J Höpfner

B Mayerhöfer

C Botha

D Bouillaud

J Farjon

P Giraudeau
email: patrick.giraudeau@univ-nantes.fr

M Wilhelm

Solvent suppression techniques for coupling of size exclusion chromatography and 1 H-NMR using benchtop spectrometers at 43 and 62 MHz

Keywords: benchtop NMR, polymer analysis, chromatography, solvent suppression, combined method, SEC-NMR

The characterisation of polymeric materials in their full complexity of chain length, monomeric composition, branching and functionalization is a tremendous challenge and is best tackled by tailored multi-dimensional coupled analytical and detection techniques. Herein, we focus on the improvement of an affordable but information rich 2D-method for polymer analysis: the online hyphenation of benchtop 1 H-NMR spectroscopy with size exclusion chromatography (SEC). The main benefit of this approach is correlated information of chain length (SEC) to chemical composition ( 1 H-NMR). Our setup combines SEC onflow with a benchtop NMR spectrometer at 43 or 62 MHz with chemical shift resolution as a robust detector. A detailed comparison of the two instruments is included considering, that only the 43 MHz instrument is equipped with a dedicated z-gradient enabling pulse sequences such as WET.

The main challenge of this method is the very low concentration of species of interest after chromatographic separation. At typical SEC conditions, the analyte dilution is typically more than a factor of 1000:1 in a protonated solvent.

Therefore, an efficient solvent signal suppression is needed. In this article, several suppression pulse sequences are explored like WET, WEFT, JNR and a simple one-pulse approach -some for the first time on this hardware. By choosing an optimal method, signal strength ratios of solvent to analyte of 1:1 or better are achievable on flow.

To illustrate the broad range of possible applications, three typical cases of analyte to solvent signal proximity (no overlap, partial and full overlap) are discussed using typical polymers (PS, PMMA, PEMA) and solvents (chloroform and THF). For each case, several suppression methods are compared and evaluated using a set of numerical criteria (analyte signal suppression and broadening, solvent signal suppression, remaining solvent signal width).

Introduction

Today's functional polymers are everywhere: from displays to fuel cells, from medical implants to packaging or membranes for separation applications. [1] However, each polymeric material is in itself a complex mixture of chains with different length (molecular weight distribution, MWD), different monomers at different positions (copolymer, chemical composition distribution, CCD) and topology (such as linear, branched, star etc.). Additionally, often blends of different (homo-)polymers are used. In each of these "variables", a polymeric material has a distribution of manifestations which are important for the material properties. Each dimension of the polymer needs to be investigated for a complete characterisation of the material.

The MWD of natural and synthetic polymers is, as a standard method, characterized by size exclusion chromatography (SEC). This is a solution based separation method where the underlying separation mechanism is based on an entropic mechanism utilizing the hydrodynamic radius of a dissolved molecule. [2][START_REF] Striegel | Modern size-exclusion liquid chromatography: practice of gel permeation and gel filtration chromatography[END_REF][START_REF] Pasch | Multidimensional HPLC of polymers[END_REF] The experimental setup shares many similarities with liquid absorption chromatography (LAC). However, LAC separates according to molecular interactions with the column material and is mostly insensitive to molecular weight.

SEC alone gives information only on the MWD dimension of a polymer. As it separates due to solution size and the most used detectors in SEC, such as the differential refractive index (DRI) and viscosity detectors are only concentration sensitive or multi-angle laser light scattering (MALLS) detectors which are only molar mass sensitive. [2] Using these detectors, molecules with the same solution size but different chemical composition or topology are indistinguishable.

Information on the other dimensions can be obtained using different methods such as other types of chromatography or spectroscopy. While this information can be gathered offline with dedicated methods, we propose the advantage of directly coupled methods which give a correlated picture e.g. the chemical composition at each molecular weight. Additionally, such methods can be used with more speed and routine once they are developed.

Herein, we focus on the online-coupling of SEC to 1 H-NMR spectroscopy. This method adds highly interesting information in the chemical composition dimension such as distribution of comonomers, functional groups or blended polymers in the MWD. Due to these possibilities, the hyphenation of chromatography and 1 H-NMR has been widely explored for high-field NMR instruments in LAC. [START_REF] Albert | On-line LC-NMR and related techniques[END_REF][START_REF] Hiller | [END_REF] Far fewer investigations have focused on the coupling with SEC (see below). [7][8][9][10][11] The advantage of high-field NMR coupling is high resolution / selectivity towards chemical shift differences. However, the needed spectrometers are complex and costly especially compared to typical chromatography setups, consequently the usage has been greatly limited. Therefore, we focus on the usage of benchtop NMR instruments using permanent magnets up to 100 MHz ( 1 H) as detectors.

These instruments have several advantages; (1) they are much more affordable and their footprint is in the range of a typical chromatography setup, (2) benchtop instruments are simple to maintain as they do not include liquid gas cooling. [START_REF] Striegel | Modern size-exclusion liquid chromatography: practice of gel permeation and gel filtration chromatography[END_REF] The operation can be easily paused, allowing, with the small size, a simple transfer between applications or labs. Such instruments are robust in their operation which offers an upside in routine work such quality control or sample screening.

The typical challenges faced by SEC-NMR (and HPLC-NMR) of polymers are onflow measurements, very low analyte concentrations (<< 1 g/L), protonated solvents with strong and possibly overlapping resonances of interest. So far, only the usage of highly abundant 1 H and 19 F nuclei in NMR spectroscopy have been used in coupling methods due to the low final concentration. The relevant differences of SEC to HPLC for NMR applications include the absence of solvent gradients. Moreover, the large polymer chains exhibit significant NMR relaxation time differences (T 1 and T 2 ) compared to the solvent molecules and show fairly broad NMR signals.

The listed challenges warrant a strong focus for any SEC-NMR investigation on signal-to-noise ratio (SNR) optimization, usage of solvent signal suppression schemes and keeping flow effects within limits (in-flow and out-flow effects, for explanation see e.g. Gouilleux et al.[12]) This is true for all hyphenation techniques but the hard-and software solutions are highly dependent on the specific application. The focus in this publication is on the optimum approach for coupling SEC to a medium-resolution (MR) / benchtop NMR, which is somewhat different from the one for e.g. high field NMR couplings where pulsed gradients are easier to realise.

First investigations of SEC-NMR high-field couplings were undertaken by Hatada et al. and were later extended by Albrecht et al and Hiller with Pasch et al. [7][8][9][10][11] A first attempt of SEC-NMR coupling with a benchtop spectrometer was attempted by Cudaj et al. using a 20 MHz spectrometer. [13,14] While giving some promising first results, limits in sensitivity made experiments with typical SEC concentrations difficult and chemical resolution was too limited for many applications. However, development of benchtop NMR has been swift in recent years towards better hardware (e.g. higher fields and resolution) and pulse sequences (possible due to field homogeneity). [15,16] These developments brought experiments at typical SEC concentrations to our focus. We already reported the proof of concept for the combination of a SEC system with a stateof-the-art medium resolution 1 H-NMR spectrometer at 62 MHz as a chemical sensitive detector. [17] In a following article, we reported an optimisation of SNR and SEC systems and methods to achieve highest sensitivity. [18] Work by Sabatino et al. followed a similar approach. [19] In the present article, we focus on NMR pulse sequences used in the coupling for best solvent signal suppression while retaining analyte signals.

In this work, several solvent signal suppression pulse sequences are evaluated on a 62 MHz spectrometer with only shim gradients and compared to a 43 MHz instrument featuring a (z) gradient probe, which has not been available for the 62 MHz NMR instrument so far. The general setup is depicted in Figure 1 and is the same for the 43 MHz and the 62 MHz instruments.

Figure 1: General setup for SEC-NMR measurements featuring an isocratic pump

with an external degasser unit, a manual injector, a semi-preparative SEC column, the medium resolution (MR)/benchtop NMR instrument including the flow cell and a downstream differential refractive index detector (DRI). Reproduced from Ref. [16] with permission from the Royal Society of Chemistry.

Particularly for 1 H-NMR spectroscopy in protonated solvents at medium field, a sufficient solvent signal suppression is indispensable. The methods used can be divided into two categories: (1) solvent signal suppression prior to data acquisition to make optimal use of the available analogue to digital converter (ADC). This is done by specially designed pulse sequences. ( 2 Probably the most straightforward approach to solvent suppression in static measurements is the presaturation technique by simply applying a long lowpower selective pulse on the solvent resonance. [22] For on-flow measurements, this approach is much less suitable as long repetition times are needed, limiting obtainable SNR and a strong apparent relaxation due to inflow of new solvent is observed, strongly limiting the otherwise efficient suppression. Moreover, suppressing multiple solvent signals, e.g. the two THF multiplets introduces further challenges.

A potentially faster way to saturate the respective solvent resonance is the WETapproach: [23] This pulse sequence is generally made up of four shaped pulses selective on the solvent resonance yielding an effective 90° pulse and is compensated for inaccuracies in pulse widths and mistuned probes. The use of pulsed field gradients complementing every single WET-subunit is capable of dispersing the longitudinal solvent magnetization and allows a high degree of solvent suppression even under flow conditions. [12,24] Shorter repetition times are possible with frequency selective pulse sequences of the Jump-aNd-Return (JNR) type. [25] These are combinations of constantly spaced pulses with their duration following the binomial coefficients. The simplest version referred to as Jump and Return 11 is a sequence 90°x -τ -90°x . With the solvent magnetization on-resonance, no rotation occurs during the delay and there is practically no excitation for this frequency after the return-pulse. All other nuclei maintain at least some transversal magnetization and can be detected. These methods allow an intense suppression of analyte signals but, depending on the chosen interpulse delay, only parts of the whole spectrum can be displayed and phased correctly.

