E Petersen
email: petersen@lma.cnrs-mrs.fr

P Guillemain

J Kergomard

T Colinot

The effect of the cutoff frequency on the sound production of a clarinet-like instrument

Effect of cutoff frequency on woodwinds

The input impedance of woodwind instruments is characterized by at least two bands due to the lattice of open toneholes, a stop band at low frequencies and a pass band at higher frequencies where the acoustic energy is able to propagate past the first open tonehole and into the lattice. The cutoff frequency that separates these two bands is an approximate value that is determined by the geometry of the lattice of open toneholes. It is expected that the frequency at which the stop band transitions to the pass band affects the sound produced by the instrument, but it is not known how this frequency affects the competition between self-sustained oscillation and radiation. A simplified model of a clarinet-like resonator is conceived such that the first input impedance peak and the cutoff frequency can be independently chosen.

Experimental prototypes are built and their measured impedance are compared with the simulations. Resonators with very similar low frequency behavior, but very different cutoff frequencies, are then compared using digital synthesis to evaluate the influence of the cutoff frequency on sound production. The cutoff frequency impacts the synthesized pressure and acoustic volume velocity in the mouthpiece, particularly regarding the spectral content at high frequencies.

I. INTRODUCTION

Reed instruments are typically characterized by two main components: the reed and flow rate as a nonlinear excitation mechanism and the main bore of the instrument as a passive acoustic resonator that responds to and influences the excitation mechanism. Under normal playing conditions the resonator responds to an injection of acoustical energy from the exciter by radiating sound, but also facilitates the self-sustained oscillation of the exciter. [START_REF] Benade | Fundamentals of Musical Acoustics[END_REF] The input impedance, determined by the specific geometry of the bore and tonehole network, is a standard way to characterize the acoustic behavior of the resonator. A resonator with a series of open toneholes exhibits a well-known behavior called the tonehole lattice cutoff frequency, below which a lattice wave sampled at discrete points is evanescent, while above, a lattice wave can propagate further into the bore. [START_REF] Brillouin | Wave propagation in periodic structures[END_REF][START_REF] Benade | On the mathematical theory of woodwind finger holes[END_REF] Below the cutoff frequency the resonator has an effective length that is approximately the length between the input and the first open tonehole. In the case of a cylindrical bore, due to constructive interference of the reflected waves, the input impedance is characterized by well-defined, nearly harmonically spaced maxima expected of a quarter-wave resonator. Above the cutoff frequency, the effective length of the resonator is a complicated function of frequency, so there is less organized constructive and destructive interference, and the impedance is characterized by attenuated peaks that are not harmonically related to the first impedance peak.

It is often assumed that the cutoff frequency behavior is related to the perceptual characteristics of the sound produced by an instrument. [START_REF] Benade | Fundamentals of Musical Acoustics[END_REF][START_REF] Wolfe | Cutoff frequencies and cross fingerings in baroque, classical, and modern flutes[END_REF] However, it is not known precisely in what way the cutoff frequency affects the production and radiation of sound. Therefore, it is worth investigating how the frequency at which cutoff occurs, and the severity of the cutoff behavior on the input impedance, could impact the self-sustained oscillation and the temporal and spectral characteristics of the produced sound. Similarly, the cutoff frequency may impact characteristics of sound radiation such as directivity and total radiated power with respect to frequency. The complicated behavior associated with sound production and radiation are linked to one another through the competition between the acoustical energy that is retained within the resonator, which facilitates production, and that which is radiated into the surrounding environment.

The goal of this article is to evaluate how the cutoff frequency impacts different aspects of sound production and timbre features. To reduce the number of parameters that influence the acoustical behavior of a real instrument, the current work (except Sect. III C) considers resonators composed of a cylindrical main bore terminated by a lattice of geometrically regularly spaced toneholes. This simplified version of a clarinet resonator allows direct analysis of the effects of the cutoff frequency without added complications such as irregular tonehole lattices, conical segments, and undercut toneholes, as is the case for actual woodwind instruments.

Section II contains a review of the Transfer Matrix Method (TMM) [START_REF] Plitnik | Numerical method for calculating input impedances of the oboe[END_REF][START_REF] Causse | Input impedance of brass musical instrumentscomparison between experiment and numerical models[END_REF][START_REF] Keefe | Woodwind air column models[END_REF] for simulating the input impedance of a pipe, including the related Transfer Matrix Method with external Interactions (TMMI), [START_REF] Lefebvre | External tonehole interactions in woodwind instruments[END_REF] which accounts for mutual radiation impedance when there are more than a single radiating aperture (toneholes and open termination of the main bore).

Section III details the design and experimental verification of a cylinder and tonehole lattice resonator for which it is possible to specify both the frequency of the first impedance peak and the cutoff frequency. The effects on sound production, for three different resonators, are compared using digital synthesis in Section IV. The main focus of this section is to characterize the internal sound field, as well as the transfer function between internal and radiated time averaged intensity. Examples of synthesized pressure and velocity waveforms are furnished as supplementary materials online. Conclusions and proposals for future work are covered in Section V.

II. INPUT IMPEDANCE CALCULATIONS VIA THE TRANSFER MATRIX METHOD WITH EXTERNAL INTERACTIONS

While the TMM is elegant, simple, and efficient, it ignores the effects of external interactions due to toneholes radiating simultaneously into the same space. A variant of the TMM is the Transfer Matrix Method with external Interactions (TMMI). [START_REF] Lefebvre | External tonehole interactions in woodwind instruments[END_REF][START_REF] Chaigne | Acoustics of Musical Instruments[END_REF] This method accounts for propagation within the main bore while also accounting for the mutual impedance between apertures (N toneholes and the bore end) that are assumed to radiate into the same space, and therefore have a mutual influence.

The TMMI allows the calculation of the acoustic pressure vector P and the acoustic flow vector U through each aperture of the resonator. They are related to the source term by

U = [I + Y(Z + B)] -1 U s P = Y -1 (U s -U ), (1)
where U s is the source vector, of dimension N + 1 and with only one non zero element:

U s n = δ n,1
. I is the identity matrix, and Y is a tri-diagonal admittance matrix, derived from the 2x2 transfer matrices of the main bore between toneholes. A cylinder is added to account for the section of pipe separating the mouthpiece from the first open tonehole of a real instrument. B is the impedance matrix for the acoustic mass of each tonehole: the definition includes many equations, and can be found in the appendix of Lefebvre et al. [START_REF] Lefebvre | External tonehole interactions in woodwind instruments[END_REF] Z is the radiation impedance matrix. Along the diagonal, it accounts for the classical self radiation impedance of each aperture, and for the mutual radiation impedance between two apertures, it is assumed that they radiate as a monopole in an half infinite space. [START_REF] Lefebvre | External tonehole interactions in woodwind instruments[END_REF][START_REF] Pritchard | Mutual acoustic impedance between radiators in an infinite rigid plane[END_REF][START_REF] Rucz | A finite element approach for the calculation of self and mutual radiation impedances of resonators[END_REF] The input impedance is calculated from the knowledge of the quantities P 1 and U 1 .

At low frequencies the monopoles can be assumed to radiate into a full space, however, the uncertainty of the mutual loading approximation is greater than the factor of two between half space and full space radiation.

The TMMI impedance calculation yields subtly different results when compared with that of the standard TMM, as seen in Fig. 4. The two methods are in good agreement at low frequencies. At higher frequencies, the TMMI reduces the height of the resonances and reduces the depth of the anti-resonances, which can be interpreted as a reduction in the standing wave ratio of the resonator and an increase of the radiation losses in the system.

Knowing P and U at the location of the toneholes allows for the calculation of external values such as directivity and the pressure waveform at a given external location, topics that are outside the scope of this paper but useful for future work.

III. RESONATOR DESIGN

A. Periodic lattice of holes and resulting cutoff frequency

The basic resonator considered in the current work is designed to have a desired first impedance resonance frequency at f 1 and desired cutoff frequency at f c that can be manipulated independently from one another by varying the geometry. This is achieved by concatenating a cylinder of length L, which largely determines f 1 , to a lattice of geometrically regular toneholes that imposes the cutoff behavior at f c , as shown in Fig. 1(b). This is similar to the resonator design in the dissertation by Worman. [START_REF] Worman | Self-sustained nonlinear oscillations of medium amplitude in clarinet like systems[END_REF] While not explained in great detail, it is important to note that simple geometric calculations must include low frequency length corrections, defined in Section III B, due to the lattice of toneholes as well as the more familiar radiation length correction at each open tonehole.

The analytic formulations used in this method allow only for the frequencies f 1 and f c to be independently determined (see Eq. ( 2) and ( 4)). The second impedance peak of two resonators with different cutoff frequencies occur at nearly the same frequency because there is very low inharmonicity due to the cylindrical portion of the resonator, and the use of identical toneholes. Dimensions and acoustic features of sample resonators are provided in Appendix A.

B. Geometrically regular lattice

A geometrically (and therefore also acoustically) regular lattice is constructed by repeating the basic cell shown in Fig. all identical and their number infinite, the global cutoff frequency, which can be estimated from impedance measurements or simulations, will be equal to the local cutoff frequency.

The local cutoff frequency is approximated by the Helmholtz resonance of a single cell whose main bore is closed at the extremities, 13

ω c = 2πf c ≈ c 1 c 2 m h C a + 1/3 , (2)
where C a = 2 πa 2 /ρc 2 is the acoustic compliance of the main pipe, m h = ρh/πb 2 is the acoustic mass of the tonehole, and a, , b and h are the geometric dimensions of the cell shown in Fig. 1 (b).

The exact local cutoff frequency for a cell in an infinite lattice can be calculated from

c cot(k ) = ωm h C a , (3)
which is derived by solving the pressure and flow transfer matrix equations assuming either Dirichlet or Neumann boundary conditions within the main bore. [START_REF] Chaigne | Acoustics of Musical Instruments[END_REF] This exact form differs by less than 1% from the approximate form given in Eq. ( 2) for an equivalent Helmholtz resonator of dimensions used in the current article. To impose a desired local cutoff frequency it is sufficient to find a combination of dimensions a, , b, and h, accounting for the length corrections due to the inertial effect of adjacent fluid and the radiation impedance of the tonehole, that result in the desired resonance. Therefore, it is possible to design a lattice to have a desired global cutoff frequency by manipulating the dimensions of the constituent cells to have the same local cutoff frequency. To impose the frequency of the first impedance peak f 1 , a cylinder of length L is concatenated to the lattice and treated like a quarter-wave resonator,

f 1 = c 4L a , (4)
where L a = L+∆ is an effective acoustic length that accounts for the presence of a lattice of open toneholes. The lattice modifies the radiation impedance of the cylinder as a mass-like term, and is treated as a length correction

∆ = 1 + 2 c 2 C a m h , (5)
which must be included in addition to the physical length of the main cylinder when determining the appropriate length that results in the desired frequency f 1 . 1 A strong cutoff frequency behavior is observed with as few as three open toneholes, however, the frequency at which cutoff occurs tends to be somewhat higher than that calculated in Eq. ( 2) when the lattice is comprised of a small number of holes, see Fig. constructed physically (dimensions provided in Appendix A) and compared with simulation in Section III D.

