D Habault

Ph E Friot

C Herzog

Pinhede

Active control in an anechoic room : Theory and first simulations

Noise control and source design require the measurement of sound radiation at low frequencies. Anechoic rooms, which are designed for this purpose, allow echo-free measurements at medium or high frequency but passive wall treatment is less effective at low frequency and in practice no facility provides anechoicity below 50Hz. This paper discusses the applicability of an active control algorithm which has been previously introduced to minimize the echoes from a scattering object to the cancellation of the low frequency wall echoes in an anechoic room including wall-embedded secondary sources. At first the paper discusses, in the general case then for a free half-space as a model case, the algorithm key which consists in estimating the scattered acoustic pressure from total pressure measurements. Boundary Element Method computations are secondly used to simulate estimation and active control of error signals accounting for the low-frequency scattered pressure in an anechoic room. The simulations show that control with a few dozen microphones and noise sources allows a large reduction of the noise scattered from the walls at low-frequency.

Introduction

Lower audible frequencies have received an increased interest during the last decades, especially because light structures are spreading in many application fields. Annoyance problems are then rising, requiring suitable measurement means and standards. Conversely, existing measurement facilities are barely adapted to the measurement of sources below 70-100 Hz, as conventional lining depth cannot be increased enough for technical and economical reasons. It is thus well known that, at lower frequencies, some unwanted modes of an anechoic room may appear [START_REF] Schneider | Acoustical behavior of the large anechoic chamber at the LMA in the low frequency range[END_REF]. As an example, measurement of usual audio loudspeakers is a real concern [START_REF] Melon | Comparison of four subwoofer measurement techniques[END_REF]. Although alternatives to free-field measurements have been proposed [START_REF] Melon | Evaluation of a method for the measurement of subwoofers in usual rooms[END_REF] and are becoming available as commercial solutions [START_REF] Bellmann | Holographic loudspeaker measurement based on near field scanning[END_REF], most standards still recommend the use of an anechoic room for response measurements [START_REF]International standard : Sound system equipment -Part 5 : Loudspeakers. Third edition 2003-05[END_REF].

Alternatives to thick lining involving fibrous or porous materials have thus been proposed, among which active control seemed a good candidate as its efficient frequency range complements the one of conventional materials. The use of basic active noise control for the reduction of wall reflections is an old topic, proposed several decades ago for underwater acoustics [START_REF] Beatty | Acoustic impedance in a rigid-walled cylindrical sound channel terminated at both ends with active transducers[END_REF] and audio use [START_REF] Guicking | Active impedance control for one-dimensional sound[END_REF]. This concept has then been thoroughly validated and improvements have been proposed [START_REF] Orduña-Bustamante | An adaptive controller for the active absorption of sound[END_REF][START_REF] Clark | Active damping of enclosed sound fields through direct rate feedback control[END_REF][START_REF] Thenail | Furstoss : The active control of wall impedance[END_REF][START_REF] Zhu | Active control of acoustic reflection, absorption and transmission using thin panel speakers[END_REF][START_REF] Lissek | A preliminary study of an isodynamic transducer for use in active acoustic materials[END_REF][START_REF] Yuan | Causal impedance matching for broadband hybrid noise absorption[END_REF]. Related work is still in progress, the main concerns being to implement a practical sensing method and widen the application range.

Basic active noise control is however limited to lower frequencies, and becomes less efficient when frequency increases. So-called "hybrid" absorbers have thus been studied, combining absorbing materials and transducers driven so that the resulting device exhibits the performances of a conventional passive lining at higher frequencies and suitable performances at lower frequencies. Such systems have also been thoroughly investigated, with varying combinations of passive material, actuator, and control strategies [START_REF] Guicking | An active sound absorber with porous plate[END_REF][START_REF] Beyene | A new hybrid passive/active noise absorption system[END_REF][START_REF] Thenail | Active enhancement of the absorbent properties of a porous material[END_REF][START_REF] Smith | A broadband passive-active sound absorption system[END_REF][START_REF] Cobo | Hybrid passive-active absorption using microperforated panels[END_REF][START_REF] Leroy | Experimental study of a smart foam sound absorber[END_REF][START_REF] Melon | One dimensional study of a module for active/passive control of both absorption and transmission[END_REF].

Most of the above-mentioned results deal with plane-wave field inside a waveguide, with a normal incidence angle. As pointed out in [START_REF] Leishman | On the significance of reflection coefficients produced by active surfaces bounding one-dimensional sound fields[END_REF], the absorption coefficient estimated in a planewave tube is a confusing indicator, barely representative of actual absorption performances inside a room. Some works thus addressed this aspect, considering the influence of incidence angles or the behavior of 3D volumes. For example, a speaker array was used to control an oblique incident plane wave [START_REF] Guicking | Coherent active methods for applications in room acoustics[END_REF], dipolar sources were used to damp the first modes of a small room [START_REF] Ph | Passive and active control of the low-frequency modes in a small room[END_REF], an array of hybrid cells was studied under various incidences [START_REF] Nicolas | Analogy electromagnetism-acoustics : Validation abd application to local impedance active control for sound absorption[END_REF], or an hybrid cell has been used to dampen a structural-acoustic coupled system [START_REF] Lacour | Preliminary Experiments on Noise Reduction in Cavities Using Active Impedance Changes[END_REF].

Although one might feel intuitively that lining the walls of a room with "perfect" absorbers should approximate an anechoic situation, there is no direct relation between achieving a locally reacting absorbing condition and anechoicity. This has been simulated by driving hybrid cells from a local or global error criterion, showing that maximum absorption by a local criterion does not achieve a true anechoic condition [START_REF] Friot | Improving absorption of sound using active control[END_REF]. This motivated our work aiming at the active control of the pressure reflected by the walls of a 3D volume, taking into account a non-local boundary condition. This is a problem similar to the control of the field diffracted by an object in free field -but here the "object" surrounds a volume.

The acoustic facets of active control are introduced in text-books such as [START_REF] Nelson | Active Control of Sound[END_REF] which specifically includes a chapter dedicated to the control of diffracted sound fields. This chapter discusses earlier references and control strategies that mainly focused at reducing the echoes to sonar excitation. Some recent references also address scattering by an obstacle in a fluid medium [START_REF]Bobrovnitskii : A new-impedance based approach to analysis and control of sound scattering[END_REF][START_REF] Miller | On perfect cloaking[END_REF][START_REF] Friot | Active control of scattered acoustic radiation : A real-time implementation for a 3D object[END_REF][START_REF] Han | Active control of one-dimension impulsive reflection based on a prediction method[END_REF][START_REF] Han | Active control on the scattered radiation by a rigid surface[END_REF]. A main difficulty is then to get a reliable estimate of the scattered field, a non-measurable quantity. For example, the diffracted pressure may be deduced from two series of measurements, with and without the diffracting object.

For the application to an anechoic room, the diffracted pressure is the pressure reflected by the walls of the room so that such a method cannot be applied. An integral relation may be used to estimate a quantity to be minimized (the scattered field) from the total pressure observed with usual pressure microphones [START_REF] Friot | Control of low-frequency wall reflections in an anechoic room[END_REF][START_REF] Friot | Estimation and global control of noise reflections[END_REF]. This mathematical operator is considered here as a matrix of filters to be applied to the error microphones. For practical purposes, this matrix is estimated during a calibration step, using a set of reference sources and an array of identification microphones. As a first test of the method, an experiment was carried out in a scaled mock-up of an actual room [START_REF] Ph | Towards an active anechoic room[END_REF][START_REF] Pinhede | Contrôle actif/passif de salle anéchoïque [Passive/active control for anechoic room[END_REF].

The aim of the present paper is twofold. First, some theoretical developments are provided in order to characterize the relation between the total and the diffracted pressure. Secondly, the efficiency of the method is shown on a numerical example of control. The paper is organized as follows. Section 2 presents the details of the method. Section 3 is dedicated to the study of the

Description of the method

The geometry considered in this paper is shown in Figure 1. In what follows, it is assumed that the walls of the room provide some acoustic absorption. The method is based on the assumption that it is possible to write a relation between p tot the total pressure measured on a surface Γ and p dif the diffracted pressure received at an observation point M located in the volume Ω inside Γ. This relation is expressed with the help of an operator or a "filter" H as :

p dif (S, M ) = Γ p tot (S, P )H(M, P )dσ(P ) (1)
The observation point M may represent any of the minimization microphones as defined in the previous section. The surface Γ represents the surface described by the set of the identification microphones. The relation will be applied to S the primary source, that is the source to be characterized in the room.

The existence of such a relation is not so conventional. Indeed, it is well known that the diffracted pressure may be written as an integral of the total pressure and the normal derivative of the total pressure on any surface inside the room by using integral representations. The point of interest here is to express the diffracted pressure as a function of the total pressure only. If the relation exists and if the operator H is known, the total pressure measured on Γ can be used to drive the secondary sources of the active control system and reduce the diffracted pressure at point M . In practice, the integral relation is discretized and the surface Γ is approximated by the set of identification microphones. The method includes two steps. The first one is a precalibration step in which an approximation of the filter H is evaluated. The second one is a control step in which an active control algorithm is applied to drive the secondary sources.

The precalibration step is based on the essential fact that H does not depend on the source S. It can therefore be evaluated with a set of sources denoted here identification sources which are chosen such that their sound radiation patterns are known. It is determined by minimizing the difference between the diffracted pressure at point M and its expression [START_REF] Schneider | Acoustical behavior of the large anechoic chamber at the LMA in the low frequency range[END_REF]. Once a numerical approximation of H is obtained, the approximation of the integral relation can be used to compute the diffracted pressure for any other source. Then the active control step is carried out.

