Keywords: metasurface, efficiency, bandwidth

Two methods are proposed for the accurate and fast analysis of the efficiency of arbitrarily modulated metasurface (MTS) antennas. The surface current on the MTS is expanded into entire-domain basis functions. The first method uses a Fourier-Bessel basis and relies on the Poynting theorem, while the second approach is based on the calculation of the surfacewave residue with a Gaussian-Ring expansion for the current. Both methods allow one to compute the efficiency of MTS antennas in a few minutes, which represents a drastic reduction of the computation time in comparison with any commercial software. The algorithms are then used to analyze the frequency dependence of the efficiency of MTS antennas in two different cases: 1) anisotropic MTS with uniform periodicity, 2) anisotropic MTS with non-uniform periodicity. The latter case corresponds to an active region MTS and provides a larger bandwidth. Validation with commercial software and measurements data is provided.

Metasurface Antennas: Efficiency versus Bandwidth Modeste Bodehou * , Marco Faenzi † , David González-Ovejero ‡ , Stefano Maci † , Christophe Craeye * , Enrica Martini †

I. INTRODUCTION

It is well known that metamaterials (MTMs) and metasurfaces (MTSs) generally suffer from bandwidth limitations due to the frequency dispersion of the propagating waves [START_REF] Quevedo-Teruel | Roadmap on metasurfaces[END_REF]. In many communication and radar systems, a minimum bandwidth is required to satisfy the desired data rate or to provide a sufficient resolution. Despite several advantages provided by MTS antennas (low profile, low cost, low losses), their bandwidth may be limited, mainly due to a mismatch between the wavelength of the supported surface-wave (SW) and the period of the modulation. Indeed, those antennas are designed to mimic a modulated surface impedance at a given frequency [START_REF] Fong | Scalar and tensor holographic artificial impedance surfaces[END_REF]- [START_REF] Bodehou | A Quasi-direct method for the surface impedance design of modulated metasurface antennas[END_REF]. The role of the feed consists of exciting a SW in the slab, which progressively looses its energy as a leaky-wave (LW). The LW radiation results from the interaction between the SW and the impressed impedance profile. When the frequency of operation is changed, the periodicity of the modulation rapidly becomes inconsistent with the supported SW mode, thereby limiting the bandwidth. This issue has been recently addressed in [START_REF] Faenzi | Wideband active region metasurface antennas[END_REF] by an active region radiation concept. The idea is to impose a variation of the modulation period in the radial direction such that, at a given frequency, a given annular region of the antenna radiates. This allows one to enlarge the antenna bandwidth, at the expense of a lower aperture efficiency. However, since those structures are LW antennas, one should ensure that the power carried by the SW is radiated before reaching the periphery of the antenna. Otherwise, significant diffraction/reflection effects can occur at the antenna rim, which can strongly impact the antenna performance. This calls for the definition of a conversion efficiency, which represents the ratio between the radiated power, assuming an infinite slab, and the power provided by the feed. When analyzing those antennas printed on an infinite slab as commonly done in Method of Moments (MoM) approaches [START_REF] Bodehou | Method of Moments simulation of modulated metasurface antennas with a set of orthogonal entire-domain basis functions[END_REF]- [START_REF] Verni | On the use of entire-domain basis functions and fast factorizations for the design of modulated metasurface[END_REF], scattering at the rim is not taken directly into account and a non-negligible SW power may be carried away by the infinite slab. Therefore, it is important for the antenna designer to have a rapid evaluation of the conversion efficiency, namely the ratio between the radiated power and the total power delivered by the feed, and the aperture efficiency, so as to optimize the global efficiency of the antenna before proceeding to realization. This paper provides two effective (fast and accurate) methods for the computation of the efficiency of MTS antennas. With those methods, we analyze the frequency variation of the efficiency when the bandwidth of the antenna is getting larger. The paper is structured as follows. Section II is devoted to the MTS modeling, Sections III and IV discuss the computation of the powers required for the derivation of the efficiency. Section V analyzes the efficiency versus frequency response of MTS antennas and Section VI concludes the paper.

II. MTS MODELING AND POWER BALANCE

Let us consider a circular domain MTS antenna of radius a, whose cut is represent in Fig. 1. The antenna is fed at the center with a source point. The MTS layer is modeled as a sheet impedance distribution and lays on an infinitely extended substrate. Assuming a perfect matching of the feed, the power delivered to the antenna (P tot ) is partially radiated (P vis ), partially lost in the substrate (ohmic power losses (P Ω )) and partially reflected/diffracted at the substrate rim (P dif f ). Since the losses in MTS antennas are negligible at low frequency (< 30 GHz) [START_REF] Minatti | Modulated metasurface antennas for space: synthesis, analysis and realizations[END_REF], one can simply write the power balance as

P tot = P vis + P sw dif f . (1)
This paper aims at accurately and rapidly estimating the power contributions P tot , P vis and P dif f , as well as the aperture efficiency of the antenna. First, the equivalent surface current distribution on the MTS is computed with Fourier-Bessel basis functions (FBBF) [START_REF] Bodehou | Method of Moments simulation of modulated metasurface antennas with a set of orthogonal entire-domain basis functions[END_REF] or Gaussian-Ring basis functions (GRBFs) [START_REF] González-Ovejero | Gaussian ring basis functions for the analysis of modulated metasurface antennas[END_REF] using the MoM. The first method (based on the Poynting theorem) enables the computation of P vis , P tot and then derives P dif f from equation ( 1). The second method, based on the residue theorem, computes P vis , P dif f , and then deduces P tot from (1). In both methods, P vis is computed directly in spectral domain using the close form spectrum of the FBBFs or GRBFs [START_REF] Bodehou | Power balance and efficiency of metasurface antennas[END_REF]. In the following, the computation of P tot and P dif f are detailed.

III. TOTAL POWER CALCULATION BASED OF THE POYNTING THEOREM

From the Poynting theorem and in absence of losses, the power delivered by the feed can be calculated after integrating the flux of the Poynting vector along any closed surface including the feed. We suggest the surface represented in red line in Fig. 1. This surface comprises the MTS layer, the rim as well as the ground plane. The total power delivered by the feed can then be written as:

P tot = 1 2 Re S1 E t × H * t . ẑ dS + S2 E × H * . ρ dS = P M T S + P rim , (2)
where P M T S and P rim are illustrated in Fig. 1. t means the tangential part. E and H are the electric field and the magnetic field respectively. ρ is the radial unit vector and Re is the real part operator.

A. MTS contribution

The MTS contribution is written as:

P M T S = 1 2 Re S1 E x . H * y -E y . H * x dS , (3)
To compute the electric field, we calculate the tested electric field on the surface from the MoM matrix [Z E G ] and the MoM excitation vector [V E ] as:

[E t ] = [Z E G ] [I] -[V E ], (4)
Since the FBBFs are orthogonal on the MTS aperture (see orthogonality relation in [START_REF] Bodehou | Method of Moments simulation of modulated metasurface antennas with a set of orthogonal entire-domain basis functions[END_REF]), the electric field in the Fourier-Bessel basis is derived from the tested field by simple renormalization. A similar procedure is used for the computation of the magnetic field. This allows one to express integral (3) as a double summation over all the basis functions:

P M T S = 1 2 N n=-N M m=1
K(m, n)Re {e x mn h y * mn -e y mn h x * mn } .

(5) where index mn identifies each basis function, e mn and h mn correspond to the electric field and the magnetic field coefficients respectively and K(m, n) is the normalization factor.

B. Rim contribution

For a quasi-TM mode, the rim contribution can be developed as

P rim ≈ - 1 2 Re E z (ρ = a) H * φ (ρ = a) dφ dz (6)
where ρ and φ are the classical cylindrical coordinates.

E zx = 1 4π 2 m,n i x mn G EJ zx R mn e -j(kxx+kyy) dk x dk y (7)
The spectral Green's function G EJ zx and the FBBF Fourier transform R mn can be structurally written as g E zx (k ρ ) cos α, and F mn (k ρ )e -jnα respectively. Inserting the latter expressions in [START_REF] Li | Polarization-insensitive holographic surfaces with broadside radiation[END_REF], and using a cylindrical spectral coordinates (k ρ , α), expression (7) can be rewritten as

E zx = 1 4π 2 m,n i x mn ∞ 0 g E zx F mn 2π 0 cos α e -jnα e -jkρρ cos(φ-α) dα k ρ dk ρ (8)
Then, one can integrate in closed form along α:

E zx = 1 4π 2 m,n i x mn ∞ 0 g E zx F mn j f 0 n (k ρ ρ, φ, 1)- f 0 n (k ρ ρ, φ, -1) k ρ dk ρ (9)
where we defined the function f 0 n as

f 0 n (k ρ ρ, φ, i) = πj n e -j(n+i)φ J n+i (-k ρ ρ) (10)
and J n (x) represents the first kind Bessel function of order n. We simplify the notations by defining the following functions:

f a mn (k ρ ρ, φ, -1) = -j f 0 n (k ρ ρ, φ, -1) g E zx F mn (11)
f b mn (k ρ ρ, φ, 1) = j f 0 n (k ρ ρ, φ, 1) g E zx F mn (12)
Then, according to [START_REF] González-Ovejero | Gaussian ring basis functions for the analysis of modulated metasurface antennas[END_REF], and ( 12), E zx can be structurally written as

E zx = 1 4π 2 m,n i x mn ∞ 0 [f a mn (k ρ ρ, φ, -1)+ f b mn (k ρ ρ, φ, 1) k ρ dk ρ (13)
Expression ( 13) is a harmonic representation of the fields in the the azimuthal direction. The same thing is done for all the other components of the electric and magnetic fields. This enables a harmonic azimuthal representation of each component of the total electric field, and hence an analytical integration along the azimuthal direction. Moreover, integrals along k ρ are separable and can be pre-computed, which significantly accelerates the computation. Integration along z is carried out with very few sampling points (about 4) since the integrand along z is relatively smooth. Details regarding the formulation can be found in [START_REF] Bodehou | Power balance and efficiency of metasurface antennas[END_REF], [START_REF] Bodehou | Modulated metasurface antennas: analysis and direct numerical synthesis[END_REF].

