Thus, if we can find mask C associated to a contractive scheme and normalized as (C * (1)

) d d = 1, such that H A is a C d -convergent subdivision scheme, then we can compute B * (z) as B * (z) = 2 p I d z -1 -1 p+1 C * (z) I d z -1 -1 -p-1 = 2 p C * 11 (z) 1 z -2 -1 p+1 C * 12 (z) (z -1 -1) p+1 C * 21 (z) 1 z -1 +1 p+1 C * 22 (z)
. The construction of C has been pointed out in [START_REF]Generalized Taylor operators and polynomial chains for Hermite subdivision schemes[END_REF]. More precisely, given any symbol h * d d of a mask h such that the univariate scalar stationary subdivision scheme S h is contractive, then there exists a recursive scheme [10, eq. (65) in the proof of Theorem 5] to compute h * d ,d -1 , . . . , h d ,0 such that for any h * j k , k = 0, . . . , j -1, j = 1, . . . , d -1, the upper triangular symbol This also concludes the proof of Theorem 9.

C * (z) =          z -1 -1 2 (z -1 -1) 2 h * 10 (z) (z -1 -1) 2 4 . . . . . . . . . (z -1 -1) d h * d -1,0 (z) . . . (z -1 -1) d h * d -1,d -2 (z) (z -1 -1) d 2 d c * d 0 (z) . . . c * d ,d -2 (z) c * d ,d -1 (z) c * d d (z)          with c * d j (z) = z -1 -1 d -j h * d j (z -1 ), j = 0, . . . ,
A * (z) = 1 2 d T * (z) -1 C * (z) T * (z 2 ) = 1 2 p+d T * d ,p (z)
We finish with revisiting one example from [START_REF]Generalized Taylor operators and polynomial chains for Hermite subdivision schemes[END_REF] where already a mask with a supercomplete Taylor factorization was constructed. z 2 32 (z-1) 6 (5-4 a-3 z+4 a z) z 7 0 4 (z-1) 2 z 2 -8 (z-1) 5 (15-20a+16a 2 -18 z+32 a z-32 a 2 z+7 z 2 -12 a z 2 +16 a 2 z 2 )

z 7 0 0 1+z 2 z  
as well as

A * (z) =    - ( 1+z
) (-20a+16a 2 -23 z+48 a z-32 a 2 z+15 z 2 -28 a z 2 +16 a 2 z 2 ) 2 z 3 -(z-1) (1+z) (11-8 a-7 z+8 a z) z 3 2 (z-1) 2 (1+z) (5-4 a-3 z+4 a z) z 4 30 a-40 a 2 +32 a 3 +31 z-46 a z+56 a 2 z-32 a 3 z+24 z 2 -78 a z 2 +72 a 2 z 2 -32 a 3 z 2 -45 z 3 +94 a z 3 -88 a 2 z 3 +32 a 3 z 3 4 z 3 (z-1) (-31+40a-32a 2 -24 z+16 a z+43 z 2 -56 a z 2 +32 a 2 z 2 ) 4 z 3 -(z-1) 2 (-15+20a-16a 2 -12 z+8 a z+19 z 2 -28 a z 2 +16 a 2 z 2 ) 2 z 4 a+z+7 a z+8 z 2 +22 a z 2 +30 z 3 +42 a z 3 +72 z 4 -183 a z 4 -119 z 5 +63 a z 5 32 z 5 (z-1) (-1-8z-30z 2 -72 z 3 +119 z 4 +32 a z 4 ) In Fig. 2, we have plotted the "limit" function and its first and second derivatives. Since the process does not compute the next derivatives, the approximations for higher derivatives in Fig. 3 have been determined using the successive finite differences of f (2) .

Conclusions

Convergent Hermite subdivision schemes have a limit function that belongs to C d , at least if the scheme converges in the sense proper for Hermite subdivision. We have shown that this regularity can be raised to an arbitrary order and provided an explicit recipe to determine such schemes. The approach uses factorizations and contractivity, but in contrast to the scalar univariate case the factor mask B must satisfy additional, nontrivial conditions, and the main task in the construction is to satisfy these conditions.

Definition 6

 6 d , the operator clearly annihilates all polynomials of degree at most d . Moreover, it enables us to give a sufficient criterion for the convergence of Hermite subdivision schemes by means of factorization. The masks B , B ∈ r ×r (Z) are called a Taylor factorization and a complete Taylor factorization of A, respectively, if they satisfy

Figure 1 :

 1 Figure 1: The different schemes: S A , S B and H A n

e r - 1 B

 1 which becomes < 1 for m sufficiently large. Since ρ(S C ) ≤ S mn C 1/(mn) for any choice of m, n ∈ N, this completes the proof that S C is a contraction. Next, note that the second condition on B in Corollary 10 ensures that B * in (22) is a Laurent polynomial while the normalization yields (2α) e d = α∈Z B (2α + 1) e d = e d , which is the necessary condition for the limit function to be of the form f c e d . Combining the two factorizations into one, we arrive at the following definition. Definition 12 The (generalized) supercomplete Taylor operator of order d and extra regularity p is of the form T d ,p :=

- 1 B

 1 * (z) T * d ,p (z 2 ) defines a C d convergent Hermite subdivision scheme by Theorem 7 and has a p-supercomplete Taylor factorization with factor B . The symbol B* is a lower triangular matrix with the same diagonal structure as C * and thus also defines a contraction. Hence, by Theorem 9 and Corollary 10, the last component of the limit φ, φ , . . . , φ(d ) of H A belongs to C p (R) which eventually verifies that φ ∈ C d +p (R).

Example 13

 13 In the case n = 5 with the functions from [10, Example 5], we can get a factorization with p = 4 and obtain B

32 z 5 -

 5

Figure 2 :

 2 Figure 2: The function and its first and second derivatives

  d , defines a Taylor factor with a contractive associated subdivision scheme by Lemma 11. Note, in particular, that C * 12 = 0 and that c * d d = h * d d . If, in addition, we choose

	h * d d (z) =	(z + 1) p+1 2 p+1 a(z),	a(1) = 2,
	in a B-spline fashion, then B

* (z) is a matrix Laurent polynomial and S A , defined by


Luis Filipe
email: luisfilipevf@tecnico.ulisboa.pt

V Ferreira

Zouhair Ait-Touchente
email: zouhair.aittouchente@gmail.com

Ahmed M Khalil
email: akhalil75@yahoo.com

Serap Simsek
email: tuncayserapsimsek@gmail.com

Sami Boufi
email: sami.boufi@fss.rnu.tn

Filipe V Luis

Manuel Ferreira

Rachid Rei Vilar

Mohamed M Touzani

Chehimi
email: chehimi@icmpe.cnrs.fr

Manuel Rei Vilar
email: manuel.rei-vilar@univ-paris-diderot.fr

Rachid Touzani
email: r.touzani@ump.ac.ma

Mohamed M Chehimi

Ultrasonic effect on the photocatalytic degradation of Rhodamine 6G (Rh6G

Keywords: cotton fabrics, TiO 2 nanoparticles, adhesion, sonication, Rhodamine 6G, photodegradation of dyes.  )) )