C. Acoustic regularity

Although the analysis based on digital synthesis in Section IV A is limited to a tonehole network composed of geometrically regular cells, Moers and Kergomard [START_REF] Moers | On the cutoff frequency of clarinet-like instruments. geometrical versus acoustical regularity[END_REF] show that it is possible to define an acoustic regularity for which each constituent cell has the same Helmholtz resonance, but not necessarily the same geometry. In that article, they consider the inverse problem of determining the division of an acoustically regular lattice that results in the expected global cutoff frequency. In the present paper, the problem is solved directly by designing each cell to be geometrically different, except the bore radius, but have a common Helmholtz resonance, and therefore have an acoustic regularity that results in the desired global cutoff frequency. The individual cells have identical main bore internal radii a, and the tonehole radii and heights b, h taken from the clarinet dimensions provided in Appendix A1 of Moers and Kergomard. [START_REF] Moers | On the cutoff frequency of clarinet-like instruments. geometrical versus acoustical regularity[END_REF] Once a desired cutoff frequency is set, the length of pipe on each side of the symmetric cells is deduced by Eq. ( 2). This acoustically regular lattice can then be constructed with a leading pipe to create the desired first resonance peak described in Section III B. Figure 3 shows that a global cutoff frequency is attained using an acoustically (but not geometrically) regular lattice. As with a geometrically regular lattice, the low frequency behavior of the acoustically regular lattice is largely unchanged between resonators with different cutoff frequencies. Specifically, the height and frequency of the first two impedance peaks are negligibly different for different resonators. This similarity between geometrically and acoustically regular resonators suggests that the choice to use geometrically regular resonators in this work is a reasonable simplification of real instruments, which exhibit substantial acoustic regularity. Next, the effect of the order of acoustically regular cells is examined by calculating the input impedance for randomized permutations of the cell locations. Figure 3 demonstrates that the order of the cells in the lattice does not greatly influence the frequency at which cutoff occurs and, as above, has a negligible influence on the height and frequency of the first two impedance peaks. This is a result of work by Fürstenberg 14 demonstrating exceptions to the nonexistence of pass bands in an infinite one-dimensional random medium. [START_REF] Moers | On the cutoff frequency of clarinet-like instruments. geometrical versus acoustical regularity[END_REF][START_REF] Anderson | Absense of diffusion in certain random lattices[END_REF] It is worth noting that, above the cutoff frequency and up to approximately 2000 Hz, the impedance of each resonator, regardless of the cell order, follows similar and slowly diverging paths. This is unexpected and suggests that, for frequencies just above cutoff, the input impedance behavior is determined by global features of the lattice. This is curious, but less relevant to the case of real clarinets, which typically have a fairly regular progression from small, closely spaced holes near the top of the instrument, to large and distantly spaced holes near the bell.

D. Experimental results of a simple resonator

The simplified pipe-lattice resonator described in the previous section was tested experimentally by measuring the input impedance of two polyamide 6 tubes where the lengths L and tonehole network geometries are designed to have first impedance resonances at 170 Hz and cutoff frequencies at 1000 Hz and 2000 Hz (R 1.0 , R 2.0 ), according to Eq. ( 2) and ( 4).

The dimensions are summarized in Appendix A. The input impedance was measured using an impedance measurement device developed at CTTM. [START_REF] Roux | A new impedance sensor for industrial applications[END_REF] Figure 4 shows that the target first impedance resonance and cutoff frequencies are attained by both simulation and measurements. This demonstrates the relevance of the model and accuracy of the measurement methodology. The better agreement between simulation and measurement for the

f c = 1000
Hz resonator is a consequence of the greater distance between holes, and not a general trend related to the cutoff frequency of a resonator. A second stop band in the impedance of R 1.0 is visible around 2500 Hz both in the simulation and measurement. The discrepancy between simulation and measurements at high frequencies may be due to slight imprecision of the physical geometries of the experimental pipes, inaccuracies of the measurement hardware, or higher order phenomena not included in the simulation. These differences do not affect the conclusions, which are based on the location of the cutoff frequency and not the precise high frequency behavior.

IV. EFFECTS OF CUTOFF FREQUENCY USING DIGITAL SYNTHESIS

Digital sound synthesis is a convenient method to evaluate the playing characteristics of a real or hypothetical resonator, and to compare two or more different resonators, because it can synthesize time domain acoustic pressure and volume velocity waveforms inside the mouthpiece for a wide range of control parameters. One can then compute descriptors of the waveform that are used to quantify playing characteristics, with respect to the control parameters, such as playing frequency, spectral centroid, and attack time. These playing descriptors are functions of the dimensionless control parameters excitation pressure γ, and embouchure ζ (defined below), but also are dependent on the reed parameters and input impedance of the resonator. [START_REF] Guillemain | A digital synthesis model of double-reed wind instruments[END_REF][START_REF] Coyle | Predicting playing frequencies for clarinets: A comparison between numerical simulations and simplified analytical formulas[END_REF] It is assumed that γ and ζ control timbre features of a given instrument, and are therefore useful parameters to evaluate the connection between sound production and sound descriptors. It is important to note that synthesized waveforms do not necessarily match those of a real instrument because the values of physical parameters, particularly regarding the reed dynamics, are not precisely known. However, digital synthesis is a reasonable tool to compare the behavior of different resonators. The following sections are based on synthesis for resonators R 1.0 , R 1.5 and R 2.0 as defined in Section III B.

A. Digital synthesis model

The digital synthesis model used in this work combines the discretization scheme of Guillemain, 17 with the use of the reflection function to model the response of the resonator. [START_REF] Gazengel | Time domain simulation of single-reed wind instrument. From the measured input impedance to the synthesis signal. Where are the traps?[END_REF][START_REF] Guillemain | Real-time synthesis of clarinet-like instruments using digital impedance models[END_REF] Its purpose is to solve, in the domain of time sampled signals, a system of three equations and three unknowns giving the values at sample number n as functions of the past known values of the three variables. For that purpose, we start with the continuous time model describing the resonator, the reed dynamics, and the nonlinear relationship between pressure and flow in the mouthpiece.

The continuous time reed instrument model used in this work is comprised of three main equations (Eq. ( 7), (9), and ( 10)). The three variables are the dimensionless pressure, flow rate, and reed displacement, respectively

                     p(t) = p(t)/p M ũ(t) = Z c u(t)/p M , x(t) = x(t)/H, (6)
where Z c is the characteristic impedance, p M is the pressure difference needed to close the reed completely in a static situation, and H is the distance between the tip of the reed and the mouthpiece lay at equilibrium. The value x = -1 corresponds to the closure of the reed.

The first equation corresponds to the movement of the reed which consists of a damped spring-mass driven by the pressure differential between the mouth of the musician and the mouthpiece

           1 ω 2 r d 2 x(t) dt 2 + qr ωr dx(t) dt + x(t) = p(t) -γ if x > -1, dx(t) dt = 0 if x < -1, (7)
where ω r is the resonance angular frequency of the reed, q r is the damping coefficient, and the parameter γ = p m /p M is the dimensionless pressure in the mouth of the musician.

Following values similar to other articles, [START_REF] Coyle | Predicting playing frequencies for clarinets: A comparison between numerical simulations and simplified analytical formulas[END_REF] the synthesis results presented in this paper assume ω r = 2πf r = 2π1500 (rad/s) and q r = 0.4 (dimensionless). The second line of Eq. ( 7) corresponds to the limitation of the reed displacement by the mouthpiece lay. There exist more refined models of the reed that include dynamical behavior and the collision of the reed and the lay. [START_REF] Avanzini | Modelling the mechanical resonse of the reed-mouthpiecelip system of a clarinet. part i. a one-dimensional distributed model[END_REF][START_REF] Walstijn | Modelling the mechanical resonse of the reed-mouthpiecelip system of a clarinet. part ii: A lumped model approximation[END_REF][START_REF] Bilbao | Numerical modeling of collisions in musical instruments[END_REF] These models, however, are unnecessary for the current work which aims to efficiently compare the response of different resonators, not predict absolute behavior. The nonlinear relationship relating the input flow u(t) as a function of the pressure differential and the reed position is

ũ(t) = u b (t) + u r (t), (8)
where

           u b (t) = ζ(1 + x(t))sign(γ -p(t)) |γ -p(t)|, u r (t) = λ c dx(t) dt . (9)
The variable ζ = Z c wH 2/ρp M is the dimensionless embouchure parameter, depending on the width w and height H of the reed channel cross section when the reed is at rest, and is proportional to the square root of H because p M is a function of H. The total flow rate is the sum of the flow computed from the stationary Bernoulli equation u b and the flow due to the movement of the reed u r , depending on the speed of the reed and a parameter λ c that characterizes the effective vibrating surface of the reed. [START_REF] Dalmont | Influence of the reed flow on the intonation of the clarinet[END_REF] The third principle equation represents the passive role of the resonator and is expressed by its input impedance. Following McIntyre 26 and Gazengel, 19 a straightforward change of variables in the Fourier domain, using wave variables rather than Kirchhoff variables, leads to the expression of the backward pressure wave at time t in the temporal domain

p -(t) = (r * p + )(t), (10)
where p -(t) = (p(t)ũ(t))/2 and p + (t) = (p(t) + ũ(t))/2 are the backward and forward pressure waves in the resonator. The operator * represents continuous time convolution and r is the reflection function of the resonator, obtained from the input impedance by

r = F -1 Z(ω) -Z c Z(ω) + Z c , (11)
where F -1 is the inverse Fourier transform. The advantage of using the reflection function rather than the impulse response is that it is assumed to decrease much faster in time, and can therefore be truncated to save computation time.