IV. SW POWER COMPUTATION BASED ON THE CALCULATION OF THE SW RESIDUE This approach is based on the evaluation of the residual (i.e. not radiated) SW power through the calculation of the residue of the current at the SW pole of the bare grounded slab. This calculation is performed in an extremely efficient way through a closed-form expression involving the spectrum of the current. The derivation starts from the following expression for the zcomponent of the spectrum of the aperture electric field:

E z (k ρ , α) = n E zn (k ρ )e -jnα (14)
which contains both the contribution of the currents in the MTS and the direct contribution of the feed, which only contributes to the 0-indexed term, due to its axial symmetry. It is noted that this form is automatically available if GRBFs or FBBFs are used for MTS current representation. The spatial field is obtained by taking the inverse Fourier transform of the expression in [START_REF] Bodehou | Modulated metasurface antennas: analysis and direct numerical synthesis[END_REF]. For ρ > a, where a is the radius of the MTS, this can be divided into two contributions: the radiated field and the SW field. This latter is given by the contribution of the residue at the SW pole of the bare grounded slab, β sw 0 , and reads

E SW z = - j 2 e -j π 4 2β sw 0 πρ e -jβ T M 0 ρ n R n e -jnφ (15)
where we have assumed β sw 0 ρ 1 and

R n = lim kρ→β sw 0 (k ρ -β sw 0 ) E zn = J T M n (β sw 0 ) R GF +δ n,0 R f eed (16)
In ( 16), J T M n (β sw 0 ) = 2π 0 J T M (β sw 0 , α) e jnα dα, R GF is the residue of the Green's function relating the TM currents on the MTS to the vertical electric field on top of the slab and R f eed is the residue of the field radiated by the feed on the bare slab. From [START_REF]FEKO: Field computations Involving Bodies of Arbitrary Shape[END_REF], taking into account the local structure of the SW electromagnetic field, the real part of the relevant Poynting vector is derived, and integrated over a cylindrical surface surrounding the radiating aperture and extending from the ground plane to infinity, to obtain the following expression for the residual SW power

P sw dif f = ω 0 4 r tan (k zd h) k zd + h cos 2 (k zd h) + r α za n |R n | 2
(17) where 0 is the free-space permittivity, r is the relative permittivity of the substrate, ω = 2πf is the angular frequency, h is the slab thickness,

k zd = k 2 0 r -(β sw 0 ) 2 and
α za = (β sw 0 ) 2 -k 2 0 .
Notice that the evaluation of (17) only requires the knowledge of the residues in [START_REF] Faenzi | Metasurface antennas: New models, applications and realizations[END_REF], which can be calculated for any given current spectrum. However, the general expression can be also specialized to the particular case of a given current representation. For instance, if GRBFs are used to represent MTS currents, it results

J T M n (β sw 0 ) = πj n M m=1 ji x m,n+1 + i y m,n+1 Φ m,n+1 (β sw 0 ) + i y m,n-1 -i x m,n-1 Φ m,n-1 (β sw 0 ) (18)
where the Φ functions are reported in [START_REF] González-Ovejero | Gaussian ring basis functions for the analysis of modulated metasurface antennas[END_REF].

V. RESULTS

A. Validation of the two methods

This section provides a numerical validation (with the commercial software FEKO [START_REF]FEKO: Field computations Involving Bodies of Arbitrary Shape[END_REF]) of the two methods detailed in Section III and Section IV. We consider a MTS antenna radiating a broadside pencil beam, which parameters are described in Section V.A of [START_REF] González-Ovejero | Gaussian ring basis functions for the analysis of modulated metasurface antennas[END_REF]. Fig. 2 shows the efficiency results for various modulation indices. One can see that, although those two methods are based on quite different formalisms and that they have been demonstrated here with two different sets of basis functions, the efficiency results are very similar and follow well the FEKO solution. The slight difference with respect to FEKO may be explained by the feeder modeling. In terms of computation time, the Poynting theorem based method and the residue based method take respectively 2.5 min and 1.1 min on a conventional computer, while FEKO simulations require 2.4 hours. This represents a drastic reduction of the computation time.

B. Efficiency versus frequency analysis

A MTS antenna with uniform periodicity and designed to generate a broadside beam at 29.75 GHz is now considered. The modulation index of this antenna has been optimized to provide a quasi-uniform aperture field illumination [START_REF] Faenzi | Metasurface antennas: New models, applications and realizations[END_REF]. seen that the antenna bandwidth is not very large. Defining the bandwidth as the one corresponding to a drop of half of the maximum efficiency, the bandwidth of this antenna is about 6.5%. It is interesting to see that the global efficiency is mainly impacted by the aperture efficiency, the conversion efficiency being almost constant over a very large bandwidth.

As demonstrated in [START_REF] Faenzi | Wideband active region metasurface antennas[END_REF], the bandwidth can be improved by modulating the period of the reactance surface. simulations can be attributed to the feeding modeling and the surface impedance implementation process with patches. In comparison with the uniform period MTS case, the global efficiency is lower due to the active region nature of the antenna. On average, the efficiency is 13% over a 22% bandwidth.

VI. CONCLUSION

We have presented two techniques for the analysis of the efficiency of MTS antennas. These two methods are based on completely different formalisms, but provide very similar results and a very good agreement with commercial softwares and measured results. The impact on the efficiency when the bandwidth of the antenna is enlarged has been illustrated. It has been observed that the reduction of the global efficiency when one moves away from the center frequency is mainly due to the aperture efficiency, while the conversion efficiency has little impact. The algorithms presented in this paper can be used for the rapid and accurate optimization of the efficiency of MTS antennas designed with any synthesis technique, i.e the methods can handle arbitrarily modulated surface impedances. As such they provide to the antenna designer a reliable and efficient analysis tool.

Fig. 1 .

 1 Fig. 1. Power contributions for a finite MTS over an infinite grounded substrate.

Fig. 2 .

 2 Fig. 2. Validation of the efficiency results for different modulation indices. The global efficiency is the product between the aperture (tapering) efficiency tap and the conversion efficiency conv .

Fig. 3 (Fig. 3 .

 33 Fig. 3. Analysis of the efficiency for a uniform period MTS assuming that the frequency dispersion of the sheet is negligible. (a) cut of the impedance profile. (b) efficiency results. The global efficiency is the product between the aperture (tapering) efficiency tap and the conversion efficiency conv .

Fig. 4 .

 4 Fig. 4. Analysis of the efficiency for an active region MTS. (a) cut of the impedance profile. (b) efficiency results.

  The z component of the electric field is split into three contributions corresponding to each Cartesian direction of the current (E zx , E zy , and E zz ). In the same way, we defined H xx , H xy , H xz , H yx , H yy , and H yz . 1) Evaluation of E zx : The z component of the electric field due to the x-directed current (known in the FBBF basis) is computed spectrally with the pertaining spectral Green's function G EJ zx through inverse Fourier transform.

ACKNOWLEDGMENT

This research has been partially funded by Région Wallonne through the LUMINET project under contract number 1810114.


Marco Faenzi
email: faenzi@diism.unisi.it

David González-Ovejero
email: david-gonzalez-ovejero@univ-rennes1.fr

Stefano Maci
email: macis@diism.unisi.it

Modulated Metasurface Antennas with Enhanced Broadband Response

Keywords: metasurface antennas, broadband, leaky waves, surface waves. I

Modulated Metasurface Antennas with Enhanced

Broadband Response

INTRODUCTION

Aperture antennas based on modulated metasurfaces (MTSs) [START_REF] Fong | Scalar and tensor holographic artificial impedance surfaces[END_REF]- [START_REF] González-Ovejero | Multibeam by metasurface antennas[END_REF] have recently received considerable attention from the antenna community. In this class of antennas, radiation arises from the beating between a TM surface wave (SW) and a periodically modulated impedance boundary condition (IBC). This way, the -1 Floquet mode can enter the visible region, becoming radiative. The main advantages of modulated MTS antennas are their lowprofile and low mass, a simple fabrication by PCB technology [START_REF] Fong | Scalar and tensor holographic artificial impedance surfaces[END_REF]- [START_REF] Faenzi | Metasurface Antennas: New models, applications and realizations[END_REF] or additive manufacturing [START_REF] González-Ovejero | Additive Manufactured Metal-Only Modulated Metasurface Antennas[END_REF] and the absence of protruding feeds, as opposed to reflect-arrays and transmit-arrays. The IBC modulation is typically achieved by a dense textured layer composed of many small metal patches printed on a grounded dielectric slab [START_REF] Fong | Scalar and tensor holographic artificial impedance surfaces[END_REF]- [START_REF] Faenzi | Metasurface Antennas: New models, applications and realizations[END_REF]. These patches are arranged on a periodic lattice, and they take different shapes and dimensions to provide the requested impedance modulation.

Modulated MTS antennas are especially appealing for space, where strict mass and volume constraints apply. Moreover, they have demonstrated the capability of coping with a variety of challenging requirements such as contoured beams [START_REF] Minatti | Synthesis of modulated-metasurface antennas with amplitude, phase, and polarization control[END_REF]- [START_REF] Faenzi | Metasurface Antennas: New models, applications and realizations[END_REF], high gains [START_REF] Minatti | Synthesis of modulated-metasurface antennas with amplitude, phase, and polarization control[END_REF]- [START_REF] Faenzi | Metasurface Antennas: New models, applications and realizations[END_REF], [START_REF] Minatti | Efficiency of metasurface antennas[END_REF], multi frequency operation [START_REF] Faenzi | Dualband self-diplexed modulated metasurface antennas[END_REF]- [START_REF] Li | Multiwavelength multiplexing hologram designed using impedance metasurfaces[END_REF], dual polarization [START_REF] Pereda | Dual circularly polarized broadside beam metasurface antenna[END_REF]- [START_REF] Li | Polarization-insensitive holographic surfaces with broadside radiation[END_REF], and multiple beams [START_REF] González-Ovejero | Multibeam by metasurface antennas[END_REF]. It is important to note that such features do not alter the MTS low mass and low envelope; a careful design of the patch texture suffices to achieve an accurate control of the aperture field in terms of amplitude, phase and polarization. Nonetheless, MTS apertures are typically constrained by a limited gain-bandwidth. Indeed, it was shown in [START_REF] Minatti | Bandwidth of gain in metasurface antennas[END_REF] that broadside apertures provide relative bandwidths (Δf/f0) ranging from 3% to 9% for decreasing gains going from 40 to 28.5dBi. Such detrimental effect originates from a progressive mismatch between the SW wavelength λsw and the periodicity d of the modulation as frequency shifts from the central frequency, f0. The SW wavelength variation is mainly due to the dispersive behaviour of the dielectric substrate, which causes a progressive beam depointing and thus a rapid gain reduction.