Previous experimental results showed that such a method provides efficient control results [START_REF] Friot | Estimation and global control of noise reflections[END_REF]. However it is still necessary to prove the validity of formula (1) and to check the existence and the properties of the operator H. Indeed the quality of the control relies on the properties of this operator. This is the aim of the analysis presented in the next paragraph.

3 The integral relation between the total pressure and the diffracted pressure

By using conventional Green's representations it is possible to relate the sound pressure inside a volume to the values of this pressure and of its first normal derivative on the boundary of the volume [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF]. Writing a relation such as [START_REF] Schneider | Acoustical behavior of the large anechoic chamber at the LMA in the low frequency range[END_REF] with no normal derivative term is a bit more challenging.

A simple idea is to choose the specific Green's function defined in the volume inside Γ which satisfies a homogeneous Dirichlet boundary condition on Γ. This leads to a relation of the expected form [START_REF] Schneider | Acoustical behavior of the large anechoic chamber at the LMA in the low frequency range[END_REF]. However this Green's function is not uniquely defined for frequencies equal to the Dirichlet resonance frequencies of the volume. Since the aim of the study is to evaluate the mathematical properties of the operator H it is more efficient to introduce only welldefined (uniquely defined) Green's functions to construct the operator. Therefore the propagation problem is transformed into a transmission problem [START_REF] Filippi | Problème de transmission pour l'équation de Helmholtz scalaire et problèmes aux limites équivalents : application à la transmission gaz parfait -milieu poreux[END_REF][START_REF] Kleinmann | On single integral equation for the problem of transmission in Acoustics[END_REF] between two domains : The domain Ω 1 corresponding to the inside of the surface Γ of the identification microphones and the domain Ω 2 located between Γ and the walls of the room. The sound pressures defined in each domain are related by continuity conditions on the interface Γ. The method consists in using two Green's functions defined for any real frequency and applying integral representations to the two sound pressures. Developments are given for the most general case and then for the half-space case for which some analytical expressions based on 2D-Fourier transforms are obtained.

The general case

The transmission problem is considered for a volume with absorbing conditions on the walls. The room is divided into two domains, an interior domain Ω 1 and an exterior domain Ω 2 as shown in Figure 2. The surface Γ is the boundary between the two domains and Σ represents the outside boundary of Ω 2 that is the walls of the room. Two unit normal vectors n are defined, both directed towards the interior of Ω 2 . One is normal to the boundary Γ and the other one is normal to the boundary Σ.

Both domains are characterized by the same sound speed c and density ρ. An acoustic source is placed in Ω 1 and emits a sinusoidal signal denoted by e -ıωt . The sound pressure is denoted p 1 in Ω 1 and p 2 in Ω 2 . The problem of sound transmission between the two domains is described by the following system :

           (∆ + k 2 )p 1 (M ) = F in Ω 1 (∆ + k 2 )p 2 (M ) = 0 in Ω 2 p 1 = p 2 on Γ ∂ n p 1 = ∂ n p 2 on Γ A(p 2 , ∂ n p 2 ) = 0 on Σ
where k = ω/c is the wavenumber and ∂ n p i denotes the normal derivative of p i on the boundary Γ. At this stage, the operator A does not need to be specified but it must include an absorbing condition (in a simple case, A could correspond to an impedance condition for example). Because of this absorbing condition, the solution (p 1 , p 2 ) to this system exists and is unique [START_REF] Kleinmann | On single integral equation for the problem of transmission in Acoustics[END_REF][START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF][START_REF] Angell | L 2 -boundary integral equations for the Robin problem[END_REF]. Let G 1 be the conventional Green's function defined in the infinite space :

transformed into a term which relates the total pressure on Γ to the diffracted pressure in Ω 1 . This is done by writing another relation to link p 1 and its normal derivative on Γ.

Let the Green's function G 2 be defined in Ω 2 as the solution to the system of equations :

   (∆ + k 2 )G 2 (M, M ′ ) = δ(M, M ′ ) in Ω 2 G 2 = 0 on Γ A(G 2 , ∂ n G 2 ) = 0 on Σ
Because the condition on Σ is an absorbing condition, G 2 exists and is unique (see for example [START_REF] Angell | L 2 -boundary integral equations for the Robin problem[END_REF]). Let us recall that explicit analytic expressions of G 2 are known for a few number of simple conditions and geometries [START_REF] Ph | Methods of Theoretical Physics[END_REF].

The Green's formula applied to G 2 and the pressure p 2 in Ω 2 leads to :

p 2 (M ) = - Γ p 2 (P ′ )∂ n ′ G 2 (M, P ′ )dP ′ (4)
because of the boundary condition of G 2 on Γ. The next step is to apply a normal derivative to this expression and take the limit when M tends to a point P on Γ. This requires special attention since it leads to the derivative of a double layer potential which is a highly singular integral but can still be defined properly (see [START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF][START_REF] Angell | L 2 -boundary integral equations for the Robin problem[END_REF]) :

This is a way to obtain an expression of the operator H. Let us point out that the method used here is based on two Green's functions which are defined and unique for any real frequency. In the general case it is not so easy to determine the mathematical properties of the operator. Their study will provide the necessary information on the minimization problem but stands beyond the purpose of the present article. It could be based on the theory of pseudo-integral operators [START_REF] Antoine | Integral Equations and Iterative Schemes for Acoustic Scattering Problems[END_REF]. As shown in the next paragraph, in the simple case of a half-space geometry and an impedance condition, an explicit expression of the operator H can be obtained. This is of interest since as far as the mathematical non-smoothness of the operator is concerned the properties obtained for the half-space case and for the general case are of the same order.

The half-space case

In the half-space case the total and diffracted sound pressures can be calculated in the Fourier domain in order to get a closer idea of the characteristics of the operator H. Here the surfaces Σ and Γ corresponding to the walls of the room and to the array of the identification microphones are chosen as the infinite planes z = 0 and z = a respectively. The problem is solved for a point harmonic source located at S = (0, 0, s > a). The minimization points are in the domain (z > a). The sound pressure p in the domain z > 0 is the solution to the following system :

   (∆ + k 2 )p(M ) = δ(S, M ) in z > 0 A(p, ∂ n p) = 0 on Σ(z = 0) Sommerfeld conditions at infinity
Since the propagation medium is homogeneous and isotropic, the pressure only depends on the radial coordinate ρ = x 2 + y 2 and may be written p(ρ, z). The 2D transverse Fourier transform and its inverse are defined by :

p(ξ, z) = 2π ∞ 0 p(ρ, z)J 0 (ρξ)ρdρ and p(ρ, z) = -ı 4 ∞ -∞ p(ξ, z)H (1) 0 (ρξ)ξdξ
where J 0 is the Bessel function of zero order and H

(1) 0 the Hankel function of zero order and of the first kind. In the second integral the lower limit (-∞) must be understood as (∞ e ıπ ) for a correct definition of the Hankel function [START_REF] Abramowitz | Handbook of mathematical functions[END_REF].

The solution p(ξ, z) may be written for z > 0 :

p(ξ, z) = ptot (ξ, z) = e ıK|z-s| 2ıK + Â(ξ) e ıK(z+s) 2ıK
where K = k 2 -ξ 2 chosen such that Im(K) > 0 and Â(ξ) is the plane wave reflection coefficient. The second term on the right-hand side is the diffracted pressure pdif (ξ, z).

The operator Ĥ is equal to the ratio between pdif (ξ, z) and ptot (ξ, z = a) :

Ĥ(ξ, z) ≡ pdif (ξ, z) ptot (ξ, a) = Â(ξ)e ıKz e -ıKa + Â(ξ)e ıKa (6)
Let us remark that Ĥ depend on ξ and a and does not depend on the source position s as expected.

For the particular case of an impedance condition on (z = 0) Â(ξ) is explicitly obtained. The impedance condition is written ∂ n p + ıkp/ζ = 0 where ζ represents the reduced specific normal impedance and n is the unit vector normal to the plane (z = 0) and pointing towards the domain (z > 0). Therefore :

Â(ξ) = Kζ -k Kζ + k and Ĥ(ξ, z) = e ıKz Kζ -k 2Kζ cos Ka -2ık sin Ka
The inverse Fourier transform of Ĥ can be formally expressed as a sum of layer potentials located in the (z = a) plane, following steps and results presented in [START_REF] Habault | On the resolvent of the Pekeris operator with a Neumann condition[END_REF]. The details of the calculation are given in Appendix. The result may be written :

H(Q) = -1 2 ∂ ∂z e ıkr(O,P ) 4πr(O,P ) -ı 2 ∂ ∂z z ′ =a n c n H 0 (λ n ρ(P ′ )) ∂ ∂z ′ e ıkr(Q,P ′ ) 4πr(Q,P ′ ) dP ′ (7)
where

c n = λ n h b (λ 2 n )/D ′ (λ 2 n
) with the notations defined in Appendix. Finally the diffracted pressure is expressed as :

p dif (Q) = z=a p tot (P ) H(Q -P )dP (8)
Replacing H by formula [START_REF] Guicking | Active impedance control for one-dimensional sound[END_REF] leads to an expression similar to the one obtained in the general case with integrals on Γ. In particular the first term of ( 8) is :

-1 2 z ′ =a p tot (P ) ∂ ∂z e ıkr(Q,P ) 4πr(Q, P ) dP

An example of control

As said in the introduction, the method proposed here was first tested experimentally on a scaled model (see [START_REF] Ph | Towards an active anechoic room[END_REF][START_REF] Pinhede | Contrôle actif/passif de salle anéchoïque [Passive/active control for anechoic room[END_REF]). The aim of this section is to present a detailed numerical example which illustrates the way the method applies and to examine its efficiency. The efficiency criterion is taken from the standard rules for anechoic rooms (ISO 3745 and ANSI S12.35 norms). The comparison is made between the sound pressure levels obtained with control along one axis in the room and the corresponding exact incident pressure levels. The anechoicity criterion is said to be fulfilled if the deviations stay within ± 1.5 dB.