Introduction

Currently, there is a growing demand for imparting textiles a technical property in addition to their intrinsic function of clothing. Indeed, functionalized textiles (Horrocks and Anand 2000; [START_REF] Blaser | Estimation of cumulative aquatic exposure and risk due to silver: contribution of nano-functionalized plastics and textiles[END_REF], are used for different applications such as comfort (odour elimination) [START_REF] Yuranova | Self-cleaning cotton textiles surfaces modified by photoactive SiO2/TiO2 coating[END_REF][START_REF] Budavari | Improvement of odor elimination and anti-bacterial activity of polyester fabrics finished with composite emulsions of nanometer titanium dioxide-silver particles-water-borne polyurethane[END_REF]), biomedical (prostheses, implants, bandages ...) [START_REF] Li | Silk-based biomaterials in biomedical textiles and fiber-based implants[END_REF], civil engineering (composites, geotextile...) [START_REF] Fangueiro | Fibrous and composite materials for civil engineering applications[END_REF] and protection of individuals (protective clothing [START_REF] Scott | Textiles for protection[END_REF], clothes for military [START_REF] Wilusz | Military textiles[END_REF] or firemen [START_REF] Hristian | Study of the tensile properties of materials destined to manufacture protective clothing for firemen[END_REF]. Protection can also be provided for public ready-towear clothes using textiles with protective TiO 2 nanoparticles. These nanomaterials protect the skin from the UV-B (290-320 nm) and UV-A (320-400 nm) regions of the solar spectrum [START_REF] Popov | TiO2 nanoparticles as an effective UV-B radiation skin-protective compound in sunscreens[END_REF] [START_REF] Bozzi | Self-cleaning of modified cotton textiles by TiO2 at low temperatures under daylight irradiation[END_REF]. It is well-known that titanium dioxide in its crystalline anatase form is considered as the semiconductor oxide having the highest photocatalytic activity and the best quantum yield (reaction rate in moles per second) [START_REF] Hadjiivanov | Surface chemistry of titania (anatase) and titania-supported catalysts[END_REF].The photocatalytic activity of TiO 2 rests on its excitation with light the energy of which hν≥Eg (band gap energy) to create an electron pair (e -)/hole (h + ) by the transfer of an electron of the valence band to the conduction band. This electron can be ejected and will react with oxygen adsorbed on the surface of TiO 2 , while the hole h + created can react with surface water, forming highly oxidizing hydroxyl radicals (OH°) responsible for the degradation of pollutants [START_REF] Min | Time-resolved chemiluminescence study of the TiO2 photocatalytic reaction and its induced active oxygen species[END_REF][START_REF] Salvador | On the nature of photogenerated radical species active in the oxidative degradation of dissolved pollutants with TiO2 aqueous suspensions: a revision in the light of the electronic structure of adsorbed water[END_REF][START_REF] Tariq | Heterogeneous photocatalytic degradation of an anthraquinone and a triphenylmethane dye derivative in aqueous suspensions of semiconductor[END_REF][START_REF] Ochiai | Photoelectrochemical properties of TiO2 photocatalyst and its applications for environmental purification[END_REF]. In addition to these salient chemical reactivity features, TiO 2 is industrially employed due to its chemical stability, non-toxicity and low cost.

Either dyes or pigments employed in the textile industry undergo numerous testing;

probably the most important one is colour fastness [START_REF] Mohammad | High-energy radiation induced sustainable coloration and functional finishing of textile materials[END_REF][START_REF] Rehman | Modulation of marigold based lutein dye and its dyeing behaviour using UV radiation[END_REF].

It is thus important to investigate the adhesion of TiO 2 NPs to cotton textile in order to understand the shelf life of the textile and its long-term use to catalyse the degradation of dyes. Dealing with technical textile loaded with TiO 2 immobilized for photocatalytic purpose, we have envisioned testing the stability of the immobilized nanocatalysts in terms of their (i) adhesion, and (ii) photoactivity. In an academic laboratory scale, ultrasonication remains a simple procedure to test the shaking off of the nanoparticles away from the textile support.

However, it was also necessary to investigate the reminiscent photoactivity of the textiles after the adhesion testing.

In the recent literature, laboratory scale adhesion testing experiments, or in technical wording, "colour fastness" were conducted by (i) light exposure (Hinsch and Robinson 2018), (ii) washing and flexibility [START_REF] Monzavi | A Novel Polyester Fabric Treated with Nanoclay/Nano TiO2/PAMAM for Discoloration of Reactive Red 4 from Aqueous Solution Under UVA Irradiation[END_REF], and (iii) rubbing [START_REF] Zou | Functionalization of Silk with In-Situ Synthesized Platinum Nanoparticles[END_REF].

In this context, we wished to contribute to this topic by bridging the gap between adhesion phenomena and photocatalysis. In the course of the investigation of adhesion of TiO 2 to the textiles by an ultrasonic test ("ultrasonic colour fastness"), we surprisingly discovered that the ultrasonic waves increased the photocatalytic activity of TiO 2 NPs. This good example of serendipity has encouraged us to deepen the study of the ultrasonic effect on the photocatalytic properties of TiO 2 -immobilized textiles.

In this paper, we describe the use of a cotton fabric that has been coated with TiO 2 nanoparticles for the photocatalyzed destruction of Rhodamine 6G (Rh6G) taken as model organic dye as well as the frequently investigated Methylene Blue. The rationale for choosing these model compounds for testing textile photocatalyst is the following: Rhodamine 6G is a product with well-known toxicity. Its LD50 lethal dose is 89.5 mg/kg for rats. Rh6G is specifically used as a tracer dye in water to determine volumes, flow rates and directions of flow and transport. Methylene Blue is most often used as a disinfectant and can be found in natural waters through medical discharges. The presence of these two compounds in the aquatic environment can be harmful for the environment [START_REF] Budavari | Improvement of odor elimination and anti-bacterial activity of polyester fabrics finished with composite emulsions of nanometer titanium dioxide-silver particles-water-borne polyurethane[END_REF] The TiO 2 NPs were prepared in situ in the presence of cotton fabric and activated by a hydrothermal treatment. The resulting hybrid textile has shown good photocatalytic activity for the degradation of Remazol Brilliant Blue R [START_REF] Abid | Hybrid cotton-anatase prepared under mild conditions with high photocatalytic activity under sunlight[END_REF]). This method of preparation of such specific fabrics is reliable and easy to produce. Although, much progress has been made in the production of TiO 2 -modified smart textiles, the examination of the photocatalytic properties of the final product after adhesion test of TiO 2 remains sparse despite the technological interest of this kind of investigation to the textile industry, hence the motivation for this in-depth study.

Nanostructured cotton samples were characterized by several techniques (SEM, XPS, UV-Vis and FTIR spectroscopy, GSDR and LIL). The final fabric/TiO 2 was used to degrade Rhodamine 6G. The photodegradation of the dye molecules in fabric/TiO 2 -containing solutions exposed to a sun simulator was monitored by the colour change, observed by naked eye and UV/Vis absorption spectroscopy.

Experimental Chemicals

Chemical reagents were provided by Sigma Aldrich were of analytical grade and used without further purification: Ti(OBu) 4 , glacial acetic acid, Rhodamine 6G (~95%), tertbutanol 99.7%. Cotton fabrics were common gauze bandages with more than 99% of cellulose (purchased from Mega, Médicale Gaze S.A.).

Preparation of cotton fabrics/TiO 2

The cotton fabrics/TiO 2 samples were prepared as previously reported (Abid et al.

2016).

It is a non-hydrolytic sol-gel process followed by a mild hydrothermal treatment at a temperature below 140 °C. We used samples prepared at different temperatures (50, 110, 120 and 140 °C).

Sonication of cotton fabrics

The samples were placed in vials containing deionized water and subjected to ultrasonication in a sonocleaner (Bandelin, Sonorex Digitec model; 860 W, 35kHz) for 5, 15 or 30 min. After sonication, they were washed with water, and dried for further studies.

Characterization

Scanning electron microscopy images were recorded using a Field Emission Scanning Electron Microscope (FE-SEM) ZEISS SUPRA40, previously described [START_REF] Abid | Hybrid cotton-anatase prepared under mild conditions with high photocatalytic activity under sunlight[END_REF]).

X-ray photoelectron spectroscopy (XPS) measurements were recorded using a K Alpha system (Thermo Fisher Scientific) fitted with Al X-ray source (hν=1486.6 eV; spot size=400 μm) and an electron flood gun. Surface composition was determined using the manufacturer's sensitivity factors.

FTIR spectra were acquired using a Bruker Tensor 27 IR spectrometer in the range of 400-4000 cm -1 spectral region. Samples were first washed 3 times with distilled water and dried before analysis.

Ground state diffuse reflectance (GSDR) absorption spectra were recorded using an OLIS 14 spectrophotometer with a diffuse reflectance attachment. Further details are given elsewhere [START_REF] Vieira Ferreira | Surface Photochemistry: Organic Molecules within Nanocavities of Calixarenes[END_REF][START_REF] Errokh | Controlled growth of Cu2O nanoparticles bound to cotton fibres[END_REF]).

The set-up for the laser induced luminescence (LIL) was described in the literature [START_REF] Errokh | Controlled growth of Cu2O nanoparticles bound to cotton fibres[END_REF]). Time-resolved emission spectra were performed in the nanosecond to second time range with a N 2 laser (excitation wavelength=337 nm, PTI model 2000, ca. 600 ps FWHM, about 1 mJ per pulse). Emission spectra were obtained at 77 K, due to the fact that, at the room temperature, luminescence intensity is negligible for the anatase form of TiO 2 .

Photocatalytic measurements

The irradiation tests were carried out using UVACUBE400 photoreactor, producing a simulated sunlight; this device was used to test the photocatalytic activity of the cotton fabrics/TiO 2 samples. All measurements were performed with 30 mg (per step) of fabric/TiO 2 in 6 ml of a Rh6G solution (5 mg/L) stored in the dark for 45 min to reach the adsorption equilibrium. The absorbance of the Rh6G solution was monitored using a Varian Cary 50 Bio UV-visible spectrophotometer controlled by the CaryWinUV software.