A method was proposed in [START_REF] Faenzi | Wideband active region metasurface antennas[END_REF] for enhancing the gainbandwidth of circular apertures. It is based on adopting a radially-dependent periodicity function d(ρ) that grows exponentially with the radius. Then, as frequency shifts from lower to higher frequencies, the matching between λsw and d(ρ) occurs in an active annular region that progressively shifts from the rim of the antenna to its center (see Fig. 1). However, the gain responses obtained in [START_REF] Faenzi | Wideband active region metasurface antennas[END_REF] still show a quite high fluctuation with respect to the average gain value. In this paper, we propose an optimization scheme to eliminate these gain fluctuations. To this end, a more general shape of d(ρ) is considered in the optimization algorithm, which also manages the frequency ωp at which the MTS texture is designed to obtain an efficient aperture illumination.

This paper is structured as follows. Section II presents the problem at hand and the initial algorithm setup. Section III details the algorithm scheme, and numerical results relevant to a specific broadband design are presented in section IV. Finally, conclusions are drawn in Section V.

II. PROBLEM DESCRIPTION AND ALGORITHM SETUP

The circular apertures dealt with are characterized by a radius a, while the texture lays at z=0 plane of a cylindrical reference system (see Fig. 1). This texture is modelled by a penetrable IBC as ( )

0 0 ĵ t z z + - = = = ⋅ × - E X H H t t z ( 1
)
where

0 t z + =
( ) ( ) ( ) ( ) { ( ) ( ) ( ) } ˆˆˆˆ cos ˆˆˆˆ sin X m m m ρ ρ ϕ ρ    = + Φ ρ +ϕ ρ - ρ       - ρ Φ ρ +ϕ +    X I ρρ ϕϕ ρϕ ϕρ (2)
where X ρ is the average reactance value, , m m ρ ϕ are radius-dependent modulation indexes, and Φ(ρ)

( ) ( ) ( ) 0 2 0 d d ρ π ρ ρ ρ ′ Φ = +Φ ′  (3)
is the phase of the modulation calculated from the periodicity function d(ρ), where Φ(0) in ( 3) is a fixed initial phase. In this work, we adopt a function d(ρ) that results from the optimization scheme in Fig. 2 to get rid of the gain fluctuations observed in [START_REF] Faenzi | Wideband active region metasurface antennas[END_REF].

The optimization process starts by selecting the desired bandwidth, delimited by the angular frequencies ωmin and ωmax, and with ω 0 center frequency. The grey box in Fig. 2 represents this step. The objective in-band average gain Gave is then selected and the aperture radius a is found using the formula [START_REF] Faenzi | Flat gain broad-band metasurface antennas[END_REF] , 0 0 0 28.7 28.7

g ave ave v f a a G f c λ λ Δ ≈ < ( 4
)
where λ0 is the free space wavelength at ω 0 and vg,ave is the integral average of the group velocity associated to ( )

X ρ ω
within the objective frequency band. Of course, one may also determine Gave starting from a fixed radius. The preliminary choice of the MTS substrate (in the following we use a Rogers RO3006 laminate with εr=6.15 and thickness h=0.635mm) affects the selection of the texture; we have adopted here elliptical elements. The definition of the texture allows one to fix the average reactance value ( )

0
X ρ ω at f0, which is maintained constant at any optimizer iteration. Once ( ) 0 X ρ ω has been defined, the dispersion of the wavenumber βsw of a SW propagating on the average texture reactance ( ) X ρ ω is characterized within the band of interest by using the full-wave solver in [START_REF] Maci | FSS-based EBG surface[END_REF]. The obtained results are used to build the dispersion database (DB), shown as step A in Fig. 2. A set of N equispaced inband phase control frequencies fn are selected and the SW wavelengths λsw,n=2π/βsw,n (obtained from dispersion curve) are associated to each frequency fn. Each λsw,n is then linked to a radial control point ρn so to build the function d(ρ). The ρn ordering is done to preserve a monotonic growth of d(ρ) for increasing ρ.

Therefore, we assume that to every fn corresponds the center of the nth active region located in ρn. The operations above are carried out in step 1 in Fig. 2.

In order to get a flat gain response, it is also important to select an amplitude control frequency ωp at which the amplitude of the -1 indexed Floquet is imposed as constant along ρ. In the optimization process, this frequency can vary within the design band. The coordinates ρn and the frequency ωp are iteratively managed by the algorithm using as control the value C of a cost function calculated at the end of any iteration, and defined next. In the first iteration, radially equispaced control points ρn are assumed, and ωp coincides with the middle frequency ω 0. The cost function C is defined as ( ) . S(ρ,ω) in ( 6) represents the surface power density distribution and Ψ is the phase of the -1 mode in the adiabatic Floquet expansion [START_REF] Minatti | Flat optics for leaky-waves on modulated metasurfaces: Adiabatic Floquet-wave analysis[END_REF] defined as

( ) 0 ( , ) ( ) sw sw d ρ ρ ω ρ β ρ β ρ ρ ′ ′ Ψ =Φ - -Δ  (7)
whre Δβsw represents a deviation from βsw due to the reactance modulation, and can be found by solving the local canonical problem [START_REF] Caminita | New wine in old barrels: The use of the Oliner's method in metasurface antenna design[END_REF], the phase Φ(ρ) is instead related to d(ρ) through [START_REF] Faenzi | Metasurface Antennas: New models, applications and realizations[END_REF].

The gain G depends on ω by means of S(ρ,ω) and Ψ(ρ,ω), which frequency dependence is due to the MTS dispersion. At lower frequencies (active rings are located in the vicinity of the rim), it is particularly important to obtain a uniform illumination of the aperture. To this end, the optimizer adaptively changes the frequency ωp at which a uniform profile of S(ρ,ωp) is imposed, with rapid zero dropping close to the source and the rim (see [START_REF] Faenzi | Wideband active region metasurface antennas[END_REF] for S(ρ,ωp) expression). Once S(ρ,ωp) has been defined, one can retrieve the local attenuation constant α(ρ,ωp) for the -1 Floquet mode from

( ) 0 ( , ) / 2 , 1 ( ', ) ' ' 2 p p sw p S P S d ρ ρ ρ ω α ρ ω ρ ω ρ ρ π = - (8)
where Psw is the power associated to the SW. Next, by solving the generalized local canonical problem that matches the local periodicity d(ρ) and average reactance X ρ [START_REF] Caminita | New wine in old barrels: The use of the Oliner's method in metasurface antenna design[END_REF] it is possible to find the modulation index functions (step 2 in Fig. 2) that locally match the profile of α in (8) (here we consider mρ(ρ) equal to mϕ(ρ), see Fig. 3). At this point, the reactance tensor ( )

, p X ρ ω
is completely characterized at frequency ωp.

Fig. 3 The surface represents the solution of the canonical problem [START_REF] Caminita | New wine in old barrels: The use of the Oliner's method in metasurface antenna design[END_REF] for the attenuation constant α for a set of modulation indexes and periodicities at frequency ωp. Local modulation index for any ρn (dn) is found by selecting on on the modulation index axis the value that provides the desired α(ρn,ωp).

The successive steps consist in determining Δβsw(ρ,ω) and α(ρ,ω) as functions of ω. With Δβsw(ρ,ω), one can obtain Ψ(ρ,ω), while S(ρ,ω) may be computed using α(ρ,ω) in the inverse formula of (8)

( ) ( ) ( ) 0 1 ( , ) 2 , exp 2 ', ' . 2 SW P S d ρ ρ ω α ρ ω αρ ω ρ ρ π = - (9)
Now, G(ω) is fully defined.

The calculation of Δβsw(ρ,ω) and α(ρ,ω) above, implies the characterization of the reactance tensor ( ) , X ρ ω for any frequency ω. Knowing ( ) , p X ρ ω , we can synthesize the surface by metal elements (step 3 in Fig. 2). This operation is carried out using the periodic spectral MoM in [START_REF] Maci | FSS-based EBG surface[END_REF], which uses Rao-Wilton-Glisson basis functions, to calculate a database that associates reactance values to the geometry of metallic patches. Such database (B in Fig. 2) is produced for any frequency ω. Then, the surface synthesis is performed at ωp by a least-square error (LSE) minimization for any unit cell, and the local reactance values are also stored for the other frequencies ω. To speed-up the iteration, the surface synthesis is only performed along direction ϕ=0. Given the linear dependence of X on ϕ , the surface reactance can be easily reconstructed by algebraic relations for any ϕ and ω pair.

At the end of this process we obtain a family of reactance curves which represents the radial variation of the reactance with respect to ω, as represented in Fig. 4. By using these curves, it is easy to retrieve the actual frequency variation of the modulation indexes mρ(ρ,ω), mϕ(ρ,ω) and the average reactance value ( )

X ρ ω
of the synthesized reactance function. The iteration is then carried on (step 4 in Fig. 2) by determining the relation between the sinthesyzed modulation indexes and average reactance and the leaky-wave parameters α(ρ,ω) and Δβsw(ρ,ω), which represent the perturbation on the SW wavenumber due to the modulation. This link is provided for a generic point (ρ0,ϕ0) on the surface using the synthesized functions mρ(ρ,ω), mϕ(ρ,ω) and ( ) X ρ ω to set the generalized canonical problem [START_REF] Caminita | New wine in old barrels: The use of the Oliner's method in metasurface antenna design[END_REF] that locally matches the modulation features mρ(ρ0,ω), mϕ(ρ0,ω), d(ρ0) for any ω. The information for α(ρ,ω) and Δβsw(ρ,ω) is stored in database C, as shown in Fig. 2. At this point, one can proceed to the calculation of the gain versus frequency response (step 5 in Fig. 2) and of the cost function (step 6 in Fig. 2). The proposed procedure is very fast, as the runtime for one iteration is around 1 min to evaluate the gain at 25 frequency points for a Ka-band antenna of radius a=11.11cm on an I7-Intel core machine. Fig. 5 Gain versus frequency responses for the 11.11 cm radius optimized antenna. The solid lines stand for full-wave analysis of the homogenized impedance with the MoM tool in [START_REF] González-Ovejero | Gaussian ring basis functions for the analysis of modulated metasurface antennas[END_REF], while the circular markers correspond to the FMM simulation [START_REF] Faenzi | Metasurface Antennas: New models, applications and realizations[END_REF] of the apertures synthesized by elliptical patches. The top inset shows the final periodicity as a function of the radial distance.