Another frequency selective method is the so-called excitation sculpting or its enhancement the WATERGATE sequence. [26] This sequence is a quite popular approach and its simplest form is made up of a spin-echo sequence with two symmetrical gradients. An undesirable side-effect is the so-called J-modulation around the suppressed solvent resonance, which potentially prohibits quantitative peak analysis near the solvent signal. This was the main reason for not pursuing this approach in this work. Nevertheless, WATERGATE has been applied to coupled LAC-NMR systems at high fields. [START_REF] Hiller | [END_REF] A very appealing circumstance in NMR spectroscopy of macromolecules is the significant difference in the longitudinal relaxation time T 1 between polymers (200-500 ms at 62 MHz) and the small solvent molecules (typically 1-2 s at 62 MHz under SEC-conditions) as the solvent is degassed and triplet oxygen content substantially reduced. This fact can also be exploited for the use of solvent signal suppression. The simplest way to do so is keeping the repetition times very short, allowing the polymer FID to decay completely whereas the residual solvent FID is destroyed by a spoil gradient after acquisition before the next transient begins. This was implemented in the 90-spoil pulse sequence described in detail later.

The WEFT sequence, is a method for solvent suppression in static samples exploiting the zero-intersection of the longitudinal solvent magnetization similar to an inversion recovery experiment. [27] Theoretically, the experiment would have to be conducted in a way that an effective 180° pulse is followed by a waiting time corresponding to the effective T 1 relaxation time of the solvent molecules to achieve a fully suppressed solvent signal when exciting the system with a 90° hard pulse. However, it is possible to achieve almost full solvent suppression with shorter repetition times, since the apparent T 1 of the solvent resonance decreases with decreasing recycle delay. Exploiting this fact makes WEFT a very interesting approach for on-flow solvent suppression. To the best of our knowledge, this pulse sequence has not been reported for measurements under flow conditions, yet even though the so-called in-flow effect causes faster T 1 relaxation due to the shortened residence time. [28] Both T 1 selective approaches are not expected to yield a high degree of analyte signal suppression.

In addition to these pulse sequences, the acquired raw data can be afterwards subjected to mathematical post-processing (see Höpfner et al and Botha et al. [17,18]). The nature of SEC-measurements allows the acquisition of pure solvent spectra in the first few minutes of every run. For constant solvent signals, a numerical subtraction of the full solvent signal should be straight forward.

However, due to variations in the chemical environment of the protons in the presence of analyte molecules, slight changes in the peak shape prevent effective subtraction in this area of the elugram. Additionally, long-and shortterm fluctuations of the strong solvent signal can cause additional difficulties. It was found, that a combination of both concepts, pulse sequence and numerical subtraction, is most efficient as discussed in the main part. This work is focused on finding the best pulse sequences for this coupled method.

These pulse sequences should be able to gather as much signal to noise per time as possible and reduce unwanted solvent signals efficiently. Both must be possible in an onflow condition with a high repetition rate. The high repetition rate is used to maximize the sensitivity but sacrifices the quantifiability of the recorded spectra. This would have required the use of a significantly longer repetition time, typically 5 times the longest T 1 . [29] Special emphasis is given to compare several pulse sequences that need a designated z-gradient (WET-type) to those which don't need this gradient for typical polymer samples (PS, PMMA, PEMA) in SEC. Consequently, two spectrometers are used: a 43 MHz with a designated z-gradient and a 62 MHz without a z-gradient but with better sensitivity and resolution. An extensive set of criteria is presented to evaluate the pulse sequence performance in typical analytical problems of the method.

Materials and Methods

Used materials

As eluents tetrahydrofuran (THF, SEC-grade, stabilized with butylhydroxytoluol (BHT), Scharlau, Hamburg, Germany) and chloroform (SEC-grade, stabilized with 2-methyl-2-butene, VWR, Bruchsal, Germany) were used as received. The polymers were narrow dispersity reference materials and obtained from different sources as given in Table 1. Static NMR spectra of the polymers are shown in the supporting information Figure SI1.

NMR-Spectrometer

The NMR and SEC-NMR measurements were carried out on two different benchtop NMR spectrometers from the Spinsolve series (Magritek, Aachen, Germany). A Spinsolve 60 with a 1 H Larmor frequency of 62 MHz was equipped with an optimized single channel 1 H probe head but no gradient unit. A Spinsolve 40 with a 1 H carrier frequency of 43 MHz was also equipped with a single channel probe head but had a dedicated z-gradient with a maximal strength of approximately 160 mT/m. Both instruments carry a permanent magnet in form of a Hallbach array with a bore that allows the usage of standard 5 mm outer diameter (O.D.) tubes or flow cells. Both feature a 15-channel electrical shim system using shim terms up to the 3 rd order. The pulse sequences are freely programmable using the PROSPA language. Data acquisition on the Spinsolve 40 is performed using the PROSPA V354 software while on the newer Spinsolve 60 the Spinsolve Expert V1.21 software is available and was used. The pulse programs cannot be exchanged directly between the two software systems as some commands are inconsistent and the instructions need to be recompiled. A cross-compiler function is available.

SEC-Setup

In connection with both NMR spectrometers, an Agilent 1260 Infinity SEC system (supplied by PSS, Mainz, Germany) is used for SEC measurements. It consists of an in-line degasser, isocratic pump, manual injector (Rheodyne 7725i, 500 µL sample loop, 6 ports) and differential refractive index (DRI) detector. The chromatographic separations were performed with PSS SDV linear M semipreparative columns (300 x 20 mm I.D., 10 µm particle size, mixed bed), one for THF and one for chloroform. One NMR spectrometer at a time was coupled as an additional detector to the SEC system between the column and the DRI detector using a custom-made flow cell (see Figure 1). The flow cell has an active region in the area of the NMR coil with 3.4 mm I.D. over 15 mm length. The flow cell has been described in more detail elsewhere. [18] The flow rate is 1.0 mL/min, leading to a run time of 85 min for one experiment. The sample concentrations were 2 and 4 g/L for Spinsolve 60 and 40, respectively to account for differences in sensitivity as discussed in the evaluation. This corresponds to an injected mass of either 1 or 2 mg (0.5 mL injected volume). A column loading of 2 mg is already close to the overloading limit of the semi-preparative column (depending on the exact conditions). [18]

SEC-NMR measurements

Before the combined measurements, the magnet was shimmed on the flowing eluent giving a typical line width for chloroform of 0. dB (832 Hz) and -30.4 dB (1189 Hz) and a duration of 82 ms for each as given in previous work for this instrument. [12] The frequency of the soft pulses was slightly offset from the solvent peak by -8.5 Hz and -10 Hz for chloroform and THF, respectively (see as well application case 3 for the optimization of the offset). The z-gradient pulses followed each soft pulse with a duration of 2 ms and halving the power for each consecutive gradient pulse starting from 85 % of maximum gradient strength. The WET-180 block, Fig. 2 (c), is similar to the WET block but after the last soft pulse a 180° hard pulse is added, and the strength of the last soft pulse is reduced to -39.6 dB (412 Hz).

The excitation block can be a simple 90° hard pulse, see Figure 2 (e). The second option is a composite 90° pulse consisting of four 90° pulses with an interpulse spacing of 5 µs, see Figure 2 (f). The last option is a 1D-NOESY block: two 90°h ard pulses (spacing 10 µs) are followed by the mixing delay adjusted in this work to 6 ms and a final 90° hard pulse, see Figure 2 (g).

In the JNR1331 pulse sequence, see Figure 2 (d), the preparation and excitation part are combined by a train of four hard pulses with their respective length given by the binomial sequence 1a, 3a, 3a, 1a where 8a is the duration of one 90° pulse or 12 µs at 62 MHz. All interpulse delays τ JNR are the same and adjusted to focus the area of interest in the spectrum. [25] In this work, focusing on the aliphatic protons in chloroform a delay of 2 ms is used.

Read-out blocks were either a simple FID with 100 µs acquisition delay, see Figure 2 (h), 2k data points and 200 µs dwell time giving a length of 410 ms. For the weaker suppression techniques, a 90-spoil block, Fig. 2 (i), was used as the FID had not decayed to noise level during the acquisition period. For the spoil block, after the FID as described above, a 5 ms waiting time was added followed by a crusher gradient for 20 ms using the linear terms of the shim coils and further 5 ms of waiting time. The strength of the crusher gradient was 5 000 a.u.

which roughly corresponds to 0.5 mT/m.

Numerical data processing

The 2D SEC-NMR spectra evaluation was automated using a self-written MATLAB script. The principle evaluation procedure has been disclosed in previous publications, [17,18] but some specific procedure was found necessary for data produced by the new pulse sequences employed in this work. The 2D raw data is in the form of FIDs which are in the first step apodized (Gaussian lb 1.8 Hz) and zero filled to 16k. After Fourier transform, the spectra are corrected in a second step for 0 th and 1 st order phase offsets. For the pulse sequences with a strong, undistorted solvent signal remaining (90-spoil) the 0 th order phase was corrected by minimizing the sum of disperse signal of the solvent peak(s). In the other cases, a constant 0 th order phase was used. In both cases, a constant 1 st order phase correction was used, which was mostly dependent on the solvent used.

The next step depends on the pulse sequence.