Results and discussion

The sequential steps of the work are presented in Fig. 1. In the first step of the preparation of cotton-TiO 2 , woven cotton samples were immersed in a solution of Ti(OBu) 4 in tert-butanol for 12 hours at room temperature. Then, after drying, the cotton fabrics were subjected to mild hydrothermal treatment for 3 hours at indicated temperature to crystallize the TiO 2 layer generated on the surface of the cotton fibers [START_REF] Abid | Hybrid cotton-anatase prepared under mild conditions with high photocatalytic activity under sunlight[END_REF]). These samples have been tested to determine optimal conditions for high performance photocatalytic activity.

Cotton-TiO 2 samples were then sonicated at room temperature (Fig. 1(a)). 30 mg of each sample was placed in a beaker containing 10 ml of distilled water and sonicated for 30 minutes.

Scanning electron microscopy (SEM)

The SEM images of the different cotton samples are shown in Fig. 2. The morphology of cotton cellulose fibres prior to hydrothermal treatment is shown in Fig. 2(a). In situ synthesis of TiO 2 nanoparticles the latter for a very thin layer that is topping the cotton fibres.

The mean size of the TiO 2 NPs is less than 20 nm for the hydrothermal treatment at 120 °C.

X-ray photoelectron spectroscopy (XPS)

Fig. 3 displays XPS spectra of untreated cotton (Fig. 3 (a,c,e,g)) and TiO 2 loaded cotton fabric-120 °C (b, d, f, h). Survey regions are displayed in Fig. 3 (a (untreated cotton) and b (Fabric/TiO 2 -120 °C)); and the main peaks are C1s (285 eV), Ti2p (459 eV) and O1s (532 eV). The untreated cotton does not contain TiO 2 NPs, which was confirmed by the shape of Ti2p line obtained in Fig. 3(g), whereas fabric/TiO 2 -120 °C spectrum exhibits a peak at 459 eV corresponding to Ti2p (Fig. 3

2013).

The O1s XPS region presented in Fig. 3 (e (50, 110 and 140 °C). Table 1 shows the chemical composition of the untreated and TiO 2 loaded cotton fabrics. The cotton fabric used is contaminated even if it is a sterile sample. In fact, hydrocarbons contribute 33.2% to the composition of untreated cotton (regardless of hydrogen). Once TiO 2 has been deposited, this contribution is substantially reduced. We note also that the atomic percentage of TiO 2 is higher for the refluxed cotton fabric sample at 120 °C; this variation may be due to the difference in reflux temperatures. In order to study the effect of the adhesion test (sonication) on the fabric/TiO 2 as well as the effect of the exposure to simulated sunlight on the stability of the photocatalytic activity of TiO 2 NPs, 4 samples of 30 mg of cotton fabric/TiO 2 -120 °C were taken for post-treatment under ultrasonic waves for 5, 15 and 30 min, respectively; and the 4th sample was placed in the photoreactor for an insolation of 4h30. These samples were then analysed in XPS, and the ratio Ti/C3 (Fig. 4) was compared, Ti being the marker element of titanium oxide and C3 that of cellulose. One can remark that the atomic percentage of Ti increases from a non-posttreated fabric/TiO 2 -120 °C to a sonicated sample and the sonication time. This may be due to the fact that sonication leaves the distribution of TiO 2 NPs more homogeneous on the surface of the cotton fibres, improving the screening of the fabric. The ratio Ti/C3 for the insolated sample for 4 hours 30 min still reveals a high percentage of Ti, confirming that titanium oxide remains on the fabric surface despite the long treatment in the photoreactor.

Stability of the dye and degradation time

Prior to any photocatalytic test, a 5 mg/L aqueous solution of the model dye Rhodamine 6G (Rh6G) was exposed to simulated sunlight in the absence of the fabric. Similarly, spectra of Rh6G solutions exposed to the photoreactor in the presence of fabric/TiO 2 -120 °C were recorded to check the stability of the dye before and after contact with the photocatalytic textile. Fig. 5 shows the UV-vis spectra for this assay as well as the chemical formula of the Rh6G dye. It should be noted that the aqueous solution of Rh6G gives a peak at 525 nm and after exposure to the simulated sunlight, the absorbance remains the same, that is to say that the Rh6G was not degraded under irradiation at photoreactor for 120 minutes. This indicates that this dye is stable and can be used for testing the photocatalytic activity of the textile catalytic samples as it does not give any false positive result.

Dual effects of hydrothermal and post-sonication treatments on the catalytic fabrics

The treated fabric/TiO 2 hybrid nanocomposites were refluxed to 50, 110, 120 and 140 °C then introduced to Rh6G aqueous solution before exposure to simulated sunlight.

After sunlight irradiation for 120 minutes, all vials turned colourless which denotes the effective photocatalytic activity of the immobilized TiO 2 NPs. The sharp decrease in absorbance after exposure of the dye/textile mixtures to light in Fig. 7(a) emphasizes the successful photocatalytic degradation of Rh6G in the presence of the refluxed fabrics after being UV irradiated for 120 minutes.

Furthermore, the hydrothermal treatment carried out at different temperatures undergone by the fabric/TiO 2 influences the catalytic power of TiO 2 NPs. It is noted that the optimum temperature for photodegradation of Rh6G is 120 °C (Fig. 7(a)). At higher temperature; namely 140 °C, we have a higher absorbance value than that obtained for the fabric prepared at 120 °C indicating a decrease in the photocatalytic activity at higher temperatures than 120 °C as previously reported [START_REF] Abid | Hybrid cotton-anatase prepared under mild conditions with high photocatalytic activity under sunlight[END_REF]).

An approach by means of ultrasonication (US) has been endeavoured with fabric/TiO 2 to test the adhesion of TiO 2 to the cotton fabric. Three vials; each containing (30 mg of treated fabric/TiO 2 ) were ultrasonicated for 5, 15 and 30 min. Then, the samples were washed 3 times with distilled water. This was followed by placing 6 ml of the prepared Rh6G solution in each vial containing the ultrasonicated treated fabrics and subjecting them to simulated sunlight exposure for 120 minutes. The effect of post-sonication is shown in Fig. 7(a) (red bars). As one can observe, there is a significant decrease in absorbance compared to non-ultrasound samples. This means that despite partial removal of the catalytic nanoparticles, the fabrics retain photocatalytic activity. Surprisingly, the decrease in the spectral peak intensity of the dye is systematically lower for post-sonicated hydrothermally treated fabrics, up to 120 °C.

This difference is significant and is not contained within the error bars, therefore testifying a true synergetic effect of ultrasonication on the fabric. There is an exception for the TiO 2modified fabric that has been treated hydrothermally at 140 °C: we noted slight increase of absorbance, mostly due to slight adhesion loss of TiO 2 nanoparticles. Indeed, XPS analysis of cotton-TiO 2 samples reported in Table 1 shows that annealed sample at 140 °C has a Ti at.% of 3.01% , slightly lower compared to 3.14% for 120 °C annealing temperature. Sonication does not improve the photocatalytic character of the fabric annealed at 140 °C; for this reason we further examined the performances of the fabric annealed at 120 °C (see Fig. 7b). To confirm that sonication increases the photocatalytic effect of TiO 2 NPs, we repeated a similar experiment with a fabric that has been hydrothermally treated at 120 °C. Three samples of this fabric were post-sonicated for 5, 15 and 30 min and then used to catalyse the photodegradation of Methylene Blue, MB (Fig. 7(b)). It is clear that the fabric continues to act as a true photocatalyst after sonication since the maximum absorbance of the MB decreases by 60% compared to the pure MB solution (4 mg/L) after a short sonication period (5 minutes). On the other hand, after a sonication of 30 minutes, one reaches to attain 90% of degradation of MB.

The results displayed in Fig. 7 demonstrate the important improvement of the photocatalytic performances of TiO 2 loaded cotton fabrics. This unexpected result, accounts for the morphology displayed in the SEM images, which show that after sonication the surface of the cotton becomes more homogeneous. Possibly, with an even distribution of the cotton fabric, a larger specific surface was achieved by the TiO 2 NPs that remained on the textile.

The adhesion between the cotton fabric and TiO 2 after sonication has been studied by FT-IR spectroscopy in transmission mode. Hence, the treated fabrics were washed 3 times with distilled water and dried before analysed by FTIRS. Fig. 8 illustrates the infrared spectra of treated fabric/TiO 2 , showing no changes in the wave numbers of the characteristic peaks of the three treated samples after 5, 15 and 30 minutes of sonication. Cellulosic chains are characterized by different bands, a broad one corresponding to the O-H stretching located at 3320 cm -1 , the C-H stretching at 2886 cm -1 with the corresponding C-H bending at 1430 cm -1 , -O-H bending at 1630 and at 1418 cm -1 stretching vibrations. In addition, the peak at 1031 cm -1 refers to symmetric and asymmetric stretching C-O-C bonds. The peak that appears at 690 cm -1 is assigned to Ti-O-O bond and accounts for the presence of TiO 2 after sonication.