IV. NUMERICAL RESULTS

The optimization scheme detailed in Sections II and III has been used to design a broadband aperture with radius a=11.11cm. After selecting the upper and lower frequencies fmax=29.5GHz and fmin=20.5GHz, we choose the average reactance value at center frequency f0=25GHz as ( ) 0 260.7 X ρ ω = -Ω . Therefore, using (4) one can predict an in-band average gain Gave=25.4dBi. Fig. 5 presents the gain response of the aperture designed by the proposed optimization algorithm, which converged to the optimum solution in about one thousand iterations. The so-obtained layout has been tested by a full-wave MoM solver for the ideal MTS reactance [START_REF] González-Ovejero | Gaussian ring basis functions for the analysis of modulated metasurface antennas[END_REF] (solid lines in Fig. 5), and by a fast multipole method (FMM) [START_REF] Faenzi | Metasurface Antennas: New models, applications and realizations[END_REF] of the same aperture synthesized by elliptical patches elements arranged in a Cartesian lattice with unit cell side of 1.6mm (circular markers in Fig. 5). The FMM exploits the analytical entire domain functions in [START_REF] Mencagli | Surface wave dispersion for anisotropic metasurfaces constituted by elliptical patches[END_REF]. The agreement between the two curves is excellent, and the calculated average gain value is close to the expected one. The oscillation of the gain response around Gave is also limited to around 1.5 ± dB.

Besides the reduced gain oscillation, the bandwidth is considerably extended with respect to the prototype in [START_REF] Faenzi | Wideband active region metasurface antennas[END_REF], which is based on an exponential profile of the periodicity function d(ρ). Finally, the inset in Fig. 5 represents the output optimal function for the periodicity function d(ρ), whereas the optimized amplitude control frequency was ωp=21.85GHz.

V. CONCLUSION

A novel methodology has been introduced for the design of broadside modulated MTS antennas with a broadband response. The presented method relies on an optimization scheme, which outcome is a layout able to provide a flat gain response over broad bandwidths. An accurate formula for the gain-bandwidth product is also introduced to establish a bound for the performance of this kind of antennas. In the first instance, the gain bandwidth and the gain oscillations are controlled by adopting a general piecewise shape of the periodicity function d(ρ). Each periodicity value is related to the surface wave (SW) wavelength that provides broadside pointing at a given frequency among a set of points in the band of interest. The radial position for each periodicity value is one of the optimization parameters. Second, the amplitude of the -1 mode is also controlled by imposing a quasi-constant profile at the frequency ωp, which is also adapted at each iteration of the optimization loop. This way, it is possible to design flat and low-mass apertures that are able to provide a very stable gain performance over considerably large bandwidths, overcoming a typical limitation for this class of antenna.

Fig. 1 .

 1 Fig. 1. Reference coordinate system and active regions (colored areas) at three different frequencies (f1 < f2 < f3), which radiation patterns are shown in the top insets.

H

  are the tangential components of the magnetic field infinitesimaly close to the IBC for positive and negative z values. t E is the averaged tangential electric field component, and the tensor X models the texture reactance. A generic point on the aperture is defined in cylindrical

Fig. 2

 2 Fig. 2 Flow diagram for the optimization process.

  ALGORITHM WORKFLOWComputing the cost function C requires the evaluation of the inband gain G(ω) as[START_REF] Minatti | Efficiency of metasurface antennas[END_REF]

Fig. 4

 4 Fig. 4 Representation by colored curves of the Xρρ reactance profile variation at several different frequencies ω within the antenna bandwidth along ϕ=0.


Evdokia K Oikonomou

Grazia M L Messina

Laurent Heux

Giovanni Marletta

Jean-François Berret
email: jean-francois.berret@u-paris.fr

Adsorption of a fabric conditioner on cellulose nanocrystals: Synergistic effects of surfactant vesicles and polysaccharides on softness properties

Keywords:

Concentrated fabric conditioners are water-based formulations containing 10 wt. % of cationic surfactants that are deposited on textile fibers during the rinse cycle in a washing machine to make them smoother and softer to touch. In topical formulations, the concentration of cationic surfactants is reduced by half, this reduction being compensated by the addition of environmentfriendly polysaccharides. Using atomic force microscopy (AFM), quartz crystal microbalance with dissipation (QCM-D) and ellipsometry, the deposition of formulations with or without polysaccharides on model cellulose substrates is studied. We found that 180 nm long cellulose nanocrystals (CNCs) deposited by spin-coating on amorphous silicon dioxide or on quartz crystal sensors provide a good model of cellulose surfaces. QCM-D results reveal strong electrostatic interactions of the surfactant vesicles and polysaccharides with the CNC layer. In the presence of polysaccharides, the adsorbed quantities are less (by 60%) than the sum of the respective amounts of each component, the structure of the interface being however more homogenous and rigid. This outcome suggests that surface techniques coupled with CNC coated substrates are promising for studying interactions of current formulations with cellulose surfaces.

I -Introduction

In soft matter, soft interfaces are defined as transition zones between two phases, in general a liquid and a solid [START_REF] Sigel | Concepts for soft interfaces Soft Matter[END_REF]. Such interfaces result from the adsorption of (macro) molecules from the dispersed phase, leading to the formation of a layer a few to several hundred nanometers thick. Prominent examples of soft interfaces are self-assembled monolayers, assemblies of surfactants comprising Langmuir-Blodgett films and micelles, lipid membranes, vesicles and polymerlike chains in various conformations, or combinations thereof. The primary goal of soft interfaces is to modify the surface properties of a solid, and provide materials with controlled, switchable or stimuli-responsive interactions with respect to its (liquid) environment.

One area where soft interfaces are important is that of the deposition of actives on textiles. On cellulose, relevant applications are dealing with laundry and conditioning [START_REF] Mishra | Ester quats: The novel class of cationic fabric softeners[END_REF][START_REF] Murphy | Fabric softener technology: A review[END_REF]. The actives are thereby low molecular-weight compounds such as surfactants and hydrophobic fragrance molecules or polymers. Fabric conditioners (or softeners) are water-based formulations used to impart softness and confer soft handle and smoothness to the clothes. Concentrated softeners contain around 10 wt. % of double tailed cationic surfactants, the surfactants assembling into micron sized vesicles. The vesicles ensure the stability of the dispersion and act as carriers for oils and fragrances onto the cotton. Designed for more than 50 years [START_REF] Murphy | Fabric softener technology: A review[END_REF], these formulations are used in the last rinse of the washing cycle during which they adsorb onto cellulose (cotton) fibers. Recently, efforts have been devoted to the design of conditioners based on more environment-friendly materials, with the aim of significantly reducing the quantity of surfactants used, or replacing them by natural compounds. Pertaining to mechanistic softness models, it is accepted that the deposition is driven by electrostatics and the softness effect is due to the formation of lubricating surfactant layers on the fibers, facilitating thereby their mutual contacts and sliding. In most models, the performance of a conditioner is assumed to be correlated with the amount of surfactants adsorbed on the fabrics during the deposition cycle. In the present work, we aim to address this issue and evaluate the deposition of a conditioner formulation on cellulose substrates using surface analytical techniques, such as atomic force microscopy (AFM) [START_REF] Reimhult | Simultaneous Surface Plasmon Resonance and Quartz Crystal Microbalance with Dissipation Monitoring Measurements of Biomolecular Adsorption Events Involving Structural Transformations and Variations in Coupled Water[END_REF], ellipsometry [START_REF] Stroumpoulis | A kinetic study of vesicle fusion on silicon dioxide surfaces by ellipsometry[END_REF]) and quartz crystal microbalance with dissipation monitoring (QCM-D) [START_REF] Cho | Quartz crystal microbalance with dissipation monitoring of supported lipid bilayers on various substrates[END_REF][START_REF] Duarte | Study of in situ adsorption kinetics of polyelectrolytes and liposomes using quartz crystal microbalance: Influence of experimental layout[END_REF][START_REF] Richter | Pathways of Lipid Vesicle Deposition on Solid Surfaces: A Combined QCM-D and AFM Study[END_REF][START_REF] Vermette | Liposome Characterization by Quartz Crystal Microbalance Measurements and Atomic Force Microscopy[END_REF]. QCM-D, in particular, is appropriate as it allows to monitor the kinetics of adsorbed and released masses, whereas the energy dissipation properties provide knowledge on the viscoelasticity of the adsorbed layer [START_REF] Reviakine | Hearing What You Cannot See and Visualizing What You Hear: Interpreting Quartz Crystal Microbalance Data from Solvated Interfaces[END_REF]). To our knowledge this combination of techniques was not attempted for the evaluation of the deposition of surfactant vesicles on cellulose.

An important factor to consider in the deposition is the type of cellulose substrate that should be used. Regarding the QCM-D technique, crystal quartz resonators coated with microfibrillated cellulose [START_REF] Henriksson | An environmentally friendly method for enzyme-assisted preparation of microfibrillated cellulose (MFC) nanofibers[END_REF], also referred to cellulose nanofibrils have been designed and are available commercially. To increase the sensitivity in the measurements, it is desirable to increase the specific surface area and/or the electrostatic net charges of cellulose, which can be achieved in using cellulose nanocrystals (CNCs). CNCs are the crystalline part of cellulose fibers and are available as bulk dispersions [START_REF] Elazzouzi-Hafraoui | The shape and size distribution of crystalline nanoparticles prepared by acid hydrolysis of native cellulose[END_REF]. Their shapes are mostly rod-like and their sizes around 100 nm. In recent years, CNC coated substrates have received increasing attention with regard to surface techniques, mainly for testing adsorption properties of polymers and proteins [START_REF] Reid | Comparison of polyethylene glycol adsorption to nanocellulose versus fumed silica in water[END_REF][START_REF] Villares | Kinetic aspects of the adsorption of xyloglucan onto cellulose nanocrystals[END_REF][START_REF] Vuoriluoto | Effect of Molecular Architecture of PDMAEMA-POEGMA Random and Block Copolymers on Their Adsorption on Regenerated and Anionic Nanocelluloses and Evidence of Interfacial Water Expulsion[END_REF]. Engström et al. for instance studied the deposition and adhesion of latex nanoparticles on surfaces of cellulose nanofibrils by QCM-D, and have found that the particle softness, which can be modulated via the glass transition temperature of the polymers, was a key parameter for producing strong and tough materials of high nanocellulose content [START_REF] Engström | Tailoring adhesion of anionic surfaces using cationic PISA-latexes -towards tough nanocellulose materials in the wet state[END_REF][START_REF] Engström | Soft and rigid core latex nanoparticles prepared by RAFT-mediated surfactant-free emulsion polymerization for cellulose modification -a comparative study[END_REF]. Similar QCM studies have been carried out on CNC substrates for food packaging [START_REF] Saini | Nisin anchored cellulose nanofibers for long term antimicrobial active food packaging[END_REF], biomimicking wood [START_REF] Pillai | Towards Biomimicking Wood: Fabricated Free-standing Films of Nanocellulose, Lignin, and a Synthetic Polycation[END_REF]) and enzymatic hydrolysis applications [START_REF] Lin | Effect of lignin-based amphiphilic polymers on the cellulase adsorption and enzymatic hydrolysis kinetics of cellulose[END_REF]. Finally, studies combining AFM and QCM have been performed using proteins, antibodies or enzymes [START_REF] Raghuwanshi | Cellulose Nano-Films as Bio-Interfaces[END_REF], showing again the importance of nanocellulose substrates in producing model substrates for material science or biochemistry studies. Despite the high potential of CNC soft interfaces, we are not aware of recent reports dealing with interaction with surfactant vesicles. CNCs and vesicles have been studied in bulk dispersions however. Navon et al. [START_REF] Navon | pH-Sensitive Interactions between Cellulose Nanocrystals and DOPC Liposomes[END_REF] analyzed the mixtures of CNC-1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) vesicles in aqueous bulk solutions to understand the interaction between cellulose and a cell membrane mimicking system [START_REF] Navon | Deposition of Cellulose Nanocrystals onto Supported Lipid Membranes[END_REF]. In addition, this group demonstrated by cryo-TEM that the vesicles remain stable during CNC adsorption. On our side, we have shown that the micron sized cationic surfactant vesicles present in fabric conditioners interact strongly with anionic CNCs, forming mixed aggregates and phase separations [START_REF] Mousseau | Design and Applications of a Fluorescent Labeling Technique for Lipid and Surfactant Preformed Vesicles[END_REF][START_REF] Oikonomou | Design of eco-friendly fabric softeners: Structure, rheology and interaction with cellulose nanocrystals[END_REF][START_REF] Oikonomou | Fabric Softener-Cellulose Nanocrystal Interaction: A Model for Assessing Surfactant Deposition on Cotton[END_REF].