Discrete model

The continuous time model of the complete instrument is discretized so that temporal synthesis can be simulated by a computer. Therefore, a strictly causal formulation must be found so that every variable at a given time sample can be computed from the previous values of all variables. Equation ( 7) is discretized using the finite difference scheme, 20 such that

x[n] = b 1r p[n -1] + a 1r x[n -1] + a 2r x[n -2], (12)
where the tilde notation is omitted for dimensionless variables in the discrete formulation (p[n] is the n th sample of the dimensionless pressure signal). The coefficients b 1r , a 1r and a 2r are given in Guillemain et al. [START_REF] Guillemain | Real-time synthesis of clarinet-like instruments using digital impedance models[END_REF] In order to ensure the limitation of the reed displacement, the following condition is added

if x[n] < -1, then x[n] = x[n -1]. ( 13
)
Once the position of the reed at sample n is known, the reed flow is computed using the 253 finite difference scheme

254 u r [n] = λ(x[n] -x[n -1]). ( 14
)
The parameter λ = λ c f s is fixed at the value -0.7 as in Coyle et al. [START_REF] Coyle | Predicting playing frequencies for clarinets: A comparison between numerical simulations and simplified analytical formulas[END_REF] based on the measurements available in the literature. [START_REF] Dalmont | Influence of the reed flow on the intonation of the clarinet[END_REF][START_REF] Mcintyre | On the oscillations of musical instruments[END_REF][START_REF] Dalmont | Some aspects of tuning and clean intonation in reed instruments[END_REF][START_REF] Dalmont | Nonlinear characteristics of single-reed instruments: Quasistatic volume flow and reed opening measurements[END_REF] The third equation involving the reflection function can be reformulated using the discrete convolution product

(r * p + )[n] = ∞ i=0 r[i]p + [n -i], (15)
where r is the discrete version of the reflection function, which is causal. For the numerical application, the reflection function is deduced from the computed input impedance, which is windowed down to the characterstic impendance by 8 kHz, well below the first nonplanar mode and following recommendations from Gazengel. [START_REF] Gazengel | Time domain simulation of single-reed wind instrument. From the measured input impedance to the synthesis signal. Where are the traps?[END_REF] Above this frequency the impedance is completed by the characteristic impedance until the Nyquist frequency, before computing the reflection function by inverse Fourier transform. Only the first D = 3000 elements of the reflection function are kept for the synthesis, chosen to be long enough so that it does not disturb the low frequency content of the resonator's response. Specifically, the impulse response is a sum of exponentially damped sinusoids and this choice of D ensures that the time response of the lower mode, with the highest quality factor, is not truncated. The backward pressure wave corresponds is written as

p -[n] = D i=0 r[i]p + [n -i]. (16)
Separating this equation into a strictly causal and an instantaneous part and substituting for p and u, p

-[n] becomes 1 2 (p[n] -u[n]) = 1 2 r[0] (p[n] + u[n]) + 1 2 D i=1 r[i](p[n -i] + u[n -i]), (17)
where r[0] is the first sample of the discrete reflection function. This yields p[n] as a function of u[n] and the past of p and u p

[n] (1 -r[0]) = u[n](1 + r[0]) + D i=1 r[i](p[n -i] + u[n -i]), (18)
The final pressure expression is as follows:

p[n] = 1 + r[0] 1 -r[0] u[n] + 1 1 -r[0] D i=1 r[i](p[n -i] + u[n -i]). (19)
Using Eq. ( 19), it is possible to implement the temporal synthesis scheme described in Guillemain et al., [START_REF] Guillemain | Real-time synthesis of clarinet-like instruments using digital impedance models[END_REF] because it is in the form p[n] = b co u[n] + V where V is a known quantity at sample n.

B. Results

Synthesized time and frequency domain waveforms of the internal pressure and flow for resonators R 1.0 , R 1.5 , and R 2.0 are shown in Fig. 5, and available to listen to online [START_REF] Petersen | Synthesized mouthpiece waveforms as audio for pressure SuppPubmm1-3, acoustic flow SuppPubmm4-6, and time derivative of pressure plus flow SuppPubmm7-9 (an approximation of raidated sound), for resonators with[END_REF] from the three resonators have been arbitrarily shifted in time by less than one cycle, so that they overlay one another, facilitating visual comparison. The timing difference of the pressure and flow signals is small, and may result from a small change in playing frequency.

There is a subtle difference between both the pressure and volume velocity for resonators R 1.0 , R 1.5 , and R 2.0 . The biggest visual difference is in the volume velocity waveform, in which resonator R 1.0 has larger amplitude and lower frequency rippling when compared with resonators R 1.5 and R 2.0 . A simple interpretation, examined in Section IV B 1, is that there is more energy in the high frequency harmonics in the waveforms computed from resonators with higher cutoff frequencies. The height of the peaks that are the main feature in the velocity waveform, corresponding to the closing of the reed, also changes between the different resonators.

Frequency domain characteristics

Frequency domain characteristics are computed directly from the steady state portion of the synthesized waveforms p and u inside the mouthpiece, discarding the attack transient.

The playing frequency, f p , of resonators R 1.0 , R 1.5 , and R 2.0 , as a function of γ with constant ζ = 0.45, are displayed in Fig. 6. The values are presented in cents, relative to the frequency of the first impedance peak, such that

f cents = 1200log 2 f p f 1 . ( 20
)
It is seen that resonators with higher cutoff frequencies also have playing frequencies that are higher by a few cents. In the beating reed regime, approximately γ > 0.5, the difference between resonators R 1.0 and R 2.0 is greater than five cents, therefore possible large enough to be perceived by a human. [START_REF] Shower | Differential pitch sensitivity of the ear[END_REF] This is not due only to the difference in inharmonicity between the first and second peaks, verified through analytical formulations, [START_REF] Coyle | Predicting playing frequencies for clarinets: A comparison between numerical simulations and simplified analytical formulas[END_REF] the inharmonicity of these resonators results in a maximum of 2.5 cents difference between resonators R 1.0 and R 2.0 . However, the tonehole lattice also changes the inharmonicity of higher resonances of the input impedance below the cutoff frequency. This could influence the playing frequency, particularly at high values of γ and ζ.

To define a single value descriptor, the spectral centroid 30 is calculated as

centroid = M m=1 f (m)G(m) M m=1 G(m) , (21)
where G(m) is the single sided power spectrum corresponding to the Fast Fourier Transform frequency bins f (m) up to f (M ) = 8 kHz, at which point the assumptions made in the impedance simulation regarding thermo-viscous losses and radiation are no longer valid.

This definition results in a single number for a given spectrum with units of Hz, which follows the 'center of mass' interpretation of the spectral content. The spectral centroid values, normalized by the playing frequency at each combination of control parameter and for each resonator, are shown in Fig. 7. The spectral centroid is higher for resonators with higher cutoff frequencies, even after adjusting for the playing frequency. A peak detection scheme was applied to the spectra of the internal pressure where the square and circular data markers correspond to the odd and even harmonics. The results are plotted in Fig. 8 for resonators of type R 1.0 , R 1.5 and R 2.0 in top, middle, and bottom, respectively, and vertical lines denote the cutoff frequency of the resonators used in each figure.

For all resonators shown here, regardless of their cutoff frequency, the odd harmonics that are below the cutoff frequency have approximately the same amplitude climbing the semitone scale. Around and above the cutoff frequency the amplitudes of the odd harmonics decreases, and the strong organization disappears as also seen in Fig. 5, which implies that the complicated input impedance above cutoff frequency has a randomizing effect on the production of odd harmonics. This can be interpreted as the system no longer behaving as a quarter-wave resonator at these frequencies. Additionally, the even harmonics generally grow in amplitude as a function of frequency, and display a marked increase in amplitude for frequencies higher than the cutoff frequency. This implies that below the cutoff frequency, the spectral content of the waveforms produced by resonators with different cutoff frequencies is relatively similar, but that the high frequency spectra, and hence the perceived sound, can be quite different above the cutoff frequency. Specifically, the plots show more energy in even harmonics above the cutoff frequency.

The same 60 resonators are compared in terms of the time averaged intensity at input of the resonators and the the time averaged intensity summed over each radiating aperture, shown in Fig. 9. The time averaged intensity is defined as

< I in > = 1 2 Re[P 0 U * 0 ], (22)
< I rad > = M n=1 1 2 Re [P n U * n ] , (23)
where M = N + 1 is the number of toneholes N plus the open end of the resonator. This provides a direct comparison between the energy that remains inside the resonator and contributes to the auto-oscillation and the total energy that exits the resonator. A comparison of the left and right panels shows that the time averaged intensity of the fundamental frequency at the input is on the order of 40 dB higher than the output, suggesting that the resonators are inefficient sound sources, regardless of cutoff frequency. Note that the level of the fundamental frequency input intensity is not horizontal as it is for pressure in the mouthpiece. The output intensity of the fundamental frequency increases proportional to ω 2 . This is consistent with known monopole radiation efficiency and bolsters the claim that, below the cutoff frequency, the main radiating source can be treated as a monopole at the location of the first open tonehole. Above the cutoff frequency the time averaged intensity is approximately the same for the input and output. This implies that the resonators are more efficient sources above the cutoff frequency.

It is concluded that the cutoff frequency affects the spectral content of waveforms within the mouthpiece by both demarcating the frequency at which even harmonics develop, as well as shifting the spectral centroid higher for resonators with higher cutoff frequencies. follows the curve of the first half of a Hann window. In the current work the duration of the transition is 201 samples, corresponding to about five milliseconds for a sampling rate of 44100 samples per second. To define the attack time, the time domain pressure signal is decomposed into its constituent harmonics, each of which evolves as a function of time. To include frequencies above the cutoff frequency of all resonators, 16 harmonics are considered.

The harmonics are calculated by an inverse Fourier transform on windowed versions of the spectrum around each harmonic. This estimates the instantaneous amplitude and frequency of each harmonic for the duration of the signal, including during the attack transient. The sum of the squared amplitude of these bands is proportional to the total energy in the signal at a given instant, and the energy in the steady state portion of the signal can be determined by choosing this value well into the steady state portion of the signal. The beginning of the attack is determined when the signal first passes a threshold that is a specific, arbitrary fraction of the final value, and the end of the signal attack is defined as the last time that the sum is outside a threshold from the steady state value. Onset threshold values of 5 and 10 percent, and ending threshold values of 90 and 95 percent have been investigated. Figure 10 shows a typical pressure waveform onset (in this case for R 1.0 ), with the sum of the first 16 harmonics, normalized to have a maximum value of unity, superimposed. The threshold at 5% and 95% are marked in bold lines. It is found in this work that the attack time is linearly related to the choice of threshold, and therefore any of these four choices (5-90, 5-95, 10-90, 10-95) is reasonable for comparing the attack time of different resonators. Figure 11 shows the attack time normalized by the period of the playing frequency as a function of γ with constant ζ = 0.45. Thresholds of 5% and 95% are used to define the onset and saturation of the waveform. The attack time of resonators R 1.0 , R 1.5 , and R 2.0 are sufficiently similar as to be difficult to discern. The difference between attack times of different resonators is less than 10% of a period for γ > 0.5, and up to 2.5 periods for γ < 0.5. However, for low blowing pressures the attack time is of the order of tens of periods, so the difference of 2.5 periods is still a small percentage of the total attack time. It is concluded that the cutoff frequency does not strongly impact the attack time of a given resonator which, in a musical context, may be related to the ease of playing.