These results show that despite this strong sonication test, the characteristic peak of TiO 2 NPs is always present. The presence of the same peak at about 312 nm after sonication shows that this test does not detach the TiO 2 NPs from the cotton substrate, i.e. the bond is strong between the TiO 2 NPs and cotton fibres. However, a small amount of TiO 2 is detached from the substrate, as the decrease of the remission function value for the sonicated sample clearly shows.

Laser induced luminescence

The laser induced luminescence spectra ( exc =337 nm, ~1.3 mJ per excitation pulse), at 77 K, for the fabric/TiO 2 samples are illustrated in Fig. 10(a). The initial curve was obtained after laser pulse, and the following curves were obtained with the step of 1000 ns. The gate width was 10 ms. The laser induced emission spectra of non-sonicated fabric/TiO 2 are displayed with maxima at 533 nm. The maximum emission of non-sonicated fabric/TiO 2 , was obtained with a maximum of about 533 nm (Fig. 10(a)) and a luminescence intensity of about 150000 (a.u.). These spectra reflect a simultaneous emission of cotton and TiO 2 with a greater contribution of the latter. After ultrasonic treatment with fabric/TiO 2 -120 °C (Fig. 10(b)), the luminescence intensity increases to about 230000 (a.u.). This large difference between the two spectra is certainly due to the effect of sonication on the distribution and the organization of TiO 2 NPs immobilized on the fabric. Only the better attached TiO 2 NPs remain on the surface and possibly the number of surface defects decreases with sonication, meaning that the number of traps for the excited TiO 2 to be deactivated by a non-radiative pathway also decreases. Therefore, the luminescence intensity of TiO 2 increases for the sonicated sample and also the half-life is about 1.5 s, while the non-sonicated sample exhibits a shorter halflife, at least in the initial times of its decay. These data also support the interpretation that the remaining TiO 2 NPs (after sonication) contain less defects (traps), therefore larger intensities for irs luminescence and larger lifetimes for the TiO 2 NPs are observed. after sonication. The excitation wavelength was 337 nm and the step was 1 s in both cases. The used gate width was 10 ms

Conclusion

Woven cotton textile samples loaded with TiO 2 NPs were hydrothermally treated at temperatures ranging from 50 °C to 140 °C. The presence of TiO 2 on the samples was verified by several characterization techniques and the photocatalytic activity was found optimized for a hydrothermal treatment at 120 °C, as judged from the results obtained with a model dye Rhodamine 6G (Rh6G). In our quest of photocatalytic pigment fastness, we subjected the samples to ultrasonication to check their ability to remain attached to the surface. In a serendipitous way, we found out that ultrasonication boosted the photocatalytic effect despite a slight partial removal of TiO 2 nanoparticles. The study was repeated with Methylene Blue dye, and a similar effect was observed as the photocatalyzed degradation of MB led to a loss of 90% after 30 min of ultrasonication of the fabric primarily hydrothermally treated at 120 °C. Compared to un-sonicated fabric the photocatalyzed degradation using a sonicated catalytic fabric is 4 times better as the non-sonicated fabric degraded MB but left 40% of the dye.

This work conclusively shows that ultrasonication clearly boosts up the photocatalytic activity of TiO 2 , and the process could be extended to other types of nanocatalysts such as mixed oxides or noble metal loaded TiO 2 . The process is simple and could be generalized for pilot testing.

  photocatalytic properties imparted by immobilized TiO 2 nanoparticles (Bozzi et al. 2005; Meilert et al. 2005; Xue et al. 2008; Abidi et al. 2009; Abid et al. 2016) for the elimination of dyes, fats, bacteria and viruses (Bozzi et al. 2005). It is well-known that titanium dioxide in

Fig. 1

 1 Fig. 1 Illustration of the different steps followed to : (a) prepare different cotton-TiO 2 NPs samples and (b) sonication and photodegradation of Rh6G.

Fig. 2

 2 Fig. 2 SEM images: (a) top view of pristine cotton substrate; (b) cotton fabric/TiO 2 -120 °C; (c) sonicated cotton fabric/TiO 2 -120 °C/5 min; (d) sonicated cotton fabric/TiO 2 -120 °C/30 min and (e) cotton fabric/TiO 2 -120 °C after 4 hours and 30 min exposure to simulated sunlight

  (h)), which proves that the immobilization step of TiO 2 NPs on cotton samples was successfully accomplished. The C1s region presented in Fig.3(c, d) was fitted with four components centred at 285.0, 286.6, 288.3 and 289.5 eV, assigned to: C-C/C-H bonds derived from the hydrocarbon contamination or treatment of cotton fabrics (sizing); C-O-C/C-OH, O-C-O bonds corresponding to the hemi-acetal group characteristic of cellulose; and finally, an oxidation peak which corresponds to an ester or an adsorbed carbonate. Since a cellulose unit has only one hemiacetal carbon atom, we will choose this atom as a marker element for cellulose in the following[START_REF] Belgacem | Surface characterization of cellulose fibres by XPS and inverse gas chromatography[END_REF][START_REF] Attia | One-step UV-induced modification of cellulose fabrics by polypyrrole/silver nanocomposite films[END_REF]

Fig. 3

 3 Fig.3XPS spectra of untreated cotton fabric (a, c, e, g) and cotton fabric/TiO 2 -120 °C(b, d, f, h)

Fig. 4

 4 Fig. 4 Atomic ratios (Ti/C3) of samples untreated or treated (sonication and insolation) undergone by a fabric/TiO 2 -120 °C sample

Fig. 5 Fig. 6

 56 Fig.5UV-Vis absorption spectra of Rh6G before and after irradiation in the UVACUBE400 photoreactor

  Fig. 7 (a) UV-Vis absorption spectra of Rh6G after photocatalytic degradation in the presence of non-sonicated and sonicated fabric/TiO 2 refluxed at (50, 110, 120 and 140 °C) subjected to UV irradiation. (b) Photodegradation spectrum of methylene blue with fabric/TiO 2 -120 °C with and without sonication

Fig. 8

 8 Fig. 8 FTIR spectra of treated Fabric/TiO 2 -120 °C ultrasonicated during 5, 15 and 30 minutes

Fig. 9

 9 Fig. 9 Ground State Diffuse Reflectance (GSDR) spectra of Fabric/TiO 2 sample before and after sonication

Fig. 10

 10 Fig. 10 Laser induced luminescence spectra, obtained at 77 K, of Fabric/TiO 2 -120 °C samples (a) before sonication and (b)

Table 1

 1 Surface chemical composition of untreated and TiO 2 loaded woven cotton fabrics.

	Samples	C1 (C-C/C-H)	C2 (C-O-C/C-OH)	C3 (O-C-O)	C4 (CO x )	O1 (TiO 2 )	O2 (org)	Ti
	Untreated cotton fabric	33.20	25.90	8.86	3.51	-	28.60	-
	Fabric/TiO 2 -50 °C	11.23	29.75	12.74	2.52	3.72	38.11	1.92
	Fabric/TiO 2 -110 °C	14.64	30.16	10.06	2.13	7.56	33.10	2.45
	Fabric/TiO 2 -120 °C	11.01	32.89	10.20	2.10	7.42	12.30	3.14
	Fabric/TiO 2 -140 °C	18.75	25.34	9.54	3.11	6.24	34.10	3.01

Acknowledgements

The authors would like to thank NATO for financial support through the CATALTEX project No SfP 984842.


Jean Pinson

Fetah I Podvorica

Z Wang

Q Li

Z Chen

J Liu

T Liu

H Li

S Zheng

Enhanced Comprehensive

Surface modification of materials: electrografting of organic films

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

Introduction

Upon dediazonation (spontaneous or through reduction) diazonium salts provide highly reactive aryl radicals that bind very strongly to substrates surface through covalent bonds to give composite materials as shown on Scheme 1:

Surface-Aryl

Scheme 1. Grafting of diazonium salts, the basic reaction.

Since the first described electrografting of diazonium salts in 1992 [1], their use has included i) more and more materials to which they have been attached, ii) more and more methods available for the reaction, iii) improved descriptions of the structure and formation mechanism of the films, iv) involvement in new reactions and v) an increased number of possible and real-life applications. In this opinion we review recent advances in the field. This reaction has been the subject of many reviews [2] including two Current Opinions [3,4].