Here, we used CNC as a cotton fiber model, assessing the adsorption of topical formulations of double-tailed surfactants and polysaccharides [START_REF] Oikonomou | Fabric Softener-Cellulose Nanocrystal Interaction: A Model for Assessing Surfactant Deposition on Cotton[END_REF] . The polysaccharides are cationic guar gum (C-Guar) and a hydroxypropyl guar (HP-Guar), both extracted from the seeds of cyamopsis tetragonalobus plant, and used as additives. According to independent panelists in double-blinded tests as part of an important survey by Solvay, surfactant formulations modified with C-Guar and HP-Guar were shown to display remarkable softness performances as compared to the additive-free benchmark (Supplementary Information S1). At the same time the surfactant content of concentrated fabric conditioners could be reduced by about half, from 10.5 wt. % to 6 wt. % [START_REF] Zhang | Composition comprising a quaternary ammonium compound, a cationic polysaccharide and a nonionic polysaccharide[END_REF][START_REF] Al | Compositions Comprising Quat and Polysaccharides[END_REF]. In this work, we use AFM, QCM-D and ellipsometry to monitor the adsorption of surfactant vesicles and polysaccharides onto model cellulose substrates and to search for a correlation between the structure of the deposited layer and the softness properties.

II -Materials and Methods

II-1 -Materials and sample preparation Materials

The surfactant used in the softener formulations (abbreviated TEQ in the following) is the esterquat ethanaminium, 2-hydroxyN,N-bis(2-hydroxyethyl)-N-methyl-esters with saturated and unsaturated C16-18 aliphatic chains. TEQ was provided to us by Solvay and used as received. The counterions associated to the amines are methyl sulfate anions. Small-angle X-Ray scattering performed on TEQ aqueous dispersions (𝑐 = 4 wt. %) have revealed the TEQs self-assemble into bilayers of thickness 4.6 nm, whereas wide-angle X-ray scattering disclosed the existence of an hexagonal order in the bilayer at room temperature. This order was found to melt progressively with increasing temperature, indicating a gel-to-fluid transition at the temperature of 60 °C. The ordered surfactant phase thus ensures great stability of the vesicles under the conditions of use. CNCs were prepared according to earlier reports using catalytic and selective oxidation. Briefly, cotton linters provided by Buckeye Cellulose Corporation were hydrolyzed according to the method described by Revol et al. [START_REF] Revol | Helicoidal self-ordering of cellulose microfibrils in aqueous suspension[END_REF]) treating the cellulosic substrate with 65% (w/v) sulfuric acid at 63 °C during 30 min. The suspensions were washed by repeated centrifugations, dialyzed against distilled water until neutrality and sonicated for 4 min with a Branson B-12 sonifier equipped with a 3 mm microtip. The suspensions were then filtered through 8 µm and then 1 µm cellulose nitrate membranes (Whatman). At the end of the process, a 2 wt. % aqueous stock suspension was obtained. The polysaccharides put under scrutiny are a cationic (C-Guar) and a hydroxypropyl (HP-Guar) guar, both synthesized by Solvay. C-Guar and HP-Guar were extracted from the seeds of cyamopsis tetragonalobus plant and their molecular weights were estimated at 0.5×10 6 g mol -1 and 2×10 6 g mol -1 respectively. C-Guar was obtained by introducing positively charged trimethylamino(2-hydroxyl)propyl into the backbone. Water was deionized with a Millipore Milli-Q Water system (resistivity 18.2 MΩ cm). All the products were used without further purification.

Formulation preparation

Concentrated softener formulations were prepared using MilliQ filtered water at 60 °C. TEQ surfactants were first melted at 60 °C and added dropwise to water. The final pH was adjusted at 4.5. The dispersion containing guar polysaccharides were produced by first dissolving the guars in water at 60 °C and then adding the melted TEQs. The pH was again adjusted at 4.5 after each step. Mixed dispersions of surfactant vesicles and guars were prepared following the method of continuous variation [START_REF] Courtois | Probing oppositely charged surfactant and copolymer interactions by isothermal titration microcalorimetry[END_REF][START_REF] Fresnais | Nanoparticle aggregation controlled by desalting kinetics[END_REF][START_REF] Renny | Method of continuous variations: Applications of job plots to the study of molecular associations in organometallic chemistry[END_REF]) applied to study phase behaviors of oppositely charged colloids (Mousseau and Berret 2018;Mousseau et al. 2018;[START_REF] Mousseau | Polyelectrolyte assisted charge titration spectrometry: Applications to latex and oxide nanoparticles[END_REF]. Τhe following formulations were prepared: TEQ 6 wt. %, TEQ 6 wt. % + C-Guar 0.3 wt. % + HP-Guar 0.3 wt. %, C-Guar 0.3 wt. % + HP-Guar 0.3 wt. %. These dispersions were then diluted 60 times, leading to the three dispersions: TEQ 0.1 wt. %, TEQ 0.1 wt. % + Guars 0.01 wt. %, Guars 0.01 wt. % (HP-Guar 0.005 wt. %, C-Guar 0.005 wt. %).

Cellulose coated substrates

For the deposition of surfactants and polysaccharides on cellulose, two different substrates were investigated. We first considered the Q-Sense sensor (QSX 334, Biolin Scientific) which is described as a model surface of native cellulosic fibers. The type of cellulose used is microfibrilated cellulose, also referred to as nanofibrillar cellulose in literature [START_REF] Henriksson | An environmentally friendly method for enzyme-assisted preparation of microfibrillated cellulose (MFC) nanofibers[END_REF][START_REF] Pääkkö | Enzymatic Hydrolysis Combined with Mechanical Shearing and High-Pressure Homogenization for Nanoscale Cellulose Fibrils and Strong Gels[END_REF]. Its adhesion to the quartz crystal is mediated via a poly(ethylene imine) layer. As recommended by the supplier, Q-Sense sensors were gently rinsed before use with water and dried with nitrogen gas. An AFM image of the microfibrilated cellulose is provided in Supplementary Information S2. The second type of substrate was achieved through spincoating technique of a cellulose nanocrystal dispersion on model amorphous silicon dioxide or quartz surfaces. Spincoating was selected as particularly suitable for obtaining reasonably uniform coatings. Several protocols have been tested to evaluate the optimal coating conditions, by modifying the quantity of CNC dispersions, the spinning speed and the spinning time. Good results have been obtained by depositing and drying, onto amorphous silicon dioxide or quartz substrates, a single dose of 100 µl of a 1.4 wt. % CNC dispersion in water with a spinning speed of 3500 rpm for 30 s. Once spincoated, the solvent was allowed to evaporate and the coated substrates were stored under atmospheric conditions.

II-2 Methods

Dynamic Light scattering (DLS) and zeta potential

The scattering intensity 𝐼 ) and hydrodynamic diameter 𝐷 + were measured using the Zetasizer Nano ZS spectrometer (Malvern Instruments, Worcestershore, UK). A 4 mW He-Ne laser beam (𝜆 = 633 nm) is used to illuminate the sample dispersion, and the scattered intensity is collected at a scattering angle of 173°. The second-order autocorrelation function 𝑔 (/) (𝑡) is analyzed using the cumulant and CONTIN algorithms to determine the average diffusion coefficient 𝐷 2 of the scatterers. The hydrodynamic diameter is then calculated according to the Stokes-Einstein relation, 𝐷 + = 𝑘 5 𝑇/3𝜋𝜂𝐷 2 , where 𝑘 5 is the Boltzmann constant, 𝑇 the temperature and 𝜂 the solvent viscosity. Laser Doppler velocimetry using the phase analysis light scattering mode and detection at an angle of 16° was used for electrokinetic measurements of electrophoretic mobility and zeta potential with the Zetasizer Nano ZS equipment (Malvern Instruments, UK). DLS and Zeta potential was measured in triplicate after a 120 s equilibration at 25 °C.

Cryo-transmission electron microscopy (cryo-TEM)

For cryo-TEM, few microliters of dispersion (concentration 1 wt. % for TEQ and 0.1 wt. % for CNCs) were deposited on a lacey carbon coated 200 mesh (Ted Pella Inc.). The drop was blotted with a filter paper using a FEI VitrobotTM freeze plunger. The grid was then quenched rapidly in liquid ethane to avoid crystallization and later cooled with liquid nitrogen. It was then transferred into the vacuum column of a JEOL 1400 TEM microscope (120 kV) where it was maintained at liquid nitrogen temperature thanks to a cryo-holder (Gatan). The magnification was comprised between 3000× and 40000×, and images were recorded with an 2k×2k Ultrascan camera (Gatan). Images were digitized and treated by the ImageJ software and plugins (http://rsbweb.nih.gov/ij/).