V. CONCLUSION

The acoustic resonators developed in this work are adapted for studying the effect of the cutoff frequency on sound production and, potentially, radiation. The simple nature of these resonators, for which it is possible to independently vary the frequencies of both the first impedance peak and the cutoff frequency, is particularly useful because the low frequency behavior (below cutoff) is largely the same for resonators with very different high frequency behavior. The data resulting from either digital sound synthesis or experimental measurements can then be used to directly assess the effect of the cutoff frequency on the sound production and resulting external sound field.

It is found that the cutoff frequency does affect the synthesized waveform within the mouthpiece, particularly the spectral characteristics. Resonators with a higher cutoff frequency have higher playing frequencies, and higher spectral centroids, when compared with resonators with lower cutoff frequencies. Furthermore, even harmonics tend to be excited above the cutoff frequency of a given resonator. However, the differences between internal waveforms in resonators with different cutoff frequencies is subtle. Furthermore, the cutoff frequency of a resonator is not found to have a significant influence on the attack time,

showing that one measure of the "playability" of a clarinet-like instrument relies more on the low frequency response of the resonator. This implies that an instrument maker may be able to make modifications to the geometry of an instrument that results in a different cutoff frequency, without the risk of greatly changing the sound production qualities of the instrument.

In contrast to the relatively small effect of the cutoff frequency on sound production, the cutoff frequency has a larger impact on the high frequency spectral characteristics and radiation. The radiation efficiency, calculated as the ratio of the time averaged intensity at the input of the instrument with that summed over the radiating apertures, is higher above the cutoff frequency than below. Above the cutoff frequency the resonators produce more even harmonics and radiate more efficiently, implying that the cutoff frequency could have a substantial impact on the perceived timbre of different instruments, or different notes on the same instrument.

This, along with the small impact on sound production, indicates that the cutoff frequency may be used to change the timbre characteristics of an instrument without altering its sound production qualities.

Future work includes extending the synthesis analysis to the external sound field to evaluate the effect of the cutoff frequency on radiated sound, for which Fig. 9 provides an initial answer. This includes directivity of the source due to multiple radiating apertures. A characterization of the external sound field can then be compared with the internal sound field to evaluate the competition between sound production and radiation of clarinet-like musical instruments.

FIG. 1 .

 1 FIG. 1. Pipe and lattice resonator (a) constructed from identical cells (b).

2 .FIG. 2 .

 22 FIG. 2. Simulated impedance curves of resonators with f 1 = 170 Hz, f c = 1350 Hz, and varying

FIG. 3 .FIG. 4 .

 34 FIG. 3. Multiple simulated input impedances for resonators with acoustically regular lattices and

FIG. 5 .

 5 FIG. 5. Synthesized (dimensionless) pressure (top) and flow (bottom) inside mouthpiece for res-

FIG. 6 .

 6 FIG.6. Synthesized playing frequency in cents, normalized by the first impedance peak frequency,

FIG. 8 .

 8 FIG. 8. Synthesized (ζ = 0.45, γ = 0.55) spectra of the mouthpiece pressure for resonators

FIG. 9 .FIG. 10 .FIG. 11 .

 91011 FIG. 9. Time averaged intensity passing through the entrance of the resonator (left) and the

4 10 3 12 4 12 5 20 6 21 7

 1012122021 holes; thick dashed black: 10 holes; thick solid black: 20 holes). The cutoff frequency f c = 1350 Hz was chosen to occur at an impedance minimum so that the effect on the next maximum can be more easily distinguished. Lattices with 10 holes exhibit very strong cutoff behavior. . . . . . . . . . . . . . . . . . . . . . . . . Multiple simulated input impedances for resonators with acoustically regular lattices and random cell orders (50 permutations). The target frequencies are f 1 = 170 Hz and f c = 1350 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Input impedance of R 1.0 (a) and R 2.0 (b). Grey: TMM simulation; dashed black: TMMI simulation; solid black: experimental. . . . . . . . . . . . . . . . . . . . . . . Synthesized (dimensionless) pressure (top) and flow (bottom) inside mouthpiece for resonators R 1.0 (solid black), R 1.5 (dash-dot gray), and R 2.0 (dashed black) in the time (left) and frequency (right) domains. Control parameters are γ = ζ = 0.45, and reed resonance f r = 1500 Hz. . . . . . . . . . . . . . . . . . . . . . . Synthesized playing frequency in cents, normalized by the first impedance peak frequency, for resonators R 1.0 , R 1.5 , and R 2.0 with solid black, solid grey, dashed black lines, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Synthesized spectral centroid, normalized by the playing frequency, for resonators R 1.0 , R 1.5 , and R 2.0 with solid black, solid grey, dashed black lines, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

  in which FIG.7. Synthesized spectral centroid, normalized by the playing frequency, for resonators R 1.0 , R 1.5 , and R 2.0 with solid black, solid grey, dashed black lines, respectively.

	Centroid [normalized]	1.1 1.2 1.3
		1
		0.4	0.5	0.6	0.7	0.8	0.9

TABLE II .

 II Frequencies of the first two impedance peaks and inharmonicity of resonators R 1.0 , R 1.5 , and R 2.0 . Simulated impedance curves of resonators with f 1 = 170 Hz, f c = 1350 Hz, and varying number of toneholes (thin solid black: cylinder; thick solid grey:

	Resonator	f c [Hz]	f 1 [Hz]	f 2 [Hz]	inharm [cents]
	R 1.0	1000	170	511.6	5.4
	R 1.5	1500	170	513.0	10.2
	R 2.0	2000	170	513.1	10.5

ACKNOWLEDGMENTS

This work has been partly supported by the french Agence Nationale de la Recherche (ANR16-LCV2-0007-01 Liamfi project). The authors would also like to thank Patrick Sanchez and Guy Rabau for their help with the impedance measurements.

Two resonators were realized to experimentally verify the feasibility of designing resonators with independently chosen first input impedance peaks and cutoff frequencies. The resonators were made by drilling holes in polyamide 6 tubes, the dimensions of which are summarized in Table I. The input impedance was then measured using the impedance measurement device developed at CTTM in hemi-anechoic conditions. [START_REF] Roux | A new impedance sensor for industrial applications[END_REF] Table II summarizes the first two impedance peak frequencies of resonators R 1.0 , R 1.5 , and R 2.0 used for the digital synthesis. The inharmonicity, in cents, is defined as


Erik Alan Petersen
email: petersen@lma.cnrs-mrs.fr

Romane Rosser

External sound field simulations and measurements of woodwind resonators

Keywords: Musical acoustics, radiation, cutoff frequency, clarinet

The input impedance is a useful quantity to characterize the acoustic response of woodwind instruments and is efficiently simulated using the Transfer Matrix Method (TMM). However, the TMM does not calculate intermediate variables that are convenient for simulating the external sound field. Recent work by Lefebvre et al proposes the Transfer Matrix Method with external Interactions (TMMI) which accounts for the mutual radiation impedance of toneholes radiating into the same space and uses the acoustic flow through each aperture as the reference variable. Treating the flow through each tonehole as a source in a linear array, it is possible to calculate the external sound field such as the waveforms at a given receiver location and directivity patterns. Additionally, the efficiency of a resonator can be calculated as the ratio of the power at the input of the resonator and the sum of the power radiating from each aperture. This is a useful step towards understanding the competition between the energy that is retained within a resonator and facilitates the auto-oscillation of the reed, and that which radiates from the resonator. This topic is explored through simulation and measurements using simplified cylindrical resonators.

INTRODUCTION

The tonehole lattice cutoff frequency is a well-known property of the clarinet and is generally considered to have an influence on the "character" of a given instrument [START_REF] Benade | Fundamentals of Musical Acoustics[END_REF]. However, it is not precisely known how the cutoff frequency influences the sound production and radiation. Recent work has shown that the cutoff frequency has a relatively small effect on the production of sound, primarily changing the spectral content of internal waveforms above the cutoff frequency [START_REF] Petersen | The effect of the cutoff frequency on sound production of clarinet-like instruments[END_REF]. It is hypothesized that the cutoff frequency will also change the radiation of a given resonator, an effect that could also determine the timbrel properties of an instrument. An early development of this idea is provided in the current paper where simplified resonators are excited at acoustically linear levels by a compression chamber. The pressure inside the resonators and at external points are measured in anechoic conditions. Transfer functions are then calculated and used to compare resonators with different cutoff frequencies and validate simulations.

SIMULATION OF EXTERNAL SOUNDFIELD

In order to predict the radiating characteristics of a resonator, the transfer function between a source inside the resonator and an external observation point is simulated. The simulation relies on the Transfer Matrix Method with external Interactions (TMMI) [START_REF] Lefebvre | External tonehole interactions in woodwind instruments[END_REF], a variant of the classic Transfer Matrix Method (TMM). The TMMI is designed to simulate the input impedance of a resonator by assuming a fictitious flow source U s , and calculates the resulting pressure and flow P n and U n at each n th radiating aperture, accounting for the mutual radiation load between sources that radiate into the same space. Therefore, the transfer function between the source flow U s and each hole is simply

H u,n = U n /U s , (1)
where U s has equal amplitude and phase for all frequencies.

The TMMI is readily exploited for radiation calculations because, in the process of simulating the input impedance, it provides the flow from each hole, already accounting for external interactions. Ignoring the presence of the body of the resonator, the holes can then be treated as an array of monopole sources. The pressure at some observation point M is proportional to the time derivative of the summation of the contributions of each U n , accounting for a propagation delay and spherical spreading:

P M = d dt N ∑ n=1 H u,n U s r n e -jωr n /c , (2)
where r n is the distance between the n th hole and M, ω is the angular frequency, and c is the speed of sound in air. This model assumes anechoic conditions and ignores absorption in air.

Because this is a linear model, at each step we have the option to choose the acoustic variables P or U, related by the impedance. It is numerically equivalent to consider pressure sources P s and P n , from which the external pressure P M is calculated. Therefore, equations 1 and 2 can just as easily be written in terms of pressure sources, which is less physical but more convenient when comparing with measurements because a microphone measures pressure inside resonator.

MEASUREMENTS

EXPERIMENTAL RESONATORS

The resonators used in this work are designed to have the same first impedance resonance, but different tonehole lattice cutoff frequencies [START_REF] Petersen | The effect of the cutoff frequency on sound production of clarinet-like instruments[END_REF]. The global cutoff frequency is fixed by choosing the geometry of each cell to have an equivalent local cutoff frequency [START_REF] Moers | On the cutoff frequency of clarinet-like instruments. Geometrical versus acoustical regularity[END_REF]. Three resonators R 1.0 , R 1.5 , and R 2.0 are used in the current work.

All have first impedance peaks at 185 Hz, and cutoff frequencies at 1000, 1500, and 2000 Hz, respectively. The distance between toneholes is the same for all three resonators, and the cutoff frequency is changed only by varying the radius of the toneholes.

