Jérémy Charriot
email: j-charriot@chu-montpellier.fr

Zysman Maeva

Laurent Guilleminault

Mathilde Volpato

Aurelie Fort-Petit

Isabelle Vachier

Jeremy Patarin

Carey Suehs

Ahmed Engi

Nicolas Molinari

Jeremy Charriot

Maeva Zysman

Engi Ahmed

Arnaud Bourdin

Sputum-Rheology-Based Strategy for Guiding Azithromycin Prescription in COPD Patients with Frequent Exacerbations: A Randomized, Controlled Study ("COPD CARhE")

Keywords: COPD, exacerbations, sputum rheology, azithromycin, mucus, eosinophil

Introduction

Chronic obstructive pulmonary disease (COPD) is a devastating and currently irreversible chronic disease that affects approximately 10-13% of the European population [START_REF] Adeloye | Global and regional estimates of COPD prevalence: Systematic review and meta-analysis[END_REF][START_REF] Terzikhan | Prevalence and incidence of COPD in smokers and non-smokers: The Rotterdam Study[END_REF]. Its prevalence and incidence are increasing [START_REF] Burgel | An attempt at modeling COPD epidemiological trends in France[END_REF]. Chronic bronchitis (chronic cough and sputum) is a feature of COPD which is predictive of acute exacerbations of COPD (AECOPD) [START_REF] Kim | Chronic Bronchitis and Chronic Obstructive Pulmonary Disease[END_REF]. These events are part of the natural history of this disease and a marker of poor prognosis [START_REF] Çolak | Prognosis of COPD depends on severity of exacerbation history: A populationbased analysis[END_REF]. For instance, in France, more than 8% of patients hospitalized with acute AECOPD die during their stay [START_REF] Molinari | Hospitalizations for COPD Exacerbations: Trends and Determinants of Death[END_REF]. Currently, therapeutic strategies aim to decrease their annual rate.

For a long time [START_REF] Anthonisen | Antibiotic Therapy in Exacerbations of Chronic Obstructive Pulmonary Disease[END_REF], several elements have been analyzed in sputum to determine the stable or exacerbated character of COPD, including its volume, its color, and its purulence Sputum is classically attributed to excessive mucus production in the respiratory tract. This hyperproduction of mucus is now even thought to play a major role in distal airway (bronchiolar) obstruction, which is then complicated by emphysema [START_REF] Mcdonough | Small-Airway Obstruction and Emphysema in Chronic Obstructive Pulmonary Disease[END_REF]. While mucus production is a classic symptom of exacerbations, this symptom persists in the chronic form in slightly more than half of patients, with a rather negative significance, even if this remains controversial [START_REF] Kim | Chronic Bronchitis and Chronic Obstructive Pulmonary Disease[END_REF][START_REF] Lahousse | Epidemiology and impact of chronic bronchitis in chronic obstructive pulmonary disease[END_REF]. (Chronic bronchitis is defined by the presence of cough and sputum for 3 consecutive months over a period of at least 2 years.) This excess of mucus is responsible for chronic sputum that constitutes a considerable dimension of the burden carried by patients in this disease. However, at the moment, in France, there is no treatment reimbursed by health insurance that specifically targets this symptom, mostly due to lack of evidence or efficacy.

Multiple components vary in this sputum, and they are independent of the intrinsic quality and quantity of mucus [START_REF] Kesimer | Airway Mucin Concentration as a Marker of Chronic Bronchitis[END_REF] produced by the epithelium: microorganisms, epithelial cellular debris, polynuclear DNA, and other inflammatory cells. These changes are likely to modulate the rheology of the mucus [START_REF] Fahy | Airway mucus function and dysfunction[END_REF]. Mucus is a complex, viscoelastic component consisting essentially of water, ions, and proteins, some of which (with the mucins MUC5AC and MUC5B being the most commonly represented) can modulate the biological and biophysical properties of mucus. Goblet cell hyperplasia and metaplasia would thus be the primum movens of chronic bronchitis [START_REF] Charriot | Methods of Sputum and Mucus Assessment for Muco-Obstructive Lung Diseases in 2022: Time to "Unplug" from Our Daily Routine! Cells[END_REF][START_REF] Hogg | The nature of small-airway obstruction in chronic obstructive pulmonary disease[END_REF]. This airway remodeling is partially explained by an increased production of certain cytokines (IL-4, IL-13, etc.) and mediators of epithelial or systemic origin [START_REF] Boucherat | Bridging Lung Development with Chronic Obstructive Pulmonary Disease. Relevance of Developmental Pathways in Chronic Obstructive Pulmonary Disease Pathogenesis[END_REF]. The complexity of the pathophysiology of COPD is difficult to portray; nevertheless, it may be approached through the elaboration of biological network models that focus on cell proliferation, cell fate, or inflammation [START_REF] Boue | Enhancement of COPD biological networks using a web-based collaboration interface[END_REF]. All of these processes are implied in the hyperproduction of mucus. Interestingly, azithromycin has been shown to decrease neutrophil chemotaxis in the lower respiratory tract [START_REF] O'reilly | Sputum PGP is reduced by azithromycin treatment in patients with COPD and correlates with exacerbations[END_REF], to improve the phagocytic capacity of alveolar macrophages [START_REF] Hodge | Azithromycin increases phagocytosis of apoptotic bronchial epithelial cells by alveolar macrophages[END_REF], and to reduce some markers of systemic inflammation. However, equally convincingly, azithromycin has been shown to inhibit the hypersecretion of mucus by airway epithelial cells [START_REF] Shimizu | Azithromycin inhibits mucus hypersecretion from airway epithelial cells[END_REF], which may explain why this macrolide engenders fewer exacerbations. Indeed, in clinical trials, the use of long-term azithromycin at a reduced-dosage regimen (either 250 mg/d or 500 mg/3 times per week) significantly reduced the annual rate of AECOPD, respectively, by 27% and 42% in two large, randomized, double-blind, controlled trials against placebo [START_REF] Albert | Azithromycin for Prevention of Exacerbations of COPD[END_REF][START_REF] Uzun | Azithromycin maintenance treatment in patients with frequent exacerbations of chronic obstructive pulmonary disease (COLUMBUS): A randomised, double-blind, placebo-controlled trial[END_REF]. Moreover, it also proved efficient in improving the cough-specific health status of patients with COPD [START_REF] Berkhof | Azithromycin and cough-specific health status in patients with chronic obstructive pulmonary disease and chronic cough: A randomised controlled trial[END_REF] and the Leicester Cough Questionnaire (LCQ) scores of patients with resistant cough [START_REF] Hodgson | The Effects of Azithromycin in Treatment-Resistant Cough: A Randomized, Double-Blind, Placebo-Controlled Trial[END_REF]. As a result, azithromycin is now proposed in international and French guidelines as a therapeutic tool for the prevention of exacerbations, in addition to optimal standard management [START_REF]Global Strategy for the Diagnosis, Management and Prevention of COPD[END_REF][START_REF] Zysman | Optimisation du traitement médicamenteux des patients atteints de BPCO en état stable. Position de la Société de pneumologie de langue française[END_REF]. However, the pathophysiological link between azithromycin administration and the reduction of COPD exacerbations has not been established. As macrolides may cause infrequent, but significant, adverse effects (e.g., deafness; cardiac conduction disorders) and may increase microbial resistance, prescriptions of this type of molecule must be properly targeted. Nevertheless, the COPD patient population receiving azithromycin (or macrolides in general) is poorly defined and far from specifically "targeted".

The objective selection of a subpopulation of COPD patients, based on altered or unaltered sputum rheology (now easily quantifiable within minutes), would provide a more effective treatment algorithm than the current standard management. In patients with COPD, asthma, and non-CF bronchiectasis, our team recently demonstrated that sputum rheology (elastic modulus G', viscous modulus G", and critical stress c), was correlated with MUC5AC and MUC5B mucin content and sputum eosinophilia [START_REF] Volpato | Rheology predicts sputum eosinophilia in patients with muco-obstructive lung diseases[END_REF]. We have also shown that sputum rheology can predict sputum eosinophilia at a rate superior or equal to 1.25%, a value that correlated with exacerbations requiring corticosteroids and a reduction in FEV1 in the SPIROMICS cohort [START_REF] Hastie | Association of sputum and blood eosinophil concentrations with clinical measures of COPD severity: An analysis of the SPIROMICS cohort[END_REF].