In this example the "measured" data (incident and diffracted pressure at all microphones) are computed using a BEM code called FELIN [START_REF] Friot | Active control of scattered acoustic radiation : A real-time implementation for a 3D object[END_REF]. The code is based on the Green's formula and the solutions are obtained by using piecewise-constant approximations on triangular elementary domains. In what follows these data are called "exact".

The next paragraphs present the geometry of the problem, the precalibration step and the control step. The source S represents the source to be characterized (such as the loudspeaker in example [START_REF] Melon | Evaluation of a method for the measurement of subwoofers in usual rooms[END_REF]). For the precalibration step, comparisons are presented between the exact diffracted pressure levels computed at a set of minimization points M ℓ and those obtained from the identification of the operator H by using Equation (1). To illustrate the control, comparisons are presented between the exact incident pressure levels emitted by a source S at points M ℓ and those obtained after control.

Let us point out that for this example, both steps are carried out in the frequency domain. In practical situations, the precalibration will be carried out in the frequency domain and the active control will be implemented in real-time using the Fx-LMS algorithm [START_REF] Elliott | Signal processing for active control[END_REF].

The geometry

The dimensions of the room are those of the scaled model that is a room of dimensions (2 m × 1.2 m × 1.1 m). The computations described here are carried out for the 50-450 Hz frequency band. This means that the maximum dimension of the room goes from 0.3λ to 2.7λ.

The precalibration step is achieved with a set of N m = 32 identification microphones P i and a set of N s = 32 positions of identification sources S j . For the control, 32 loudspeakers V k are mounted in the walls so that they are flush with the walls.

The six walls of the room are described by the same impedance values. Figure 3 presents the curve of the reduced specific normal impedance (real and imaginary parts) as a function of frequency. The choice of these impedance values was not meant to reproduce the behaviour of the walls in an anechoic room but rather to provide a reflection coefficient with very high values at low frequency going down to low values around 450Hz. Here the amplitude reflection coefficient goes from 0.98 at 50 Hz to 0.47 at 450 Hz. This is a way to explore several kinds of regimes, to get modes at low frequency and increasing absorption at medium frequency. Indeed, at very low frequency, some modes may be observed in anechoic rooms. A specific study of the effect of the impedance parameter on the identification results shows, not surprisingly, that the best identification of the diffracted pressure is obtained for resonance frequencies that is where the amplitude of the diffracted pressure is large compared with the incident pressure. On the contrary if the diffracted pressure level is low, the identification of the pressure is not that good but in that case an accurate control is not necessary. So this is a positive aspect for the active control. The source S to be characterized is here a monopole located at (0.48, 0.46, 0.47) and 7 minimization points M ℓ are located on a straight line in the plane z = 0.59 at a distance from the source going from 0.22 (point 1) to 1.17 (point 7). They are presented in Figure 4. All distances and lengths are given in meters.

The identification or precalibration step

This step consists in estimating the diffracted sound pressure p dif ( S, M ℓ ) from the identification of the operator H. Since the operator H does not depend on the source an approximation is obtained by minimizing the linear system :

F (S j , M ℓ ) = ||p dif (S j , M ℓ ) - Nm i=1 p tot (S j , P i )H(M ℓ , P i )||
The identification sources S j are used one by one in turn. Their radiation pattern in free-field is supposed to be known. Here they are chosen as monopoles.

Let us define the two vectors H and E and the matrix P by :

H i = H(M ℓ , P i ) i = 1, N m E j = p dif (S j , M ℓ ) j = 1, N s P ij = p tot (S j , P i ) i = 1, N m and j = 1, N s (9)
The element P ij is the total pressure emitted by the source S j alone and measured at microphone P i . E j is the diffracted pressure due to the source S j alone and received at point M . This quantity cannot be measured in a straightforward way ; it is computed as the difference (p tot (S j , M )p inc (S j , M )) where the first term is measured and the second term is known as soon as S j is known.

This optimization problem is solved by a Singular Value Decomposition method (SVD) [START_REF] Hansen | Discrete Inverse Problems -Insight and algorithms[END_REF]. Once H is known, the estimate denoted p dest of the diffracted pressure can be computed for any source S inside the boundary Γ by formula (1) applied to the source S.

The positions of the identification microphones and sources are shown in Figure 5. The microphones P i are regularly spaced on a rectangular array which is 0.15 m away from the walls of the room. The shortest distance between two microphones is 0.56 m (except close to the Table 1 -Error index as a function of frequency corners of the room where it goes down to 0.37 m). The identification sources S j are located inside the volume bounded by the antenna of microphones. These sources represent a rectangular volume with boundaries situated at minimum distances of 0.33 m, 0.50 m and 0.23 m from the walls (along the x, y, z-axis respectively). They are regularly spaced inside the volume because it is a convenient geometry for experimental conditions. However the spaces may differ of a few cms in order to avoid any specific data redundancy. Figures 6 to 8 present the curves obtained for 3 frequencies 130, 250 and 330 Hz. The abscissa corresponds to the 7 positions of the minimization microphones. In each figure, the upper part presents the comparison of the exact and estimated diffracted pressure levels. The continuous line represents the exact diffracted pressure level p dif ( S, M ℓ ) and the crosses represent the estimated pressures p dest ( S, M ℓ ). The lower part presents a comparison of the exact and estimated phase of the diffracted pressure. The same scales are used for the 3 figures.

For the sound levels, at 130 Hz, the estimation is quite close to the exact value. At 250 Hz, the error goes from 0 to 2.5 dB. At 330 Hz, the error goes up to 10dB. In order to quantify the error, a global index has been defined as the mean value, over all points M ℓ , of the amplitude of the difference between the exact diffracted pressure and its estimated value. It increases from 7.10 -3 at 50 Hz to 1.6 at 450 Hz as shown in Table 1. As far as the phases are concerned, the comparison is correct at 130Hz. At 250Hz, the difference goes from 0 to 1.1 at the end points. At 330Hz, the difference is small at points 4 to 7 but goes up to π for point 2. For both levels and phases, the errors increase with frequency as expected.

Sound levels (dB) 0

The control step

The active control consists in estimating the control commands u of the loudspeakers V k by minimizing the quantity :

J(M ℓ ) = ||p dif ( S, M ℓ ) + C t (V k , M ℓ )u(V k )|| 2 + α||u(V k )|| 2 (10)
where C t is the transfer matrix between the secondary sources V k and points M ℓ . The second term adds a condition on the norm of the solutions u in order to get a well-posed problem even for very small α. The minimization of J(M ℓ ) can be carried out with the exact value of the diffracted pressure p dif or with its estimated value p dest . For the commands this leads to two minima called u ex and u est from which two approximations of p dif are defined as follows :

pdex = -C t u ex and pdest = -C t u est
The comparison between the two terms is a way to point out the influence of the identification step. The corresponding incident pressures are then estimated from the substraction between the exact total pressure and the estimated diffracted pressure. For example, piest = p tot -pdest . This is not a conventional way to present active control results. The idea here is to show the efficiency of the active control in relation with the respective levels of the incident and the diffracted pressures. As said before in the cases for which the diffracted pressure is quite small compared with the incident pressure, there is little interest for a high reduction of the diffracted pressure.

Figures 9 to 11 present the results obtained for the same 3 frequencies. The continuous curve represents the exact incident pressure p inc ( S, M ℓ ) as a function of the 7 points M ℓ . For all frequencies the incident pressure piex is equal to the exact incident pressure which means that the control step itself works efficiently. This is not surprising since it consists in solving a problem with more unknowns than equations. The crosses + and × respectively represent the values of the estimated incident pressure piest and the exact total pressure p tot . The total pressure levels are shown in order to give the relative values of the diffracted pressure levels. Again the higher the diffracted pressure level is, the higher its estimate is crucial. In each figure, the two dotted lines are located 1.5dB below and above the exact incident levels. This interval corresponds to the anechoicity criterion defined previously.

Figure 9 shows a good agreement between the sound levels of p inc and piest . The difference between the total and the incident pressure levels points out that the diffracted pressure is a large part of the total pressure. This shows that the method (identification+control) provides a convenient tool at low frequency. At 250 Hz (figure 10) the agreement is still good except at point 7 (with a 5dB deviation) and the diffracted pressure level is small compared with the total pressure. At 330 Hz (Figure 11) the method is not that efficient however the diffracted pressure level is much lower so that the accuracy on its estimation is not so critical. This shows that even if the accuracy of the results obtained in the identification step is not good the final result of the control step may still be satisfying. This points out that the identification criterion (used for the precalibration step) must be adapted to the final criterion.