Quartz crystal microbalance with dissipation monitoring

Measurements of adsorption kinetics were performed by using a QCM-D instrument (Q-Sense AB, Sweden) with AT-cut gold crystals sensors. The measurement chamber was operating in water at 25 ± 0.1° C. The simultaneous measurements of frequency 𝑓, and energy dissipation, 𝐷, were performed for the fundamental resonance frequency (𝑛 = 1, i.e. 𝑓 = 5 MHz) and the six overtones (n = 3,5,7,9,11 and 13 corresponding to 𝑓 = 15,25,35,45,55 and 65 MHz,respectively). The resolution in 𝑓 and 𝐷 is ± 0.1 Hz and 1×10 -7 , respectively. Regarding the analysis and discussion of QCM-D data, we will focus on the third overtone Δ𝑓 > /3 because, in contrast to higher harmonics it probes the bulk solution above the sensor [START_REF] Feiler | Adsorption and viscoelastic properties of fractionated mucin (BSM) and bovine serum albumin (BSA) studied with quartz crystal microbalance (QCM-D)[END_REF][START_REF] Giamblanco | Serum Protein-Resistant Behavior of Multisite-Bound Poly(ethylene glycol) Chains on Iron Oxide Surfaces[END_REF]). In the cases of rigid, evenly distributed and sufficiently thin adsorbed layers, the frequency to mass conversion is assumed to be given by Sauerbrey equation [START_REF] Sauerbrey | Verwendung von schwingquarzen zur wagung dunner schichten und zur mikrowagung[END_REF]) : ∆𝑚 = -𝐶 ∆𝑓 C 𝑛 ⁄ , where 𝑓 is the decrease in resonant frequency, ∆𝑚 is the mass uptake at the sensor surface, 𝐶 is a constant depending on the intrinsic properties of quartz slab (in our case 𝐶 = 17.7 ng cm -2 Hz -1 at 𝑓 = 5 MHz) and 𝑛 (= 1, 3, …) is the overtone number [START_REF] Rodahl | Simultaneous frequency and dissipation factor QCM measurements of biomolecular adsorption and cell adhesion Faraday[END_REF]. Amorphous silicon dioxide substrates and quartz crystal sensors prepared for the surface experiments were obtained by spincoating a 1.4 wt.% CNC aqueous dispersion, followed by the evaporation of the solvent. Each QCM-D experiment started with the sensor running in water (outgassed with 30 min of sonication), then the addition of vesicles and softener and, after 30 minutes the exchange of the solution being measured with water, to check both desorption and stability of the adsorbed layer. All the experiments were performed in water at 25 °C and the flow rate was 150 µL min -1 . Three replicas for each experiment were performed for data reproducibility.

Atomic force microscopy

Topographical images were obtained using a commercial Nanoscope IIIa Multimode AFM Instrument (Digital Instruments, Santa Barbara, CA, USA). The device was equipped with a <J> calibrated scanner using grating manufacturers. All samples were observed in tapping mode using 0.5 -2 Ω cm phosphorous n-doped silicon tips mounted on cantilevers with a nominal force constant of 40 N m -1 and a resonant frequency of 300 kHz. The applied force was varied over a range from several nanoNewtons up to tens of nanoNewtons in contact mode. Measurements in liquid have been performed using liquid cell and Milli-Q ultrapure water has be injected with a syringe. The surface was analyzed using triangular silicon nitride tip having a nominal force constant of 0.12 N m -1 and a resonant frequency of 20 KHz. The glass surfaces were examined ex situ after gently dried. Image analysis was carried out using DI software v4.23r6. The images were flattened to remove background slopes. Film roughness was measured on tapping mode images obtained at a scan speed of 1 Hz over scanned areas of 500×500 nm 2 for a minimum of three separate images for each sample obtained from different regions.

Spectroscopic Ellipsometry

The coated samples were characterized at room temperature in M2000 ellipsometer (rotating compensator ellipsometer, J.A. Woollam Co. Inc.) in a wavelength range of 400 -1600 nm with a step size approximately 1.5 nm at two different incidence angles of 65° and 75°. The acquired data were analyzed using WVASE32 software module. This technique measures the ratio of the Fresnel reflection coefficients 𝑅 F and 𝑅 G obtained respectively for p-polarized and s-polarized light reflected from the surface sample. Measured values, in the form of the ellipsometric angles 𝛹 and 𝛥, are described from the following equation: 𝑃 = 𝑅 F /𝑅 G = 𝑡𝑎𝑛Ψ𝑒 NO . VASE is an indirect method used to extract the optical constants as well as the film layer thickness and it is based on the fitting of experimental data with the modelled data and generally the quality of fit is given by an estimated value which is a positive quantity and tends to zero when the experimental data approaches or exactly matches the calculated data. Cauchy model was used to match the acquired data with the generated theoretical data

III -Results and discussion

III.1 -Surfactant, polysaccharides and cellulose nanocrystals Fig. 1a displays the molecular structure of the TEQ surfactant and a representative image of a 1 wt. % dispersion studied by cryo-TEM. The image shows uni-and multivesicular vesicles of size between 50 nm and 500 nm in a crowded environment. Morphologies characteristic for double-tailed surfactant self-asssembly, including deflated vesicles and doublets are also observed. At the concentration studied (0.1 wt. %) and at the concentration of use (0.024 wt. %), the TEQ vesicles remain stable, the proportion of multivesicular structures compared to the unilamellar ones reversing under dilution. Electrophoretic mobility measurements on dilute dispersions resulted in zeta potential 𝜁 = +65 mV, indicating strongly cationic vesicles. The cellulose nanocrystals were investigated via electron microscopy and light scattering. Fig. 1b show a cryo-TEM image of CNCs in the form of thin anisotropic fibers [START_REF] Oikonomou | Design of eco-friendly fabric softeners: Structure, rheology and interaction with cellulose nanocrystals[END_REF]. Their average length, width and thickness were derived from image analysis and were found equal to 180 ± 30 nm, 17 ± 4 nm and 7 ± 2 nm, respectively. Using light scattering, CNC dispersions (pH 4.5) prepared at 0.1 wt. % display a single relaxation mode in the autocorrelation function 𝑔 (/) (𝑡). This function is seen to decrease rapidly above delay times of 10 3 µs, suggesting the absence of CNC aggregates. The hydrodynamic size retrieved by DLS reveals an intensity distribution centered around 120 nm (Fig. 1b), in agreement with the size obtained from electron microscopy. Electrophoretic mobility experiments resulted in zeta potential 𝜁 = -38 mV and negatively charged CNCs. Figs. 1c and1d show the molecular structures of C-Guar and HP-Guar respectively, together with and the 𝑔 (/) (𝑡)functions obtained by DLS from dispersions at 0.02 wt. % (after filtration at 0.2 µm). The intensity distributions in the insets reveal a single peak around 200 nm for C-Guar and a double peak for HP-Guar at 50 and 350 nm [START_REF] Oikonomou | Design of eco-friendly fabric softeners: Structure, rheology and interaction with cellulose nanocrystals[END_REF]. These hydrodynamic diameters are larger than those expected from polysaccharides of molecular weight 𝑀 R = 0.5 and 2×10 6 g mol -1 , as provided by Solvay, suggesting that the chains are likely associated in water, forming hydrocolloid particles [START_REF] Rubinstein | Polymer physics[END_REF]. Electrophoretic mobility measurements on dilute guar dispersions give zeta potential of +30 mV and 0 mV respectively. The dynamic light scattering and 𝜁-potential data are summarized in Table I.

Table I:

Hydrodynamic diameter (𝐷 + ) and 𝜁-potential determined for TEQ surfactant vesicle at 0.1 wt. % [START_REF] Oikonomou | Fabric Softener-Cellulose Nanocrystal Interaction: A Model for Assessing Surfactant Deposition on Cotton[END_REF], guar dispersions at 0.02 wt. % and cellulose nanocrystals at 0.1 wt. %. The dispersity indexes (pdi) for the 4 compounds were in the range 0.2 -0.3 and the uncertainty in 𝜁-potentials found at ± 5 mV.

Compounds

Abbreviation [START_REF] Oikonomou | Design of eco-friendly fabric softeners: Structure, rheology and interaction with cellulose nanocrystals[END_REF]. c) and d) Chemical structure and dynamic light scattering characterization of the cationic and hydroxypropyl modified guars, respectively [START_REF] Oikonomou | Design of eco-friendly fabric softeners: Structure, rheology and interaction with cellulose nanocrystals[END_REF].

𝑫 𝑯 (nm) 𝜻-potential (

III.2 -Cellulose nanocrystals on silica substrates

Fig. 2a displays AFM tapping mode images in air of cellulose nanocrystals spincoated onto a quartz crystal. It can be seen that a uniform coverage of the substrate is achieved, consisting of densely packed CNCs. The average thickness and roughness of the CNC layer were determined by ellipsometry at 34 nm and 3.9 nm, respectively (Supplementary Information S3). These data, compared with those obtained from cryo-TEM experiments in Fig. 1b, suggest that the layers consist primarily of side-on lying crystallites, arranged with the longitudinal axes parallel to the surface [START_REF] Navon | Deposition of Cellulose Nanocrystals onto Supported Lipid Membranes[END_REF]). It should be noted that the coating locally exhibits correlations in the CNC orientations. These correlations, which extend over a length scale of a few microns, attest of strong steric interactions between the nanocrystals in the dry state. To assess the CNC layer stability, an important step in monitoring repeatible QCM-D testing, the previous coated substrate was rinsed with Milli-Q ultrapure water and studied in air with AFM, again after solvent evaporation. Fig. 2b shows the AFM images in air of the CNC after rinsing. It can be seen that the rinsing treatment tends to randomize the CNC orientations, most likely through capillary forces, but it does not modifty the uniformity or the thickness of the adsorbed layer. AFM measurements were then conducted in liquid onto a CNC coating layer immersed in water. The AFM images in Fig. 2c have characteristics similar to those found in the previous conditions, confirming the high stability of the CNC layer under wet and dry conditions, and its relevance for QCM-D. Fig. 2d provides a schematical representation of the CNC layer obtained using the spincoating technique described previously.

III.3 -Adsorption of surfactants and guar polysaccharides on cellulose using QCM-D

To study the interaction of the cationic double tailed surfactants (TEQ) with cotton fibers, QCM-D experiments have been carried out with two types of cellulose substrates: the commercially available Q-Sense sensor coated with microfibrilated cellulose, and a substrate obtained by spincoating a CNC dispersion on a quartz crystal sensor. For QCM-D, the same procedure was applied with both surfaces.