Here, we hypothesize that sputum rheology could guide the prescription of azithromycin in a more appropriate manner than does clinical evaluation, so as to reduce both AECOPD and exposure to antibiotics.

Materials and Methods

Objectives

Primary Objective

The primary objective of this trial is to compare the exacerbation rates over 12 months between the comparator arm, i.e., a group of COPD patients with frequent exacerbations treated according to clinical evaluation using the Cough and Sputum Questionnaire (CASAQ; see below for details), and the experimental arm, comprising a similar group in which azithromycin is prescribed according to sputum rheology.

Secondary Objectives

To compare the two arms, we will study: -the exacerbation rates by severity (as observed throughout the study); -the evolution of symptoms, sputum rheology, and lung function (via repeated measurements collected every 3 months); -the medication usage and adverse events (follow-ups for the duration of the study); -the patient trajectories during follow-ups; -the overall clinical improvement achieved by the end of the study and changes in quality of life (via repeated measurements every 3 months); and -the changes in biomarkers of interest (baseline versus end of study). -We will compare the exacerbation rates, throughout the study, of the experimental arm without azithromycin prescription, and of the comparator arm without azithromycin; and between the experimental arm with azithromycin prescription and the comparator arm with azithromycin. -We will evaluate patients in the experimental arm who would have been managed differently using the CASA-Q versus rheology.

Study Design

COPD CaRhe is a parallel-group randomized trial (36 vs. 36 patients with COPD and frequent exacerbations). It will compare a group of patients treated according to clinical evaluation (and azithromycin will be prescribed according to the Cough and Sputum Assessment Questionnaire (CASA-Q), if patients score < 70) versus a similar group of patients in which azithromycin prescription will be guided by sputum rheology (if the c > 39). The chosen cutoffs of 70 for the CASA-Q and 39 for the c correspond to the expected medians of these variables in our population of interest. (See Figure 1).

The CASA-Q is a questionnaire validated in French which aims to rate cough and sputum and their respective impacts on quality of life. According to the national [START_REF] Zysman | Optimisation du traitement médicamenteux des patients atteints de BPCO en état stable. Position de la Société de pneumologie de langue française[END_REF] and international [START_REF]Global Strategy for the Diagnosis, Management and Prevention of COPD[END_REF] recommendations in force, the prescription of azithromycin is conditioned upon the specialist's free, subjective appreciation of respiratory symptoms, including cough and sputum, in patients with COPD and frequent exacerbations. These symptoms will be assessed here by the CASA-Q [START_REF] Monz | Responsiveness of the cough and sputum assessment questionnaire in exacerbations of COPD and chronic bronchitis[END_REF][START_REF] Crawford | Development and validation of a cough and sputum assessment questionnaire[END_REF][START_REF] Deslee | Impact of current cough on health-related quality of life in patients with COPD[END_REF][START_REF] Tagaya | Effect of tiotropium on mucus hypersecretion and airway clearance in patients with COPD[END_REF].

For the variable "critical stress _c", the threshold of 39 was determined based on a previously published ancillary study [START_REF] Volpato | Rheology predicts sputum eosinophilia in patients with muco-obstructive lung diseases[END_REF].

Setting and Participant Eligibility

All patients consecutively attending the outpatient clinic in the Respiratory Diseases Departments of the Montpellier, Bordeaux, and Toulouse University Hospitals (France) who meet the eligibility criteria will be prospectively enrolled. Adults aged 40-85, with a confirmed diagnosis of COPD according to international guidelines, are eligible. COPD patients must have a "frequent exacerbation" phenotype, i.e., at least three exacerbations over the past 12 months before inclusion, despite optimal treatment according to the Global Initiative for Chronic Obstructive Lung Disease [START_REF]Global Strategy for the Diagnosis, Management and Prevention of COPD[END_REF] guidelines. All inclusion and exclusion criteria are summarized in Table 1. Basic demographic (age, sex, weight, height, and body mass index) and comorbidities (specifically, the starting dates for renal insufficiency, diabetes, diffuse intestinal pneumonia, or ischemic heart disease), if present, will be used to describe the population at baseline.A listing of any concomitant treatments will be maintained for each patient throughout the study.

Interventions

In both the comparator arm and the experimental arm, patients will receive optimal background treatment for COPD according to current national and international guidelines.

Comparator Arm

Patients randomized to the comparator arm of the study will receive azithromycin in case of severe sputum symptoms, here defined by a CASA-Q sputum symptom score of <70, so as to homogenize practices between centers. The CASA-Q will be scored every 3 months.

-If the patient presents a sputum symptom score of <70, azithromycin will be initiated for 3 months. This prescription will be renewable every 3 months during the 12 months of followup planned in this study if the patient continues to obtain a sputum symptom score of <70.

-If the patient has a sputum symptom score of >70, management will not be changed.

Experimental Arm

Patients randomized to the experimental arm of the study will receive azithromycin according to sputum rheology. Sputum rheology will be quantified every 3 months using a Rheomuco device.

-If sputum has a critical stress c >39, azithromycin will be initiated for 3 months. This prescription will be renewable every 3 months during the 12 months of follow-up planned in this study. -In the absence of sputum, azithromycin will not be prescribed.

Randomization and Blinding

To optimize the comparability between the two groups, a blocked randomization with randomly selected block sizes will be used to assign patients to treatment groups using a 1:1 allocation ratio. Due to the sample size, stratification will only be performed according to each center. The other variables of interest (e.g., the number of exacerbations in the 12 months prior to inclusion, gender, background treatment, etc.) will serve as adjustment variables in the statistical analysis.

Given the scope of this first study in the field, a level of blinding involving a placebo would not be appropriate. Nevertheless, by blinding the sputum rheology and CASA-Q results for the duration of the study, the blinding allowed by randomization will be ensured. A "guess the arm" question will be asked of investigators and patients at the end of the study to assess the level of blinding.

Data required at the baseline visit, namely sputum rheology and CASA-Q responses, will be promptly entered into the eCRF by a designated person at each center, as these data will condition the allocation of the treatment.

A web-based application within the eCRF (Ennov Clinical) will be created for the purposes of the study and will ensure the randomization of patients. The creation of the eCRF, the sequence, and the allocation modules will be supervised by the Department of Clinical Research and Epidemiology of Montpellier University Hospital. In addition to randomization, the eCRF will also be parameterized to automatically apply the treatment algorithm specific to each study arm. The randomization of patients will be performed by the participating investigators through specific modules in the eCRF. The randomization will be performed after the data entry corresponding to the sputum rheology or the CASA-Q.

Predicted and Explanatory Variables

Patient-specific measures and time frames are summarized in Table 2.

Sputum Collection

A spontaneous or induced, unselected sputum sample will be collected for each individual patient. All samples will be rigorously processed following the same procedure, as previously described. Within 2 h of collection, rheology parameters will be measured, and then cytology assessments will be performed (see the following paragraph). In parallel, a part of the sample will be sent to routine, direct bacterial examination and culture.

Sputum Rheology

Rheology will be assessed on 720 mL of 1 fresh sputum sample, as previously described by Patarin et al. [START_REF] Patarin | Rheological analysis of sputum from patients with chronic bronchial diseases[END_REF]. Briefly, after gentle vortexing at low velocity for 1 min, rheometric analysis will be performed using the Rheomuco® rheometer (Rheonova, Grenoble, France) at 37 °C, with an oscillatory frequency of 1 Hz and an increasing strain. Measurements of the critical stress threshold (c) will be obtained as needed to determine the prescription of azithromycin. Other parameters will also be collected (such as elastic modulus (G'; Pa), viscous modulus (G"; Pa), etc.).

Pulmonary Function Tests

Spirometry and plethysmography will be carried out as recommended by the European Respiratory Society using BodyBox equipment (MediSoft BodyBox 5500, Sorinnes, Belgium). With the exception of the FEV1/FVC ratio, which is calculated as liters/liters, all outcomes (forced expiratory volume in 1 s (FEV1), forced vital capacity (FVC), total lung capacity (TLC), and residual volume (RV)) will be calculated as both % predicted values and as liters.