In this example, most levels lie within the 3dB interval around the exact incident pressure. This means that the method can be useful regarding the anechoicity standard rules. However this will not be sufficient in more specific studies such as the case of characterization measurements Figure 11 -Incident pressure levels at 330Hz -exact (continuous line) and estimated (+), total pressure levels (×), the dotted lines represent a 3dB interval around the exact incident pressure for the radiation of a source for which higher accuracy should be required. Further numerical studies must also be carried out to thoroughly determine the influence of the parameters and for more complex cases. For example, in realistic applications, active control may lead to errors if the number of minimization microphones is larger than the number of secondary sources.

Conclusion

In this paper, it has been shown that the total pressure measured on a closed surface is directly related to the diffracted pressure at any point inside this boundary. The proof of its existence relies on the mathematical study of a transmission problem. It has been shown that in the simple case of a half-space with an impedance condition, a formal explicit expression of the operator H can be obtained. This example is a convenient tool to further study the properties of non-smoothness of H which in turn provide information on the properties of the inverse problem and its regularization.

The relation between the total and the diffracted pressure provides a method to develop an active control system. The numerical tests presented were carried out for a scaled model on the frequency band 50-450 Hz in order to illustrate the method in the case of an anechoicity criterion.

Many questions still arise on numerical and experimental aspects. On the numerical side, the main points are the discretization of the formula and the regularization of the inverse problem. The parameters are the number and positions of the identification sources and microphones. Some results on compressive sensing [START_REF] Chardon | Near-field acoustic holography using sparse regularization and compressive sampling principles[END_REF] could provide some ideas on other choices of the sources location in order to obtain a better conditioning of the identification matrix. Possibly the directivity of the identification sources are also to be considered in order to improve the well-posedness of the problem. Let us underline that the essential criterion to be taken into account is the influence of the regularization of the identification problem on the active control performance rather than on the identification of the operator itself. On the experimental side, a first full-scale test is now carrying out in a semi-anechoic room with one highly reflecting vertical wall and five absorbing walls. The experiment will provide information on the best ways to associate identification and control and on the efficiency of the control.

Let us finally point out that this method can be useful for rooms other than anechoic and therefore for sound field synthesis.

APPENDIX

A The half-space case -calculation of the operator H

The method follows [START_REF] Habault | On the resolvent of the Pekeris operator with a Neumann condition[END_REF]. The operator is obtained by inverting the Fourier transform through an integration in the complex plane. The integral to be calculated is given by :

H(ρ, z) = -ı 4 ∞ -∞ Ĥ(ξ, z) H (1)

(ρξ)ξdξ

It is well defined since the function to be integrated exponentially decreases at infinity and is properly defined at ξ = 0 [START_REF] Abramowitz | Handbook of mathematical functions[END_REF]. Following the principle of limit absorption (equivalent to Sommerfeld conditions [START_REF] Vainberg | Principles of radiation, limit absorption and limit amplitude in the general theory of partial differential equations[END_REF]) it is assumed that the wave number k has a small positive imaginary part and the final expression of H will be the limit of the result obtained by letting the imaginary part of k tend to zero. The calculation below is therefore done with k non real in which case the denominator of H has no zeros on the real axis.

In order to obtain layer potentials in the (z = a) plane the first step is to introduce a term e ıK|z-a| with : Ĥ = e ıK|z-a| F (ξ 2 ) with F (ξ 2 ) = e ıKa (Kζ -k) 2Kζ cos Ka -2ık sin Ka = e ıKa Kζ -k 2Y (ξ 2 )

Because of the square root K, the integration in the complex plane involves branch integrals. However as remarked in [START_REF] Habault | On the resolvent of the Pekeris operator with a Neumann condition[END_REF], these branch integrals disappear if the function to be integrated is even in K.

Therefore the function F is first written as : For the first one let us remark that :

I = ∂ ∂z
e ıK|z-a| 2ıK [START_REF] Zhu | Active control of acoustic reflection, absorption and transmission using thin panel speakers[END_REF] so that the inverse Fourier transform is given by :

∂ ∂z -e ıkr 4πr ( 12
)
For the second term let us similarly remark that :

-ıK 2 e ıK|z-a| = ∂ 2 ∂z∂a e ıK|z-a| 2ıK which is the Fourier transform of :

∂ 2 ∂z∂a -e ıkr 4πr ( 13
)
The Fourier transform of the function h b /D is obtained by the method of residues. The poles of this function, denoted λ n , are the zeros of the denominator D. Let us assume for simplicity that these zeros are simple zeros, the integral can be written :

- ı 2 n h b (λ 2 n ) D ′ (λ 2 n )
λ n H 0 (λ n ρ(P ′ )) [START_REF] Guicking | An active sound absorber with porous plate[END_REF] where D ′ (λ 2 n ) = ∂D(ξ 2 )/∂(ξ 2 ) at ξ = λ n . The inverse Fourier transform of II is the convolution product of the terms in Equations ( 13) and [START_REF] Guicking | An active sound absorber with porous plate[END_REF]. It is given by :

- ı 2 ∂ ∂z z ′ =a n h b (λ 2 n ) D ′ (λ 2 n ) λ n H 0 (λ n ρ(P ′ )) × ∂ ∂z ′ e ıkr(Q,P ′ ) 4πr(Q,P ′ ) dP ′ (15)
with Q = (x, y, z), P ′ = (x ′ , y ′ , z ′ ).

The inverse Fourier transform of Ĥ is the sum of Equation( 12) and ( 15) which leads to the final expression [START_REF] Guicking | Active impedance control for one-dimensional sound[END_REF].

Figure 1 -

 1 Figure 1 -Active control of the sound pressure in an anechoic room

Figure 2 -

 2 Figure 2 -Geometry of the problem

Figure 3 -

 3 Figure 3 -Reduced specific normal impedance versus frequency -Real part (continuous line) and imaginary part (+)

Figure 4 -

 4 Figure 4 -Source S (×) and minimization points (+) in the (z = 0.59) plane

Figure 5 -Frequency

 5 Figure 5 -Identification sources (×) and microphones (•)

Figure 6 -

 6 Figure 6 -Diffracted pressure levels and phases at 130 Hz -exact (line) and estimated (+)

Figure 7 -Figure 8 -

 78 Figure 7 -Diffracted pressure levels and phases at 250 Hz -exact (line) and estimated (+)

Figure 9 -Figure 10 -

 910 Figure 9 -Incident pressure levels at 130 Hz -exact (continuous line) and estimated (+), total pressure levels (×), the dotted lines represent a 3dB interval around the exact incident pressure

F (ξ 2

 2 ξ 2 ) with B = -k cos Ka + ıKζ sin Ka 2Y where B is uneven in K. To obtain an even function one more step is introduced by using the following equalities : B(ξ 2 )e ıK|z-a| = where D = KY and h b = ı(-k cos Ka + ıKζ sin Ka). The function h b /D is an even function in K. The next step is to compute the inverse Fourier transform of the two terms I and II defined by Ĥ = I + II with :

G 1 (M, M ′ ) = -e ıkR(M,M ′ ) 4πR(M, M ′ )where R(M, M ′ ) is the distance between two points M and M ′ . The Green's representation of p 1 in Ω 1 writes :p 1 (M ) = p inc (M ) -(2)Γ [∂ n ′ p 1 (P )G 1 (M, P ) -p 1 (P )∂ n ′ G 1 (M, P )]dPwhere p inc is the incident sound pressure. The term ∂ n ′ is used for the normal derivative at the integration point P and dP is shortly written for the element of integration. The integral on Γ is equal to the diffracted pressure p dif (M ). It can be formally written as :p dif (M ) = V 1 [∂ n ′ p 1 ] + U 1 [p 1 ](3)with the definitions, for any function µ on Γ :V 1 [µ](M ) ≡ -Γ µ(P )G 1 (M, P )dP U 1 [µ](M ) ≡ Γ µ(P )∂ n ′ G 1 (M, P )dPIn this representation, U 1 is a double layer potential operator which operates on the value of the total pressure on Γ, similarly to the operator H.The operator V 1 corresponds to a simple layer potential and operates on the derivative of the total pressure on Γ. In order to obtain a formula similar to (1) this first term must be

∂ n p 2 (P ) = -∂ n(P ) Γ p 2 (P ′ )∂ n ′ G 2 (P, P ′ )dP ′Introducing the continuity conditions between p 1 and p 2 in this equation leads to :∂ n p 1 (P ) = -∂ n(P ) Γ p 1 (P ′ )∂ n ′ G 2 (P, P ′ )dP ′ ≡ [T [p 1 ]](P ) (5)where P is a point on Γ. The second equality defines the operator T which relates the total pressure p 1 to its derivative ∂ n p 1 both defined on Γ. The next step is to replace ∂ n ′ p 1 (P ) in the first term of the right-hand side in Equation (3) by T [p 1 ]. This leads to :V 1 [∂ n p 1 ](M ) = V 1 [T [p 1 ]](M ) = Γ ∂ n(P ) Γ p 1 (P ′ )∂ n ′ G 2 (P ′ , P )dP ′ G 1 (M, P )dPFollowing[START_REF] Colton | Inverse acoustic and electromagnetic scattering theory[END_REF] it is possible to show that the two integrals in this expression can be interchanged. This gives :V 1 [T [p 1 ]] = Γ p 1 (P ′ )∂ n ′ Γ G 1 (M, P )∂ n ′ G 2 (P ′ , P )dP dP ′Using this last equation and Equation[START_REF] Melon | Evaluation of a method for the measurement of subwoofers in usual rooms[END_REF] shows that the operator can be formally expressed :H = -∂ n ′ G 1 + ∂ n ′ Γ G 1 ∂ n ′ G 2


Catherine Picart
email: catherine.picart@medecine.u-strasbg.fr.