Adsorption on microfibrilated cellulose Q-Sense sensor

QCM-D data obtained with a microfibrilated cellulose coated sensor are presented in Supplementary Information S2. Experiments were first conducted in triplicate with TEQ and guar dispersions at 0.1 and 0.01 wt. % respectively, but the QCM-D signal was too low to be exploited quantitatively. In a second attempt, the concentrations were increased tenfold, to 1 and 0.1 wt. %, leading to the adsorption kinetics shown in Figs. S2c andS2d. For TEQ alone, the frequency shift exhibits a decrease followed by a saturation at Δ𝑓 > /3 = -6 Hz, while for guars this decrease reaches -15 Hz. These Δ𝑓 > /3 values are relatively low, indicating a weak deposition of each of the two components studied TEQ and Guars, the densities of adsorbed materials being around 200 ng cm - 2 . When the two compounds were combined in the same sample, the adsorption level remains of the same order (Δ𝑓 > /3 = -16Hz). It should be noted that the concentrations used here are much higher than those used in real conditions (0.024 wt. %), and for this reason these substrates were not considered further as model for the assessment of actives on cotton. A possible reason for such a weak adsorption could stem from the low density of nanofibers and/or electrostatic charges at the surface sensor. In particular, it should be noted that independent measurements of the zeta potential on these dispersions of nanofibrils gave values of -10 mV [START_REF] Stenstad | Chemical surface modifications of microfibrillated cellulose[END_REF], against -34 mV for the CNCs.

Adsorption on cellulose nanocrystal (CNC) coated sensor

In this section, the QCM-D results obtained using standard quartz sensors preliminarily coated ex situ with a 34 nm CNC layer. Fig. 3a displays the adsorption profiles of a 0.1 wt. % TEQ dispersion obtained at room temperature. In the figure, the traces corresponding to the different overtones (𝑛 = 3, 5, 7, 9 and 11) and the related dissipations, Δ𝐷 C are reported. Upon injection (arrow at 𝑡 = 10 min), the frequency exhibits a rapid decrease and then a saturation for all overtones. In Fig. 3a, the frequency shift measured for the third overtone is large, at Δ𝑓 > /3 = -225 Hz. Similarly, Δ𝐷 > increases rapidly and reaches a plateau at 55×10 -6 . This adsorption kinetics, and in particular the adsorption saturation behavior, are consistent with those reported by QCM-D upon adsorption of close-packed layer of intact lipid vesicles [START_REF] Keller | Surface Specific Kinetics of Lipid Vesicle Adsorption Measured with a Quartz Crystal Microbalance[END_REF][START_REF] Richter | Pathways of Lipid Vesicle Deposition on Solid Surfaces: A Combined QCM-D and AFM Study[END_REF][START_REF] Serro | Formation of an intact liposome layer adsorbed on oxidized gold confirmed by three complementary techniques: QCM-D, AFM and confocal fluorescence microscopy: Formation of an intact liposome layer adsorbed on oxidized gold[END_REF][START_REF] Viitala | Structure of Anionic Phospholipid Coatings on Silica by Dissipative Quartz Crystal Microbalance[END_REF]. Rinsing with water (arrow at 𝑡 = 40 min) has no effect on the frequency shifts, suggesting that the vesicles are irreversibly adsorbed on the surface. These results confirm findings obtained on vesicle/CNC interaction in the bulk phase and the persistence of the vesicular structures upon adsorption on nano-and microcellulose fibers (see Supplementary Information S4 and Ref. [START_REF] Oikonomou | Design of eco-friendly fabric softeners: Structure, rheology and interaction with cellulose nanocrystals[END_REF]) for more details). A schematic representation of the adsorption mechanism on the CNC coated crystals is depicted in Fig. 3a.

The role of the guar polysaccharides was then examined by performing two complementary experiments involving i) the sequential adsorption of a C-guar and HP-guar mixture (0.005 wt. % each), followed by the adsorption TEQ surfactants at concentration 0.1 wt. % (Fig . 3b), and ii) the simultaneous adsorption from a formulation containing TEQ surfactants at 0.1 wt. % and guars at 0.01 wt. % (Fig. 3c). Fig. 3b shows that guar molecules are at first strongly adsorbed at the sensing substrate, with an immediate adsorption peak reaching Δ𝑓 > /3 = -83 Hz and a large dissipation (Δ𝐷 > = 18×10 -6 ), immediately followed by a partial mass loss up to a plateau at Δ𝑓 > /3 = -49 Hz and a dissipation decrease. The subsequent injection of the TEQ surfactant dispersion (𝑡 > 57 min in Fig. 3b) produces a small and reversible adsorption of nonruptured vesicles, suggesting that the precoating of the CNC layer by the guars prevented further immobilization of the vesicles. Pertaining to the data in Fig. 3b, it is interesting to note that the overall process reminds the characteristic two-step mechanism observed for lipid vesicles interacting with quartz crystal sensors [START_REF] Reimhult | Intact vesicle adsorption and supported biomembrane formation from vesicles in solution: Influence of surface chemistry, vesicle size, temperature, and osmotic pressure[END_REF][START_REF] Reviakine | Adsorbed liposome deformation studied with quartz crystal microbalance[END_REF]. This process involves first the adsorption of vesicles, immediately followed by their spontaneous rupture and leading to the formation of a supported lipid bilayer and to a significant loss of the water initially trapped in the lipid structures. To justify this analogy, one has to recall that the guar dispersed in aqueous medium behave as hydrocolloids (i.e. highly solvated crosslinked polymer networks) rather than ideal linear polymer chains in good solvent. For the cationic and hydroxypropyl guars (Fig. 1), the hydrodynamic diameters derived from DLS are 200 and 350 nm respectively, that is much larger than the expected diameters (estimated at 40 and 90 nm) retrieved from their molecular weights 0.5×10 6 g mol -1 and 2×10 6 g mol - 1 respectively. The frequency shift variation in Fig. 3b may then be explained in terms of the initially strong adsorption of hydrocolloids, carrying their solvation water on the CNC surface and producing the observed large frequency shift. This peak is then followed by the release of the water, leaving the polymer chains tightly bound to the fibers. For the cationic guar, this interaction is driven by electrostatics, as shown in bulk studies [START_REF] Oikonomou | Design of eco-friendly fabric softeners: Structure, rheology and interaction with cellulose nanocrystals[END_REF]). This picture is further supported by results obtained on cationic polyelectrolytes adsorbed onto CNC coated substrates, in which the frequency shift variation has been attributed to the water release from the interface upon charge neutralization [START_REF] Vuoriluoto | Effect of Molecular Architecture of PDMAEMA-POEGMA Random and Block Copolymers on Their Adsorption on Regenerated and Anionic Nanocelluloses and Evidence of Interfacial Water Expulsion[END_REF].

The narrow distribution of the frequency overtones, along with the lack of further desorption upon rinsing (arrow at 𝑡 = 40 min), confirms the existence of strong interactions between guars and CNCs, and suggests that following their adsorption, the guars are likely to interpenetrate the cellulose nanocrystal network, as illustrated in Fig. 3b. This hypothesis is also supported by the peculiar decrease of the dissipation to negative values, i.e., below the initial dissipation baseline. To account for this effect, it must be reminded that in Fig. 3 the prior-to-injection baselines in Δ𝑓 and in Δ𝐷 represent the reference states that was previously characterized in pure water. These baselines reflect the mechanical properties of the CNC layer, i.e. in particular the connectivity of the CNC network and the related CNC layer organization. The seemingly negative dissipation may be then understood in terms of the binding effect induced by the positively charged guar molecules connecting the negative cellulose nanocrystals in the layer. These interactions produce a global reorganization of the polymer/CNC interface, driving the dissipation to negative values with respect to the reference state. Effects implying change of the substrate dissipation to negative values have been also reported for bacterial membranes disrupted by aminoglycosides [START_REF] Joshi | Real-time examination of aminoglycoside activity towards bacterial mimetic membranes using Quartz Crystal Microbalance with Dissipation monitoring (QCM-D)[END_REF], phospholipid layers damaged by membrane-active peptides [START_REF] John | The Quartz Crystal Microbalance with Dissipation Monitoring (QCM-D) Technique Applied to the Study of Membrane-Active Peptides[END_REF], and disruption of layers of fatty acid friction modifiers [START_REF] Zachariah | Correlation Between the Adsorption and the Nanotribological Performance of Fatty Acid-Based Organic Friction Modifiers on Stainless Steel[END_REF]. To highlight the role of guars on the deposition, we then examined a formulation that contains TEQ vesicles at 0.1 wt. % and guars at 0.01 wt. %. As displayed in Fig. 3c, the injection of the mixed formulation resulted in a large frequency shift at Δ𝑓 > /3 = -162 Hz and in a dissipation of Δ𝐷 > = 31× 10 -6 . It must be emphasized here that the frequency shift is stable over time, also after the rinsing treatment, showing thus that no mass loss is involved in the vesicle and guar adsorption onto the CNC layer. This in turn suggests that vesicles do not break during the adsorption step, and remain firmly bound to the CNCs. On the other hand, the fact that the overall TEQ surfactant/guars adsorbed mass is about 60% lower than the sum of TEQ and guars taken separately indicates that the adsorbed the guar biomolecules interfere with the vesicle adsorption, screening the adsorption sites on the CNC layer. This picture is also supported by the observation that there is a significant spread of the overtones for TEQ surfactants/guars. This later outcome suggests that a lower amount of mobile vesicles are adsorbed together with random guar aggregates, as pictured in Fig. 3b. The QCM-D experiments in Fig. 3 were repeated and provide similar results (Supplementary Information S5). [START_REF] Dixon | Quartz crystal microbalance with dissipation monitoring: enabling real-time characterization of biological materials and their interactions Journal of biomolecular techniques[END_REF][START_REF] Reviakine | Hearing What You Cannot See and Visualizing What You Hear: Interpreting Quartz Crystal Microbalance Data from Solvated Interfaces[END_REF].

Fig. 4 shows the plot of Δ𝐷 > versus -Δ𝑓 > /3 during the adsorption of TEQ vesicles, guars and mixture of vesicles and guars on CNC coated substrates. The data are those of Fig. 3a, 3b and 3c respectively for which only the first initial 30 minutes have been retained. These Δ𝐷-Δ𝑓 plots allow to overcome the temporal effects pertaining to the deposition kinetics and to directly compare the ratio between Δ𝐷 and -Δ𝑓, that is, the induced energy dissipation per coupled unit mass. A low Δ𝐷/(-Δ𝑓) value indicates a mass addition without significant dissipation increase, which is characteristic of a fairly rigid layer, whereas a large Δ𝐷/(-Δ𝑓) value signals a soft and dissipative film. The figure also shows the Sauerbrey limit: Δ𝐷/(-Δ𝑓) = 4×10 -7 Hz -1 (straight line in grey) that is typical of homogeneous and rigid films. Data in Fig. 4 highlight that for TEQ vesicles and guars, the Δ𝐷/(-Δ𝑓)-responses lie above the Sauerbrey limit at the adsorption onset. The measured responses are characterized by a large dissipation, and both Δ𝑓 and Δ𝐷 display overtones dependencies that are typical for acoustically non-rigid films, probably due to the vesicle intrinsic properties and the nature of their attachment to the surface. In contrast, mixture of vesicles and guars display low Δ𝐷/(-Δ𝑓) ratios over the entire time range, indicating an acoustically rigid layer likely due to the presence of polysaccharides adsorbed on CNC and the presence of only few vesicles on surface [START_REF] Cho | Quartz crystal microbalance with dissipation monitoring of supported lipid bilayers on various substrates[END_REF]. A change in structure of the water-cellulose interfacial layer induced by the guars could explain the enhanced performances found for these novel formulations (Supplementary Information S1).