If the pulse sequence produced a (residual) solvent signal of stable intensity the data was smoothed along the elution dimension as a third step and a reference spectrum is subtracted as the forth step. This was used for all pulse sequences except WEFT-90-spoil and JNR1331. For JNR and WEFT-based sequences the third step was subtraction of a reference spectrum (with variable amplitude) and the forth step was smoothing along the SEC elution dimension. This order of steps was found to be beneficial due to strong fluctuations in solvent intensity but not spectral composition of the polymer.

The smoothing along the SEC dimension is done by applying a Gaussian filter with a bandwidth (sigma) of 32 s at each frequency / spectral point. For the reference subtraction, a reference is computed by averaging analyte free spectra in the beginning of each run. The reference is subtracted from each spectrum with an adjusted intensity. The solvent intensity of the reference spectrum is adjusted to the highest solvent signal in the current work spectrum.

As the fifth step, the analyte peaks are automatically detected using a threshold method. The sixth step is a baseline correction along the elution (SEC) dimension performed by adjusting a 2 nd order polynomial to the analyte peak free region for each frequency / spectral point. This corrects for slow drifts in the residual solvent signals and other constant signals. Finally, statistics for the solvent and analyte, are calculated as detailed below and the data is visualized.

-14 -

Results and discussion

Evaluation criteria for solvent suppression techniques

The theoretically best pulse sequence would eliminate all solvent signals completely while not hampering the sensitivity and selectivity of the analyte's signals of interest in all parts of the spectrum. While a "perfect" method is not known and probably does not exist, it is worthwhile to discuss criteria to benchmark the existing methods against these ideals not only in a qualitative way but also by a quantitative approach. The criteria proposed are solvent signal reduction (desirable), analyte signal reduction (undesirable), analyte signal broadening (undesirable) and available baseline (desirable) as introduced herein.

The available baseline is a concept inverse to solvent signal width but a better handled concept in our case as discussed below. These criteria are in general universal, but to be quantified they need to be assigned and measured for every given case of solvent, analyte and setup and more importantly the weighting is largely dependent on the analytical problem at hand. Here, we focus on the application to the hyphenation of medium resolution NMR to SEC and will describe where we see the optimum in this case.

The first desired criterion is a low intensity solvent signal to prevent receiver The selectivity in the NMR dimension is benchmarked by the analyte's peak width. The analyte's peak width is quantified by the full width at half height of an analyte (FWHM(A)) peak well separated from the solvent signals. In this publication, we are only concerned with the signal width in the spectroscopic dimension, for an in-depth discussion for the width in the chromatographic dimension see Botha et al. [18] As a counterpart, the peak width of the solvent is monitored as well and is to be optimized to be as low as possible. Typically, solvent peak FWHM decreases when the intensity is suppressed. However, it was found that many effective suppression methods such as WET based approaches distort the solvent signal strongly, resulting in a multitude of irregular peaks. Determination of FWHM on these residual (multiplet) signals is ill defined and very error prone, consequently a different approach was utilized. As basic idea, a good solvent suppression retains as much as possible of the baseline without solvent artefacts and fluctuation. This region where analyte peaks can clearly be identified and quantified as they develop and vanish over time is called available baseline here.

A method is needed to quantify how much of the baseline is obscured by solvent fluctuations too high to analyse the analyte peaks there.

Therefore, we propose a fluctuation based criterion relying on the 2D nature of the obtained data (Figure 3). First, the standard deviation (SD) of each NMR elugram, i.e. all 1-D cuts along time at one NMR resonance frequency of the 2D data set, is calculated as shown on the top of Figure 3. Then, the SD values are plotted as a function of chemical shift (other 1-D dimension). This yields a spectrum-like plot, where the solvent peaks and constant background are displayed as their fluctuation over time (bottom of Figure 3). The regions with high fluctuations are improper for any quantitative analysis. Consequently, a high threshold of 5 times the SD in the signal free region is defined and only the parts where the SD is higher are defined as unusable, denoted by the remaining (solvent peak) width, W rem . This value should be as low as possible. The threshold value is arbitrary but was found to work reliably for the given problem. If an analyte peak is close to the solvent peak it can obscure this evaluation. In this case, W rem is calculated only from the solvent peak centre to the far side of the peak and doubled (for CHCl 3 as solvent) or calculated on the other solvent peak (for THF as a solvent). These four criteria have been evaluated automatically for all pulse sequences and samples to guarantee reliability. In this method, a strong importance is given to the strength of analyte signals (SNR A ) as their intensities due to dilution are already low. The SR S is somewhat less important as will become clear in the forthcoming results. W rem is important for some possible applications and also decisive in identifying the best method.

Comparison of the performance of 43 and 62 MHz spectrometers

In the scope of this work, two different spectrometers were used as explained in the introduction due to their individual benefits for using solvent suppression pulses sequences. The 62 MHz spectrometer provided better spectral resolution while the 43 MHz spectrometer offers more flexibility in terms of pulse sequences (e.g. WET) due to a dedicated z-gradient. In this section, the results of the two spectrometers are compared by measuring SEC-NMR on PMMA30k in CHCl 3 using the simple 90-spoil sequence. The results are shown in Figure 4 and the results

for the numerical criteria are given in Table 2. or PC hardware which was expected to result in a SNR by factor (500/650) 0.5 = 0.88 worse for the Spinsolve 40. [START_REF] Skoog | Principles of Instrumental Analysis[END_REF] The doubling of injected mass for the Spinsolve 40 instrument will also double SNR as shown in previous work. [17] Therefore, the expected SNR (at higher injected mass) should be approximately similar for the Spinsolve 40 and for the Spinsolve 60 in the scope of the experiments for the higher loading as described. Comparing the two spectrometers, the SNR A for the Spinsolve 40 is 16 % higher than expected. This might be a feature of the different magnet design or material or achievable receiver gain. The NMR peak width is similar, the separation on the baseline is improved for 62 MHz NMR as is easily visible in Figure 4. The suppression is better on the Spinsolve 60 in both the residual solvent strength and the recovered baseline. These values serve as a reference to benchmark against all further pulse sequences and solvent / analyte combinations discussed in the next sections.

Application case 1: Measurements of PMMA in chloroform

The first and easiest case to be discussed is a full baseline separation between the solvent and analyte's signals. Typical examples would be the spectra of PMMA in chloroform where all resonances are well separated or the aliphatic resonances of PS in chloroform and the aromatic resonances of PS when recorded in THF. We consider the first case in more detail, while some aspects of the other cases will be included at the end of this section.

Results for SEC-NMR data of PMMA30k in chloroform measured on Spinsolve 40 are shown in Figure 5 for the pulse sequences 90-spoil (same dataset as in previous section) and WET-CP. The results of the numerical evaluation are given in Table 3. When comparing SNR values for the two data sets, it has to be noted that the repetition time for the WET-CP pulse sequence is longer (1 000 ms) than for the 90-spoil sequence (650 ms) due to the duration of the four (long) soft pulses. The obtainable SNR A per unit time is predicted to be lower for WET-CP by a factor of (1000/650) 0.5 = 1.24. This is in very close agreement to the measured SNR values which show a reduction of 25 % for WET-CP. The solvent suppression is already very good for the WET-CP sequence before numerical subtraction (SNR of solvent signal is half of the highest analyte peak) and consequently only improved by a smaller amount (factor 2) by the numerical methods. The width of the distorted baseline W rem is only one third for WET-CP when compared to the 90-spoil sequence. The influence on the initial solvent peak width is negligible. In conclusion, WET-CP offers a slightly reduced sensitivity but a great increase in solvent suppression capability.

Application case 2: Measurements of PS in chloroform

The given numbers for SNR (Table 4) are given for different analyte concentrations.

The injected mass was doubled for JNR1331 to obtain a higher (doubled) SNR. WET-CP approach apparently is the most powerful method suppressing the solvent intensity by a factor of 590 without affecting the analyte resonances significantly more than the other WET sequences. However, even though the raw SNR S of WET-180-NOESY is an order of magnitude higher than for the other WET sequences, the numerical data treatment yields a solvent SNR S , which is even better. This is a result of stable remaining solvent signal. WET and its derivatives are suitable pulse sequences to implement instrumental solvent signal suppression in SEC-NMR at 43 MHz using a pulsed z-gradient. For analyte signals in close spectral proximity to the strong solvent signal, WET-180-NOESY is the best technique for unperturbed analysis. However, the 62 MHz spectrometer is not equipped with a gradient coil, which leads to the necessity to find suppression approaches for this instrument without a need for it.

WEFT is a modified version of the Inversion Recovery pulse sequence using a combination of a 180° and a 90° pulse separated by a waiting time τ WEFT , with the waiting time adjusted to the apparent T 1 relaxation time of the solvent. The apparent T 1 relaxation time in the WEFT experiment is shortened by measuring in rapid sequence far from equilibrium and, therefore, different from T 1 times in equilibrium. In the case of chloroform suppression, it is necessary to implement a spoil or crusher gradient after acquisition similar to the one in the 90-spoil sequence as the solvent signal FID has not decayed at the end of the acquisition period. Otherwise, the suppression is not sufficient and subsequent mathematical data treatment is severely disrupted. A comparison of SEC-NMR data at 62 MHz for PS in CHCl 3 is given in Table 6. 0.24 0.17 (*) SR S (CHCl 3 ) for WEFT-spoil is calculated using raw solvent SNR with a repetition time of 0.5 s instead of 1 s and is therefore slightly overestimated here.

The WEFT-spoil requires a repetition time twice as long as the 90-spoil, which makes the resulting SNR (without respect to time) not directly comparable. For WET, no useful information on the aliphatic protons could be extracted and the width of unusable baseline was similar as above (0.7 ppm).