Blood Tests

A blood cell count, and especially a blood eosinophil count, will be performed at baseline and 12 months, as routinely performed in each university hospital. Identically, an assessment of the serum club cell secretory protein (CCSP; ng/mL) level will be performed, as determined by an enzyme-linked immunosorbent assay, according to the manufacturer's instructions (Biovendor, Brno, Czech Republic). A liver blood test will be performed at baseline to rule out severe hepatic insufficiency.

Quality-of-Life Questionnaires

The following questionnaires will be administered to each patient at each visit during the follow-up: the SF-36, the EQ-5D-5L, and the St. George's Respiratory Questionnaire.

Prescription of Azithromycin

The prescription of azithromycin will be standardized in both arms: 500mg (250mg x 2/d) 3 days per week for 3 months, with possible renewals during the 12 months of follow-up. Azithromycin tablets may be taken with or without meals.

As the information contained in the marketing authorizations is likely to change, it is advisable that clinicians make sure at the time of prescription that the contraindications, warnings and precautions for use, and drug interactions are respected by referring to the information available on the public drug database, accessible on the Internet at the following address: http://base-donnees-publique.medicaments.gouv.fr/ (accessed on 22 February 2008).

Ethics

This study was approved by the Comité de Protection des Personnes IDF X-Ile de France X (N° EudraCT 2019-004182-41) and approved by the ANSM (Agence National de Sécurité du médicament et des produits de Santé) on 29 November 2021, and it has been registered on Clinicaltrials.gov (NCT04339270). In accordance with French regulations for observational studies, verbal and written consent will be required, and patients can participate as long as they are correctly informed and do not oppose the data analysis.

Sample Size and Statistics

In the comparator arm, we expect an exacerbation rate of 3.59/year [START_REF] Wedzicha | Indacaterol-Glycopyrronium versus Salmeterol-Fluticasone for COPD[END_REF]. The choice of standard deviation is most conservative in GOLD classes C-D, i.e., SD = 2.5 [START_REF] Han | disease severity classification in COPDGene: A prospective cohort study[END_REF]. To detect a 50% reduction (the reduction of interest for this first-of-its-kind study) in the exacerbation rate in the experimental arm, with a two-sided, first-species risk of 0.05 and 1-beta power = 0.80, a total of 62 patients is required (31 per group). Taking into account a 10% loss of data (due to screening failures and study dropouts), we propose the inclusion of a total of 72 patients in this study (36 in each arm). Descriptive statistics will be provided as medians, followed by their interquartile ranges for quantitative variables, and as numbers (%) for qualitative variables. Pairwise comparisons between groups will be performed using Conover's tests or Fisher's exact tests, corrected for false discovery rates (FDRs). Multivariable, stepwise regression will then be performed to explain potential differences between groups.

Study Schedule

-Duration of the regulatory process: 6-9 months; -Duration of the inclusion period: 24 months; -Duration of the participation of each participant: 12 months; -Duration of the statistical analysis and writing of the final report: 12 months; -Anticipated duration of the study (inclusion and follow-up): 36 months; -Anticipated total study duration: 48 months; -Number of visits from the beginning of the study: 7 per patient (see Table 2).

Discussion

COPD CaRhe will be the first clinical study to use sputum rheology to guide the prescription of a medication known to reduce the rate of exacerbation in COPD. This trial might lead to the implementation of sputum rheology in routine clinical practice. Thus, it could regularly assist the respiratory physician in making therapeutic choices and help in the reduction (or avoidance) of patients' exposure to antibiotics. It will also give us insights into the evolution of patients' sputum rheology over a long period of time. Moreover, we will have the opportunity to analyze multiple variables, such as serum eosinophils and CCSP, to explain the determinants of this rheology.

Cough and chronic sputum are symptoms that are still poorly understood and particularly poorly managed at present. Finely modulating this feature in a supervised manner therefore seems of utmost interest.

This study could have great implications for the clinical routine since it could introduce a new objective tool to assess the sputum of COPD patients; indeed, until now, sputum has been most often assessed on a daily basis by pulmonologists worldwide via simple questions (Do you cough? Do you produce phlegm?), sometimes by bacteriological examination (often negative), more rarely by cytology.

Figure 1 .

 1 Figure 1. Study Design.

Acknowledgments:

We thank Rheonova (Grenoble, France) for providing us with the Rheomuco device.

Funding:

This research was funded by Programme Hospitalier de Recherche Clinique Inter-Régional "GIRCI-SOHO".

Author Contributions:

Conceptualization, A.B., J.C., C.S., N.M. and I.V.; methodology, C.S. and N.M.; software, C.S.; formal analysis, C.S.; investigation, M.Z., L.G., J.C., A.F.-P., I.V., M.V. and E.A.; resources, J.P.; writing-original draft preparation, J.C.; writing-review and editing, All authors; supervision, A.B., I.V. and J.C.; project administration, I.V.; funding acquisition, J.C. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement:

The study will be conducted in accordance with the Declaration of Helsinki and has been approved by the Ethics Committee of ILE DE FRANCE X (N_ EudraCT 2019-004182-41) on 20 January 2020.

Informed Consent Statement:

Informed consent will be obtained from all subjects involved in the study.

Conflicts of Interest:

The authors declare no conflict of interest.


Alain Valtz

Frédérick De Meyer

Christophe Coquelet
email: christophe.coquelet@mines-albi.fr

Density, viscosity and excess properties of aqueous solution of 1,3-dimethyl-2-imidazolidinone (DMI)

A complete study of thermophysical properties concerning the 1,3-dimethyl-2imidazolidinone (DMI) + water binary system is realized. Density, speed of sound, dynamic viscosities, and refractive index measurements have been performed at atmospheric pressure, using a vibrating tube densitometer, a falling ball viscosimeter, and a refractometer for pure DMI, pure water, and for aqueous solutions of DMI, from 278.15 to 323.15 K. The thermal expansion was calculated from density data. Excess Gibbs energy of flow and the corresponding excess entropy of flow were also calculated considering dynamic viscosity and density data. Excess molar properties (volume, thermal expansion, Gibbs energy of flow and square of refractive index) were calculated and the Redlich-Kister equations were applied to correlate the data. The analysis of the behavior of the excess properties allows to identify the most relevant molecular interaction in mixtures rich in DMI and in mixtures rich in water. Our results show that DMI can be classified as a liquid structure maker and that molecules of water are certainly inserted in the DMI network at finite dilution.

Introduction

The knowledge of thermophysical properties of solvents is essential for the optimization or design of new efficient operation units. Solvents are used in liquid-liquid extraction processes or gas absorption processes. For example, in the context of gas processing and particularly for acid gas removal using a gas-liquid absorption column, the choice of the solvent is an essential step. These solvents are mainly composed of water and an alkanolamine like MDEA (methyldiethanolamine), DEA (diethanolamine) or MEA (monoethanolamine). Because natural gas contains several species: methane, other hydrocarbons, CO 2 , H 2 S, mercaptans and, to respect the specification in terms of composition for liquefied natural gas (LNG) or gas circulating in pipeline, a co-solvent can be added to improve the gas treatment. A similar comment can be formulated in the context of utilization of supercritical fluids such as supercritical water, as solvent for extraction or purification of high value molecules. The co-solvent can also be added to change the thermophysical properties of the solvent and modify the selectivity. Consequently, it is important to know and understand how the co-solvent will alter the thermophysical properties such as density, viscosity, solubility of non-reactive species, and so on.

In this paper, we have studied the densities, speed of sound, viscosities and refractive index of mixtures composed of water and one potential co-solvent. The co-solvent considered in this study is the cyclic amide 1,3-dimethyl-2-imidazolidinone (DMI). DMI is a very stable aprotic solvent with a high dipole moment (µ=4.09 D, a high dielectric constant (=37. 6) and an excellent ability to solvate inorganic and organic compounds [START_REF] Lazarev | Structure of hydration shell of 1,3-dimethyl-2-imidazolidinone according to FT -Raman spectroscopy Russ[END_REF] . DMI is very polar and thus completely miscible with water. DMI is widely used in the manufacture of polymers, medicines, dyestuffs, agrochemicals, electronic parts, and so on. It is often utilized as a substitute for the carcinogenic hexamethylphosphoramide (HMPA).