III.4 -Deposition thickness in the dry state studied by ellipsometry

An important question related mechanics involved in the softness effect concerns the drying step, in which the surfactant vesicle adsorbed on the fabrics transforms into a dense film through water evaporation. Here we study with AFM and ellipsometry the topographical features of a CNC coated adsorbing surfaces, with and without TEQ surfactant or guars. In the dry state, the CNC layer spincoated on a quartz crystal sensor shows a densely packed in CNCs associated with a dry average thickness ℎ )] = 34.3 ± 0.1 nm and with a root mean square roughness 𝑅 ^ = 3.9 ± 0.3 nm (Fig. 5a). The roughness value reflects the CNC layer porosity and it is related also to the dimensions of the cellulose nanocrystallites. The CNC substrate topography, after the adsorption and drying of a 0.1 wt. % dispersion in water of TEQ vesicles is shown in Fig. 5b. The AFM images, in this case, show the appearance of large flake-like structures, randomly distributed onto the CNC coating. The overall process is associated to the net increase in ℎ )] from 34.3 ± 0.1 nm to 50.3 ± 0.1 nm, roughly corresponding to 3.5 times the thickness of TEQ bilayers determined by smallangle X-Ray scattering [START_REF] Oikonomou | Fabric Softener-Cellulose Nanocrystal Interaction: A Model for Assessing Surfactant Deposition on Cotton[END_REF]. Accordingly, the roughness is almost doubled, from 𝑅 ^ = 3.9 ± 0.3 nm to 7.6 ± 0.3 nm. Fig. 5c, finally, shows the effects of the adsorption and drying of a mixture of TEQ surfactant at 0.1 wt. % and guars at 0.01 wt. % on the CNC layer. The topography is somewhat intermediate between the pristine one and that found with TEQ alone (Table II). Furthermore, the AFM image suggests a large reduction of the CNC orientational correlations, which are attributed to the partial penetration of the guars inside the CNCs. The 30% decrease in layer thickness with guars (from 16.0 nm to 11.1 nm, Table II) is in line with the decrease of deposited materials found with QCM-D in wet conditions.

IV -Conclusion

We report the deposition of surfactants and polysaccharides present in topical fabric conditioners on model cellulose substrates. The conditioner performances for these novel formulations were assessed in relation to several properties, including their softness effect and wettability. In the present context, the emphasis is put on the relationship between the softness effect and the deposition of actives on cotton surfaces. To tackle this issue, we use a combination of surface techniques, such as atomic force microscopy, quartz crystal micro-balance and ellipsometry to study the deposition of formulations with or without guars on amorphous silicon dioxide substrates or on quartz crystal sensors. To model cotton fiber surfaces, these substrates were previously coated with cellulose nanofibers, either microfibrilated cellulose or CNCs. The data obtained with CNCs are the most interesting, as they display good reproducibility under wet and dry conditions and great sensitivity to the deposition of cationic vesicles and polymers. The QCM-D technique reveals strong electrostatic interactions between the surfactant vesicles, or the cationic guars and the CNC layer. Upon successive rinsing, we do not observe any loss of mass, indicating a strong adhesion onto the cellulose nanocrystals. An interesting point is that in the wet state the vesicles remain intact during the deposition, while during evaporation of the solvent and subsequent drying the vesicles rupture and produce a uniform film. The fact that vesicles are adsorbed on CNCs without modification of their structure is supported by additional experimental results (Oikonomou et al. 2017 120): in bulk solutions, we have shown that due to their strong attractive forces, TEQ vesicles and CNCs associate and form large aggregates (> 1 µm). Within these aggregates, vesicles can be visualized using fluorescence microscopy. Furthermore, using wide-angle X-ray scattering as a function of temperature, it has been shown that the surfactants are in a hexagonal gel phase up to 60 ° C, this gel phase ensuring high stability of the vesicles in various conditions. As shown by ellipsometry, this mechanism ensures a uniform distribution of the surfactants on the cellulose surface, the layer being then a few nanometers thick. In this work, we also sought to test the hypothesis that the softness performance depends on the mass adsorbed at the cellulosic interfaces, and on the possibility that the guars improve this deposit. In the presence of guars, the adsorbed quantities measured by QCM-D are less (by 60%) than the sum of the respective amounts of each component, but the structure of the soft interface is more homogenous and rigid. This outcome suggests synergistic effects between surfactants and these particular polysaccharides, and that the guars could also contribute to the softness mechanism. These results finally show that surface techniques coupled with robust CNC coated substrates are promising for studying interactions of the many components of current conditioner formulations with cotton.

Figure 1

 1 Figure 1: a) ΤΕQ double-tailed surfactant chemical structure and cryo-transmission electron microscopy image (cryo-TEM) of 1 wt. % aqueous dispersion. b) Cryo-TEM image and dynamic light scattering (DLS) characterization of 0.1 wt. % CNC dispersion[START_REF] Oikonomou | Design of eco-friendly fabric softeners: Structure, rheology and interaction with cellulose nanocrystals[END_REF]. c) and d) Chemical structure and dynamic light scattering characterization of the cationic and hydroxypropyl modified guars, respectively[START_REF] Oikonomou | Design of eco-friendly fabric softeners: Structure, rheology and interaction with cellulose nanocrystals[END_REF]).

Figure 2 :

 2 Figure 2: AFM tapping mode height (left panel) and phase (right panel) images of cellulose nanocrystals deposited on a quartz crystal sensor in different preparation conditions. a) In air after spin-coating; b) in air after rinsing the substrate shown in a) with Milli-Q ultrapure water; c) in water. The z-values in the height images represent the vertical distance between the tip and the sample, whereas the z-values in the phase images denote the phase shifts expressed as an angle in degrees and converted in bright and dark regions in the phase images. d) Schematic representation of the CNC layer.

Figure 3 :

 3 Figure 3: a) Real time binding curves and dissipation obtained by QCM-D during the adsorption of a) TEQ surfactants at 0.1 wt. %, b) a mixture of C-guar and HP-guar at 0.005 wt. % each followed by TEQ surfactants at 0.1 wt. % and c) a formulation containing ΤΕQ surfactants at 0.1 wt. % and C-guar and HP-guar (each at 0.005 wt. %). In these measurements, the quartz sensors were previously coated with cellulose nanocrystals. In each panel, the first arrow at 𝑡 = 10 min denotes the time at which the sample is injected. The second arrow denotes the time at which Milli-Q ultrapure water is introduced for rinsing. Illustrations of the adsorption processes involved are provided on the sketches on the right.

Figure 4 :

 4 Figure 4: Plot of the dissipation ∆𝐷 > as a function of the frequency shift, -∆𝑓 > 3 ⁄ , during the adsorption of TEQ surfactant vesicles, guars and mixture of surfactants and guars. The QCM-D data are those of the first 30 minutes shown in Fig. 3. The straight line represents the Sauerbrey limit (slope 4 × 10 Z[ Hz -1 ) valid for homogeneous and rigid films[START_REF] Dixon | Quartz crystal microbalance with dissipation monitoring: enabling real-time characterization of biological materials and their interactions Journal of biomolecular techniques[END_REF][START_REF] Reviakine | Hearing What You Cannot See and Visualizing What You Hear: Interpreting Quartz Crystal Microbalance Data from Solvated Interfaces[END_REF].

Figure 5 :

 5 Figure 5: AFM tapping mode height (left panel) and phase (right panel) images of a) cellulose nanocrystals obtained by spin coating on a quartz crystal sensor; b) TEQ surfactants at concentration 0.1 wt. % adsorbed on the substrate shown in a); c) a softener formulation containing ΤΕQ surfactants 0.1 wt. % and guars 0.01 wt. % adsorbed on the substrate shown in a). The z-values in the height images represent the vertical distance between the tip and the sample, whereas the z-values in the phase images denote the phase shifts expressed as an angle in degrees and converted in bright and dark regions in the phase images. The images were recorded after adsorption and drying. The scale of the AFM pictures is 2×2 µm 2 .

Table II :

 II Root mean square (𝑅 ^), average (𝑅 _ ) and maximum (𝑅 `_a ) roughness obtained from AFM image analysis. ℎ )] denotes the average layer thickness determined from spectroscopic ellipsometry.

	Sample		Roughness		Layer Thickness
			(nm)		(nm)
		𝑅 ^	𝑅 _	𝑅 `_a	ℎ )]
	CNC 1.4 wt. %	3.9 ± 0.3 3.1 ± 0.3 36.1 ± 2.8	34.2 ± 0.1
	CNC 1.4 wt. % + TEQ wt.	7.6 ± 0.3 6.0 ± 0.3 51.7 ± 6.6	50.3 ± 0.1
	0.1%
	CNC wt. 1.4% + TEQ wt.	7.3 ± 1.2 5.8 ± 0.9 47.7 ± 3.4	45.4 ± 0.1
	0.1% + Guars 0.01 wt. %

Acknowledgements G. M. L. Messina gratefully acknowledges the financial support by CSGI. ANR (Agence Nationale de la Recherche) and CGI (Commissariat à l'Investissement d'Avenir) are gratefully acknowledged for their financial support of this work through Labex SEAM (Science and Engineering for Advanced Materials and devices) ANR 11 LABX 086, ANR 11 IDEX 05 02. We acknowledge the ImagoSeine facility (Jacques Monod Institute, Paris, France), and the France BioImaging infrastructure supported by the French National Research Agency (ANR-10-INSB-04, « Investments for the future »). This research was supported in part by the Agence Nationale de la Recherche under the contract ANR-13-BS08-0015 (PANORAMA), ANR-12-CHEX-0011 (PULMO-NANO), ANR-15-CE18-0024-01 (ICONS), ANR-17-CE09-0017 (AlveolusMimics) and by Solvay.

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The authors declare that they have no conflict of interest.