Kheya Sengupta

Joerg Schilling

Gjertrud Maurstad

Guy Ladam

Andreas R Bausch

Erich Sackmann

Microinterferometric Study of the Structure, Interfacial Potential, and Viscoelastic Properties of Polyelectrolyte Multilayer Films on a Planar Substrate

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Surface modifications of materials are of primary importance for biomedical applications. [1][2][START_REF] Ratner | Biomaterials Science: An Introduction to Materials in Medicine[END_REF] Among the different techniques used to modify surfaces, the deposition of polyelectrolyte multilayers (PEMs) has emerged as an easy to handle and versatile tool. [START_REF] Decher | [END_REF][5][6] It involves alternate deposition of polycations and polyanions and allows the buildup of films with tunable properties by adjusting parameters such as the chemical nature of the polyelectrolytes, pH and ionic strength of the medium, and immersion and rinsing times. It is thus possible to obtain an almost infinite variety of architectures with different permeabilities, 7 optical properties, 8 or even cell adhesion properties. 9,10 One of the most investigated polyelectrolyte systems is based on poly(styrene sulfonate)/poly(allylamine hydrochloride) (PSS/PAH). These nanometer-thick multilayers have been characterized by a wide range of techniques. Both the mass and the thickness of the films have been shown to exhibit, in the dried state, a linear growth with the number of deposition steps. 11,12 PSS/PAH films were also used as a reference system for following in situ, under a liquid environment, the process of PEM buildup by various optical biosensors, such as scanning angle reflectometry (SAR), 13 optical waveguide lightmode spectroscopy (OWLS), 14,15 and a colorimetric resonant biosensor. 16 Films made of synthetic polyelectrolytes such as PSS/ PAH are attracting interest as new biomaterial coatings, because they are highly biocompatible in vitro and favor the growth of primary endothelial cells and primary fibroblasts. 17,18 For many applications, e.g., when the films are subjected to flow, deformation, shear stress, aging, or when they are used as a substrate for promoting or inhibiting cell growth, 10,17 it is important to control the mechanical properties of the film. 19 However, the viscoelastic properties of these layer-by-layer polyelectrolyte films are still poorly understood. This is partly due to the lack of available techniques capable of probing the rheology at length scales relevant to these films whose thickness range from nanometers to micrometers. The few studies undertaken so far were focused on the elasticity and mechanical stability of hollow capsules of the polyelectrolytes. 20,21 Several techniques based on colloidal probes have recently been developed to study soft thin films. These include surface force measurements with colloidal beads attached to atomic force microscopy (AFM) tips, 22,23 total internal reflection microscopy (TIRM), 24,25 and reflection interference contrast microscopy (RICM). [26][27][28] Using conventional RICM, it is possible to measure the instantaneous height (i.e., the distance from the underlaying substrate) of colloidal beads hovering over transparent planar substrates in a liquid medium and to reconstruct the bead/surface interaction potentials from the height distribution functions. As in any interferometric technique using monochromatic light, the height determination has an inherent ambiguity of approximately half the wavelength of the light used, which is usually of the order of 200 nm. Recently, the RICM technique was further modified to include a second wavelength (dual wavelength, or DW-RICM), which enables the measurement of absolute heights up to ∼800 nm with an accuracy of 3 nm. 29 With this technique, the interaction potentials and the effective surface friction coefficients can be measured as a function of the absolute distances of the beads from the substrate. DW-RICM is thus ideal for investigating the surface properties of thin nanostructured films. However, in all the techniques involving colloidal probes, including DW-RICM, it is important to first characterize the probe beads, in terms of their surface properties, because the bead surfaces are typically not smooth and hard but exhibit some roughness, fuzziness, and hairiness, as a result of the polymer nature of polystyrene, which is the material that typically comprises colloidal beads. 30,31 In this paper, we report the characterization of thin films made of PEMs, in terms of their thickness, surface potentials, and surface friction coefficients using DW-RICM, with special attention to the influence of bead/film interaction on the measured parameters. The beads to be used as probes were first characterized in terms of the fuzziness of their surface. Next, the DW-RICM technique was validated for absolute height measurements by comparing the heights of multilayer films of controlled thickness, as measured by DW-RICM, with OWLS data. The influence of having different coatings on the beads and of different polyelectrolytes (PSS or PAH) as the topmost layer on the behavior of the beads was investigated. Finally, the bead/film interaction potential and the surface friction coefficient of the films was obtained by monitoring the thermal fluctuations of colloidal beads placed on the films. The interaction potential was characterized by its double derivative (i.e., the curvature) at the minimum (i.e., around the equilibrium height) and was compared to previous studies of colloidal beads on glass substrates and on hyaluronic acid cushions. 27,28 An unexpected universal power law dependence of the curvature of the interaction potential on the bead height is reported.

Materials and Methods

Substrates. Glass slides (24 mm × 24 mm, VWR Scientific) that were cleaned by sonication in a 2% v/v detergent solution (Hellmanex, Hellma, Germany), followed by repeated washing and sonication in ultrapure water (Milli Q-Plus system, Millipore, Bedford, MA), were used as bare glass substrates. Similar slides that were cleaned with 10 mM sodium dodecyl sulfate (SDS) and 0.1 N HCl and extensively rinsed with pure deionized water were used as substrates for the polyelectrolyte multilayers.

Polyelectrolyte Solutions. Poly(sodium 4-styrene sulfonate) (PSS, 70 kDa), poly(allylamine hydrochloride) (PAH, 70 kDa), and poly(ethyleneimine) (PEI, 700 kDa) were purchased from Sigma-Aldrich. All solutions were prepared using ultrapure water (Milli Q-Plus system, Millipore) with a resistivity of 18.2 MΩ cm. Polyelectrolyte solutions were prepared by direct dissolution in a filtered 0.15 M NaCl solution at pH 6 in order to have a final concentration of 5 mg/mL.

Beads. Carboxyl-modified polystyrene beads with a diameter of 10.15 ( 0.06 µm were purchased from Duke Scientific (Palo Alto, CA). For experiments on glass, the beads were used after washing in ultrapure water. For experiments on PEM films, beads were coated with bovine serum albumin (BSA) (Sigma-Aldrich, incubation for 30 min), followed by three rinsing steps. Beads were also coated with PSS or PAH following the same procedure. Sulfate-modified polystyrene beads with a diameter of 9.6 ( 0.7 µm were purchased from Interfacial Dynamics Corporation (Molecular Probes, Eugene, OR). Prior to use, the beads were washed with ultrapure water.

Experiments on bare glass with Duke and IDC beads were performed in potassium chloride (KCl) solutions of various ionic strengths.

Automatic Buildup of the Polyelectrolyte Multilayered Films. In the following, films ending with a PAH (positive) layer are called PEI-(PSS/PAH) i films and films ending with a PSS (negative) layer are called PEI-(PSS/PAH) i -PSS. The subscript "i" refers to the number of PSS/PAH layer pairs.

All the PEI-(PSS/PAH) i films were prepared with an automatic dipping machine (Dipping Robot DR3, Kirstein GmbH, Germany). First, a precursor layer of PEI was deposited on the glass substrates. The glass slides were arranged vertically in a homemade holder that was immersed for 10 min in a first polyelectrolyte solution (PEI). The slides were subsequently rinsed successively in three different beakers, containing 0.15 M NaCl solution (pH 6), of volumes of 350, 200, and 200 mL for 2, 6, and 6 min, respectively. They were then dipped into the oppositely charged polyelectrolyte solution (PSS) followed by the same rinsing procedure. The entire procedure was repeated with the PAH solution and so on. Rinsing solutions were changed after the deposition of every tenth layer. Slides were stored in 6-wells culture plates at 4 °C.

Analysis of Film Growth and Zeta Potential. The PEI-(PSS/PAH) i film buildup process was followed in situ by OWLS. 32,33 Briefly, OWLS is sensitive to the penetration depth of an evanescent wave (∼200-300 nm) traveling through a polymer film deposited on a waveguide surface and gives access to the optical properties of the film. The effective refractive indexes of the waveguidesNTE and NTM for the transverse electric and transverse magnetic waves, respectivelysare measured. From these, the film refractive index (n A ) and thickness (d A ) can be deduced by solving the phase equations. This method has already been experimentally applied to polyelectrolyte multilayers. 15,34 One hundred microliters of the polyelectrolyte solutions was injected (alternately polycation and polyanion), left at rest for 15 min, and rinsed under constant flow (rate 7 mL/h) for 12 min with the NaCl solution.

The zeta potential of the layers adsorbed on a capillary were measured using a homemade streaming potential measurement apparatus. 35 The apparatus and the experimental procedure have been previously described. 13,36 The adsorption of PSS and PAH was performed in a 0.15 M NaCl solution, and measurements were performed after a rinsing step, following the deposition of each layer. The rinsing solution (also present during the measurements) contained only 0.5 mM Tris at pH 6.

Dual-Wavelength Reflection Interference Contrast Microscopy (DW-RICM) Technique. The DW-RICM technique has been described in detail elsewhere. 29 In conventional RICM, monochromatic light is incident on the bead under study, which typically hovers over a bare glass substrate or a transparent polymer film (deposited on a glass substrate) in a liquid medium. The incident light is reflected from the glass/ buffer interface and again from the buffer/bead interface. These two reflected rays interfere and result in the characteristic RICM interference pattern that consists of concentric dark and bright circular fringes in case of spherically symmetric objects such as beads. The major drawback of the conventional RICM technique is that the information about the absolute phases of the beams reflected by the various interfaces of a film is not known. Therefore, absolute distances cannot be measured, because there is an ambiguity of a factor λ/(2n) (where λ is the wavelength of the light used and n is the refractive index of the medium). This drawback can be overcome by comparing the interferograms obtained simultaneously with two different wavelengths. In other words, an additional periodicity and boundary condition is introduced by observing a second wavelength. This enables the measurement of absolute heights of the beads above a planar surface.