The WEFT sequence (Figure 9, bottom left) yielded better results. A strong suppression of both THF multiplets by more than a factor of 1000 is achieved while the PS signal can still be partly recovered. The PS signal is not smeared out and has an intensity profile identical to the other PS peaks. This is supported by the much smaller width of unusable baseline (0.3 ppm). However, even with the WEFT method, signals that directly coincide with the strongest part of the THF multiplet are lost. While a full recovery by this method is theoretically possible, in practice however, the THF signal fluctuations are much stronger than the signal of interest and the numerical methods also have limitations. In this case, no spoil pulse addition to the WEFT sequence was found to be necessary as the FID quickly decayed to zero within the acquisition time.

As a second example for the strong overlapping case PEMA in THF is considered.

This polymer features a complex spectrum in the aliphatic region with resonances over the range of 0.5 to 2 ppm across the full width of the THF peak The results for both methods in Figure 10 look similar but some differences are found. In both cases, the remaining solvent signals are quite narrow. In the case of WEFT however, the remaining solvent signal intensity is much weaker.

Additionally, the obscured baseline is smaller for WEFT. For WEFT, less and The online coupling of SEC to 1 H-NMR allows for an uniquely full picture of size distribution and chemical entities in a sample of large molecules (e.g. polymers).

The method has been established and optimized in terms of chromatography equipment in previous work, [17,18] while this paper serves as a guideline to use the best pulse sequences for typical analytical problems and shows results which can be achieved. Strong solvent signal suppression can be achieved while retaining a considerable signal-to-noise ratio of the analyte peaks. In future work, the usage of a thermoset column will reduce solvent fluctuations and allow better numerical subtraction. The even higher fields which are available now in benchtop hardware can be used together with gradients. While the evaluation of pulse sequences herein will still be valid, the increased sensitivity and better suppression will allow more information on the compounds under investigation to be extracted.

  ) post acquisition data treatment by numerical solvent signal subtraction in the data. The use of solvent suppression pulse sequences will be discussed first. It is crucial to suppress the solvent signals to some extent prior to acquisition due to several reasons.[20] A non-suppressed solvent resonance is often more than 1000 times more intense then the resonances of interest which leads to poor utilization of the receiver electronics and the ADC limiting the relative sensitivity. Moreover, effects like radiation damping or the so-called faraway water effect cause peak broadening which shrinks the quantifiability of signals near the solvent resonance.[21] Suppression techniques can be grouped by their working principle. The first group of methods relies on selective excitation and saturation of the solvent signal prior to acquisition by methods such as presaturation or Water suppression Enhanced through T 1 effects (WET). The second group of pulse sequences uses frequency specific editing of the signals of interest by removing certain frequencies by, e.g. Jump-and-Return pulse sequences or WATER suppression by GrAdient Tailored Excitation (WATERGATE) or excitation sculpting. The third group considered is the exploitation of differences in T 1 relaxation between solvent and analyte such as 90-spoil (see below), progressive saturation or inversion recovery / Water Eliminated Fourier Transform (WEFT).

7

 7 Hz and 0.4 Hz for Spinsolve 60 and 40, respectively. The transmitter frequency is set to the highest solvent peak and the receiver phase is optimized manually. The values are kept constant over the course of one SEC-NMR run. The NMR data is acquired with one of the pulse sequences described below. The receiver gain is optimized for each combination of pulse sequence and instrument (62 or 43 MHz) separately. A noninteractive program collects and co-adds 4 transients into one FID and continuously records and saves FIDs over the course of the SEC run without any interdelay. Over the length of one SEC experiment between 2600 and 1300 FIDs are collected depending on the length of the pulse sequence (see below). The NMR spectra are, therefore, independent measurements, but give a 2D data set where chemical shift/FID time is one dimension and eluent time the second. The full 2D data set is collected during the experiment and evaluated later as described below.NMR pulse sequencesSeveral different pulse sequences were used to test the solvent suppression efficiency of each sequence. The sequences were built by combining different preparation, excitation and read-out blocks as shown in Figure2to obtain the tested sequences: 90-spoil, JNR1331, WEFT, WEFT-spoil, WET-90, WET-CP and WET-180-NOESY. All pulse sequences were used inside a similar master program that recorded continuously as described above. The 90° hard pulse length was determined to be 12 µs and 6.35 µs for Spinsolve 60 and 40, respectively, at 0 dB pulse damping (full power, 1 mW). The 180° hard pulses were double the length of the 90° pulse.

Figure 2 :

 2 Figure 2: The building blocks that were combined in this work to obtain solvent suppression pulse sequences can be grouped in the categories of preparation, excitation and read-out blocks where (d) combines features of the former two). The blocks are named as follows: (a) WEFT-block, (b) WET-block, (c) WET-180block, (d) JNR1331-block, (e) 90°-pulse read block, (f) composite pulse read block, (g) 1-D-NOESY-block, (h) FID-block and (i) FID-spoil-block. For details onthe blocks see text.

  overloading and other detrimental effects discussed above. However, beyond a certain threshold, e.g. same/lower intensity as analytes' signals, a further reduction of solvent signals are not always needed. This goal can be quantified directly by the solvent signal-to-noise ratio (SNR S ) given for the major solvent peak. Here, two values are given for each dataset: the raw SNR S is the value before numerical solvent suppression (but including suppression by the pulse sequence) and the final SNR S is the value including numerical suppression. To compare different methods, the suppression ratio SR S is defined as the ratio of unperturbed solvent signal intensity divided by suppressed intensity. The unperturbed intensity is taken from a reference dataset with the same repetition time and injected mass using the 90-spoil sequence. The SR S value should be as high as possible.As the counterpart to the first criterion, the analytes' signals should be suppressed as little as possible. This is a very important point in our application as the signals are already weak. For quantification of the suppression of analyte signals, the final signal-to-noise ratio of the analyte (SNR A ) is given for the peak(s) of interest. This value should be as high as possible.

Figure 3 :

 3 Figure 3: Schematic of the evaluation of the unobstructed baseline by the solvent signal here at 7.25 ppm. The parameter W rem is calculated from the area in the lower graph where the standard deviation is above the threshold (red lines). For further explanation see the text.

Figure 4 :

 4 Figure 4: SEC-NMR results of PMMA30k in CHCl 3 on the 62 MHz (left) and 43 MHz (right) spectrometers recorded by using the 90-spoil sequence (see Figure 2

(

  e+i)). On top, contour plots of the full region of polymer elution are given with 15 evenly spaced intensity lines (from 0 to maximum analyte peak height) and below the spectra extracted at the highest points of the polymer signals are shown (highest analyte peak is normalized to 1). The same scaling is applied to all subsequent plots as well. For static spectra of the compounds under investigation see the supporting information.Since the two spectrometers feature different magnet setups, field strengths and electronics, it was necessary to evaluate the reachable SNR first. To counter the expected drop in SNR, the injected mass was doubled to 2 mg PMMA for the Spinsolve 40. Several factors influence the reachable SNR. According to a simple approximation, SNR increases with an exponent of 1.5 with respect to the B 0 field for a constant number of scans. The SNR for the Spinsolve 60 should consequently be higher by (62/43) 1.5 = 1.73. Additionally, it was not possible to reach the same repetition time on the Spinsolve 40 due to slower spectrometer

Figure 5 :

 5 Figure 5: SEC-NMR results of PMMA30k in CHCl 3 recorded on the 43 MHz spectrometer using the 90-spoil sequence (left, see Figure 2 (e+i)) and WET-CP (right, see Figure 2 (b+f+h)).

  second case of interest is partial overlap between the solvent resonances and the signals of the analyte. For polystyrene in chloroform, the aromatic signals are two broad multiplets with the upfield one centered at 7.0 ppm close to the chloroform resonance at 7.26 ppm which has a width at 0.55 % of ca. 15 Hz at 62 MHz and onflow conditions. For polystyrene in chloroform, the aliphatic signals (0.9 -2.1 ppm) are baseline separated from the solvent and can be treated in a similar fashion to the PMMA signals described in the section above. Here, first pulse sequences without gradient (62 MHz spectrometer) are presented: Jump and Return and 90-spoil and then those with gradient (43 MHz spectrometer) WET type sequences. The results of the Jump-and-Return pulse sequence, more precisely the 1331 version,[25] are given in Figure 6 and compared to the results of 90-spoil sequence.

Figure 6 :

 6 Figure 6: SEC-NMR results of PS30k in CHCl 3 recorded on the 62 MHz spectrometer using the 90-spoil sequence (left, see Figure 2 (e+i)) and Jumpand-Return 1331 (right, see Figure 2 (d+h)). Please note that due to a change of the column, the same analyte does not elute at exactly the same volume in both measurements.

W

  rem (CHCl 3 ), width of distorted baseline [ppm] 0.24 0.31 A suppression of the chloroform resonance by a factor of 490 was possible considering the pure pulse sequence and the resulting solvent intensity is on the same order of magnitude as the 13 C-satellites. The eluent spectra in JNR1331 exhibit a very narrow residual linewidth and gave the impression that this method yields a very good peak base width. Objectively considered, this criterion does not prove useful here because signals in parts of the spectrum, other than the adjusted optimized positions, are out of phase and not usable. In combination with a suppression of the analyte signals by SR A of maximum 0.29 for the well displayed signals, the JNR1331 approach does not fulfil the requirements of SEC-NMR. Increasing the interpulse delay allows signals closer to the suppressed resonance to be displayed.[25] In the case of PS, τ JNR = 30 ms allows displaying the aromatic PS signals whereas the aliphatic signals are not visible any more. However, the SR A in this case is around 20 and the repetition time increases strongly in these experiments (measurements not shown here). The long repetition time also decreases the time normalized suppression efficiency for the solvent, which results in a SR s of roughly 10. Overall, the performance of the JNR1331 pulse sequence requires a lot of knowledge about the sample to allow parameteroptimization in advance and is not a viable option for this application. The resonances of the aromatic PS protons in chloroform are an excellent example to investigate the case of analyte signals partially overlapping with the feet of the unperturbed solvent resonance.