New material modified by reduction of Diazonium Salts

Carbon, metals and oxides, semiconductors, polymers have been modified with diazonium salts. Boron nitride is particularly non-reactive, however with prior hydroxylation under very harsh conditions, it has been functionalized with diazonium salts, albeit with a low surface coverage [5]. The surface modification of polymers has been described in a book, including reactions with diazonium salts [START_REF] Pinson | Surface Modification of Polymers: Methods and Applications[END_REF]. During the last years there has been an increased interest in nanomaterials such as nanoparticles, nanotubes but also layered materials such as graphene, MXenes, black phosphorus.

2D Materials such as MoS 2 , WS 2 , MoSe 2 , WSe 2 , Sb 2 S 3 and Bi 2 S 3 can be exfoliated into nanosheets or nanoribbons and modified by diazonium salts (under sonication) [START_REF] Li | Covalent chemical functionalization of semiconducting layered chalcogenide nanosheets[END_REF]. Black phosphorous has been exfoliated with n-BuLi and the negatively modified nanosheets modified with 4-nitrobenzenediazonium to improve their stability under ambient conditions; the mechanism has been discussed including either an homolytic or an heterolytic dediazonation leading respectively to a radical or a carbocation [START_REF] Zhang | Negatively charged 2D black phosphorus for highly efficient covalent functionalization[END_REF].

New methods for surface modification.

Many different methods have been used for the modification of surfaces: electrochemistry, reducing reagents or surfaces, sonication,…. In spite of its wide use in the organic chemistry of diazonium salts, photochemistry has been seldom used for surface modification. Recent results include the photochemical cleavage of iodonium salts Ar-I + -Ar' [START_REF] Dard | Patterning Surfaces through Photografting of Iodonium Salts[END_REF] (under visible light in the presence of a sensitizer) and azosulfones Ar-N=N-SO 2 CH 3 [START_REF] Dard | Felidj Multi-functionalization of lithographically designed gold nanodisks by plasmon-mediated reduction of aryl diazonium salts[END_REF] (under UV light). With the first reaction it is possible to pattern gold surfaces by using a mask, with the second one it is possible to attach both the aryl group and the sulfone to the surface. So called "hot electrons" can be produced on structured surfaces or nanoparticles by photochemical excitation of surface plasmons. These hot electrons are able to reduce diazonium salts and provide modified nanoparticles and nanodisks [11]. When nanodisks are irradiated by a polarized laser beam along one direction grafting by a diazonuium salt provides modified spots at the extremities of one diameter, by rotating the beam by 90° another diazonium salt can be grafted, therefore this makes possible a regioselective functionalization of the nanodisk.

Controlling the structure of organic films

One of the drawbacks of aryl diazonium salts compared to SAMs is the formation of oligomeric films so-called "multilayers" due to the ability of aryl radicals to attack not only the surface but also already attached aryl layers. There have been a lot of attempts to control the growth of the aryl layer and in particular to obtain monolayers. This topic has been the subject of a Current Opinion [3]. More recently, a novel approach was devised to prepare a monolayer by use of a redox mediator (compounds that present a fast, outer-sphere, reversible redox couple). The method is based on a fast redox cross-reaction in the diffusion layer between the diazonium compound and the reduced form of the selected inhibitor. This reaction favors the formation of aryl radicals far enough from the surface so that they do not attack the first layer [12]. With this method the measured thickness of a film of iodophenyl groups (0.6 ± 0.2 nm) on an atomically flat carbon surface (PPF) fits excellently with that expected for a monolayer (0.55 nm).

The use of diazonium salts with bulky substituents on the benzene ring such as 3,5-bis-t-butyl 13, calix[4]arene [14], bulky Ru(II) complex [START_REF] Nguyen | Unprecedented Self-Organized Monolayer of a Ru(II) Complex by Diazonium Electroreduction[END_REF] permits to obtain monolayers, and even structured organized monolayers in the last example. In the case of 3,5-bis-tbutylbenzenediazonium, a thickness was measured by ellipsometry (1 ± 0.3 nm) [3]. High resolution STM images [16] confirm the formation of a uniform monolayer on HOPG (Figure 1). However, if 4-nitrobenzenediazonium is grafted on the same substrate, a very different image is obtained where small oligomers are dispersed on the surface of HOPG, this indicates a faster attack of the radical on the first grafted group than on the substrate. In this case even if the film appears as a monolayer on a µm scale by AFM or ellipsometry, it is not a real monolayer. In addition, grafting of the same 4-nitrobenzenediazonium cation on a different type of carbon (PPF) provides a real monolayer as shown by AFM [START_REF] Anariba | Mono-and Multilayer Formation by Diazonium Reduction on Carbon Surfaces Monitored with Atomic Force Microscopy "Scratching[END_REF]. Therefore, the formation of real monolayers is a complex problem, the result depends both on the diazonium salt and the substrate and there does not seem to be a general method. The above methods: redox mediators or bulky substituents permit to limit the growth of the film by either controlling the concentration or limiting the reaction of aryl radicals with already attached aryl groups but these very reactive species are able to abstract hydrogen atoms and longer reaction times would certainly lead to multilayers. A different method permits to obtain a monolayer, it is based on the simultaneous transfer of an octadecylamine (ODA) monolayer Langmuir film on a gold surface and its oxidative electrografting [START_REF] Gabaji | From Langmuir-Blodgett to grafted films[END_REF]. In this case, the formation of the monolayer Au-ODA, with a local order, is "by construction" and the formation of "multilayers" impossible.

Extending the scope of diazonium grafting

Aryl radicals react with surfaces, but they can undergo other reactions among which atom abstraction (hydrogen, bromine, iodine) from organic molecules (RH, RX). This results in the formation of a new radicals R . that are able to react with surfaces. If the aryl radical is prevented from reacting on the surface, one will obtain a surface modified with R groups only. This has been achieved using 2,6-dimethylbenzenediazonium (2,6-DMBD); due to the two methyl groups ortho of the radical, the steric hindrance prevents its reaction with surfaces, but does not prevent atom abstraction from different species in solution [START_REF] Combellas | Steric effects in the reaction of aryl radicals on surfaces[END_REF] (Scheme 2). By electrochemical reduction of 2,6-DMBD in CH 3 CN it was possible to obtain cyanomethyl radicals issued from the solvent after the H atom abstraction. These radicals react on surfaces to give Au, Cu, Si-[CH 2 -C(NH 2 )] n surfaces [20]. If a 40% aqueous solution of CH 3 NH 2 is used as a solvent, one obtains highly aminated surfaces Au-[CH(NH 2 )] n [START_REF] Médard | Electrografting of methylamine through C-H activation or oxidation to give highly aminated surfaces[END_REF].

Alkyl iodides and bromides (RBr, RI) are reduced at potentials more negative than -2 V/SCE to give GC-, Au-, Cu-R electrodes (GC = Glassy Carbon). The reaction can be performed at much less negative potential (with a gain > 1.7 V) in the presence of 2,6-DMBD following the atom abstraction mechanism of Scheme 2. This reaction leads to different modified surfaces GC-C 6 H 13 or GC-(CH 2 ) 2 C 6 F 13 [START_REF] Hetemi | Electrografting of Alkyl Films at Low Driving Force by Diverting the Reactivity of Aryl Radicals Derived from Diazonium Salts[END_REF], Au, PMMA, PE-C 5 H 10 -COOH [START_REF] Hetemi | Surface Functionalization of Metals by Alkyl Chains through a Radical Crossover Reaction[END_REF][START_REF] Hetemi | Surface Modification of Polymers by Reaction of Alkyl Radicals[END_REF] (PMMA= polymethylmethacrylate, PE = polyethylene). The activation of C-I bonds is also possible with aryl iodides at the reduction potential of 2,6-DMBD [START_REF] Combellas | Indirect Electrografting of Aryliodides[END_REF]. When using other aryldiazonium salts (4-nitrobenzenediazonium, for example) in the presence of an alkyl halide (I-(CH 2 ) 2 C 6 F 13 , for example) bifunctionnal alkyl-aryl layers with a complex structure are obtained [START_REF] Hetemi | One-Step Formation of Bifunctionnal Aryl/Alkyl Grafted Films on Conducting Surfaces by Reduction of Diazonium Salts in the Presence of Alkyl Iodides[END_REF]. These reactions expand the field of the molecules that can be grafted on surfaces, in fact, most organic molecule with a C-H bond can be grafted on surfaces, not taking in account specific problems such as solubility, stability and steric hindrance.