Excess properties are also very useful to understand the mixing state in terms of intermolecular interactions. Indeed, the information concerning molecular interactions is required for the design of solvents for industrial applications. In our case, the examination of excess properties will be helpful to better understand the molecular interaction of water with the -CON-R group in DMI.

First, new experimental measurements of density and excess volume for the binary mixture composed of water and DMI are presented. Then, these new experimental excess volumes are correlated by the Redlich-Kister equation 2 . Also, the same type of correlation will be used to correlate excess dynamic viscosity and refractive index data.

Experimental details

Materials purities and suppliers

Table 1 presents the chemical species, suppliers, purity and refractive index. Compositions are prepared gravimetrically. Water was purified by a Millipore Direct-Q5UV system with a 0.22 μm membrane.

Experimental method -Density

Density measurements have all been performed with the DSA5000M Anton Paar digital vibrating tube densimeter. The oscillating period or frequency, measured by the densitometer, is depending on the tube mass and therefore on the fluid density. Eq. 1 is used for relating the period of vibration, , to density, :

2   b a   (1)
where a and b are constants to be adjusted. For these purposes we have used bi-distilled and degassed water, and dry air, at 293.15 K. Supplier accuracy (a) on measured density is estimated lower than 0.01 kg.m -3 .

We have considered the recommendation given by Chirico et al. [START_REF] Chirico | Improvement of quality in publication of experimental thermophysical property data: Challenges, assessment tools, global implementation, and online support[END_REF] concerning the estimate of relative density uncertainty (

) with  the presumed difference in density between the compound and the impurity (the authors recommend =0.1) and x s the purity. Details on uncertainty calculation is given by Eq. 2. One platinum resistance thermometer with 0.01 K accuracy is used for temperature measurements. The sample densities are then measured at thermal equilibrium for various temperatures. This densitometer permits also the measurement of speed of sound with an accuracy of 0.5 m.s -1 .

(

) 2
-Viscosity

The viscosity data have been obtained by an Anton Paar LOVIS 2000 ME viscometer. It is a modular instrument to be coupled with the densimeter DSA5000M. The solution sample is filled into a glass capillary (diameter of 1.59 mm) which is introduced into a temperature controlled capillary block.

This block can be inclined at a variable predefined angle according to the calibration performed. We can note that the capillary had already been calibrated with appropriate standard fluids (see Reference Manual of Lovis 5 ).

Briefly, the experimental method of LOVIS 2000 ME viscometer is based on the falling ball method.

The principle consists of measuring the falling time of a steel ball in the capillary tube filled with the sample fluid and inclined at different angles. It automatically provides the viscosity value for the specified temperature level. The equation governing this principle (i.e. the calibration curve of the instrument) is given by Eq. 3.

(

) 3
where is the dynamic viscosity of the sample solution in mPa.s; is the calibration constant of the measuring system; is the ball (steel) density in g.cm -3 ; is the sample density in g.cm -3 , measured by the instrument, and is the rolling time in seconds. More details are given in Chirico et al. [START_REF] Chirico | Improvement of quality in publication of experimental thermophysical property data: Challenges, assessment tools, global implementation, and online support[END_REF] . The estimate of viscosity uncertainty depends on the uncertainty of density. The calculation of the uncertainty is given by Eq. 4.

(4)

, linked to the accuracy of the instrument, is equal to the 0.5% of the measured value for the LOVIS viscometer (given by Manual of Lovis 5 ). The second term is the standard deviation ( ) of the average of three different measurement cycles (we have considered 3 cycles). When just one cycle of measurements is performed, the uncertainty of the experimental data corresponds to the uncertainty of the instrument used ( ). More details concerning estimation of uncertainty are given in Supporting Information of the paper from Cremona et al. [START_REF] Cremona | Density and Viscosity Measurements and Modeling of CO2-Loaded and Unloaded Aqueous Solutions of Potassium Lysinate[END_REF] Also, following the recommendation of Huber et al. -Preparation of mixture To prepare the mixtures, an empty 20 cm 3 glass bottle is air-tight closed with a septum and then put under vacuum using a vacuum pump wherein a needle is introduced through the septum. The empty bottle is weighed, and then the less volatile component, freshly degassed, is introduced by means of a syringe. After weighing the bottle loaded with the first component, the more volatile one is added similarly and then the bottle is weighed again. All weightings are performed using an analytical balance with 10 -4 g accuracy and hence the uncertainty is estimated to be lower than 2x10 -5 for mole fractions.

Results and Discussions

Pure component properties

The density values of pure chemical measured using the DSA 5000M Anton Paar densitometer are presented as a function of temperature in

Excess properties

Excess molar volume

The measured density of aqueous DMI solutions over the entire range of mole fractions x 1 and temperature range between (283.15 and 343.15) K are listed in Table 2. Density data were used to calculate the excess volume. The excess molar volume v E , is calculated using Eq. 5.

* 2 2 * 1 1 v x v x v v E    (5)
where x 1 and x 2 represent mole fractions and * 1 v and * 2 v are the molar volumes of components 1 and 2, respectively. v stands for the molar volume of mixture. Using the measured density , Eq. 5 can be rewritten as:  are the densities of components (1) and (2), respectively.  stands for the density of the mixture. Maximum uncertainty resulting in the calculation of v E is estimated to be less than 0.0011 cm 3 .mol -1 using Eqs. 7-9.

* 2 2 2 * 1 1 1 2 2 1 1    M x M x M x M x v E           (6)
2 2 x m u u u     (7)     2 2 2 1 2 1 1 1 1 m m u x x u m x   (8) 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 2 1 1 1 2 2 2 2 1 1 x m m m v u M M M M u M x M x M x M x u E                                                        (9)
With M i the molar mass of component i, m i the mass of component i,  i the density of component i,  m density of the mixture.

Thermal expansion

Thermal expansion coefficient data are calculated using density data measured for different composition and temperature. A second order polynomial expression for the molar volume for each composition is considered (parameters are given in Table S3 in Supporting Information). Considering the definition of thermal expansion and the derivative of molar volume equation with respect to temperature for each composition, the thermal expansion was calculated. Like excess molar volume, the excess molar thermal expansion  E , is calculated using Eq. 10. (10) where x 1 and x 2 represent mole fractions and and are the molar thermal expansivity of components 1 and 2 respectively. The analysis of excess thermal expansion is very important as it takes into account the influence of temperature in the variation of molar volume.

The uncertainty in thermal expansion is linked to the uncertainty in density, the accuracy of the derivative with respect to temperature, and the uncertainty in molar fraction. Eq. 11 gives the expression.

(

) 11
The uncertainty in composition is given by Eq. 8. For the thermal expansion, uncertainty is given by Eq. 12.

(

) 12
The value of is equal to 1×10 -5 K -1 . The uncertainty in excess thermal expansion is given by Eq. 13. The value obtained is also close to 1.5×10 -5 K -1 .

(

) 13

Refractive index

According to the theory remind by Reis et al. [START_REF] Reis | Refractive index of liquid mixtures : theory and experiment[END_REF] , the square of the refractive index is approximately equal to the relative permittivity ( ). This property, also known as the dieletric constant, characterises the capability of the solvent to form electrolytes (and to stabilize electrolytes). The refractive index is linked to the molar volume by the Lorentz-Lorenz equation (Eq. where n is the refractive index, v the molar volume, N A Avogadro's number,  0 the permittivity of vacuum and  the mean molecular polarizability. Eq. 14 was applied to estimate the excess squared refractive index (n²) E . Volume fractions instead of mole fractions are considered. Eq.

15 is considered for the calculation of excess squared refractive index.

(

) 15
Where is the volume fraction of component i and the squared refractive index of component i.

The uncertainty in refractive index is linked to the uncertainty in refractive index measured and the uncertainty in molar fraction. Eq. 16 gives the expression.

The uncertainty in composition is given by Eq. 8. The value of is equal to 6×10 -5 . The uncertainty in excess squared refractive index is given by Eq. 17. The value obtained is also close to

2×10 -4 . ( 17
)

Viscosity

It is more convenient to consider the kinematic viscosity instead of the dynamic viscosity. According to Eyring theory 10 , kinematic viscosity can be correlated using an Arrhenius type law. An energy of activation of the flow is considered. This energy is required to activate the flow or to produce a hole of requisite size for the translation to occur. The kinematic viscosity is linked to the dynamic viscosity by Eq. 18.