Figure 7 :Fig. 2 (

 72 Figure 7: SEC-NMR results of PS30k in CHCl 3 recorded on the 43 MHz spectrometer using the 90-spoil (top left, see Figure 2 (e+i)), WET-90 (top right, Fig. 2 (b+e+h)), WET-CP (bottom left, Fig. 2 (b+f+h)) and the WET-180-NOESY pulse sequence (bottom right, Fig. 2 (c+g+h)).

Figure 7 5 .

 75 Figure 7 features SEC-NMR results of PS30k in chloroform for 90-spoil (same data set as above) at 62 MHz, WET-CP, WET-90 and WET-180-NOESY at 43 MHz using a pulsed z-gradient for the WET derivatives. For comparison of SNR values, all SEC-NMR runs, in Figure 7, were performed using a repetition time of 1 s and 2 mg injected sample mass. The corresponding numerical results are given in Table 5. From the solvent signal suppression point of view of the raw data, the -25 -

  this case, other evaluation criteria like the residual solvent peak base width, gain further importance as a measure of the impact of the solvent resonance on the analyte signal intensity. The WET-180-NOESY sequence achieves a limited solvent signal suppression in the raw data compared to the other tested WET derivatives but the residual chloroform peak width is drastically reduced. This results in an excellent retention of the analyte signals without major distortions. The effect of the signal intensity of the broad feet of the solvent resonance can be observed in the 1D-spectra shown in Figure7. Due to the short repetition times (much shorter than 5 times T 1 ), it is not possible to obtain the expected integral ratio of 2:2:3 for the respective protons of PS. All our approaches have been designed to yield maximum SNR and displayable analyte signals rather than quantitatively integrable spectra.

Figure 9 :

 9 Figure 9: SEC-NMR results of PS30k in THF recorded for different pulse sequences and setups: on the left data is shown which was recorded on the Spinsolve 60 and on the right for Spinsolve 40. The two examples on top were recorded using the 90-spoil sequence (see Figure 2 (e+i)) and on the bottom left using WEFT (Fig. 2 (a+e+h)) and WET-CP suppressing the THF signal at 1.84 ppm only (bottom right, Fig. 2 (b+f+h)). For static spectra of the compounds under investigation see the supporting information.

  The results for 90-spoil with signals close to the THF peaks (Figure9, top left and right) are unsatisfactory on both spectrometers: the signals of the aliphatic protons can hardly be found: shoulders at the THF peak at 1.8 ppm can be seen, which seem to be indicating the presence of the polymer signals but in the contour plot the signals are smeared towards higher elution time. This is less a representation of the initial polymer signals but probably an effect of the slight increase of solvent relaxation caused by the eluting polymer. This is shown as well by the large part of unusable baseline of almost 0.7 ppm. Therefore, these signals are concluded to be unusable for identification or quantification. The aromatic protons are well recovered as they correspond to case 1 and can serve as an estimate of the possible SNR of each sequence. Surprisingly, the usage of WET-CP (Figure9, bottom right) suffers from a similar problem, as the polymer signals are not well visible. The THF peak at 1.8 ppm is highly suppressed by this pulse sequence but the polymer resonances in close proximity (by frequency) are suppressed as well and only a weak, elongated artefact remains. Different soft pulse offsets (as discussed in the experimental section) were used but none was found to suppress the solvent with less impact on the PS signals.

at 1 .Figure 10 : 2 (

 1102 Figure 10: SEC-NMR results of PEMA78k in THF recorded for different pulse sequences recorded on the Spinsolve 60: on the left by 90-spoil (see Figure 2 (e+i) and on the right by WEFT (Fig. 2 (a+e+h).

  schemes for several narrow molecular weight distributed polymers and typical solvents are shown. The examples are clustered into three groups according to the closeness of the analyte peak of interest to solvent peaks obscuring it (baseline separated, partial overlap, strong overlap). For each group, case studies are conducted and focus on selecting the best pulse sequence for this challenge. The selection is based not only on inspection of the spectra but on a set of objective criteria. These criteria evaluate numerically the solvent suppression in terms of solvent signal reduction, baseline distortion and best possible analyte signal-to-noise ratio. As pulse sequences for solvent suppression, high-field approaches like WET, WEFT and JNR1331 are compared to a simple and efficient one-pulse scheme. A new method is presented to evaluate quantitatively the area of distortion free baseline, which uses the advantage of a 2D data set. High analyte SNR is most important in this method due to the very low concentrations encountered after chromatographic separation.In the case of baseline separation between signals of interest and solvent signals, a simple one-pulse approach, including a gradient spoil pulse, was found to yield the best results. Solvent suppression before acquisition is sufficient to avoid ADC overflow and this simple method gives the highest repetition rates and consequently the highest analyte signal-to-noise ratio. Upon partial or full overlap between solvent and analyte signals, better suppression by the pulse sequence is required which can be achieved using WET or WEFT schemes. For overlapping signals, WET faces problems at low fields due to its broad suppression window which affects the analyte signals in proximity as well. When using large molecules WEFT has benefits as it can exploit the T 1 relaxation time difference to the solvent small molecules. A direct comparison of WET/WEFT was not included as a 62 MHz spectrometer with z-gradient was not available. At higher fields the advantage can go towards WET, which will be interesting to investigate as better equipment becomes available.The favourite suppression schemes found in literature for high-field couplings of SEC-NMR or LAC-NMR are dominated by WET. However, for the special application under investigation here, different sequences are utilized with success as well. Different considerations are found to be important at lower fields. Here, obtaining maximum signal for the analyte signals is paramount and the often already broad signals profit less from frequency specific suppression methods.

Table 1 :

 1 Designations, sources and characteristics of the polymer samples used in this study.

	Designation Polymer	Sources	M w / kg/mol	Dispersity Đ / ---
	PS30k	Polystyrene	Polymer Labs	30.9	1.02
		(PS)
	PS90k	PS	In-house synthesis 90.0	1.10
	PMMA30k	Polymethyl	In-house synthesis 31.0	1.08
		methacrylat
		e (PMMA)
	PEMA78k	Polyethyl	In-house synthesis 78.0	1.12
		methacrylat
		e (PEMA)

Table 2 :

 2 Evaluation criteria for solvent suppression in SEC-NMR of PMMA30k.

	Spectrometer ( 1 H)	62 MHz	43 MHz
	Inj. mass polymer [mg]	1.00	2.00
	Repetition time [s]	0.50	0.65
	SNR A (-O-CH 3 )	127	150
	FWHM A (-O-CH 3 ), [Hz]	6.42	6.23
	ratio of solvent SR S (CHCl 3 ), suppression	194	62.0
	(Raw and) Final SNR S (CHCl 3 )	(32 800) 169	(35 500) 570
	distorted baseline [ppm] W rem (CHCl 3 ), width of	0.26	0.60

Table 3 :

 3 Evaluation criteria for solvent suppression in SEC-NMR of PMMA30k inCHCl 3 (well separated case) comparing two pulse sequences for solvent suppression used on the 43 MHz spectrometer.

	Pulse sequence	90-spoil @ 43	WET-CP @ 43
		MHz	MHz
	Inj. mass polymer [mg]	2.00	2.00
	Repetition time [s]	0.65	1.00
	SNR A (-O-CH 3 )	150	121
	FWHM A (-O-CH 3 ), [Hz]	6.23	6.54
	ratio of solvent SR S (CHCl 3 ), suppression	62.0	987
	(Raw and) Final	(35 500) 570	(55.0) 29.0
	SNR S (CHCl 3 )
	distorted baseline [ppm] W rem (CHCl 3 ), width of	0.60	0.19

Table 4 :

 4 Evaluation criteria for solvent suppression in SEC-NMR of PS30k in

	CHCl 3 (well separated case for aliphatic PS protons) comparing 90-spoil and
	JNR1331 at 62 MHz and 0.5 s repetition time.
	Pulse sequence	90-spoil @ 62 MHz JNR1331 @ 62 MHz
	Inj. mass polymer [mg]	1.00	2.00
	SNR A (PS aliphatic)	39.0	23.0
	ratio of solvent SR S (CHCl 3 ), suppression	142	549
	(Raw and) Final SNR S (CHCl 3 )	(36800) 260	(67) 67

Table 5 :

 5 Evaluation criteria for solvent suppression in SEC-NMR of PS30k in

	CHCl 3 (partially overlapping case) at 43 MHz, comparing the WET-90, WET-CP,
	WET-180-NOESY and the 90-spoil pulse sequence (repetition time 1 s for all). SNR
	values for the solvent denoted by * are only shoulders on the analyte signals and
	therefore difficult to quantify.
	Pulse sequence 90-spoil	WET-90	WET-CP	WET-180-NOESY
		@ 43 MHz	@ 43 MHz	@ 43 MHz	@ 43 MHz
	Inj. mass polymer	2 .00	2 .00	2.00	2.00
	[mg]
	SNR A (PS aliphatic)	42.0	36.0	35.0	23.0
	suppression ratio SR S (CHCl 3 ),	43.0	592	327	1070
	of solvent
	(Raw and) Final	(32 000) 736	(309)
	SNR S (CHCl 3 )

Table 6 :

 6 Evaluation criteria for solvent suppression in SEC-NMR of PS30k in

	CHCl 3 (partially overlapping case) at 62 MHz, comparing the WEFT-spoil and the
	90-spoil pulse sequence.
	Pulse sequence	90-spoil @ 62 MHz	WEFT-spoil @ 62 MHz
	Inj. mass polymer [mg]	1 .00	1 .00
	Repetition time [s]	0.50	1.00
	SNR A (PS aliphatic)	39.0	33.0
	ratio of solvent SR S (CHCl 3 ), suppression	142	466(*)
	(Raw and Final)	(36800) 260	(1900) 79.1
	SNR S (CHCl 3 )
	W rem (CHCl 3 ), width of distorted baseline [ppm]

Table 7 :

 7 Evaluation criteria for solvent suppression in SEC-NMR of PS30k in THF

	-30 -

(strong overlap case) comparing three pulse sequences for solvent suppression used on the 43 MHz and 62 MHz spectrometers. H A and H B represent the aromatic ortho-and para-and the meta-protons of polystyrene, respectively.