N + N CH 3 CH 3 CH 3 CH 3 . CH 3 CH 3 . + N 2 S X H-CH 2 R I-CH 2 R Br-CH 2 R . CH 2 R S -CH 2 R S growth

Conclusion.

Nearly 30 years after its discovery, the surface reaction of diazonium salts is still a very active field. Future challenges involve the grafting of complex molecules such as proteins and nucleic acid with a perfect control of the positions to be attached on the surface and the development of industrial applications in addition to the existing ones: inks and paints [START_REF] Belmont | Patents and Industrial Applications of Aryl Diazonium Salts and Other Coupling Agents in Aryl Diazonium Salts: New Coupling Agents in Polymer and Surface Science[END_REF],

copper

Figure 1 .

 1 Figure 1. STM images after grafting of 3,5-bis-tert-butylbenzenediazonium and 4-nitrobenzenediazoniumon salts on HOPG.

  interconnects and TSV in microelectronics [28], drug eluting stents (BuMA ® )[29], polymer brush coating and adhesion technology [30], molecular junction overdrives [31]. Papers of particular interest, published within the period of review, have been highlighted as: * Paper of special interest (Ref. 3,4, 10, 12, 15, 19). ••Paper of outstanding interest. (Ref. 1,11,[START_REF] Gabaji | From Langmuir-Blodgett to grafted films[END_REF][START_REF] Hetemi | Electrografting of Alkyl Films at Low Driving Force by Diverting the Reactivity of Aryl Radicals Derived from Diazonium Salts[END_REF]

	References and recommended reading

http://www.sinomed.com/ last accessed April

2020 30 https://www.radisurf.com last accessed April 30 2020

http://www.chem.ualberta.ca/~mccreery/ last accessed April 30 2020


Michael Lobb
email: mlobb@yorkat.ac.uk

Tony Brown
email: tony.brown@soton.ac.uk

Jules Leyland
email: j.leyland@soton.ac.uk

Vincent Bernard
email: vincent.bernard@univ-rennes1.fr

Marie-Yvane Daire
email: marie-yvane.daire@univ-rennes1.fr

Loïc Langouët
email: loic.langouet@wanadoo.fr

Mike Lobb

An estuarine tidescape of production: terrestrial laser scanning of fixed fishing structures and a tide mill in the Léguer Estuary, Brittany

Keywords: fish weirs, terrestrial laser scanning, tidal mill, estuarine resources, tidal zone excavation Research Region: Western Europe

Background

Terrestrial laser scanning TLS has been used to record artefacts from waterlogged wooden stakes [START_REF] Lobb | A new approach to recording and monitoring wet-preserved archaeological wood using threedimensional scanning[END_REF] to complete boats [START_REF] Tanner | 3D Laser Scanning for the Digital Reconstruction and Analysis of a 16th Century Clinker Built Sailing Vessel[END_REF], and archaeological structures from Megalithic monuments [START_REF] Abbott | Stonehenge laser scan: Archaeological Analysis Report[END_REF] to Postmedieval interiors (English Heritage 2011). It has also been used for palaeo-landscapes [START_REF] Entwistle | Three-dimensional (3D) visualisation: the application of laser scanning in the investigation of historical Scottish farming townships[END_REF]) and caves -for which it has proven particularly valuable [START_REF] Zlot | Three-dimensional mobile mapping of caves[END_REF]. However, it has not yet reached its full potential as it can provide highly accurate 3D models which both record excavation and can be used to answer questions of function and context through environmental modelling. We show here how The inter-tidal environment is particularly challenging for conventional methods of survey and excavation, but also the most threatened archaeological zone worldwide [START_REF] Pontee | A review of coastal risk management in the UK[END_REF]. There are also structures/artefacts typically found in the coastal zone, such as fixed fishing-structures (or fish weirs, basket racks) and tidal mills, that have long histories and about which we have very limited knowledge (Jecock 2011). Erosion, or erosion management, of the coastal zone typically exposes these structures and prompts recording (e.g. Rapid Coastal Zone Surveys in the UK) but rarely excavation.

Fixed fishing-structures (or mass fish capture devices) and tidal mills are particularly valuable as they provide information on wood technology, fish processing and storage, environmental cognition, cultural beliefs, delayed return economics, and social organisation. They are also intimately connected to environmental factors including relative sea level, river discharges, storminess and fish populations. Because of this they should not be seen in isolation and require setting within their environmental -or sea/landscape context. Recently several coastal surveys have targeted fixed fishing structures along the Shannon estuary, Ireland (O'Sullivan 2005), Wales (Moore-Scott 2007), Southern England [START_REF] Brown | Medieval fishing structures and baskets at Sudbrook Point, Severn Estuary, Wales[END_REF][START_REF] Royall | Rapid Coastal Zone Assessment Survey for South-West England -South Coast Dorset Component One National Mapping Programme[END_REF][START_REF] Cooper | A Saxon Fish Weir and Undated Fish Trap Frames Near Ashlett Creek, Hampshire, UK: Static Structures on a Dynamic Foreshore[END_REF]) Brittany [START_REF] Langouët | Ancient maritime fish-traps of Brittany (France): a reappraisal of the relationship between human and coastal environment during the Holocene[END_REF] and Normandy (Billard and Bernard 2016).

Methodological Considerations

The TLS survey used a Leica ScanStation C10 with scans registered together using High Definition Survey (HDS) targets, with a mean registration error of 4 mm. GNSS was used to survey in the target locations, creating a final registered accuracy of c.15 mm. The registered scans from each survey were incorporated into a single database. Standard challenges encountered with TLS are exacerbated by the environment of the inter-tidal zone. Reflectivity, the ability of a surface to return the laser signal, is reduced due to the presence of moisture or water, resulting in inaccurate or negative data. 'Shadows' or voids in the data are caused not just by the multi-faceted nature of the features of interest, creating the need for numerous setups, but also by the null or negative responses from where the laser hits standing water. The stability of the equipment is problematic as the intertidal substrate becomes unstable with the rising tide, as the freshwater table rises in advance of salt water incursion. Logistical considerations are also exacerbated within the coastal zone as the tidal range creates a limited time frame in which to access and record archaeological remains. In many cases this also necessitated cleaning or re-excavating sites which have been inundated by sediment in every tidal cycle. In addition, in coastal environments wind and rain can be particularly challenging for equipment and the surveyor. Nonetheless, TLS offers a potential solution to the logistical difficulties associated with recording inter-tidal structures due to its relative speed, portability and high spatial resolution and accuracy.

The Sites

The sites are all located within the estuary of the Léguer River 5 km west of Lannion within the Côtes-d'Armor, Brittany (Figure 1). The Léguer river is 61.3 km long, the last 9 km of which is tidal. The river discharges to the west into the Bay of Lannion, and is thus sheltered from the stronger currents of the Channel to the north. Downstream of Le Yaudet promontory the estuary widens significantly and it is here that the majority of the intertidal structures are located. Enclosing the Baie de la Vierge is a large curved wall which is known locally as 'le mur du pêcherie' -'the wall of the fishery' but which has been interpreted as a tidal mill [START_REF] Cunliffe | Les Fouilles Du Yaudet En Ploulec'h, Cotes-D'armor[END_REF]. To the west of the Baie de la Vierge a large rectangular line of stones is located at the higher end of the tidal range. Further to the west the short curved line of stones at Poull Mad Dogan has been tentatively identified as a fishing weir (6), along with two further V-shaped weirs to the west at Dourven (1 & 2). Along the right (north) bank of the river are the large V-shaped structure at 'Le Petit Taureau' which were excavated in 2012 and 2013 (by Daire and Langouët). The excavations have revealed four distinct phases of weir built in stone (7), with an earlier timber phase identified through excavation in 2012 and 2013 (5).

Further downstream another smaller V-shaped structure, Corps de Garde, is evident (3).

The Léguer estuary is a Site du Réseau Natura 2000, a designation applied to maintain fish stocks. Today the estuary has migratory species of sea trout (Salmo trutta), Atlantic salmon (Salmo salar), and European eel (Anguilla anguilla), and was likely home to the same species when the structures within the estuary were in use. Historical evidence Mur du Pêcherie' [START_REF] Cassard | Les Bretons de Nominoë[END_REF]) -an event notable for being unusual, rather than representing the intended purpose of the structure.

Results

Structures within an Intertidal Landscape

In 2006 the Association Manche Atlantique pour le Recherche Archaéologique dans les ĺles (AMARAI) commenced the 'Maritime Fish-traps of Brittany' project, to quantify and characterise fishing structures around the Breton coastline [START_REF] Langouët | Ancient maritime fish-traps of Brittany (France): a reappraisal of the relationship between human and coastal environment during the Holocene[END_REF].