A reflection interference contrast inverted microscope (Axiomat Zeiss, Germany) that was equipped with an antiflex objective (63×, oil immersion, N.A. ) 1.3) was used. In this setup, the sample is illuminated through the objective by a bichromatic light beam (λ ) 546 and 436 nm) generated by passing the light from a 100 W mercury vapor lamp (Osram, Germany) through an interference filter (IF filter, 85% transmission, 12 nm waveband; AF-Analyzetechnik, Germany). Just before recording the image, the images corresponding to the two wavelengths are separated by a dichroic mirror and recorded with two charge-coupled device (CCD) cameras (models 4880-85 (10 bit) and 4742-95 (12 bit), Hamamatsu, Japan).

The heights of the beads were determined, within the λ/(2n) ambiguity, from the radii of the interference fringes obtained using λ ) 546 nm. In principle, for a bead of radius R, the height h(r l ) and the radius of the lth fringe r l are related by However, in this equation, the curvature of the bead has not been taken into account. Therefore, a more accurate empirical relation 29 was used to determine the heights of the beads. The ambiguity of λ/(2n) in the height was removed by comparing the interferograms obtained with the two wavelengths. Using this technique, the absolute and relative heights of a bead can be determined respectively with an accuracy of ∼3 nm about ∼0.4 nm.

Determination of the Interfacial Potential and the Surface Viscosity. The Brownian motion of a colloidal bead in an interfacial potential V(h) can be described by the Langevin equation:

where h is the distance of the bead from the substrate, m the mass of the bead, t the time variable, and f stoch the stochastic force due to thermal noise. In the case of a colloidal bead, the inertial term (the first term on the left-hand side) can be neglected. The second term accounts for the viscous force, where γ is an effective friction coefficient. The third term accounts for the interfacial interaction potential V(h) between the bead and the polymer film. For small deformations, V(h) can be expressed in terms of a harmonic potential V(h) ) V′′(∆h) 2 , where V′′ ) ∂ 2 V/∂h 2 and ∆h ) h -〈h〉. V′′ is a measure of the stiffness of the interfacial potential.

The effective friction γ experienced by a sphere exhibiting steady-state motion at a distance h from a wall is determined by a modified Stoke's law: 37,[START_REF] Happel | Low Reynolds Number Hydrodynamics[END_REF] where the function F(R/h) accounts for the fluid drainage between the bead and the wall and η eff is the effective surface viscosity.

The two viscoelastic parameters V′′ and γ can be determined by calculating the time correlation function of the fluctuation of the distance between the bead and the substrate, according to

The interaction potential can also be obtained from the probability distribution of the heights using Boltzmann's law: 26 For our measurements, we have confirmed that both methods yield approximately the same value for V′′(h) (data not shown).

Results and Discussion

Characterization of Probe Beads. The absolute equilibrium height of colloidal beads hovering over glass substrates were measured as a function of the ionic strength of the medium, which was varied from 0.5 mM to 0.3 M by adding KCl. Measurements were performed with carboxyl-modified (Duke) and sulfate-modified (IDC) beads.

In Figure 1, the measured equilibrium heights of the beads are plotted against the ionic strength of the buffer. Each data point corresponds to the average equilibrium height calculated from at least five different beads; the error bars indicate the standard deviations of the distributions.

The interfacial interaction potential experienced by the beads is expected to be the sum of three contributions: the electrostatic double layer repulsion, gravity, and the van der Waals force. When the separation distance is greater than several Debye lengths, the van der Waals attraction is expected to be severely retarded and screened 24 and can, therefore, be neglected. Also, the double-layer repulsion is expected to be well-modeled using

h(r l ) ) λ 2n -R + R 2 -r l 2 m ∂ 2 h ∂t 2 + γ ∂h ∂t + ∂V ∂h ) f stoch (1) γ ) 6πRη eff F ( R h ) (2)
〈h(t)h(t + τ)〉 ) k B T V exp ( -V′′ γ τ ) (3) V(h) ) -k B T ln ( P(h) P 0 ) (4)
linear superposition and Derjaguin's approximation. This interaction potential is given by where and 0 and R are, respectively, the permittivity and relative dielectric constant of water, R is the radius of the sphere, e is the elemental charge, and ψ B and ψ G are the Stern potentials of the bead and the glass wall. The reciprocal screening length is given as where I the total ionic strength, k B the Bolzmann constant, and F B and F F are the densities of the bead and the medium, respectively. The surface potentials (of both glass (G) and polystyrene bead (B)) are related to the surface charge densities (σ B ) by the Graham equation: [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF] with Equation 5 can then be minimized to obtain the theoretically expected equilibrium height (h EQ ), given by Contrary to expectations from theoretical consideration, heights of ∼20 nm were measured for the carboxyl-modified Duke beads at high salt concentrations (g50 mM). At these ionic strengths, the Coulomb repulsion is expected to be screened and the beads are expected to "salt out" or sediment out of solution, onto the substrate. However, when the beads were placed on a positively charged surface (glass slides coated with aminosilane), the measured height was of the order of a few nanometers. It was also possible to push the beads closer to a bare glass surface by applying an external downward mechanical force. All these observations are consistent with the hypothesis that a fuzzy polymer layer is present around the bead, 31,[START_REF] Van De Ven | [END_REF] and this layer is too dilute to reflect light but nevertheless marks the mechanical boundary of the particle. When an external force is applied, either in the form of Coulomb attraction from a positively charged surface or as a downward mechanical force, the dilute fuzzy layer gets compressed and the bead is driven closer to the substrate. The thickness of this fuzzy layer depends strongly on the surface modification of the bead and can vary, even for the same chemical surface modification, from one manufacturer to the other (data not shown). Thus, in experiments using optical techniques that necessarily rely on the optical density of the bead, the bead appears to be higher than it really is, because of the presence of the fuzzy layer.

For fitting the experimental heights shown in Figure 1 with eq 6, the surface charge of the glass substrate was taken to be σ G ) 0.0007 C/m 2 , according to von Gru ¨nberg. 41 The only free parameter in the fitting was the surface charge density of the bead, which was varied in order to optimize the fit. It was found that the measured heights were always consistently higher than the expected height. However, adding a constant height h fuzz to the h EQ value, to account for the fuzzy layer, improves the fit significantly (the thick solid line in Figure 1). The best values for h fuzz and the bead surface charge density (σ B ) were determined to be 7 nm and 0.0005 C/m 2 , respectively. The fuzzy layer thickness estimated here is approximately the same as the surface roughness estimated from AFM observations. 24 The second type of beads (IDC) considered were sulfatemodified polystyrene. The theoretical equilibrium height curve is plotted as a thin solid line in Figure 1 (with h fuzz ) 0 and σ B ) 0.0001 mC/m 2 ) and fits the data adequately for concentrations of <50 mM. The best fit for the fuzzy layer thickness was zero. Contrary to expectations, the height of the IDC beads increases at high salt concentrations. We believe that these deviations are unlikely to result from zeta potential effects, because the zeta potential shift toward more-positive values (in the case of our beads, which have large negative charges, the charge should become less negative, that is, shift toward zero charge) when the ionic strength is increased, thus leading to an enhanced attraction (or reduced repulsion) between the bead and the glass substrate. On the other hand, the RICM interference fringes indicate that, above an ionic strength of 50 mM, the average bead size increases with the salt concentration, which, in turn, leads to a decrease in the bead density. The increase of height with salt concentration is thus attributed to the decrease in bead density.

It is also possible to extract from the height fluctuations the interfacial interaction potential experienced by the beads. The potential energy of Duke beads hovering over bare glass in media of different ionic strengths is shown in Figure 2. The data points are represented by symbols and the expected theoretical potentials (calculated from eq 5) are represented as solid lines. As can be seen, the theoretical curve agrees well with the data for low ionic strengths (up to ∼10 mM). At ionic strengths of 5 mM and above, the observed potentials are more symmetric than expected, because the slope C of the attractive portion of the measured potential is higher than that expected from gravity alone. The discrepancy between theory and experiment becomes larger as the ionic strength increases (corresponding to lower heights), and, at 50 mM or more, the theoretical and experimental curves are totally mismatched.

In the second part of this work, measurements were performed on PEMs using the monodisperse Duke beads as probes. Because the size of each bead can be determined from the RICM interference fringes, IDC beads could, in principle, also be used. However, beads of different sizes have different weights and, hence, compress the films differently. Therefore, although the Duke beads exhibit a fuzzy layer, they were preferred over the IDC beads, because they are more monodisperse.