Table 8 :

 8 Evaluation criteria for solvent suppression in SEC-NMR of PEMA78k in

	THF (full overlapping case) at 62 MHz, comparing the WEFT-spoil and the 90-spoil
	pulse sequence.
	Pulse sequence	90-spoil @ 62 MHz	WEFT-spoil @ 62 MHz
	Inj. mass polymer [mg]	2.05	2.05
	Repetition time [s]	0.5	1.0
	SNR A (PEMA aliphatic @	93(*)	69
	1.4 ppm)
	ratio of solvent SR S (THF), suppression	102	1800
	(Raw and Final)	(61000) 600	(60) 34
	SNR S (CHCl 3 )
	distorted baseline [ppm] W rem (CHCl 3 ), width of	0.9	0.3

(*) This value is overestimated due to the only partially suppressed solvent peak close by.

Acknowledgements

The authors would like to thank the German Science Foundation (DFG) for founding this work within the grant SFB 1176 "Molecular Structuring of Soft Matter," project Q1 and the AiF-iGF for funding this work within the grant 19925N. The authors would also like to thank K.-F. Ratzsch and J. Kübel (KIT) for helpful discussions, A. Bucka (KIT) for assisting in MATLAB programming and M. Heck (KIT) for providing some polymer samples. J. F. warmly thanks Sandrine Bouchet for an unfailing assistance.

According to an increase of SNR with the square root of the number of scans, it is expected to be factor a (1000/500) 0.5 = 1.41 higher for 90-spoil, which is approximately the case here, indicating no substantial analyte suppression. The great benefit of this pulse sequence is the combination of a solvent signal suppression in the raw time data prior to digitization by SR s of up to 20 and very narrow residual solvent linewidths already before mathematical post-processing.

After solvent subtraction, the residual solvent SNR is in a comparable range as is expected for the WET pulse sequences. As depicted in Figure 8 it was possible to achieve a remarkable separation of the aromatic PS signals from the CHCl 3 resonance in the resulting spectra, which is due to the pronounced reduction of the "chloroform feet". Small residual solvent signals remain as T 1, solvent slowly changes over time. This will be tackled in future by usage of a thermoset column compartment to reduce temperature fluctuations in the incoming analyte.

In contrast to the frequency selective approaches like WET and JNR, the WEFT sequence does not require prior optimization of any offsets from the carrier frequency to prevent suppression of analyte resonances close by. The discrimination here is done with respect to T 1 relaxation times, which must be present for this method to work. The ratio T 1, polymer /T 1, solvent << 1 is true for macromolecules in low molecular solvents (such as in SEC-NMR) but it should be noted that this approach cannot straightforwardly be applied to chromatography of small molecules. [START_REF] Guo | [END_REF] However, another appealing feature is the possibility of suppressing more than one solvent signal at a time as long as T 1 for all solvent signals is equivalent. This will be discussed in the following section in more detail.

Supporting Information


HEN a circuit breaker detects a fault current, it opens the electrical circuit and an arc appears. The electrical network viewed from this circuit breaker can be modeled as follows:

L s d I + R s I = U s -U arc (1)
Fig. 1. Current and voltage represented during a breaking process. The prospected current is the short-circuit current obtained without circuit breaker. The limited current is the current obtained because of the electrical arc. When the arc voltage is greater than the system voltage, the current is limited.

3-D Magnetostatic Moment Method Dedicated to Arc Interruption Process Modeling

Loic Rondot 1 , Olivier Chadebec 2 , and Gérard Meunier 2

1 Schneider Electric, Strategy and Innovation, 38TEC, Grenoble 30050, France 2 Electrical Engineering Laboratory, UMR CNRS 5269, Grenoble 38400, France

The behavior of arcs in circuit breakers is affected by interactions of several physics. To simulate this, a 3-D modeling system has already been developed. It considers the arc fluid dynamics, the radiation, the plastic vaporization, the current flow within the electrodes and the plasma, and the magnetic field generated by currents. However, to consider the influence of ferromagnetic regions, the current simulation model has to be extended. The magnetic moment method (MoM) is an integral method where only active regions (i.e., ferromagnetic regions in our context) are meshed. Thus, it is particularly well adapted for breaking modeling with a few ferromagnetic regions in comparison with the whole air region. In this paper, circuit breaker principle is explained, the arc interruption modeling is described, a nonlinear MoM model based on a point matching approach is discussed, and finally an arc interruption process modeled with the introduction of ferromagnetic regions is presented.

Index Terms-Circuit breakers, magnetostatics, moment methods, nonlinear equations, plasma simulation.

I. INTRODUCTION

W dt

where L s , R s , and U s are, respectively, the inductance, the resistance, and the voltage of the system and U arc is the arc voltage.

If the arc voltage (U arc ) rises above the system voltage (U s ), the time derivative of current becomes negative and the current can decrease. This process is so-called the current limitation [START_REF] Brice | Review of technologies for current-limiting low-voltage circuit breakers[END_REF] and is shown in Fig. 1.

Because of the limitation, the maximum current is reduced and the zero current is obtained sooner.

To increase the arc voltage, it is necessary to lengthen and split the arc into series of arcs by metallic splitter plates. Thereby, the arc voltage is increased by multiple cathode and anode falls [START_REF] Lindmayer | The process of arc splitting between metal plates in low voltage arc chutes[END_REF]. The arc displacement during a breaking process is shown in Fig. 2.

This displacement is due to hydrodynamic constraints and magnetic forces. To design a circuit breaker, it is important to have a detailed knowledge of the arc behavior.

The breaking process is fast (∼millisecond) and energizing (>kilojoules). It makes the experimentation a difficult task and it is almost always an intrusive experimentation.

Simulations also help to obtain additional information, which is usually not accessible by the experimental investigations, and thus can reduce the time for the development of new circuit breakers.

II. ARC INTERRUPTION MODELING

The core of the arc system modeling [START_REF] Chevrier | Industrial application of high-, medium-and low-voltage arc modeling[END_REF], [START_REF] Rondot | Dedicating finite volume method to electromagnetic plasma modeling: Circuit breaker application[END_REF] is based on the commercial software FLUENT. It uses a finite-volume method

H ext r = 1 4π V J(r) × ∇ r G(r, r )d 3 r ( 3
)
where J is the current density, G is the Green function for a Laplace's equation, r is the integration variable, and r is the position of the computation. All these models are shown in a simplified diagram of the modeling process in Fig. 3.

During a breaking process, the current density of the arc leads to Ohmic heating (Joule losses > energy source) and magnetic forces (Lorentz forces > momentum source), which cause gas flow and energy transport within and outside the arc.

In the case of a breaking process at low current, ferromagnetic regions can lead to substantial increase in the Lorentz forces on the plasma.

This paper presents a method to model the ferromagnetic effects because of the magnetostatic moment method (MoM) in the Schneider electric's arc system modeling.

III. 3-D MOM

The MoM is an accurate and inexpensive numerical approach dedicated to the modeling of electromagnetic devices. In this method, only active regions (i.e., ferromagnetic parts of the device) are meshed. Thus, it is particularly well adapted for the modeling of simple radiated devices with only a few ferromagnetic parts in comparison of the total air region [START_REF] Harrington | Field Computation by Moment Methods[END_REF].

A. General Equation

For a magnetostatic problem consisted of some ferromagnetic regions and coils supplied by currents, the magnetic field can be expressed as the sum of the external field (H ext ) and the reduced magnetic field (H r ), which derives from the magnetic scalar potential [START_REF] Janet | Simplified magnetic moment method applied to current transformer modeling[END_REF]

H = H ext + H r = H ext -gradφ (4)
with for the magnetic scalar potential

φ r = V ∇ r G(r, r ) • M (r) d 3 r ( 5
)
where M is the magnetization, r is the integration variable, and r is the position of the computation. This formulation strongly ensures Maxwell-Ampere and Maxwell-Thomson equations. Therefore, to determine the magnetic field H, the magnetic law behavior must also be ensured. For that, the residual is defined as follows:

R (M) = V (M -m (H)) • W r d 3 r (6)
where m(H) is the image of the magnetic field view from the nonlinear magnetic law and W is a vectorial test function.

B. Discrete Approximation

Several formulations exist with different interpolate functions of M and different test functions. In our example, functions are chosen to be constant by element. Therefore, the magnetization is uniform in each cell and residual is minimized at each cell centroid (point matching method). In this framework and for p magnetic cells.