EXPERIMENTAL DESIGN

The transfer function in pressure from inside the resonator to the external sound field was measured experimentally. The resonators were connected to a compression chamber using a 3D printed adapter piece with a 6 cm cylindrical portion in which the microphone is mounted. The pressure measured by this microphone is treated as the source pressure P s in Section 2. The pressure was measured at three external locations, the positions of which are shown in Fig. 1. Pure sinusoidal excitation signals were used to force the resonator from 100 Hz to 4000 Hz in steps of 10 Hz. This signal choice was chosen instead of a frequency sweep because the radiation from the resonator, particularly at frequencies corresponding to the impedance troughs, is very weak. The coherence function calculated using a sweep type signal was only robust in the vicinity of the impedance peaks.

The measured signals are band filtered from 50 Hz to 6000 Hz and windowed to retain only the stable portion of the signal. Peak amplitudes of the i th harmonics are extracted using synchronous detection

P M (ω i ) = |2 < p m (t) • e -jω i t >|. ( 3
)
This method is valid because the frequencies are precisely known and stable. In the following analysis only the fundamental frequency (i = 1) is considered.

MEASURED TRANSFER FUNCTION FOR THREE RESONATORS

The transfer function between internal pressure and three external observation points described in Section 3.2 was measured for all three resonators. Figure 2 shows the transfer function between the internal pressure and the microphone that is located 0.7 m above the first open hole for each of the three resonators. All three resonators radiate similarly at frequencies below the tonehole lattice cutoff frequency, with peaks that are harmonically related to the first peak. This is because at low frequencies the main acoustic source of the resonator is the first open tonehole, which radiates as a monopole. As the frequency increases up to and past the cutoff frequency of a given resonator, the transfer functions become more complicated. This transition is particularly abrupt for resonator R 1 , where the cutoff frequency happens to coincide with an existing peak.

COMPARISON WITH SIMULATION

The simulated and measured transfer functions between internal pressure and all three microphone locations is shown in Fig. 3 for R 1 . Because the simulation is normalized (ie the flow source U s = 1), the amplitude of the simulation is shifted so that the minimal point between the first two peaks is at the same amplitude as the measurement. The peak amplitude was not chosen as the reference point because the measurements (in steps of 10 Hz) do not fully resolve the peak. The simulation matches the measurements at low frequencies. At high frequencies, the shape of the simulation aligns reasonably well with the measurements, but the amplitude is not adequately captured. The simulation under-predicts the amount of radiation. This is also noticed in the impedance measurements (not shown). Around the cutoff frequency there is a marked shift in frequency between the simulation and measurement. This may be an interesting topic for further developing the model.

CONCLUSIONS

The effect on radiation of the tonehole lattice cutoff frequency is investigated. It is found that the cutoff frequency has a considerable impact on the transfer function between internal pressure and pressure at multiple points external to the resonator. This implies that the cutoff frequency could act as a filter of the sound that is radiated from an instrument, irrespective of its influence on sound production. It is also found that it is possible to extend the TMMI calculation to simulate external pressure waveforms. This use of the TMMI is adequate at low frequencies but is only approximate at high frequencies. This is a logical place to search for an improved model. Future work could involve repeating the experiment at sound pressure levels that are high enough to induce nonlinear effects. This would be done with a compression chamber or artificial blowing machine.

Figure 1 .

 1 Figure 1. Three microphones (right, in red) are positioned around the resonator. The first two microphones are at the level of the first open hole (right, in yellow) and at a distance of b = 4 m and a = 0.7 m, respectively. The third is positioned above the end of the resonator at a distance a = 0.7 m from the first open hole.

Figure 2 .

 2 Figure 2. Transfer function between interal pressure and external pressure at the overhead microphone location. The resonators with f c = 1000, 1500, and 2000 Hz are shown in red, blue, and black, respectively.

Figure 3 .

 3 Figure 3. Simulated and measured transfer function for resonator R 1 between internal pressure and external pressure at three microphone locations (see Fig. 1).

ACKNOWLEDGEMENTS

The authors would like to thank Patrick Sanchez and Clément Rabau for their help with the experimental aspect of this work, and in particular for designing the adapter piece that connects the compression chamber and the resonators. We would also like to thank Guy Rabau for piercing the resonators. Finally, we acknowledge the many helpful discussions with Marc Pachebat, Fabrice Silva, and Philippe Guillemain. This work has been partly supported by the French Agence Nationale de la Recherche (ANR16-LCV2-0007-01 Liamfi project.)


Özgül Tezgel

Sylvie Noinville

Véronique Bennevault

Nicolas Illy
email: nicolas.illy@sorbonne-universite.fr

Philippe Guégan
email: philippe.guegan@sorbonne-universite.fr

An alternative approach to create N-substituted cyclic dipeptides †

a N-Modified peptide backbones are promising peptidomimetics which offer several advantages in terms of improved biological activity and stability. They further allow the development of novel functional materials.

However, the synthesis of N-substituted peptides is very challenging with the existing methods, particularly the synthesis of peptides with larger N-substituents. In this work, we are introducing a new method to create N-polyether substituted cyclic dipeptides via anionic ring-opening polymerization (AROP). Four different cyclic dipeptides with different hydrophobic functional groups were selected to create N-substituted cyclic dipeptides. Backbone amides -NHwere deprotonated with phosphazene bases to form nucleophilic initiators. Furthermore, the effect of different phosphazene bases (tBuP 4 and tBuP 2 ) and of the addition of a Lewis acid (i-Bu 3 Al) was studied in detail towards creating N-polyether-cyclic dipeptides bearing either hydrophobic poly(butylene oxide) chains, or hydrophilic linear polyglycidol chains, thanks to the polymerization of 1,2-epoxybutane and the polymerization followed by the deprotection of t-butyl glycidyl ether monomers, respectively. Moreover, we have demonstrated the possibility of avoiding the isomerization of cyclic dipeptides during the synthesis of N-substituted analogues depending on the synthetic approach.

Introduction

In the last few decades, cyclic dipeptides, so-called 2,5-diketopiperazines (DKPs), have attracted significant attention due to their exceptional biological activity as antiviral, 1,2 antimicrobial 3,4 and anticancer 5,6 agents. Beyond their therapeutic properties, DKPs also have the ability to form supramolecular structures [7][8][9] which could find a wide range of applications, such as hydrogels, 10,11 nano-devices, 12 and sensors. 13 In addition, due to their structural rigidity, N-substituted DKPs are also used as rigid motifs on the polypeptide backbone to imitate protein secondary structures. [START_REF] Zhao | Peptidomimetics II[END_REF] Moreover, in the field of peptide-based drugs, N-substituted peptides often demonstrated superior outcomes in terms of biological activity and stability. [START_REF] Ovadia | [END_REF][16][17] Peptides with N-substitution could also offer the creation of a novel class of materials, for which N-substitution will potentially change the peptide conformation and induce new properties. For example, polypeptoids are one of the most popular classes of peptidomimetics which are basically short N-alkyl substituted glycine analogues with remarkable properties. [18][19][20] Indeed, the antimicrobial peptoid derivatives with long N-alkyl chains create lipophilic peptoids which are shown to be more selective against bacteria and fungi than their non-lipophilic analogues. 21 Other than functionalisation with peptoids, N-methylation is the most commonly used tool to improve the biological activity of peptide drugs. [START_REF] Ovadia | [END_REF]16,22,23 However, there is also a limited number of studies reported for other N-substituted analogues, such as peptides with N-substituted short alkyl groups, i.e. N-ethyl, 24 N-butyl, 25 and N-guanidyl butyl. 26 More recently, Albericio and coworkers have reported the synthesis of N-oligoethylene glycol (N-OEG) substituted cyclic peptides: sansalvamide A with 3 repeat units of N-OEG 27 and cilengitide with different repeat units of N-OEG such as 2, 11 and 23. 28 The synthesis of cilengitide with N-OEG 2 and sansalvamide A with N-OEG 3 by solid-phase peptide synthesis in multiple steps was reported; however, peptides with longer N-OEG substitutions were very complicated to synthesize and were reported to require extensive purification steps. 28 In the current work, we are introducing anionic ring-opening polymerization (AROP) as a novel technique to create peptides with N-polyether derivatives in a one-step reaction without sophisticated purification steps. AROP is an effective method which is frequently used to synthesize † Electronic supplementary information (ESI) available: 1 H, 13 C, 2D-HSQC, and 2D-HMBC NMR spectra, SEC traces and MALDI-ToF spectra. An additional scheme of plausible configurations. See DOI: 10.1039/c8py01552j various functional polymers readily applicable in the biomedical field. [29][30][31] Phosphazene superbases construct powerful catalytic systems for the polymerization of different heterocyclic rings through AROP as a result of their high basicity and poor nucleophilicity coming from their sterically hindered structure. 32,33 They could easily deprotonate protic precursors to create nucleophilic initiating sites for the polymerization of oxirane monomers. Recently, we have demonstrated the polymerization of 1,2-epoxybutane with a secondary carbamate 34 and a secondary amide. 35 In this study, we selected di-functional symmetric DKPs (Fig. 1) with different hydrophobic functional groups, such as glycine, alanine, valine and phenylalanine, as model cyclic peptides due to their structural simplicity and interesting behaviors. 36 The backbone -NHof symmetric DKPs was deprotonated by phosphazene bases to create nucleophilic initiating species for the polymerization of 1,2-epoxybutane and of a commercially available protected analogue of glycidol, t-butyl glycidyl ether (tBuGE). The reaction conditions were optimized with 1,2-epoxybutane. First, the use of phosphazene base mixtures with different tBuP 2 /tBuP 4 ratios has been investigated. Then the influence of a Lewis acid addition has been considered. The positive impacts of using a less basic phosphazene base, 32 tBuP 2 , and the presence of a Lewis acid, triisobutyl aluminum (i-Bu 3 Al), have been demonstrated. Both methods yielded N-substituted-DKPs with expected structures. However, we have observed by NMR and FT-IR the ring conformation differences of DKPs under different conditions of synthesis. Eventually, the optimized conditions were extended to the polymerization of tBuGE, a protected version of glycidol.

Results and discussion

We have investigated the possibility of creating N-substituted DKPs using oxirane monomers via AROP in the presence of phosphazene bases. We attempted to synthesize N-substituted analogues of four different symmetric cyclic dipeptides with varying hydrophobic functional groups (Fig. 1).