The fieldwork compiled a database of over 550 structures. One key site identified for further investigation was the large V-shaped structure in the Léguer estuary at 'Le Petit Taureau', which had at least four phases and so presented an unusual opportunity to look at the development of such structures over time. A programme of excavation in 2012 led to the discovery of a much earlier timber built V-shaped fishing structure directly underlying one phase of the stone alignments. The excavations revealed a line of wattle and post construction, supported by a finely crafted mortice and tenon sillplate with triangular stanchions similar to the 'hurls' known from Irish weirs, which would have been filled with stones to stabilise the walls of the structure and prevent their destruction in strong tides or storms. The structure represents a substantial investment of skill, with finely constructed mortice and tenon construction, dated through dendrochronology to the 615 AD [START_REF] Bernard | Early Middle Ages Fishweirs, Dendrochronology and Wood Supply in Western France: The Case of the Léguer Estuary, Servel-Lannion, Northern Brittany, France[END_REF]. The TLS survey of the timber-built fishing structure was carried out while the excavation was ongoing, in June 2013. The excavated sections of the late 6 th to early 7th century Vshaped structure were recorded, along with the extant 16th-18th century stone-built structures (Figs. 234). In addition a stone-paved trackway leading down to the beach (Figure 5), contemporaneous with the later structures, was also surveyed, as was the smaller V-shaped fishing structure at Corps de Garde to the west of the site. In November 2014 a return visit was made to scan the structure in the Baie de la Vierge, to the west of the promontory of Le Yaudet in order to survey in other features in the estuary. The georeferenced TLS survey was imported into AutoCAD to generate twodimensional sections of each structure. These sections were then plotted against each other with reference to the local mapping frame (IGN69) and zéro hydrographique (ZH), the French equivalent of chart datum (Figure 6). The cross-sections enable a direct visual comparison of the vertical placement of structures across the estuary within the tidal frame, and immediately allow some conclusions to be drawn. The most obvious is the relative heights of the structures at Le Yaudet and 'Le Petit Taureau', suggests that the structures were exploiting different ends of the tidal frame and therefore fulfilling very different functions.

The relative heights of two phases of structure at 'Le Petit Taureau' initially suggested that the later structure sat higher in the tidal frame, potentially suggesting a shift in the RSL at this point. However, the cross-section of the c. 7th weir indicates that the timber sill beam had curved over a large underlying rock, implying that the structure has been pushed downward into the sediment by the weight of the overlying stone-built phase.

This implies that the 7th century AD structure originally sat at a higher position within the tidal frame, by as much as 0.5m. The earlier phases of the structure at 'Le Petit Taureau' were first exposed due to an industrial level of sand extraction in the late 20th century, causing channel migration and displacement of sediments through the estuary.

This relationship between the structures and the sediment is important, as it suggests that the various phases of rebuilding of the structure, and the increased height at which subsequent phases were built, could therefore be attributed to the gradual increase of the level of the foreshore due to sedimentation caused naturally and/or by the structures themselves. The scan of the trackway (Figure 5) also shows ongoing erosion of the cobbled surface, and the intensity of return image clearly differentiates between sand burying the lower part of the structure and the stone.

Scanning and Volumetric Modelling 'Le Mur de Pêcherie' in the Baie de la Vierge

The first archaeological investigations of 'Le Mur du Mêcherie' (Figure 7a) carried out in the 1970s included an excavation across a sluice [START_REF] Pinot | Histoire d'un estuaire: La rivière de Lannion[END_REF]; Figure 7b). The structure was then comprehensively surveyed as part of the investigations into the site at Le Yaudet [START_REF] Cunliffe | Les Fouilles Du Yaudet En Ploulec'h, Cotes-D'armor[END_REF], and a stone by stone plan produced (Figure 7c). The excavation had uncovered the base of the presumed sluice, and so provided valuable Alignment of the Pinot drawing with an elevation generated from the laser scanning data allows the accurate placement of this significant feature within the tidal frame (Figure 7b). The results compare well with the tide mill uncovered at Nendrum, Northern Ireland, where the base of the millrace pond sat approximately 0.5 m below MHWN [START_REF] Browne | An Examination of Extractable Energy and Milliny Capacity of Mill 2[END_REF]. This supports the hypothesis that the structure at Le Yaudet is most likely the reservoir wall for a tide mill. A comparison of the sections also reveals slight differences in the morphology of the structures at 'Le Mur de Pêcherie' compared to those at 'Le Petit Taureau'. The walls of the 'Le Mur de Pêcherie' slightly inclined at an angle of around 6° from the vertical, differing in profile from the structure of 'Le Petit Taureau' which has pronounced perpendicular construction in the areas of extant blockwork. Similarly in plan the structure differs, resembling a gentle arc rather than the V-shape which defines the other fishing structures within the estuary. The structure is also perforated by several openings which appear to have functioned as sluices, and which have no comparable features with the structures at 'Le Petit Taureau'.

The placement of the structures within the tidal frame can also be used to explain their differences in size and morphology. The large fixed fishing structures at 'Le Petit Taureau' are situated between the MLWN and the MLWS levels, where 74% of low tides fall. It follows that the principal investment in the construction of fixed fishing structures would be within this area, as it would maximise the return of investment by providing the greatest tidal exposure and potentially thus the largest amount of fish.

The smaller weirs at Dourven and Corps de Garde fall below the level of the MLWS but crucially not below the level of the LAT. There is therefore no reason to assume that they were not contemporary with the large structure at 'Le Petit Taureau', as they would still have been exposed, and therefore been functional, during extremely low tides. Approximately 15% of low tides fall within this range and it may be postulated that the relatively small size of the weirs and their position within the tidal frame are not due to their relative antiquity, as originally proposed by [START_REF] Langouët | Les Barrages de Pêcheries de l'estuaire du Léguer[END_REF], but to the lesser exposure of this area of the foreshore.

As noted above, a comparison of the heights and morphology of the various structures across the estuary supports the view that structure in the Baie de la Vierge at Le Yaudet is the reservoir for a tide mill as suggested by [START_REF] Cunliffe | Les Fouilles Du Yaudet En Ploulec'h, Cotes-D'armor[END_REF]. The majority of the tidal mills in France are from the Brittany region and in 2016, a new inventory was completed and 140 tidal mills are now identified from archaeological data and historical records (accessible at: http://patrimoine.bzh/gertrude-diffusion/dossier/lesmoulins-a-maree-de-bretagne/19934f6b-e5d4-42a0-ba6d-ec8b00638295). The Baie de la Vierge is ideal for a tide mill, being an unused inlet that could easily be blocked off [START_REF] Charlier | The Rise and Fall of the Tide Mill. Ocean sciences bridging the millennia: a spectrum of historical accounts[END_REF]). The placement of the structure at the top end of the tidal frame is consistent with the siting of tide mills, which required a tidal coefficient of 65 to 70 (Wailes and Gardner 1938). In practice this meant that they were sited at a point in the tidal frame three hours above the turning tide [START_REF] Charlier | The Saga of Tide Mills[END_REF]. The barrage is referred to in the Vita S. Euflam (Life of Saint Efflam) the earliest version of which is thought to have been written down in the twelfth century (de La Borderie 1892, 289-90 cited in [START_REF] Cunliffe | Les Fouilles Du Yaudet En Ploulec'h, Cotes-D'armor[END_REF]. It is referred to as a device for trapping fish, omitting any mention of a milling function suggesting that by the 12th Century the mill and all knowledge of it had disappeared [START_REF] Cunliffe | Les Fouilles Du Yaudet En Ploulec'h, Cotes-D'armor[END_REF].