Growth of PSS/PAH Films. As a first check of the growth of the PSS/PAH films, we used the well-established in situ OWLS technique. A typical curve obtained during the successive adsorption of PSS and PAH from 0.15 M NaCl solutions is shown in Figure 3. The increase in the effective refractive index NTM (raw signal) after the addition of each new layer indicates polyelectrolyte adsorption. The charge overcompensation that

V(h) ) B exp(-κh) + Ch (5) B ) 4πR 0 R( kT e ) 2 tanh ( eψ B 4kT ) tanh ( eψ G 4kT ) C ) 4 3 πR 3 (F B -F F )g κ ) 8πIe 2 k B T 4πλ B σ G ) -2κ sinh ( ψ G 2 ) λ B ) e 2 4π 0 kT R κh EQ ) ln ( κB C ) (6)
D J. Phys. Chem. B Picart et al.

appears on the surface after each deposition constitutes the driving force for the deposition of the next layer. The zeta potential of these films alternated between negative and positive values. It was approximately -50 mV for PSS-ending layers and approximately +55 mV for PAH-ending layers (see inset in Figure 3). The thickness increased linearly with the number of bilayers, and after the deposition of 10 layer pairs, the mean thickness estimated by OWLS was determined to be ∼50 nm with a film refractive index of n A ) 1.5. BSA Beads on PSS-Ending Films: Film Thickness and Interaction Potential. The validity of the DW-RICM technique for thickness measurements of thin PEM films was checked with PSS/PAH multilayers of various well-controlled thicknesses. BSA-coated beads are known to be passivated, with respect to hydrophobic attraction, 42 and PSS is known to interact only slightly with albumin. 43 Therefore, in the first set of experiments, PSS-ending films were studied using BSA-coated beads as colloidal probes.

PEI-(PSS/PAH) i -PSS films with the number of layer pairs "i" in the range of 5-30 were studied. The absolute equilibrium heights of the beads are plotted in Figure 4A for all the PSSending films. Each data point is obtained by averaging the heights of at least 16 beads on at least 2 different samples. The error bars correspond to the standard deviation of the heights obtained for a given number of layer pairs. The absolute height of the beads, as well as the inhomogeneity in the bead height distribution, increase as the number of layer pairs increases; the highest scatter in the height data is observed for 30 layer pairs (where the height ranges from 80 nm up to 260 nm). The dotted line in Figure 4A is the regression line obtained from OWLS data from films up to 10 layer pairs, which was linearly extrapolated to 30 pairs of layers. The experimental height of the beads obtained by RICM matches the extrapolated thickness from OWLS data fairly well. A slight deviation is observed for the thinnest films comprising 5 layer pairs. This discrepancy may be related to the fact that, even on bare glass, bare beads at 150 mM NaCl exhibited an average height of ∼20 nm (see Figure 1). The error bars reflect the heterogeneity of both the PEM film and the bead surface. The standard deviation of the height fluctuations of the beads is represented versus the number of layer pairs in Figure 4B. As the number of layer pairs increases, the height fluctuations are larger and more disperse. For films containing more than 30 layer pairs, the interference patterns were highly distorted, especially for the weaker blue light, because of the high layer thickness combined with a rather high refractive index (1.5) of the polyelectolyte film (Figure 5). As a consequence, images from films containing more than 30 layer pairs could not be analyzed.

PEM films are rather soft; therefore, the relatively large colloidal beads used here compress and/or sink into the film, which leads to a decrease in the apparent thickness of the film. On the other hand, because of the fuzzy layer around the bead, the bead appears to be slightly higher than it is in reality, thus increasing the apparent thickness of the film. Moreover, in the height calculations, we have not taken into account the fact that the refractive index of the polymer films (n F ) is higher than that of water (n), the medium for which the beads were calibrated. This leads to two effects. The first effect results from the fact that the optical path traveled by the ray that gets reflected from the bead now travels through a medium with a refractive index of n F ) 1.500, instead of through water that contains salt, with a refractive index of n )1.334. This leads to an increase in the path difference between the two reflected rays and, therefore, an increase in the apparent height by a factor of n F /n ≈ 1.12. The second effect is more subtle and results from the fact that the radii of the interference fringes and, hence, the calculated heights depend on the refractive index of the film.

For the PSS/PAH films studied here, these three effectss namely, film compression, steric levitation (the fuzzy layer effect), and the refractive index effectsscompensate each other in such a way that the measured data remain close to the line extrapolated from OWLS data. The interaction potential of the beads on PSS/PAH films of various thicknesses are represented in Figure 6 (note the difference in scale of the abscissa between panels A and B). A comparison with Figure 2 shows that the potentials over the PEI-(PSS/PAH) i -PSS films are different from the potentials over glass at an equivalent ionic strength. The stiffness of the potential decreases as the number of layers increases.

PSS-or PAH-Coated Beads on PSS-Ending Films:

Influence of Bead Coating. The effect of coating the beads with different polymers (PSS or PAH) on the bead/film interaction potential was explored. Duke beads were coated either with PSS or with PAH and were placed on PSS-ending PEM films. As a polycation, PAH adsorbs onto polystyrene 10 and also onto carboxyl-modified surfaces groups, because of its positive charge. In case of PSS, the adsorption is not because of the charge but because of hydrophobic interactions with the modified polystyrene beads. As a control, the behavior of the coated beads on bare glass (buffer ionic strength 0.15 M NaCl, pH 6.5) was checked. The average height of the BSA-coated beads (22.5 ( 2.3 nm) was similar to that of the PSS-coated beads (21.9 ( 5.5 nm); however, the dispersion in the measured heights was higher for the PSS-coated beads. The height fluctuations of the BSA-and PSS-coated beads with standard deviations of 0.5 ( 0.3 nm and 0.4 ( 0.3 nm, respectively, were also similar. The positively charged PAH-coated beads were slightly lower (17.9 ( 2.2 nm), indicating an attractive interaction of the beads toward the glass substrate. Free PAH molecules from a solution are known to adsorb onto a bare glass substrate. 44 Therefore, it can be conjectured that, in the present case, the PAH molecules that are adsorbed onto the bead surface facilitate its attraction to the glass substrate.

The behavior of the coated beads placed on the PEM films ending with a PSS layer depends crucially on the nature of the bead coating. The most obvious difference is that, whereas the PSS-coated beads fluctuate in a manner similar to that of BSAcoated beads on similar films, the PAH-coated beads do not experience any fluctuation whatsoever.

The absolute heights of the beads with the two different polymer coatings, as a function of the number of pairs of layers in the film, are shown in Figure 7. A comparison with Figure 4 shows that the measured heights for the PSS-coated beads are very close to that of the BSA-coated beads. The height fluctuations are consistently larger for the PSS-coated beads (Figure 7B) than for the BSA-coated ones (Figure 4B). This is Figure 5. DW-RICM interference patterns obtained using two wavelengths (G denotes green at 546 nm and B denotes blue at 436 nm) for two 10-µm beads hovering over PSS-ending films made of 30 layer pairs (G1 and B1) and 50 layer pairs (G2 and B2). For the 50 layer pairs, the interference pattern is highly distorted and, therefore, is not amenable to further analysis. likely to be a consequence of the steric and Coulomb repulsive interactions between the PSS coating of the beads and the PSS outer layer of the film.

In case of PAH-coated beads, the bead heights and, hence, the apparent film thickness did not increase with the number of layers (Figure 7A) but remained approximately constant at ∼42 nm. The maximum height measured was ∼60 nm. The amplitude of the height fluctuation is ∼0.4 nm, which is close to the limit of resolution of the experiment (Figure 7B).

The low bead heights and low amplitude of height fluctuations of the PAH-coated beads suggest that these beads get embedded into the PEM film. A possible mechanism could be a step-bystep bead/PEM interaction, such that the positively charged PAH coating of the bead first interacts with the negatively charged outermost PSS layer of the film. At this stage, the bead gets partially covered with negatively charged PSS, which then interacts with the next underlying positively charged PAH layer, and so on. In the case of negatively charged PSS-covered beads, the similarly charged PSS on the outermost layer of the PEM film has a repulsive interaction with the bead, preventing it from undergoing further interaction with the underlying layers. Figure 8 summarizes why PSS-or BSA-coated beads do not sink into the film, whereas PAH-coated beads do.

Experiments on PAH-Ending Films. In the next set of experiments, the behavior of the coated beads on PAH-ending PEM films was studied. The BSA-and PSS-coated beads were at a height of ∼50 nm, irrespective of the number of layers in the PEM films, and did not exhibit any height fluctuations (see Table 1). Since BSA-and PSS-coated beads are negatively charged, they are expected to experience an attractive interaction with the positively charged PAH-ending films. Moreover, it has been demonstrated by SAR that albumin adsorbs strongly onto PAH-ending films but only weakly on PSS-ending films. 43,45 Therefore, the absence of fluctuations and the uniform bead height for all the samples can again be attributed to the attractive interaction of the beads with the positively charged outermost layer of the films, eventually leading to the beads getting embedded in the films (Figure 8).

Unexpectedly, PAH-coated beads on PAH-ending films did not exhibit significant height fluctuations. This may originate from an imperfect coating of the beads with PAH and/or the presence of defects in the positive topmost layer, which expose the negative layers underneath and lead to attractive bead/film interactions.

Film Viscosity and Bead/Film Interaction Potential. The effective frictionsand, hence, the effective surface viscosity (η eff )sexperienced by a bead can be calculated from the height correlation function, using eqs 3 and 4. When, as in the present case, the bead fluctuates near a hard wall (the glass substrate) with R . h, F(R/h) in eq 2 can be approximated as F(R/h) ≈ R/h. 46 Because the bead is only partially submerged in the polyelectrolyte film, the effective viscosity has contributions from both the surrounding buffer and the film. Note that the viscosity reported here is a sum of these two contributions and characterizes the entire film/buffer interface. In Figure 9A, η eff (for BSA-coated beads on PSS-ending films) is plotted as a function of the number of layer pairs in the PEM film. As can be seen, η eff decreases as the number of layer pairs increases. For a 50-nm-thick film made of 10 layer pairs, the effective viscosity is ∼10 times that of water, whereas a film containing 30 layer pairs is only four times more viscous than water. This a Beads were coated with BSA, PSS (negative), or PAH (positive). The number of beads over which an average was obtained is given in parentheses. effect probably results because the film is attached to an underlying hard substrate.