Synthesis of N-PBO-Cyclo(Ala-Ala) using phosphazene bases (tBuP 4 and tBuP 2 )

In the first part of this work, we have used Cyclo(Ala-Ala) (Scheme 1) as an initiator precursor for the polymerization of 1,2-epoxybutane in the presence of phosphazene bases (Table 1). Cyclo(Ala-Ala) was chosen in the first place due to the presence of an alanine sequence which is considered as a neutral, chemically inert peptide residue. 37 Initially, 1,2-epoxybutane was polymerized with Cyclo(Ala-Ala)/tBuP 4 as the initiator system at a ratio of 1 : 1 and 1 : 2 at 25 °C (Table 1, P1-1 and P1-2) to evaluate whether tBuP 4 works under stoichiometric conditions to deprotonate the backbone amide -NHor not. In both cases, 1 H NMR analysis indicated the disappearance of -NHprotons, meaning 100% functionalization of both backbone amide -NHs. The MALDI ToF analysis of the polymerization carried out in the presence of 1 eq. of tBuP 4 (Table 1, P1-1) shows one major population corresponding to the expected polymer structure (Scheme 1A) cationized with the Na + cation (Fig. 3b). The MALDI-ToF analysis also indicates the presence of an additional minor unidentified population. A higher amount of tBuP 4 (2 eq., Table 1, P1-2) resulted in the increase of the undesired population (Fig. 3a). 1 H NMR and 13 C NMR and ATR-FTIR analyses (P1-1 and P1-2, Fig. 2a, b and4a) clearly display the presence of ester bonds on the polymers. A macromolecular structure in agreement with these spectral analyses and corresponding to a nucleophilic attack on the DKP ring is postulated and shown in Scheme 1B. The intensity of ester peaks was increased by increasing the amount of tBuP 4 , in both the NMR and ATR-FTIR spectra (P1-1 and P1-2, Fig. 2a, b and4a). The coexistence of polymer structures A and B in P1-2 was also confirmed by 2D-NMR (2D-HSQC-NMR, Fig. S1 † and 2D-HMBC-NMR, Fig. S2 †). As a conclusion, N-substitution has been proved to be possible for the reactions performed with both 1 eq. and 2 eq. of tBuP 4 . However, undesired nucleophilic attack on the DKP ring also occurred and the frequency of this side-reaction increased with the amount of tBuP 4 . To eliminate the undesired side reactions, we have decided to decrease the amount of tBuP 4 (pK BH + , MeCN = 42.7) and eventually use a mixture of tBuP 4 and a less basic phosphazene base, tBuP 2 (pK BH + , MeCN = 33.5). When we decreased the amount of tBuP 4 to 0.5 eq. and performed the polymerization at 40 °C, we reached 100% conversion within 72 h (P1-5, Table 1) with expected molar mass and Scheme 1 Synthesis of N-PBO-Cyclo(Ala-Ala) using phosphazene bases. narrow dispersity (P1-5, Table 1). In addition, ATR-FTIR (Fig. 4a) and 1 H and 13 C NMR analyses (Fig. 2, P1-5) demonstrated only the presence of an expected polymer structure ( polymer A, Scheme 1). MALDI-ToF analysis shows only one population corresponding to the expected polymer A structure (Fig. 3c). We have tested the mixture of tBuP 4 and tBuP 2 for the polymerization of 1,2-epoxybutane by Cyclo(Ala-Ala)/ tBuP 4 /tBuP 2 at a ratio of 1 : 0.5 : 0.5. At 25 °C, the reaction did not go to completion over 96 h (P1-3, Table 1). At 40 °C (P1-4, Table 1), the polymerization achieved 100% conversion in 18 h with the desired molar mass (M n ) without the presence of ester peaks in the NMR spectra (Fig. S3 and S4 †). These results suggest that 1 : 1 and higher ratios of Cyclo(Ala-Ala)/tBuP 4 decrease the stability of DKPs. The strong tBuP 4 super-base does not necessarily need to be introduced at a stoichiometric ratio for 100% functionalization of amides on the DKP backbone. Therefore, we investigated the use of the less basic phosphazene base tBuP 2 and only a catalytic amount of tBuP 4 to obtain the desired polymer structures (P1-6 to P1-9, Table 1). The ratio of 1 : 0.3 : 0.7 led to polymerization but the obtained M n was smaller than the targeted one (P1-7, Table 1) and the polymerization of 1,2-epoxybutane by Cyclo(Ala-Ala)/tBuP 4 / tBuP 2 at a ratio of 1 : 0.25 : 0.75 did not work at all (P1-6, Table 1).

Then, we increased the amount of tBuP 2 and kept the ratio of tBuP 4 at 0.25 eq. or below (P1-8 and P1-9, Table 1). The Cyclo(Ala-Ala)/tBuP 4 /tBuP 2 ratio of 1/0.1/1 was not effective, on the other hand a 1/0.25/1 ratio of Cyclo(Ala-Ala)/tBuP 4 /tBuP 2 worked with a good efficiency to yield the polymers with the desired N-PBO-Cyclo(Ala-Ala) structure (Fig. S5 †) and targeted M n with a narrow molar mass distribution (P1-8, Table 1). An absolute M n value of 2600 g mol -1 was also determined by 1 H NMR and is in very good agreement with both theoretical and SEC values. It has been reported that 1,2-epoxybutane does not polymerize in the presence of the less basic phosphazene base tBuP 2 . 38 Here, we demonstrated that 1,2-epoxybutane can be polymerized with tBuP 2 in the presence of a catalytic amount of tBuP 4 without undesired side reactions. All the techniques used for the characterization of N-PBO-Cyclo(Ala-Ala) indicated only the presence of the expected N-PBO-Cyclo(Ala-Ala) population with no evidence of free -NHamide peaks, suggesting the growth of polymer chains at both DKP backbone amide functionalities (Scheme 1, polymer A).

Synthesis of N-PBO-Cyclo(Gly-Gly) using phosphazene bases (tBuP 4 and tBuP 2 )

After the successful synthesis of N-PBO-Cyclo(Ala-Ala), we attempted to apply a similar approach to synthesize N-substituted analogues of Cyclo(Gly-Gly). The reactions with a Cyclo(Gly-Gly)/tBuP 4 ratio of 1 : 1 and 1 : 2 (P2-1 and P2-2, Table 2) yielded polymers with expected M n . ATR-FTIR did not show a peak corresponding to ester formation confirming the integrity of the DKP ring. In addition, the MALDI-ToF analysis demonstrated a single polymer population corresponding to the N-PBO-Cyclo(Gly-Gly) (Fig. S6 †). However, both the 13 C NMR and ATR-FTIR spectra of P2-2 indicate CvC double bond formation (Fig. S7 †) which could be explained by the transfer reaction to the monomer at high concentrations of tBuP 4 . 39 13 C NMR analysis demonstrated multiple CvO amide peaks (P2-1, Fig. 5a) which could be explained by the formation of different isomers of the DKP ring following the addition of N-substituents. 40 This result could be attributed to the epimerization of cyclic dipeptides in basic solutions as already reported more than forty years ago. 41,42 A similar phenomenon is also observed for cyclic peptoids (N-substituted glycine analogues), for which each single CvO amide geometry displayed a single CvO amide peak 43 and in the presence of cis/trans isomers, multiple CvO amide peaks were observed. 43,44 Additionally, similar to the N-PBO-Cyclo (Ala-Ala) synthesis, a Cyclo(Gly-Gly)/tBuP 4 ratio of 1 : 0.5 yielded a polymer having the expected structure (Table 2, P2-5) without transfer reactions. 13 C NMR demonstrated a major single CvO amide peak (Fig. S8 †) which indicates the presence of mainly a single isomer of the DKP ring. These results suggest that the use of tBuP 4 at 1 eq. and more favors the isomerization of the DKP ring and the occurrence of side reactions. Furthermore, a mixture of tBuP 4 and tBuP 2 was incorporated in the reaction mixture by varying the tBuP 4 percen- tage from 0.25 to 0.5 and the tBuP 2 percentage from 0.5 to 1 (P2-3, P2-4, P2-6 and P2-7, Table 2). The optimal conditions were determined as P2-7 (Fig. S9 †), where the addition of tBuP 2 led to the formation of N-PBO-Cyclo(Gly-Gly) with mostly single ring configuration as confirmed by 13 C NMR (P2-7, Fig. 5a).

Synthesis of N-PBO-Cyclo(Val-Val) and N-PBO-Cyclo(Phe-Phe) using phosphazene bases (tBuP 4 and tBuP 2 )

We were able to find the common reaction conditions for the synthesis of N-PBO-DKPs with Cyclo(Ala-Ala) and Cyclo(Gly-Gly). We would like to expand the validity of our method for more hydrophobic DKPs, Cyclo(Val-Val) and Cyclo(Phe-Phe).

The synthesis of N-substituted Cyclo(Val-Val) showed similar polymerization characteristics and side reactions to that of Cyclo(Ala-Ala). When the polymerization of 1,2-epoxybutane was carried out at a Cyclo(Val-Val)/tBuP 4 ratio of 1 : 1 (P3-1, Table 3), the MALDI-ToF spectrum displayed only single Poisson distribution corresponding to N-PBO-Cyclo(Val-Val) (Fig. S11a †). However, the 1 H NMR spectra show the presence of ester function due to the nucleophilic attack of growing polymer end chains on the DKP ring (Fig. S10 †) which is also confirmed by ATR-FTIR analysis (Fig. 4b, red curve). The polymer obtained by the initiating system, Cyclo(Val-Val)/ tBuP 4 /tBuP 2 , at a ratio of 1 : 0.5 : 0.5 also demonstrated the nucleophilic attack on the DKP ring at a lower degree (P3-2, Fig. S10 †). Ester formations for Cyclo(Ala-Ala) and Cyclo(Val-Val) suggest that DKPs with aliphatic substitutions at the Cα position are more sensitive to nucleophilic attack during the synthesis in the presence of high concentrations of tBuP 4 . When we applied the optimized conditions to the synthesis of N-PBO-Cyclo(Val-Val), i.e. 1 eq. of tBuP 2 with 0.25 eq. of tBuP 4 (P3-3, Table 3), 1 H-NMR and MALDI-ToF analyses did not show the presence of side reactions (P3-3, Fig. S10 andS11c †). At the same time, four different peaks are visible in the CvO amide region in the 13 C NMR spectrum (P3-3, Fig. 5b). The ATR-FTIR spectrum also demonstrates a peak at ∼1700 cm -1 (Fig. 4b, green curve) which corresponds to the cis CvO amide stretching on the DKP ring 45 in addition to the CvO trans peak at ∼1660 cm -1 . Thus, ATR-FTIR and 13 C NMR spectra demonstrate the presence of CvO amide peaks of both cis and trans isomers, meaning that N-substitution leads to the formation of isomers of the DKP rings (Scheme S1 †). In the litera-ture, DKPs have been reported to have four different configurations corresponding to two cis and two trans isomers 41 which could be easily characterized by 13 C NMR. 46 Therefore, the presence of four different peaks for CvO amide, Cα (presented as 2, 2′ in Fig. S12 †), N-substituted -CH 2 -( presented as a′ in Fig. S12 †) and Cα substituted -CH-( presented as 3 in Fig. S12 †) in the 13 C NMR spectrum represents the four different configurations of the DKP ring (two cis and two trans isomers). The ratio of cis/trans isomers is known to be affected by the nature of the N-substituent in N-substituted peptides. 40 The synthesis of N-PBO-Cyclo(Phe-Phe) was more challenging. The optimized conditions (1 eq. of tBuP 2 with 0.25 eq. of tBuP 4 ) did not work at all (P3-6, Table 3). This lack of reactivity could pave the way to the selective functionalization of an asymmetric DKP containing only one phenylalanine residue; nevertheless, it is beyond the scope of the present work. In addition, the polymers obtained using 1 eq. of tBuP 4 at 25 °C and 40 °C yielded lower molar masses than the expected ones and broad molar mass distributions (P3-4 and P3-5, Table 3). Besides, the MALDI-ToF spectra display two different populations (Fig. S13 †); one corresponding to the expected structure of N-PBO-Cyclo(Phe-Phe), and the second one is unknown.