1) The magnetic field equation ( 4) becomes

H = H ext + [K] M (7)
where (H, H ext , M) are 3p vector and [K] is (3p) 2 matrix. The matrix [K] is a matrix of view factors between the cells computed by reduction of singularities [START_REF] Chadebec | A review of magnetostatic moment method[END_REF]. 2) The residual equation [START_REF] Janet | Simplified magnetic moment method applied to current transformer modeling[END_REF] becomes To solve nonlinear magnetostatics problem, the Newton-Raphson process is described as follows [START_REF] Saitz | Newton-Raphson method and fixed-point technique in finite element computation of magnetic field problems in media with hysteresis[END_REF]:

R (M) = m (H) -M. (8)
∂m (H) ∂H T H n-1 [K] -[I] M n = -R M n-1 (9)
where the first matrix is a diagonal bloc matrix linked with the derivative of magnetic law, M is the increment of the magnetization, R is the residual vector, and n is the iteration number.

The flowchart of the nonlinear resolution is represented in Fig. 4. To reduce the CPU time, the tangent matrix is not updated at each iteration. In other words, a fixed point (with the previous tangent matrix) and a Newton-Raphson method are alternatively used. The presented approach has been validated on several test cases to prove its effectiveness.

A line search has been added to stabilize the convergence and two convergence checks are used to stop the process.

C. 3-D Nonlinear Test Case

A ferromagnetic plate excited by an inductor supplied by a I is studied first. The test case and the magnetic field in the ferromagnetic plate obtained using the MoM are shown in Fig. 5.

To control the accuracy of the solution, the magnetic field close to the ferromagnetic plate is computed on a posttreatment line and compared in Fig. 6 with the solution obtained from FLUX (commercial finite-element electromagnetic software).

It can be observed that the relative error is very low and it increases close to the ferromagnetic plate. This increase can be further reduced by increasing the number of Gauss integration point.

The method of MoM has been used successfully on several test cases.

IV. CIRCUIT BREAKER APPLICATION

In a circuit breaker model, two ferromagnetic plates are introduced close to the breaking chamber. During the breaking process simulation, the magnetization in plates and their influence on the magnetic field seen by the arc can be observed in Fig. 7.

The comparison of the global electrical data (I and U arc arc voltage) in Fig. 8 shows the influence of ferromagnetic parts on the breaking process. With ferromagnetic parts, the arc displacement is improving and therefore the arc voltage increases faster. The arc voltage being greater quickly, the current limitation is more important. The arc voltage is slightly smaller and does not have a zero current sooner. However, this configuration is still interesting because the thermal stresses caused by the flowing energy (I 2 t) are reduced.

V. CONCLUSION

The MoM method is suitable and well adapted for the breaking modeling. Compared with other approaches [START_REF] Lindmayer | 3D simulation of arc motion including the influence of ferromagnetic material[END_REF], it avoids the meshing of the whole air region and the use of two different meshes.

The use of the compression method is in progress (hierarchical matrix and adaptative cross approximation [START_REF] Bedendorf | Hierarchical matrix: A means to efficiency solve elliptic boundary value problems[END_REF]) with which a further reduction in the computation time and the memory usage is expected.

The introduction of MoM in the arc system modeling allows considering ferromagnetic regions position and shape optimization. An improvement in the design of circuit breakers is expected.

Fig. 2 .

 2 Fig. 2. Three main steps during a breaking process in a circuit breaker are represented. (a) Ignition. (b) Growth. (c) Splitting.

Fig. 3 .

 3 Fig. 3. Diagram of the arc interruption model. T : temperature. P: pressure. σ : electrical conductivity.

Fig. 4 .

 4 Fig. 4. Nonlinear flowchart resolution where A is the solving matrix, RHS is the right-hand side vector, and ε 1 and ε 2 are the tolerance parameters.

Fig. 5 .Fig. 6 .

 56 Fig. 5. 3-D nonlinear test case with a ferromagnetic plate excited by an inductor supplied by a current (I ). The magnetic field is represented in the plate.

Fig. 7 .

 7 Fig. 7. During a breaking process, arc is represented by three isovalues of temperature and the magnetization in the ferromagnetic plate can be observed.

Fig. 8 .

 8 Fig. 8. Electrical curves during a breaking process. The current and arc voltage are represented with or without the effect of ferromagnetic plates.


Berenger Bramas
email: •@pitsianis•@himscipy•@sarats

TBFMM: A C++ generic and parallel fast multipole method library

TBFMM, for task-based FMM, is a high-performance package that implements the parallel fast multipole method (FMM) in modern C++17. It implements parallel strategies for multicore architectures, i.e. to run on a single computing node. TBFMM was designed to be easily customized thanks to C++ templates and fine control of the C++ classes' inter-dependencies. Users can implement new FMM kernels, new types of interacting elements or even new parallelization strategies. As such, it can be used as a simulation toolbox for scientists in physics or applied mathematics. It enables users to perform simulations while delegating the data structure, the algorithm and the parallelization to the library. Besides, TBFMM can also provide an interesting use case for the HPC research community regarding parallelization, optimization and scheduling of applications handling irregular data structures.

Background

The fast multipole method [START_REF] Greengard | A fast algorithm for particle simulations[END_REF] has been classified as one of the most important algorithms of the 20th century [START_REF] Cipra | The best of the 20th century: Editors name top 10 algorithms[END_REF]. The FMM algorithm was designed to compute pair-wise interactions between N particles, which belong to the class of n-body problems. It reduces the complexity from a quadratic (N elements interact with N elements) to a quasi-linear complexity. The central idea of the FMM is to avoid computing the interactions between all the elements by approximating the interactions between elements that are far enough. To make this possible, the algorithm requires the potential of the interactions to decrease as the distance between interacting elements increases. The algorithm also requires that the kernel to approximate far interactions exists; providing an approximation kernel for a physical equation can be challenging. Internally, the FMM is usually implemented with a tree that is mapped over the simulation box. A cell, i.e. a node of the tree, represents a part of the simulation box and is used by the algorithm to factorize the interactions between elements. The FMM was later extended for different types of physical simulations and different approximation kernels [START_REF] Barba | ExaFMM: An open source library for fast multipole methods aimed towards exascale systems[END_REF][START_REF] Blanchard | An Efficient Interpolation Based FMM for Dislocation Dynamics Simulations[END_REF]Blanchard, Coulaud, & Darve, 2015;[START_REF] Darve | Optimizing the Blackbox FMM for Smooth and Oscillatory Kernels[END_REF][START_REF] Darve | A fast multipole method for Maxwell equations stable at all frequencies[END_REF][START_REF] Frangi | Coupled fast multipole method-finite element method for the analysis of magneto-mechanical problems[END_REF][START_REF] Malhotra | PVFMM: A parallel kernel independent FMM for particle and volume potentials[END_REF][START_REF] Pham | Fast multipole method applied to Symmetric Galerkin boundary element method for 3D elasticity and fracture problems[END_REF][START_REF] Sabariego | Application of the fast multipole method to hybrid finite element-boundary element models[END_REF][START_REF] Vazquez Sabariego | The fast multipole method for electromagnetic field computation in numerical and physical hybrid systems[END_REF].

The FMM algorithm is based on six operators with names that respect the format X2Y, where X represents the source of the operator and Y the destination. The operators are P2M, M2M, M2L, L2L, L2P and P2P, where P means particle, M multipole and L local. The term particle is used for a legacy reason, but it represents the basic interaction elements that interact and for which we want to approximate the interactions. The multipole part represents the aggregation of potential, i.e. it represents what is emitted by a sub-part of the simulation box, whereas the local part represents the outside that is emitted onto a sub-part of the simulation box. The different operators are schematized in Figure 1. Because the FMM is a fundamental building block for many types of simulation, its parallelization has already been investigated. Some strategies for parallelizing over multiple distributed memory nodes have been developed using classical HPC technologies like MPI [START_REF] Forum | MPI: A Message-Passing Interface Standard[END_REF] and fork-join threaded libraries [START_REF] Bramas | Optimization and parallelization of the boundary element method for the wave equation in time domain[END_REF]. Using these technologies, different projects have created FMM implementations that scale on thousands of computing nodes [START_REF] Abduljabbar | Extreme Scale FMM-Accelerated Boundary Integral Equation Solver for Wave Scattering[END_REF][START_REF] Lashuk | A massively parallel adaptive fast-multipole method on heterogeneous architectures[END_REF][START_REF] Malhotra | PVFMM: A parallel kernel independent FMM for particle and volume potentials[END_REF]. However, when using a single node, it has been demonstrated that fork-join schemes are less efficient than task-based parallelization on multicore CPUs (Emmanuel [START_REF] Agullo | Task-Based FMM for Multicore Architectures[END_REF]. This is because some stages of the FMM have a small degree of parallelism (for instance at the top of the tree), while others have a high degree of parallelism. For instance, the P2P in the direct pass has a significant workload available from the early beginning of each iteration. The task-based method can interleave the different operators, hence to balance the workload across the processing units and to spread the critical parts over time. Moreover, the task-based method is well designed for handling heterogeneous architecture [START_REF] Agullo | Task-based FMM for heterogeneous architectures[END_REF] and has demonstrated a promising performance on distributed memory platforms too (Emmanuel Agullo et al., 2017).

In a previous project called ScalFMM, we have provided a new hierarchical data structure called group-tree (or block-tree), which is an octree designed for the task-based method [START_REF] Bramas | Optimization and parallelization of the boundary element method for the wave equation in time domain[END_REF]. The two main ideas behind this container are (1) to allocate and manage several cells of the same level together to control the granularity of the tasks, and (2) to store the symbolic data, the multipole data, and the local data in a different memory blocks. This allows us to move each block anywhere on the memory nodes and to declare the dependencies on each sub-part.