(

) 18
Where h is Planck's constant, N A Avogadro's number, v molar volume, T the temperature and is the activation Gibbs free energy of flow. This energy can be written as the sum of contributions from the ideal solution and the excess part . According to Eyring, this energy is linked to excess Gibbs free energy of the mixture which considers the molecular interaction . The excess Gibbs free energy given by Eq. 19 is calculated from a thermodynamic model (fugacity

coefficient ()). (19)
Eq. 20 gives the viscosity of an ideal solution. For the real mixture, the excess Gibbs free energy of flow can be expresses by Eq. 21.

(20)

(21)

Redlich-Kister data treatment

Usually the Redlich-Kister (RK) correlation 2 is chosen to correlate excess molar properties of binary systems. This equation is applied for the excess thermophysical properties. Eq. 22 presents the correlation considering mole fraction. For the refractive index, we have considered the volume fraction.

(

) 22
The coefficients (A i ) have to be determined. The method of calculation of the standard deviation of each parameter is given in Supporting Information file. The variance , corresponding to each fit, is calculated using Eq. 23.

(

) 23
where P is the number of parameters (A n ) and N represents the number of experimental data. The main difficulty of the RK treatment is the selection of the number of parameters A i . In order to do that, it is recommended to investigate apparent molar properties. In our case, we have considered

2 1 x x v E
as a function of molar composition x 1 . This quantity gives us useful information concerning volumetric properties, particularly at low concentration as suggested by Desnoyers and Perron 11 . This term is directly related to the apparent molar volume and can thus be assimilated to a thermodynamic property. Eq. 24 shows that

2 1 x x v E
is directly linked to the apparent molar volume.

2 * 1 , 1 1 * 2 , 2 2 1 x v v x v v x x v E       (24)
where  , i v is the apparent molar volume. Desnoyers and Perron indicate that the change of the

2 1 x x v E
slope can be attributed to various factors: the first one, the size and shape of molecules, the second one, the intermolecular interaction energy differences at infinite or close to infinite dilution (in effect, at infinite dilution,

2 1 x x v E
decreases like the apparent molar volume) and the third one the formation of chemical complexes containing unlike molecules. There are two contributions for the apparent molar volume of a molecule: the volume of the molecule and the free volume space. By analysis of the evolution of

2 1 x x v E
as a function of x 1 we have selected the number of RK parameters (5) for the investigation of all excess thermophysical properties.

The partial molar volume i v (cm 3 .mol -1 ) of each component i has been calculated using Eq. 25, with V, the volume of the mixture (cm 3 ).

j n P T i i n V v , ,            (25)
Differentiating Eq. 25 with respect to n i and combining the result with Eq. 22 leads to equations for the partial molar volumes of the different species (Eqs. 26 and 27).

P T E E x v x v v v , 2 2 * 1 1              (26) 13 P T E E x v x v v v , 1 1 * 2 2              (27)
Using the Redlich-Kister equation, we can obtain the expression of partial molar volumes (Eqs. 28 and 29) with respect to x i .

               1 2 2 2 2 2 2 2 * 1 1 2 1 1 2 2 1 n n n n x nA x x x A x v v (28)                   1 2 2 2 2 2 2 2 * 2 2 2 1 1 2 2 1 1 n n n n x nA x x x A x v v (29)
At infinite dilution Eqs 28 and 29 give Eqs. 30 and 31.

(30

) (31)
Concerning the uncertainty of molar volume at infinite dilution, it is given by Eq. 32.

(32)

4.

Results and discussion

Excess molar volume

Table 3 presents the values of mixture densities and excess volumes. The measured density at 293.15 and 313.15 K over the entire range of mole fractions x 1 are plotted in Figure 1. One can see that starting from x 1 =0, the density increases with increasing DMI concentration, reaches a maximum value, and then decreases. The system was also studied by Uosaki et al. [START_REF] Uosaki | Excess molar volumes of (cyclic amide + water) at 298.15 and 308[END_REF] at 298.15 and 308.15 K. 4). Figs. 2-3 present respectively,

2 1 x x v E
as a function of x 1 and v E as a function of

x 1 , at 293.15 and 328.15 K. The excess molar volume is negative and we can observe that in absolute value, the excess volume is very important. This information supports the idea of a strong associative interaction between the two molecules. The objective of the following work is to better identify the interactions between the two molecules. The analysis of apparent molar volume does not reveal any original behavior. But we can see that for DMI close to infinite dilution, the apparent molar volume decreases, certainly due to the possible existence of a complex due to H bonds between atoms of oxygen and hydrogen of water and DMI. This complex is surrounded by molecules of water due to strong hydrophobic interactions. At the opposite, in the DMI rich region, the system under study behaves in a regular manner. Hepler

According to Eq. 32, we can calculate the second derivative and observed that . This led to negative and so we can assume that DMI is a structure maker. It is not a surprise considering the molecular structure of DMI and the analysis of apparent or partial molar volume. In the rest of the paper, we will analyze the other thermophysical properties in order to confirm the structure maker effect of DMI with water.

Thermal expansion

The results concerning the thermal expansion are presented in Table 3. Table 6 presents the values of the Redlich-Kister parameters. We have considered the same number of parameters as for the excess volume correlation. According to Fig. 4, the excess thermal expansion shows a maximum value for a composition close to 0.2, similar to

2 1 x x v E
. Concerning (see Fig 5), there is no particular behaviour, maybe a short break in the slope of the curve. The analysis of excess thermal expansion supports the idea that there is a complex that could form for compositions close to 0.2. In order to confirm this point, we can analyse the refractive index curve at 293.15 K.

Refractive index

The results for the refractive index (wave length =589nm) at 293.15 K are presented in Table 7 and Fig. 6. No Redlich-Kister treatment was done. We can observe that the excess squared refractive index presents a maximum whereas the excess volume presents a minimum. The extrema for the two excess properties are not observed at the same composition. Iglesias-Otero et al. [START_REF] Iglesis-Otero | Density and refractive index in mixtures of ionic liquids and organic solvents: correlations and predictions[END_REF] have also demonstrated the strong link between refractive index and molar volume. Negative excess molar volume is an indication of important attractive interaction between the two molecules. The refractive index is linked to the relative permittivity also called dielectric constant. As the excess squared refractive index is always positive, we can assume that if we consider DMI + water as a solvent, the dissociating power of the solvent is increased compared to DMI and water taken alone, which is, in theory, a good characteristic for the stabilization of electrolytes. Moreover, a careful analysis of the Fig. 5 reveals a break in the curve for a composition close 0.2 in volume fraction. This can be attributed to a change in the molecular organisation maybe due to the formation of a stabilized complex composed by molecules of DMI and water.

Viscosity

We have considered the kinematic viscosity for the data treatment. The results are indicated in Table 8. The kinematic viscosities increase with increasing DMI concentration and reach a maximum at composition close to 0.4 and then decrease. Using eqs. 17 to 20, we have considered excess Gibbs free energy of flow and correlated it with the Redlich-Kister equation. The Redlich-Kister parameters for the determination of excess activation Gibbs free energy are presented in Table 9. Figure 7 shows the excess Gibbs free energy of flow for DMI (1) + water (2) binary system over the whole range of compositions and at 293.15 and 318.15 K. Also, we can observe an increase of the excess Gibbs free energy of flow with increasing concentration. The maximum is reached at a concentration close to 0.3 and is followed by a decrease. We can also observe that an increasing temperature decreases the excess Gibbs free energy of flow. The concentration of the maximum remains constant. The reason of this maximum can be attributed to the apparition of a complex with strong molecular interaction between water and DMI molecules. In effect, according the Eyring theory, the excess Gibbs free energy of flow is linked to the excess Gibbs free energy calculated with the activity coefficient. When a stable complex is formed, difficulty to flow might appear, and more energy is required to move the molecules by breaking the molecular interaction. As we have performed the measurement at several temperatures, it is possible to calculate the excess entropy of flow ( ) considering Eq. 35.