Increasing the polymerization temperature from 25 °C (Table 3, P3-4) to 40 °C (Table 3, P3-5) led to an increase of the proportion of the unknown population and to a concomitant decrease of molar masses. 13 C NMR analysis also demonstrated the presence of multiple peaks indicating the DKP ring with multiple isomers and side reactions (Fig. S14 †).

Synthesis of N-PBO-DKPs in the presence of tBuP 4 and a Lewis acid i-Bu 3 Al

The synthesis of N-PBO-DKPs using the DKP/tBuP 4 /tBuP 2 initiating system worked very well for Cyclo(Ala-Ala), Cyclo(Gly-Gly) and Cyclo(Val-Val). However, the synthesis of N-substituted Cyclo(Phe-Phe) was more challenging. Therefore, to overcome the challenges that we faced during the synthesis, we employed an alternative strategy. A Lewis acid, i-Bu 3 Al, was incorporated in the reaction medium to control the reactivity of the active centers and to trigger monomer reactivity. 47 In the polymerization reactions with the addition of i-Bu 3 Al, we kept the ratio of DKP/tBuP 4 constant at 1 : 1 and varied the amount of i-Bu 3 Al from 2 eq. to 3 eq. As previously indicated, Cyclo(Ala-Ala) was used as a reference to test the reaction conditions (P4-1 and P4-2, Table 4). With 3 eq. of i-Bu 3 Al (P4-1, Table 4), M n was higher than the targeted value with a broad dispersity. An absolute M n value of 4500 g mol -1 was also determined by 1 H NMR, which confirmed the value obtained by SEC. The polymerization control was better in the presence of 2 eq. of i-Bu 3 Al (P4-2, Table 4). M n was closer to the theoretical value (2700 vs. 2100 g mol -1 ) with narrower dispersity. The characterization of N-PBO-Cyclo(Ala-Ala) which was synthesized in the presence of i-Bu 3 Al did not show any difference compared to the one synthesized by the first approach. Furthermore, in the case of Cyclo(Gly-Gly), the polymerizations were performed in the presence of 2 eq. and 2.5 eq. of i-Bu 3 Al (P4-3 and P4-4, Table 4). Polymer P4-3 which was synthesized in the presence of 2 eq. of i-Bu 3 Al has a slightly broad distribution and 13 C-NMR analysis shows a main CvO peak with traces of other CvO peaks corresponding to isomeric structures (Fig. S15 †). P4-4 that was synthesized in the presence of 2.5 eq. of i-Bu 3 Al has a narrow molar mass distribution (P4-4, Table 4). Furthermore, 13 C-NMR analysis shows only one single peak in the amide-carbonyl region (P4-4, Fig. 5a), demonstrating the single isomer of the DKP ring. Therefore, using an optimized amount of i-Bu 3 Al in the reaction medium helps us to eliminate the isomerization of the DKP ring, but a slight increase in the amount of i-Bu 3 Al increases the difference between the targeted and obtained M n (P4-3 vs. P4-4, Table 4). N-PBO-Cyclo(Val-Val) synthesis in the presence of 2 eq. of i-Bu 3 Al yielded a controlled polymerization with a narrow dispersity (P4-5, Table 4). Compared to the reactions without i-Bu 3 Al, cis amide CvO peaks disappeared in the 13 C-NMR (Fig. 5b) and ATR-FTIR spectra (Fig. 4b). Moreover, there are also single peaks observed at the ring level ( peaks 2 and 3, Fig. S16b †) and two peaks observed for N-substituted -CH 2on the polymer instead of four different peaks ( peak a′, Fig. S16b †). These observations confirm that the synthesis of N-PBO-Cyclo(Val-Val) in the presence of 2 eq. of i-Bu 3 Al led to the formation of N-substituted DKP with a single configuration. This could be explained by the stabilization of the active centers by i-Bu 3 Al and their reduced basicity.

Lastly, N-PBO-Cyclo(Phe-Phe) synthesis was also performed in the presence of 2 eq. and 2.5 eq. of i-Bu 3 Al (P4-6 and P4-7, Table 4). In both cases, we successfully obtained N-PBO-Cyclo (Phe-Phe) which was confirmed by NMR and MALDI-ToF analysis (P4-6, Fig. S17 † and P4-7, Fig. 6). The polymerization kinetics and the control over the polymerization were better in the presence of 2.5 eq. of i-Bu 3 Al compared to 2 eq. (P4-6 and P4-7, Table 4). Moreover, 13 C NMR analysis demonstrated the presence of a single isomer type of Cyclo(Phe-Phe) (Fig. 6c).

Synthesis of N-PtBuGE-DKPs. In this section, we demonstrated the synthesis of N-polyether-DKP analogues having a water soluble N-substituent using a commercially available protected version of the water soluble monomer glycidol, t-butyl glycidyl ether (tBuGE). Cyclo(Ala-Ala) and Cyclo(Gly-Gly) were successfully modified to create N-substituted analogues by two different strategies described above without side reactions.

N-Poly(t-butyl glycidyl ether) (N-PtBuGE) substituted DKPs were synthesized in the presence of 1 eq. of tBuP 2 and 0.25 eq. of tBuP 4 (P5-1 and P5-2, Table 5). Both polymers P5-1 and P5-2 have a very narrow mass distribution. However, the molar mass of P5-1 obtained by GPC was larger than the targeted value. MALDI-ToF characterization confirmed the expected polymer structures (P5-1, Fig. S18 and P5-2, Fig. S19 †). Furthermore, the polymerization of tBuGE was also carried out in the presence of i-Bu 3 Al. Initially, for the synthesis of N-PtBuGE-Cyclo(Ala-Ala), we have tested two different ratios of i-Bu 3 Al (P5-3 and P5-4, Table 5). Between the two reactions, 2.75 eq. of i-Bu 3 Al yielded a better control over the polymerization (P5-4, Table 5). Therefore, N-PtBuGE-Cyclo(Gly-Gly) was also synthesized using the same conditions (P5-5, Table 5). MALDI-ToF analysis of polymers P5-4 and P5-5 confirmed the expected structures (Fig. S20 and S21 †). The synthesis of N-PtBuGE-DKPs with tBuP 4 /i-Bu 3 Al yielded slightly broad dispersities compared to the tBuP 2 /tBuP 4 system; however, the control over the molar mass was more efficient, and the obtained M n values by GPC were exactly matching with the targeted ones. Finally, P5-4 was deprotected by trifluoro acetic acid according to Erberich et al. 48 and purified by dialysis against distilled water to yield the water soluble N-polyglycerol-Cyclo(Ala-Ala) (N-PG-Cyclo(Ala-Ala)) (Fig. S23 †) which could provide new opportunities for biological applications.

Conclusions

We have successfully synthesized a library of well-defined N-polyether-DKPs via AROP. Mild polymerization conditions have been identified allowing the elimination of the side-reactions, which occurred only when tBuP 4 phosphazene base was used. N-Polyether-DKPs were carefully characterized using GPC, MALDI-ToF, ATR-FTIR, and 1 H-and 13 C-NMR. Epimerization reactions were eliminated by the addition of i-Bu 3 Al in the reaction medium, leading to the preparation of mono-isomeric N-substituted DKPs with nevertheless slightly broader dispersities. Linear polyglycidol and poly(butylene oxide) chains were substituted demonstrating the possibility of the creation of both hydrophilic and hydrophobic substituents and the development of novel materials through self-assembly for biomedical applications (Scheme 2).

Experimental

Materials

All the DKPs, Cyclo(alanine-alanine) (Aldrich), Cyclo(glycineglycine) (Aldrich), Cyclo(Val-Val) (Bachem) and Cyclo( phenyl-alanine-phenylalanine) (Bachem), phosphazene base tBuP 4 solution (0.8 mol L -1 in hexane, Aldrich), phosphazene base tBuP 2 solution (2 mol L -1 in THF, Aldrich), 2-methyltetrahydrofuran (MeTHF) (98%, Aldrich), and tri-isobutyl aluminum (i-Bu 3 Al) solution (1.0 mol L -1 in hexane, Aldrich) were used without further purification. 1,2-Epoxybutane (99%, Aldrich) and t-butyl glycidyl ether (99%, Aldrich) were distilled over CaH 2 two times prior to use. Tetrahydrofuran was dried with an MBRAUN MB SPS-800 solvent purification system under nitrogen.

Instrumentation 1 H and 13 C NMR analyses were performed on Bruker Avance 300 MHz and Bruker Avance 400 MHz spectrometers. All spectra were internally referenced to the residual proton signals of the deuterated solvent: CDCl 3 and THF-d8. Polymer molar masses were determined by size exclusion chromatography (SEC) with three PLGel Mixte C 5 µm columns (7.5 × 300 mm; separation limits: 0.2 to 2000 kg mol -1 ) maintained at 40 °C coupled with 2 modular detectors: a differential refractive index (RI) detector Viscotek 3580 and a diode array UV detector Shimadzu SPD20-AV. THF was used as the mobile phase at a flow rate of 1 mL min -1 , and toluene was used as a flow rate marker. All polymers were injected (100 μL) at a concentration of 5 mg mL -1 after filtration through a 0.45 μm pore-size membrane. OmniSEC 4.7 software was used for data acquisition and data analysis. The number-average molar masses (M ˉn), the weight-average molar masses (M ˉw), and the molar mass distributions (Đ = M ˉw/M ˉn) were determined by SEC with a calibration curve based on narrow poly(methyl methacrylate) (PMMA) standards (from Polymer Standard Services), using the RI detector.