A schematic view of the group-tree is given in Figure 2.

Statement of need

The FMM is a major algorithm but it remains rare to have it included in HPC benchmarks when studying runtime systems, schedulers or optimizers. The principal reason is that it is tedious to implement and requires a significant programming effort when using the task-based method together with the group-tree. However, it is an interesting, if not unique, algorithm to study irregular/hierarchical scientific method. For the same reason, it is difficult for researchers in physics or applied mathematics to implement a complete FMM library and to optimize it for modern hardware, especially if their aim is to focus on approximation kernels. Therefore, TBFMM can be useful for both communities.

Among the few existing FMM libraries, ScalFMM is the closer package to TBFMM. ScalF MM supports lots of different parallel strategies, including fork-join implementations, and it contains several experimental methods. Consequently, ScalFMM has around 170K lines of code, for only 50K for TBFMM. Moreover, it needs several external dependencies and does not benefit from the new standard C++ features that could improve code readability. Besides, it only works for 3D problems, whereas TBFMM's tree and algorithms supports any number of dimensions. These have been the main motivations to re-implement a lightweight FMM library from scratch that only supports task-based parallelization.

However, the interface of the kernels is very similar in both libraries, such that creating a kernel for ScalFMM or TBFMM and porting it to the other library is straightforward.

FMMTL [START_REF] Cecka | FMMTL: FMM Template Library A Generalized Framework for Kernel Matrices[END_REF] is another existing FMM library with a generic C++ design. Similar to TBFMM, FMMTL works with various types of kernels. However, it relies on a fork-join parallelization strategy on top of OpenMP and on the Thrust library to support GPUs. While this approach is elegant and makes the code clean, it is also more complex to fine tune the scheduling, the load balancing, and the data movement.

Features Genericity

TBFMM has a generic design thanks to the heavy use of C++ templates. Users must provide arguments for the various template parameters of the main classes. The tree and the kernel classes are independent of each other and from the algorithm. The algorithm class needs the type of the kernel as a template argument, and its core execute method has to be templatized with the type of the tree. The algorithm takes the elements from the tree and passes them to the kernel, such that a kernel itself never accesses the tree. This is illustrated by Figure 3.

Tree

TBFMM uses the group-tree where several cells of the same level are managed together. Users can select the size of the groups, which impacts the size of the tasks, however, TBFMM also provides a simple heuristic to automatically find a size that should provide efficient executions. Also, the tree class provides different methods to iterate on the cells/leaves as any container, such that it is possible to work on the elements of the tree with an abstraction mechanism and without knowing how it is implemented internally.

Kernel

As stated in the objectives, TBFMM is a tool for scientists from physics and applied mathematics to create new kernels. TBFMM offers a convenient way to customize the kernel and to benefit from the underlying parallelization engine automatically. With this aim, a user has to create a new kernel that respects an interface, as described by the package documentation. The current package contains two FMM kernels, the rotation kernel based on the rotation-based operators and the spherical harmonics [START_REF] Dachsel | Fast and accurate determination of the Wigner rotation matrices in the fast multipole method[END_REF][START_REF] Haigh | Implementation of rotation-based operators for fast multipole method in X10[END_REF][START_REF] White | Derivation and efficient implementation of the fast multipole method[END_REF][START_REF] White | Rotating around the quartic angular momentum barrier in fast multipole method calculations[END_REF], and the uniform kernel based on Lagrange interpolation [START_REF] Blanchard | An Efficient Interpolation Based FMM for Dislocation Dynamics Simulations[END_REF][START_REF] Blanchard | Hierarchical Randomized Low-Rank Approximations[END_REF]Blanchard, Coulaud, & Darve, 2015).

Parallelization

TBFMM has two task-based parallel algorithms based on two runtime systems: OpenMP version 4 (OpenMP Architecture Review Board, 2013) and SPETABARU (Bérenger [START_REF] Bramas | Increasing the degree of parallelism using speculative execution in task-based runtime systems[END_REF]. Both are optional, such that the library can be compiled even if the compiler does not support OpenMP or if the Git sub-module for SPETABARU has not been activated. OpenMP is an API that evolves slowly, which maintains backward compatibility and which is implemented by different libraries that respect the standard. On the other hand, SPETABARU is our task-based runtime system that we use for research, and which continuously evolves. The data accesses of the FMM operators in write are usually commutative (E. Agullo et al., 2017). While SPETABARU supports commutative write access, OpenMP only supports it from version 5 with the mutexinout data access. OpenMP version 5 is currently not fully supported by the compilers, however, when a compiler that supports this access will be used with TBFMM, the mutexinout will be activated automatically.

Periodicity

The periodicity consists of considering that the simulation box is repeated in all directions, as shown by Figure 4. Computing the FMM algorithm with periodicity is usually done in two steps. In the first one, the regular algorithm and tree are used. When the algorithm needs cells outside of the boundaries, it selects cells at the opposite side of the simulation box. While in the second step, a numerical model is used to compute a potential that represents the world outside the simulation box. Such a model could be the Ewald summation [START_REF] Rokhlin | The fast multipole method for periodic structures[END_REF]. In TBFMM, we have implemented a different approach, which is a pure algorithmic strategy [START_REF] Bramas | Optimization and parallelization of the boundary element method for the wave equation in time domain[END_REF]. The idea is to consider that the real simulation box is a sub-part of a larger simulation box, i.e. that the real tree is a branch of a larger tree. Then, instead of stopping the FMM algorithm at level 2, we continue up until the root where the multipole part of the root represents the complete simulation box. We use it by continuing the FMM algorithm partially above the root, by aggregating the cells together multiple times. By doing so, we have several advantages. This method needs nothing more than an FMM kernel, which is expected to be the same as the one used without periodicity. Therefore, the method is generic and can work with any FMM kernel. Moreover, the accuracy of the method relies fully on the FMM kernel. Figure 5 shows how the simulation box is repeated with this method.

Vectorization (Inastemp)

When implementing a kernel, some parts can rely on well-optimized numerical libraries, such as BLAS or FFTW, however, others might be implemented directly in C/C++. In this case, it usually provides a significant improvement in performance to vectorize the code, which allows benefiting from the SIMD capability of modern CPUs. With this aim, TBFMM can include a vectorization library called Inastemp (Berenger Bramas, 2017) by simply cloning the corresponding Git sub-module. Using Inastemp, it is possible to write a single code with an abstract vector data type and to select at compile time the desired instruction set depending on the CPU (SSE, AVX, SVE, etc.). In the current version of TBFMM, the P2P operator of the two kernels that are provided for demonstration is vectorized with Inastemp.

Performance

In Figure 6, we provide the parallel efficiency of TBFMM for a set of particles that are randomly distributed in a square simulation box. The given results have been computed using the uniform kernel and the two runtime systems OpenMP (GNU libomp) and SPETABARU. Both implementation are similar; they use the same input data, the same tasks, the same priorities. The only differences are coming from the implementation of the runtime systems, which impact the overhead, and the type of data access they support. We recall that OpenMP 4.5 does not support mutexinout for commutative data access. In our case, the resulting degree of parallelism is then too limited to feed all the available cores. This is already a problem with 2 threads, and there is even no reduction in the execution time with more than 8 threads, which appears as a significant drop in the parallel efficiency. Even so, we believe that it would be possible to improve our OpenMP-based implementation by inserting the tasks differently. Instead, we will wait that the OpenMP library implementations support mutexinout.

Conclusion & Perspective

TBFMM is a lightweight FMM library that could be used for research in HPC and applied mathematics. We will include it in our benchmarks to evaluate scheduling strategies, but also to validate new approaches to develop numerical applications on heterogeneous computing nodes. Indeed, we would like to offer an elegant way for users to add GPU kernels while delegating most of the complexity to TBFMM and SPETABARU. We also plan to provide an MPI version to support distributed memory parallelization shortly.

Figure 1 :

 1 Figure 1: Illustration of the FMM algorithm. (a,b,c) The building of the octree. (d,e,f,g) The FMM algorithm and its operators.

Figure 2 :

 2 Figure 2: Group-tree schematic view. Several cells/leaves are managed together inside a block. Also, the symbolic, multipole and local data are allocated separately to allow declaring the data accesses on each sub-part.

Figure 3 :

 3 Figure3: TBFMM design overview. Template arguments must be used to ensure that the algorithm knows the types of the tree and the kernel, and that the tree knows the types of the cells/particles. The algorithm has to be selected among different variants (sequential, parallel OpenMP, or parallel SPETABARU).

Figure 4 :

 4 Figure 4: In the periodic FMM, the simulation box is considered to be in the middle of an infinite volume of the same kind.

Figure 5 :

 5 Figure 5: How the simulation box is repeated when using a periodic FMM algorithm.

Figure 6 :

 6 Figure 6: Parallel efficiency for TBFMM using the OpenMP and SPETABARU runtime systems, and the uniform kernel (order = 8). Test cases: two simulations, of one and ten million particles, randomly distributed in a cube. Hardware: 2 × Intel Xeon Gold 6240 CPU at 2.60GHz with 16 cores each and cache of sizes L1/32K, L2/1024K, L3/25344K.

Bramas, B., (2020). TBFMM: A C++ generic and parallel fast multipole method library. Journal of Open Source Software, 5(56), 2444. https://doi.org/10.21105/joss.02444

Acknowledgements

Acknowledgment: Experiments presented in this paper were carried out using the PlaFRIM experimental testbed, supported by Inria, CNRS (LABRI and IMB), Université de Bordeaux, Bordeaux INP and Conseil Régional d'Aquitaine.