The second derivative of the interaction potential V′′ can also be extracted from the autocorrelation function of the bead height fluctuations (see eq 3). In Figure 9B, V′′ (for BSA-coated beads on PSS-ending films) is plotted as a function of the number of layer pairs in the PEM films. As can be seen, V′′ decreases as the number of layer pairs increases. In Figure 10A, the V′′ value and the height obtained for each given number i of layer pairs (5, 10, 20, 30) have been gathered (same data as in Figure 9B) and the averages have been plotted. The monotonic decrease of V′′, as a function of the height, can be fitted by the relation V′′ ≈ h R , with R ) -4.33.

This type of power law dependence also occurs in the case of ultrathin layers of hyaluronic acid 28 and for bare beads on bare glass. In all these cases, the exponent R is close to -4. To compare the present data with the data for beads on hyaluronic acid and bare glass, all the data have been gathered in Figure 10B. Note that two different types of beads were used for the experiments on hyaluronic acid and on bare glass: carboxylmodified beads that were purchased from Duke Scientific (the same as in the present work) and carboxyl-modified PS beads that were purchased from Polysciences.

All the experimental data approximately follow the same power law (with R ) 4.19, R ) 0.84, with R being the linear regression coefficient). It thus seems that the stiffness of the potential experienced by a bead at a given height is determined solely by its height and not by any other parameter. Changing the ionic strength of the buffer or depositing polymer films between the bead and the glass substrate changes the equilibrium height of the bead and, hence, the stiffness of the potential. However, at a given equilibrium height, the beads always experience the same potential. Note that the working ionic strengths considered here are rather large (in the range of a few millimolar to a few hundred millimolar), whereas most studies of colloidal beads are conducted at much lower ionic strengths. 24,41 Our observations seem to indicate that, at high ionic strengths, the dominant contribution to the interaction potential comes from the glass surface itself and the intervening polymer layers do not have any significant role in determining the stiffness of this potential. The origin of this power law dependence is not clear and, until this issue is resolved, the elastic parameters extracted for relatively thin films from any colloidal probe technique will remain suspect.

Discussion

In this work, DW-RICM was used to probe the structure and surface properties of PSS/PAH polyelectrolyte multilayer films, using colloidal probes, with special attention to surface properties of the probe beads. It was found that the surface of the carboxyl-modified beads, which were subsequently used as probes on the PSS/PAH films, is not hard but is comprised of an ∼10-nm-thick fuzzy polymer layer. Taking into account this fuzzy layer, the theoretically estimated heights for beads on bare glass were shown to agree well with the experiments reported here. For the PSS/PAH films, the measured heights correlate well with the thickness of the films, as estimated from OWLS measurements. By coating the beads and depositing polycations (PAH) or polyanions (PSS) as the outermost layer of the film, it was found that the relative charges of the beads and the topmost layer have an important role in determining the bead behavior. When the two were oppositely charged, the beads sunk into the films and hardly fluctuated at all. When both the beads and the topmost layer were negatively charged, the beads fluctuated in a quadratic potential and exhibited heights that were comparable to the expected film thickness. The height fluctuations of the beads were analyzed to yield the bead/film interaction potential and the effective film viscosity. Both the effective film viscosity and the stiffness of the potential decreased as the film thickness increased. In case of the stiffness of the potential, a power law dependence on the height was observed and a comparison with studies of colloidal beads on bare glass substrates and on hyaluronic acid cushions 27,28 showed that this power law dependence is universal with a common exponent.

Figure 1 .

 1 Figure 1. Experimental and theoretical equilibrium heights as a function of the ionic strength for two different types of beads hovering over a bare glass substrate. Experimental points were determined by the dual-wavelength-reflectance interference contrast microscopy (DW-RICM) technique (9) for carboxyl-modified beads from Duke and (O) for sulfate-modified beads from IDC. The lines correspond to the theoretical equilibrium heights calculated using eq 6 with the surface charge of the glass substrate given as σG ) 0.0007 C/m 2 . The thin solid line corresponds to IDC beads (σB ) 0.0001 mC/m 2 , no fuzzy layer), and the thick solid line corresponds to Duke beads (σB ) 0.0005 C/m 2 , fuzzy layer of 7 nm). Inset shows an enlarged view of the low ionic strength range (0-12 mM).

Figure 2 .

 2 Figure 2. Experimental potential energy profiles determined from the height fluctuations of the beads for 10-µm polystyrene spheres interacting with a bare glass slide at various ionic strengths: (A) (O) 0.5 mM, (0) 5 mM, and (3) 10 mM, and (B) (4) 50 mM. Each curve corresponds to one single bead. Solid curves are calculated from eq 5, using F B -FF ) 100 and R ) 5.2 µm. Changing the ionic strength changes the slope of the right-hand side of the theoretical profiles but only slightly. The slope is given by C, which depends on F B -FF, and R is the radius of the sphere.

Figure 3 .

 3 Figure 3. (A) Raw NTM signal obtained as a function of time during a PEI-(PSS/PAH)10 film buildup from a 0.15 M NaCl solution, as measured by the optical waveguide lightmode spectroscopy (OWLS) technique. Dotted lines indicate the polyelectrolyte injection steps. (B) Thickness obtained by OWLS as a function of the number of layers. Inset shows the evolution of the zeta potential of a PEI-(PSS/PAH) 5 film during its buildup. A linear fit was added to guide the eyes.

42

 42

Figure 4 .

 4 Figure 4. (A) Absolute height of BSA-coated beads on the PEI-(PSS/PAH)i-PSS multilayer films, as measured by the DW-RICM technique, as a function of the number of layer pairs (i ) 5, 10, 20, 30). Symbols represent averages over all the beads for a given number of layer pairs. The error bars represent the dispersion in the height data. (B) Standard deviations of the height fluctuation of the beads as a function of the number i of layer pairs in the film. Each symbol corresponds to one bead and represents the amplitude of fluctuation.

Figure 6 .Figure 7 .

 67 Figure 6. Potential energy profiles for 10-µm polystyrene spheres hovering over PEI-(PSS/PAH)i-PSS multilayer films made of an increasing number of bilayers (i ) 5, 10, 20, 30) in a 0.15 M NaCl solution (pH 6.5): (A) (O) i ) 5 and (0) i ) 11; (B) (3) i ) 20 and (O) i ) 30. (Note the difference in the height scale for the two graphs.)

Figure 8 .

 8 Figure 8. Scheme of the interaction of a colloidal probe with a PSS/ PAH multilayer film (PSS being the polyanion and PAH the polycation) on top of a PSS-ending film for a BSA-or PSS-coated bead and for a PAH-coated bead. For clarity, only a few bilayers have been represented; the gray zone represents many bilayers. The parameter h EQ is the equilibrium height of the bead measured by DW-RICM. In the left panel, the bead is hovering over the film, whereas in the right panel, it is stuck into the film.TABLE 1: Averages and Standard Deviations of the Height Fluctuations of the Beads on Top of PEI-(PSS/ PAH) i -PAH-Ending Films a Bead Coating (nm) number of layer pairs, i BSA PSS PAH 5 0.6 ( 0.5 (20) 0.4 ( 0.2 (10) 0.7 ( 0.5 (17) 10 0.7 ( 0.5 (12) 0.8 ( 0.7 (12) 1.0 ( 0.7 (12) 20 0.6 ( 0.5 (10) 0.5 ( 0.2 (11) 1.7 ( 0.7 (10) 30 0.5 ( 0.3 (9) 0.7 ( 0.6 (10) 1.1 ( 0.4 (10)

Figure 9 .

 9 Figure 9. (A) Effective surface viscosity as a function of the number of layer pairs i in the film for the same beads, and (B) evolution of the second derivative of the interaction potential V′′ (force constant characterizing resistance of the film against elastic deformation), as a function of the number of layer pairs i in the PEI-(PSS/PAH) i-PSS films for BSA-coated Duke beads.

Figure 10 .

 10 Figure 10. (A) Evolution of V′′ as a function of the thickness of the PEI-(PSS/PAH)i-PSS films as measured by DW-RICM. Each point represents the average of all the data (8-16 beads per film) obtained for films containing i ) 5, 10, 20, and 30 layer pairs. The standard deviations (in X and Y) are indicated by error bars. (B) Evolution of V′′ as a function of the thickness obtained for different types of samples and surfaces: (3) beads on glass (Schilling thesis); (4) Duke beads on glass (present work); (O) beads on a hyaluronic acid cushion (data from ref 27); (0) beads on a hyaluronic acid cushion (data from ref 28); (b) Duke beads on PSS-ending films (present work); and (9) Duke beads on PAH-ending films (present work).

B J. Phys. Chem. B Picart et al.

Acknowledgment. We thank Guillaume Marzoff for its technical help in the zeta potential measurements and J. Crocker for helpful discussions about surfaces of PS beads. C.P. is grateful to the INSERM and the Deutsche Forschungsgemeinschaft (DFG) for financial support through a research visit fellowship at the E22 Laboratory (GZ: 418 FRA 112/2/03). G.M. is indebted to the Norwegian Research Council for financial support (Grant No. 134674/140). This work was supported by the Deutsche Forschungsgemeinschaft (SFB 563 B12) and by the Fonds der Chemischen Industrie.