(

) 35
Using our data, we have plotted on Fig. 8 the excess entropy of flow for the DMI (1) + water (2) binary system over the whole range of concentrations. The trend is the same as excess Gibbs free energy of flow. It confirms the fact that molecular interactions between DMI are very strong especially at the composition of the maximum where it is now obvious that a complex is formed between the two molecules. x DMI

Discussion

Considering all our results it is now certain that a strong attractive interaction exists between the two molecules, this because the excess molar volume is negative and DMI is considered as a liquid structure maker. Probably due to the differences in size, there might be a packing effect. The analysis of apparent and partial molar volumes, refractive index, and thermal expansion reveals that there exists a composition where a complex is formed. It signifies that a stable structure exists. This stable structure is responsible for an increase of the viscosity until a concentration close to 0.2. It is important to notice that modification of the structure of the liquid appears when low quantities of DMI are introduced. An important excess of DMI destabilizes the structure of the complex. Lazarev and Mikhailov 1 have studied the structure of hydration considering FT-Raman spectroscopy measurements. In their publication, they have highlighted the phenomenon of solvation and identified a microstructure (called complex by us) stabilized by a network of intermolecular hydrogens bonds of type O-H…O, C-H…O, and N-H…O. They conclude that the cluster DMI+(H 2 O) 6 is the optimum structure (they called it hydrate shell of monomeric molecule). According to Lazarev and Mikhailov, there is an important energy of electrostatic interactions between the charge of the cluster and the continuum of water molecules. As a consequence, it is not a surprise to observe an important excess Gibbs free energy of flow (linked to the excess Gibbs free energy). With the data we have measured, we have confirmed the existence of the cluster and also identify the composition of formation of this cluster. But we have observed the optimal cluster formation for 2 to 3 H2O molecules and 1 DMI molecule, composition slightly higher than the one identified by Lazarev and Mikhailov. This is an important result in the case we would like to use DMI as a co-solvent in aqueous solvent (amine +water or acid + water for example): certainly, the interaction (physical or chemical) between water and the solvent might be degraded as water prefers to interact with DMI. DMI is not the only amide to present the observed behaviour. For example, Garcia-Abuin et al. [START_REF] Garcia-Abuin | Density, Speed of Sound, Viscosity, Refractive Index, and Excess Volume of N-Methyl-2-pyrrolidone + Ethanol (or Water or Ethanolamine) from T =[END_REF] have studied the compressibility, the dynamic viscosity, and the excess volumes for the binary system of N-methyl-2-pyrrolidone (NMP), a cyclic amide, and water. They have observed very similar results, with a minimum excess molar volume around 40 mol% NMP. The hydrogen bonding dynamics have been studied by dielectric relaxation spectroscopy by Jia et al. 15 . They show that the size of the water clusters decreases with increasing NMP concentration. Ivanov and Kolker 16,17 have studied the binary system composed by Hexamethylphosphoramide (HMPA) and water. Heat capacity and volumetric properties were studied. They have also analysed the apparent molar properties. The profile of apparent molar volume of HMPA is very similar to DMI (the minimum is obtained for composition close to high dilution). They conclude that HMPA behave like a strongly hydrophobic solute. They observed in parallel an important increase in apparent molar heat capacity of HMPA and negative values of excess molar heat capacity. Also, in the HMPA rich region, the system behaves in a regular manner. The analysis of apparent heat capacity of water in the HMPA rich concentration region reveals a decrease in comparison to the heat capacity of pure water. The authors have concluded that no hydrogen bonds exist between molecules of water and that they are probably inserted in the holes between HMPA molecules. It is like a packing effect where HMPA molecules from a cage-type structure around the water molecule. The interaction between HMPA and water are stronger than the one between two molecules of water. We did not study the heat capacity of aqueous DMI but considering kinematic viscosity we have shown that the excess entropy of flow (linked to excess molar entropy) was negative close to infinite dilution of water. It is obvious that the negative excess entropy we have observed also indicates an excess molar heat capacity. The conclusion mentioned for the molecule of water inserted/trapped in the cage of molecules of HMPA can be also formulated for the molecule of water at high dilution in solution of DMI.

One can compare the minimum excess molar volume of N-methyl-2-pyrrolidone (NMP), DMI and HMPA, respectively a mono-, di-and tri-amide. The values at 293 K are approximately -1.2, -1.4, and -1.6 cm 3 mol -1 for NMP, DMI, and HMPA, respectively. Their respective dipole moments at 293 K are µ=4.1 D for NMP, µ=4.09 D for DMI and µ=5.54 D for HMPA. In the presence of water this dipole moment changes, but the very high dipole moment of HMPA likely contributes to the stronger hydrogen-bond network and thus the stronger structuring effect of HMPA. The higher minimum for the excess volume as well as the shift of this minimum to a more symmetric 40 mol% for NMP (instead of 30 mol% for DMI or HMPA) also indicates a weaker structuring effect of NMP compared to DMI or HMPA.

Additional results concerning speed of sound (for each pure component and mixtures) are presented in supporting information. The obtained results seem to also confirm the particular behaviour of the DMI + water binary system.

Conclusion

The densities, viscosities, speed of sound and refractive index of the DMI + water binary system were measured over the temperature range T=(283.15 -343.15 K) using various instrument (vibrating tube densitometer, falling ball viscometer and refractometer). Excess molar volume, thermal expansion and excess Gibbs free energy of flow were determined and correlated by a Redlich-Kister correlation. From the measurements, it appears that for a given molar composition close to 0.2 in DMI, a stable cluster surrounded by molecules of water may be formed. Also, after comparison with a similar study with HMPA where molar heat capacity measurements were realized, our results seem to confirm that at infinite dilution, water molecules are inserted in the holes between DMI molecules. DMI can be classified as an aqueous liquid structure maker. It highlights the strong attractive interaction between DMI and water and thus the difficulties of using DMI as a cosolvent in an aqueous solvent. Our study shows that in order to highlight the existence of complex molecular interactions, it is preferable to have different data related to the thermodynamic properties. Density measurements combined with refractive index measurements can be used to determine excess volumes, thermal expansion, apparent molar volumes and to identify complex molecular interactions between compounds in the whole range of concentration. Kinematic viscosity measurements combined with heat capacity measurements, if available, can also provide information on the existence of these complexes. All these molecular phenomena make the prediction of solvent properties and the screening of these solvents for various gas treatment or liquid applications related to effluent treatment and purification very complicated.

Supporting information Available:

It presents comparison with literature data, the correlation used to calculate the molar volumes, the results concerning the speed of sound measurements and the method used to calculate the standard deviation of the Redlich-Kister parameters correlation.

Graphical Abstract

where 1 M and 2 M are the molar masses * 1  and * 2

 1212

14 )

 14 as presented by Iglesias-Otero et al. 9 :(14)

Figure 1 :

 1 Figure 1: Density for DMI (1) + water (2) binary system as a function of DMI mole fraction at atmospheric pressure and 3 different temperatures: (○) 293.15 K, () 308.15 K, (×) 328.15 K.

Figure 2 1 Figure 3 :

 213 Figure 2:

Figure 4 1 Figure 5 :

 415 Figure 4: for DMI (1) + water (2) system as a function of DMI mole fraction at atmospheric pressure and 2 different temperatures: (○) 293.15 K, (×) 328.15 K, solid line: Redlich-Kister correlation.

Figure 6 :

 6 Figure 6: Excess squared refractive index for DMI (1) + water (2) binary system as a function of DMI volume fraction at atmospheric pressure and 293.15 K.

Figure 7 :

 7 Figure 7: Excess Gibbs free energy of flow for DMI (1) + water (2) binary system as a function of DMI mole fraction at atmospheric pressure and 3 different temperatures. (○) 293.15 K, () 308.15 K , (×) 328.15 K, solid line: Redlich-Kister correlation.

Figure 8 :

 8 Figure 8: Excess entropy of flow for DMI (1) + water (2) binary system for whole range of mole fractions.

Table 1 : Chemical sample

 1

	Compound	CAS Number	Formula	Supplier	Purity	Refractive	nD c
					(GC) a	index*
						(293.15 K)
	Water	7732-18-5	H 2 O	-	-	1.33299	1.3325
	1,3-dimethyl-2-	80-73-9	C 5 H 10 N 2 0	Sigma Aldrich	0.995	1.47259 b	1.4704
	imidazolidinone (DMI)

*: Apparatus; Anton Paar ABBEMAT 300 (=589nm), supplier accuracy +/-0.0001, u(n)=6 ×10 -5 , a GC: Gas Chromatograph, (b): 1.472 at 20°C at =589nm given by the supplier (c): Value from Simulis Thermodynamics (Prosim, France): Bloxham et al.