The presence of 7th century AD constructions on the north side of the estuary, and the association of the structure with the Life of St Efflam, suggests that the 'Mur de Pêcherie' is of an early medieval date. In this case it would be directly comparable to the 7th century AD sites of Ebbsfleet (AD 691-2) in Kent, England [START_REF] Hardy | The Mid-Saxon Mill at Northfleet[END_REF] [START_REF] Rynne | Waterpower in Medieval Ireland[END_REF]. All of these examples are on narrow tidal-creeks and on sheltered parts of the coast, with those that also use a freshwater source placed where a small stream enters an estuary, as it is the case at Le Yaudet. There is also evidence of cultivation and several corn drying mills at Le Yaudet associated with what was probably a significant early monastic settlement [START_REF] Cunliffe | Les Fouilles Du Yaudet En Ploulec'h, Cotes-D'armor[END_REF]. The lack of evidence for a wheelhouse is not as problematic as it might seem, as most mills of an early date, such as Nendrum, Ebbsfleet or Killoteran were constructed almost entirely of wood [START_REF] Hardy | The Mid-Saxon Mill at Northfleet[END_REF]. Horizontal mill wheels, common in tide mills of most periods, would have sat above the wall. The suggestion that the structure had the primary function of a mill and millpond does not preclude its use for catching fish, indeed the tide mill at Nendrum was originally thought to have been a fishing structure prior to its extensive excavation [START_REF] Mcerlean | Harnessing the Tides : The Early Medieval Tide Mills at Nendrum Monastery[END_REF]. [START_REF] Browne | An Examination of Extractable Energy and Milliny Capacity of Mill 2[END_REF] on Nendrum mill provides a model by which this volumetric data can be used to assess the potential mill output. To calculate the volume behind the 'Mur de Pêcherie' the point cloud was interpolated into a surface by fitting a TIN (triangulated irregular network) creating a Digital Elevation Model (DEM). A flat plane was then introduced above the DEM at the height of MSL, MHWN, MHWS and HAT (Figure 8). This allows a visualisation of the different tidal states and the calculation of the potential of the reservoir by calculating the volume between the plane and the mesh. Tide mills were unusable on neap tides or when the wind held up the waters, and in any case the MHWN level falls below the level of the structure. A calculation of the volume of the reservoir during MHWS will therefore give the average volume of water on usable tides. This was calculated for the extant remains and then calculated for a range of heights which may represent the extent of the original structure. The excavations by [START_REF] Pinot | Histoire d'un estuaire: La rivière de Lannion[END_REF] show that the functional part of the structure lay at least 1m below present ground level, and the lack of a binding course on top of the current structure suggest it may have been at least one course higher than the current extant remains. Calculations were therefore made for the extant remains plus additional heights at 0.5 m intervals (Table 1). 1. Estimated potential volumetric capacity (in cu m) of 'Le Mur de Pêcherie'

MHWS HAT

The volume was then calculated using the formula by [START_REF] Browne | An Examination of Extractable Energy and Milliny Capacity of Mill 2[END_REF] to calculate the potential annual output of the mill, based on the characteristics of the mill at Nendrum.

Although the morphological data from the site is incomplete, known 7th century AD tide mills show a great deal of consistency. The modern tidal range at Nendrum is c. 3.4 m.

The Nendrum millpond has area of 2000 m 2 , which at MHWS created a volume of water of approximately 1800 cu m. According to [START_REF] Browne | An Examination of Extractable Energy and Milliny Capacity of Mill 2[END_REF] this allowed the tide mill to generate an annual power equivalent to 50 tonnes of milled barley per year. A comparison with results of the volumetric analysis at Le Yaudet suggests that it could have had a volume nearly 16 times greater than that of Nendrum, allowing as much as (assuming 20 bushels per head per year).

The Léguer Estuary: A Landscape of Production

Le Yaudet headland, which has significant earthwork fortifications dating from the Iron Age and Roman periods, continued in importance through the early Medieval and Medieval periods as a religious centre for pilgrimage linked with the lives of St. Enora and St. Efflam -prominent 6th century Irish missionaries [START_REF] Cassard | Les Bretons de Nominoë[END_REF][START_REF] Cunliffe | Les Fouilles Du Yaudet En Ploulec'h, Cotes-D'armor[END_REF]). The number, size and persistence of the fixed-fishing structures, including the creation of the rock-cut and constructed track to them (Figure 5) and the tidal mill suggest a large and stable local community under monastic or seignorial control. The TLS survey also provided the opportunity to examine the relationships of the structures within the tidal frame, as different structures require fixed areas of the tidal frame to operate. Intertidal structures have the potential to be used as benchmarks for sea level, and previous work in the Léguer estuary by [START_REF] Langouët | Les Barrages de Pêcheries de l'estuaire du Léguer[END_REF] has taken this approach and assumed that the dates of the intertidal structures within the estuary can be calculated by comparing their varying heights (Table 2). In this approach the base of each intertidal structure is assumed to be at the level of the 'plus haute basse mer de morts-eaux' (PHBMme) or lowest neap tide. The difference between this level and the modern datum is then calculated (δ(PHBMme)), and used to provide a relative date. However, it is suggested here that the structures operated, contemporaneously within the tidal frame. This makes the

Conclusions

In assessing the function of the varying sites within the Léguer estuary, the application of TLS has made a significant contribution. The TLS modelling of the 'Mur de Pêcherie' of its morphology, tidal-frame position and volumetric potential in association with an early monastic settlement all make a strong case for its usage as a tide mill. The TLS survey at a landscape level shows several structures operating for several different purposes within the tidal frame. This exploitation of different parts of the intertidal and coastal zone across this small estuary implies a control of the landscape as a whole, most likely from the settlement and monastic community at Le Yaudet, -a fortified early Medieval settlement with a sheltered landing place, one large fishweir to exploit the majority of tides, one or more smaller weirs for the more extreme tides, and a tidal mill for flour production. This would certainly be in keeping with the concept of a monastic

  step-change in recording and understanding fixed fishing and allied structures through in-situ recording, landscape analysis and functional modelling.

  the catching of a large shoal of herring (Sardina pilchardus) behind 'Le

  positioning of a diagnostic feature in relation to the tidal frame.

  barley to be milled per year supporting a population of over 1000

  tidal frame has remained roughly the same since construction. This is a difficult assumption to test as an accurate enough historical sea level curve does not exist for this location. The tidal gauge at Roscoff has been recording tidal height data since 1973, during which time the monthly mean has risen from 6834 mm to 7037 mm above RLR (revised local reference) datum. However, this short period accounts for only two full metonic cycles (period of 19 years in which the sun, moon and earth return to the same relative positions) and so cannot easily be projected backwards over several centuries. It is therefore difficult to use this as a model of the RSL over past 1500 years at the site. What can be said, however, is that none of the structures at the site lies outside the tidal frame, all are submerged for at least some of the tides, and none are below LAT. If there had been a significant shift of the relative sea level in either direction, one or more of the structures would likely lie outside the tidal frame.Additionally all of the structures are in the appropriate location within the tidal frame to fulfil their original function, as all of the presumed fixed fishing structures are situated between the MLWS and MLWN levels, whilst the 'Mur de Pêcherie' is perfectly positioned to function as a tide mill. The difference in heights between the 7th century timber weir and the later overlying structures appears to have been caused by sedimentation rather than a change in RSL. In contrast, examples of fishing structures from other locations in Brittany, as well as comparable sites in Wales and Essex, have been found below the level of LAT (O'Sullivan 2003).

Figue 1 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .Figure 7 Figure 8 .

 1345678 Figue 1. Intertidal structures: Léguer estuary 1. Dourven-A 2. Dourven-B 3. Corps de Garde 4. Baie de la Vierge and the 'Mur de Pêcherie', 5. Petit Taureau (D) 6. Poull Mad Dogan 7. Petit Taureau (A). 159x97mm (220 x 220 DPI)

Table 2 .

 2 Surveyed heights of inter-tidal structures within the Léguer Estuary.

	No Site	Nb(m)	δ(PHBMme)	Attributed Age
				(m)
	1	Trédrez, Dourven-A	0.40 ± 0.20	3.70 ±0.10	Bronze Age
	2	Trédrez, Dourven-B	0.70 ± 0.20	3.40 ± 0.10	Bronze Age
	3	Servel, Corps de Garde	1.70 ± 0.10	2.40 ± 0.10	Iron Age
	4	Ploulec'h, Baie de la Vierge	2.38 ± 0.10	1.70 ± 0.20	Gallo-Roman to early Medieval
	5	Servel, Petit Taureau (D)	2.45 ± 0.05	1.60 ± 0.10	650 AD
	6	Ploulec'h, Poull Mad Dogan	2.80 ± 0.30	1.30 ± 0.30	Early Medieval
	7	Servel, Petit Taureau (A	2.90 ± 0.10	1.20 ± 0.10	Late 15 th century

  Medieval concept of self-sustaining communities with an emphasis on fish-eating in keeping with the 6th century Rule of St Benedict which also stated that monasteries should have mills[START_REF] Wikander | Handbook of Ancient Water Technology[END_REF]. The association of Le Yaudet with 6 th century Irish monasticism, and the very similar tide mill sites at Nendrum and Little Island, provides an interesting wider geographical context.

	F o r P e e r R e v i e w community and the early F o r P e e r R e v i e w F o r P e e r R e v i e w F F o r o r P e P e e r e r R e R v e v i e i w e w
	11

Acknowledgements

Thanks to Jo Nield, Rick Turner, & Martin Bell for assistance and advice.