MALDI-ToF. Mass spectra were recorded by matrix-assisted laser desorption and ionization time-of-flight (MALDI-ToF) mass spectrometry (MS) using a Bruker autoflex III smartbeam mass spectrometer, equipped with a laser that produces pulses at 337 nm using dithranol as a matrix and NaI as a cationizing agent. Spectra were recorded in reflectron or in linear mode at an accelerating potential of 20 kV. Samples were prepared by dissolving the polymer in THF at a concentration of 5 mg mL -1 . A 10 μL aliquot of this solution was mixed with 20 μL of dithranol solution as the matrix (at 20 mg mL -1 in THF) and 10 μL of NaI solution (at 10 mg mL -1 in THF). Standards ( poly (ethylene oxide) of known structures, M ˉn = 1470 and 4250 g mol -1 purchased from Polymer Standards Service) were used to calibrate the mass scale. FTIR experiments. FTIR spectra in the attenuated total reflection (ATR) mode were recorded on a Bruker Equinox 55 spectrometer using a spectral resolution of 4 cm -1 . The spectrometer was continuously purged with dry air to avoid water vapor absorption in the spectral range of amide and ester carbonyl absorptions. 60 µL of the polymer solution in CH 2 Cl 2 was deposited on the bare ATR element made of germanium allowing 15 useful internal reflections and the solvent was allowed to evaporate before collecting 128 scans. Each recorded spectrum was transformed by ATR correction to the absorption spectrum.

Polymerization conditions for Tables 1-3, P5-1 and P5-2, Table 5 The polymerization was carried out according to the following typical procedure. The initiator precursor DKP was weighed into a reaction flask. Then, the rest of the reactant was added to a glove-box. Following the initiator, the solvent (THF) was added to the reactor. Then t-BuP 4 and t-BuP 2 were added. Finally the monomer, 1,2-epoxybutane, was added and the reactor was closed with a septum. After the closure of the reaction flask, the reaction mixture was stirred at room temperature or at 40 °C and left to react for the required period of time. A small portion of the reaction mixture was sampled through a septum at various reaction times for NMR analyses. The reaction was quenched by addition of 0.1 mL of 1% acetic acid. The polymer was dissolved in ethyl acetate and purified by passing through neutralized aluminum oxide, filtering, and removing the solvent under vacuum at 50 °C to give a colorless viscous liquid.

Polymerization conditions for Tables 4 and5 (P5-3, P5-4 and P5-5)

The polymerization was carried out according to the following typical procedure. The initiator precursor DKP was weighed into a reaction flask. Then, the rest of the reactant was added to a glove-box. After the initiator, the solvent (MeTHF) was added to the reactor. Then t-BuP 4 was added. Subsequently, i-Bu 3 Al was added and stirred until the complete dissolution of DKP (2 h to 4 h). Finally, the monomer 1,2-epoxybutane was added using a microsyringe. The reaction mixture was stirred at the indicated temperature and left to react for the required period of time. A small portion of the reaction mixture was sampled through a septum at various reaction times for NMR analyses. The reaction was quenched by the addition of 0.1 mL of 1% acetic acid in H 2 O. The polymer was dissolved in ethyl acetate and purified by passing through neutralized aluminum oxide, filtering, and removing the solvent under vacuum at 50 °C to give a colorless viscous liquid.

Fig. 1

 1 Fig. 1 Structures of di-functional symmetric DKPs.

Fig. 2

 2 Fig. 2 NMR spectra showing the effect of the tBuP 4 /Cyclo(Ala-Ala) ratio on the polymerization of 1,2-epoxybutane using Cyclo(Ala-Ala) and tBuP 4 as the initiating system. (a) 1 H NMR spectra in CDCl 3 comparing the tBuP 4 /Cyclo(Ala-Ala) ratio of 2 (P1-2-blue), 1 (P1-1, green) and 0.5 (P1-5, red). The red arrows indicate the presence of side reactions. (b) 13 C NMR spectra in CDCl 3 comparing the tBuP 4 /Cyclo(Ala-Ala) ratio of 2 (P1-2-blue), 1 (P1-1, green) and 0.5 (P1-5, red). The red stars show the presence of extra peaks occurring due to the side reactions.

Fig. 3

 3 Fig. 3 MALDI-ToF analysis of polymers initiated by the tBuP 4 /Cyclo(Ala-Ala) ratio of (a) 2, (b) 1 and (c) 0.5. Top spectra are collected in linear mode and the bottom ones are collected in reflectron mode. Major populations represent the structure of polymer A.

Fig. 4

 4 Fig. 4 ATR-FITR spectra of (a) N-PBO-Cyclo(Ala-Ala); P1-1 (green curve), P1-2 (blue curve), P1-5 (red curve) and (b) N-PBO-Cyclo(Val-Val); P3-1 (red curve), P3-3 (green curve), and P4-5 (blue curve).

Fig. 5

 5 Fig. 5 13 C NMR analysis of polymers (in CDCl 3 ) synthesized under different conditions. (a) Polymers initiated by (a) Cyclo(Gly-Gly); P2-1 (red curve), P2-7 (green curve), P4-4 (blue curve) and (b) Cyclo(Val-Val); P3-1 (red curve), P3-3 (green curve), and P4-5 (blue curve).

Fig. 6

 6 Fig. 6 MALDI-ToF and 13 C NMR analysis of P4-7. (a) MALDI-ToF spectrum of P4-7 in linear mode. (b) MALDI-ToF spectrum of P4-7 in reflectron mode. (c) 13 C NMR analysis of P4-7 in CDCl 3 at 50 °C.

aScheme 2

 2 Scheme 2 Synthesis of N-PtBuGE-DKP and N-PG-DKP.

Table 1

 1 Polymerization of 1,2-epoxybutane using phosphazene bases and Cyclo(Ala-Ala) as the initiator system in THF at a monomer concentration of 2 mol L -1

	Run	tBuP 4	a	tBuP 2	a	X n a	T (°C)	Time (h)	% conv.	M n theo	M n GPC	M w /M n
	P1-1	1		-		30	25	19	100	2300	2500	1.16
	P1-2	2		-		30	25	16	100	2300	2600	1.15
	P1-3	0.50		0.50		30	25	96	82	1900	800	1.30
	P1-4	0.50		0.50		30	40	18	100	2300	2500	1.13
	P1-5	0.50		-		50	40	72	100	3700	3900	1.11
	P1-6	0.25		0.75		30	40	66	0	-	-	-
	P1-7	0.30		0.70		30	40	47	88	2000	1300	1.18
	P1-8	0.25		1		30	40	72	100	2300	2700	1.12
	P1-9	0.10		1		30	40	69	0	-	-	-

a Reactant equivalents compared to 1 equivalent of cyclo(Ala-Ala).

Table 2

 2 Polymerization of 1,2-epoxybutane using phosphazene bases and Cyclo(Gly-Gly) as the initiator system in THF at a monomer concentration of 2 mol L -1

	Run	tBuP 4	a	tBuP 2	a	X n a	T (°C)	Time (h)	% conv.	M n theo	M n GPC	M w /M n
	P2-1	1		-		30	25	19	92	2100	2400	1.22
	P2-2	2		-		30	25	14	96	2300	2100	1.20
	P2-3	0.5		0.5		30	25	120	0	-	-	-
	P2-4	0.5		0.5		30	40	42	80	1800	1400	1.38
	P2-5	0.5		-		30	40	72	100	2300	2700	1.13
	P2-6	0.5		1		30	40	23	100	2300	2400	1.21
	P2-7	0.25		1		40	40	69	100	3100	3700	1.10

a Reactant equivalents compared to 1 equivalent of Cyclo(Gly-Gly).

Table 3

 3

		Polymerization of 1,2-epoxybutane using phosphazene bases and Cyclo(Val-Val) or Cyclo(Phe-Phe) as the initiator system in THF at a
	monomer concentration of 2 mol L -1
	Run	DKP	tBuP 4	a	tBuP 2	a	X n	a	T (°C)	Time (h)	Conv. (%)	M n theo	M n GPC	M w /M n
	P3-1	Cyclo(Val-Val)	1		-		30		25	19	96	2300	2500	1.15
	P3-2	Cyclo(Val-Val)	0.50		0.50		30		40	21	100	2400	2000	1.13
	P3-3	Cyclo(Val-Val)	0.25		1		30	40	72	100	2400	2700	1.11
	P3-4	Cyclo(Phe-Phe)	1		-		30	25	46	85	2100	1100	1.32
	P3-5	Cyclo(Phe-Phe)	1		-		30	40	17	79	2000	900	1.48
	P3-6	Cyclo(Phe-Phe)	0.25		1		30	40	70	30	-	-	-

a Reactant equivalents compared to 1 equivalent of DKP.

Table 4

 4 Polymerization of 1,2-epoxybutane using phosphazene base i-Bu 3 Al and DKPs as the initiating system in MeTHF at a monomer concentration of 2 mol L -1 at 25 °C

	Run	DKP	tBuP 4	a	i-Bu 3 Al a	X n	Time (h)	% conv.	M n theo	M n GPC	M w /M n
	P4-1	Cyclo(Ala-Ala)	1		3	30	3	100	2300	4000	1.65
	P4-2	Cyclo(Ala-Ala)	1		2	30	16	86	2100	2700	1.35
	P4-3	Cyclo(Gly-Gly)	1		2	30	15	95	2300	3000	1.37
	P4-4	Cyclo(Gly-Gly)	1		2.5	30	23	100	2300	4000	1.22
	P4-5	Cyclo(Val-Val)	1		2	30	20	100	2400	3400	1.25
	P4-6	Cyclo(Phe-Phe)	1		2	30	20	93	2300	3200	1.29
	P4-7	Cyclo(Phe-Phe)	1		2.5	30	19	100	2500	3200	1.28

a Reactant equivalents compared to 1 equivalent of DKP.

Table 5

 5 Synthesis of N-PtBuGE-DKPs under different conditions at a monomer concentration of 2 mol L -1

	Run	DKP	tBuP 4	a	tBuP 2	a	i-Bu 3 Al a	T (°C)	Time (h)	% conv.	M n theo	M n GPC	M w /M n
	P5-1 b	Cyclo(Ala-Ala)	0.25		1		-	40	96	100	4000	5200	1.12
	P5-2 b	Cyclo(Gly-Gly)	0.25		1		-	40	96	100	4000	4400	1.10
	P5-3 c	Cyclo(Ala-Ala)	1		-		2.5	25	28	75	3000	2500	1.27
	P5-4 c	Cyclo(Ala-Ala)	1		-		2.75	25	21	75	3000	3000	1.27
	P5-5 c	Cyclo(Gly-Gly)	1		-		2.75	25	21	75	3000	3100	1.28

Acknowledgements

This work was supported by the Sorbonne Universités Emergence 2016 program of the French state funds managed by the ANR within the Investisements D'Avenir programme under reference number ANR-11-IDEX-0004-02 and funding from the French National Funding Agency (ANR-17-CE18-0015-01 -VINP).

Conflicts of interest

There are no conflicts to declare.