[START_REF] Bloxham | Proper use of the DIPPR 801 database for creation of models, methods, and processes[END_REF]

Table 2

 2 order polynomial expression in the temperature range 283.15-343.15 K ( where a, b and c are the parameters adjusted on experimental data). The parameters are presented in TableS3in Supporting Information. Thermal expansion data are calculated considering the derivative with respect to temperature of molar volume correlation. Calculated values of thermal expansion are presented in Table2. The dynamic viscosity data obtained with the LOVIS 2000 ME for DMI and water are also presented in Table2.

. Comparison with literature was done and presented in Supporting Information and a good agreement is observed. The density data were used to calculate the thermal expansion (

). The molar volume data were correlated by a second

Table 2 : Density, thermal expansion and dynamic viscosity of pure components DMI and water studied at atmospheric pressure (p=101.33 kPa) a DMI Water T / K / kg.m -3  × 10 4 / K -1 / mPa.s / kg.m -3  × 10 4 / K -1 / mPa.s

 2

	283.15	8.35	3.958	1000	1.34	1.306
	288.15	8.38	3.548	999	1.75	1.138
	293.15	8.40	3.204	998	2.15	1.002
	298.15	8.43	2.910	997	2.56	0.890
	303.15	8.45	2.659	996	2.96	0.797
	308.15	8.47	2.441	994	3.36	0.719
	313.15	8.50	2.252	992	3.76	0.653
	318.15	8.52	2.085	990	4.15	0.596
	323.15	8.54	1.939	988	4.55	0.547
	328.15	8.56	1.809	986	4.94	0.504
	333.15	8.58	1.695	983	5.33	0.466
	338.15	8.60	1.594	981	5.71	0.433
	343.15	8.62	1.577	978	6.09	0.404
	a Expanded uncertainties (k=2) U(p)=0.3 kPa, U(T)=0.01 K, U(=1.0 kg.m -3 U(=2.2 10 -5 K -1 , U r ()=0.01.

  It is possible to evaluate the effect of the solute at infinite dilution on the structure of the liquid. The temperature dependence of the molar volume at infinite dilution and the second derivative is particularly meaningful because the sign of the second derivative can be used as one criterion to justify if DMI is a structure breaker or structure maker. In Table5we present the values of molar volume of DMI and water at infinite dilution as a function of temperature calculated using Redlich-Kister parameters and Eqs. 29 and 30. Considering DMI, we can correlate the molar volume at infinite dilution using a second order polynomial expression (Eq. 34).

		1080
	1070 capacity is commonly negative in the case of structured liquid and so	should be
	1060 negative. Eq. 33 presents the thermodynamic equation that linked heat capacity and the derivative
	1050 of molar volume with respect to temperature. 3
	Density/kg.m	1020 1030 1040					(33)
		1010
		1000
		990
		980
		0.0	0.2	0.4	0.6	0.8	1.0
				x 1

[START_REF] Hepler | Thermal expansion and structure in water and aqueous solutions Can[END_REF]

in 1969 has observed that it was possible to identity the effect of solute in water as structure maker or structure breaker. He indicates the partial molar heat

Table 3 : Densities ( /kg.m -3 ), excess molar volumes (v E / cm 3 .mol -1 ), molar expansion ( / K -1 ) and excess molar expansion ( E /K -1 ) for DMI (1) + water (2) binary system as a function of DMI mole fraction at atmospheric pressure (p=101.33 kPa) a

 3

	x 1		 × 10 4	v	E	 E × 10 4		 × 10 4	v	E	 E × 10 4		 × 10 4	v	E	 E × 10 4		 × 10 4	v	E	 E × 10 4		 × 10 4	v	E	 E × 10 4
			T=283.15 K			T=288.15 K			T=293.15 K				T=298.15 K			T=303.15 K
	0		1.34	-	-	999	1.75	-	-	998	2.15	-	-	997	2.56	-	-	996	2.96	-	-
	0.0501		3.87	-0.260	2.2	1025	4.16	-0.249	2.1	1023	4.45	-0.239	2.0	1021	4.74	-0.231	1.9	1018	5.03 -0.223 1.8
	0.1000		5.82	-0.583	3.8	1045	6.03	-0.560	3.6	1042	6.24	-0.538	3.5	1039	6.44	-0.519	3.3	1035	6.64 -0.500 3.1
	0.1801		7.57	-1.034	5.0	1065	7.70	-0.997	4.8	1061	7.84	-0.962	4.6	1057	7.98	-0.929	4.4	1052	8.11 -0.898 4.2
	0.2598		8.27	-1.328	5.1	1074	8.39	-1.286	4.9	1069	8.50	-1.246	4.7	1065	8.61	-1.208	4.5	1060	8.72 -1.170 4.3
	0.3390		8.52	-1.475	4.8	1077	8.62	-1.433	4.6	1073	8.72	-1.393	4.5	1068	8.82	-1.354	4.3	1063	8.92 -1.315 4.1
	0.4201		8.55	-1.501	4.3	1077	8.65	-1.462	4.1	1072	8.74	-1.426	4.0	1068	8.84	-1.390	3.8	1063	8.93 -1.353 3.7
	0.4952		8.53	-1.440	3.7	1076	8.61	-1.407	3.6	1071	8.70	-1.375	3.5	1066	8.79	-1.343	3.3	1062	8.87 -1.310 3.2
	0.5807		8.47	-1.299	3.1	1073	8.55	-1.272	2.9	1069	8.62	-1.247	2.8	1064	8.70	-1.220	2.7	1059	8.78 -1.193 2.6
	0.6594		8.41	-1.118	2.4	1071	8.48	-1.097	2.4	1066	8.55	-1.078	2.3	1062	8.61	-1.057	2.2	1057	8.68 -1.036 2.1
	0.7413		8.38	-0.891	1.8	1068	8.43	-0.875	1.8	1064	8.49	-0.862	1.7	1059	8.55	-0.847	1.6	1055	8.60 -0.831 1.6
	0.8184		8.34	-0.647	1.3	1066	8.39	-0.637	1.2	1061	8.44	-0.627	1.2	1057	8.49	-0.617	1.1	1052	8.54 -0.608 1.1
	0.8990		8.34	-0.377	0.7	1063	8.38	-0.371	0.7	1059	8.42	-0.368	0.6	1054	8.46	-0.362	0.6	1050	8.49 -0.357 0.6
	0.9488		8.34	-0.195	0.3	1062	8.37	-0.192	0.3	1057	8.40	-0.191	0.3	1053	8.43	-0.189	0.3	1048	8.46 -0.187 0.3
	1		8.35	-	-	1060	8.38	-	-	1056	8.40	-	-	1052	8.43	-	-	1047	8.45	-	-

Table 3 : Continued

 3

	x 1		 × 10 4	v	E	 E × 10 4		 × 10 4	v	E	 E × 10 4		 × 10 4	v	E	 E × 10 4		 × 10 4	v	E	 E × 10 4		 × 10 4	v	E	 E × 10 4
			T=308.15 K			T=313.15 K			T=318.15 K			T=323.15 K			T=328.15 K
	0	994	3.36	-	-	992	3.76	-	-	990	4.15	-	-	988	4.55	-	-	986	4.94	-	-
	0.0501	1015	5.31	-0.216	1.7	1013	5.59	-0.209	1.6	1010	5.87	-0.202	1.5	1007	6.15	-0.196	1.4		6.42 -0.189 1.3
	0.1000	1032	6.84	-0.483	3.0	1028	7.03	-0.466	2.8	1025	7.23	-0.450	2.6	1021	7.42	-0.435	2.5		7.61 -0.419 2.3
	0.1801	1048	8.24	-0.868	4.0	1044	8.37	-0.839	3.8	1039	8.50	-0.810	3.6	1035	8.63	-0.783	3.4		8.75 -0.755 3.2
	0.2598	1056	8.83	-1.134	4.1	1051	8.94	-1.099	3.9	1046	9.04	-1.063	3.8	1042	9.14	-1.029	3.6		9.25 -0.994 3.4
	0.3390	1058	9.02	-1.277	3.9	1054	9.12	-1.240	3.8	1049	9.21	-1.203	3.6	1044	9.31	-1.165	3.4		9.40 -1.128 3.2
	0.4201	1058	9.02	-1.317	3.5	1054	9.11	-1.282	3.4	1049	9.20	-1.245	3.2	1044	9.29	-1.209	3.1		9.37 -1.173 2.9
	0.4952	1057	8.95	-1.278	3.1	1052	9.03	-1.246	2.9	1047	9.11	-1.212	2.8	1043	9.19	-1.180	2.7		9.27 -1.146 2.5
	0.5807	1055	8.85	-1.166	2.5	1050	8.92	-1.138	2.4	1045	9.00	-1.110	2.3	1041	9.07	-1.082	2.2		9.14 -1.052 2.1
	0.6594	1052	8.75	-1.014	2.0	1048	8.81	-0.992	1.9	1043	8.88	-0.970	1.8	1039	8.94	-0.946	1.8		9.00 -0.922 1.7
	0.7413	1050	8.66	-0.815	1.5	1045	8.71	-0.798	1.4	1041	8.76	-0.781	1.4	1036	8.82	-0.764	1.3		8.87 -0.746 1.2
	0.8184	1048	8.58	-0.597	1.0	1043	8.63	-0.586	1.0	1039	8.67	-0.574	0.9	1034	8.72	-0.563	0.9		8.76 -0.550 0.9
	0.8990	1045	8.53	-0.350	0.6	1041	8.56	-0.344	0.5	1037	8.60	-0.338	0.5	1032	8.63	-0.332	0.5		8.67 -0.325 0.5
	0.9488	1044	8.49	-0.184	0.3	1040	8.52	-0.181	0.3	1035	8.55	-0.178	0.3	1031	8.58	-0.175	0.2		8.61 -0.172 0.2
	1	1043	8.47	-	-	1038	8.50	-	-	1034	8.52	-	-	1029	8.54	-	-		8.56	-	-

Table 3 :

 3 Continued

	x 1		 × 10 4	v	E	 E × 10 4		 × 10 4	v	E	 E × 10 4		 × 10 4	v	E	 E × 10 4
			T=333.15 K				T=338.15 K				T=343.15 K
	0	983	5.33	-	-	981	5.71	-	-	978	6.09	-	-
	0.0501		6.69	-0.183	1.2	997	6.96	-0.176	1.1	993	7.23	-0.169	1.0
	0.1000		7.80	-0.404	2.1	1009	7.98	-0.389	2.0	1005	8.17	-0.374	1.8
	0.1801		8.88	-0.727	3.0	1021	9.00	-0.700	2.8	1017	9.12	-0.673	2.6
	0.2598		9.35	-0.960	3.2	1027	9.44	-0.925	3.0	1022	9.54	-0.889	2.8
	0.3390		9.49	-1.090	3.1	1029	9.58	-1.052	2.9	1024	9.67	-1.013	2.7
	0.4201		9.46	-1.135	2.8	1029	9.54	-1.097	2.6	1024	9.62	-1.058	2.5
	0.4952		9.35	-1.111	2.4	1028	9.42	-1.075	2.3	1023	9.50	-1.038	2.2
	0.5807		9.20	-1.022	2.0	1026	9.27	-0.990	1.9	1022	9.34	-0.958	1.8
	0.6594		9.06	-0.897	1.6	1025	9.12	-0.871	1.5	1020	9.18	-0.843	1.4
	0.7413		8.92	-0.727	1.2	1023	8.97	-0.707	1.1	1018	9.02	-0.685	1.0
	0.8184		8.80	-0.537	0.8	1021	8.84	-0.522	0.8	1016	8.89	-0.507	0.7
	0.8990		8.70	-0.317	0.4	1019	8.73	-0.309	0.4	1014	8.76	-0.300	0.4
	0.9488		8.64	-0.168	0.2	1018	8.66	-0.164	0.2	1013	8.69	-0.160	0.2
	1		8.58	-	-	1016	8.60	-	-	1012	8.62	-	-
	a Expanded uncertainties (k=2) U(P)=0.3 kPa, U(T)=0.01 K, U(=1.0 kg.m -3 , U(x)=0.002, U(V E =0.011 cm 3 .mol -1 , U(=2.2 10 -5 K -1 , U( E =4 10 -5 K -1

Table 4 : Excess molar volume: Redlich-Kister parameters and deviation for DMI + water binary system.

 4

	T/K

Table 5 : Molar volume at infinite dilution for DMI (1) + water (2) binary system.

 5

	T/ K	/ cm 3 .mol -1 u( )/ / cm 3 .mol -1	/ cm 3 .mol -1	u( )/ / cm 3 .mol -1
	283.15	103.5	1.6	13.9	1.6
	288.15	103.7	1.5	13.9	1.5
	293.15	104.3	1.3	13.9	1.3
	298.15	104.9	1.1	14.0	1.1
	303.15	105.5	1.0	14.1	1.0
	308.15	106.1	0.8	14.2	0.8
	313.15	106.6	0.8	14.3	0.8
	318.15	107.2	0.7	14.4	0.7
	323.15	107.8	0.6	14.5	0.6
	328.15	108.3	0.6	14.6	0.6
	333.15	108.9	0.5	14.8	0.5
	338.15	109.5	0.5	14.9	0.5
	343.15	110.1	0.5	15.1	0.5

Table 6 : Thermal expansion: Redlich-Kister parameters and deviation for DMI + water binary system.

 6

	T/K

Table 7 : Refractive index and excess squared refractive index for DMI (1) + water (2) binary system as a function of DMI mole fraction at atmospheric pressure and 293.15 K (p=101.33 kPa) a

 7

					volume fraction
	x 1	n	v/ cm	3 .mol -1	 1
	0.0501	1.36981	22.32	0.240	0.0055
	0.1000	1.39567	26.52	0.400	0.0145
	0.1801	1.42292	33.30	0.568	0.0253
	0.2598	1.43962	40.20	0.678	0.0302
	0.3390	1.45023	47.18	0.754	0.0308
	0.4201	1.45725	54.45	0.813	0.0284
	0.4952	1.46176	61.27	0.855	0.0252
	0.5807	1.46531	69.10	0.892	0.0208
	0.6594	1.46764	76.35	0.921	0.0165
	0.7413	1.4694	83.94	0.945	0.0122
	0.8184	1.47065	91.12	0.964	0.0083
	0.8990	1.47163	98.64	0.982	0.0044
	0.9488	1.47213	103.30	0.991	0.0021
	a Expanded uncertainties (k=2) U(P)=0.3 kPa, U(T)=0.01 K, U(x)=0.002,	0.0002, u(n)=6 ×10 -

Table 8 : Dynamic viscosity and excess Gibbs free energy of flow ( ) for DMI (1) + water (2) binary system as a function of DMI mole fraction at atmospheric pressure (p=101.33 kPa) a

 8

	x 1	/mPa.s	/J.mol -1	/mPa.s	/J.mol -1	/mPa.s	/J.mol -1	/mPa.s	/J.mol -1	/mPa.s	/J.mol -1
		T=283.15 K	T=288.15 K	T=293.15 K	T=298.15 K	T=303.15 K
	0.0501	4.413	3.019	3.697	2.978	3.142	2.943	2.704	2.914	2.354	2.892
	0.1000	7.608	4.363	6.202	4.278	5.131	4.199	4.314	4.133	3.675	4.075
	0.1801	13.042	5.619	10.267	5.469	8.254	5.337	6.762	5.223	5.621	5.119
	0.2598	16.391	6.055	12.736	5.877	10.122	5.719	8.238	5.591	6.759	5.457
	0.3390	16.602	5.923	12.973	5.750	10.345	5.593	8.408	5.455	6.948	5.332
	0.4201	14.554	5.398	11.562	5.249	9.374	5.118	7.723	5.001	6.458	4.895
	0.4952	12.137	4.737	9.836	4.618	8.127	4.517	6.805	4.425	5.771	4.341
	0.5807	9.677	3.905	8.039	3.824	6.788	3.756	5.795	3.693	5.000	3.635
	0.6594	7.825	3.104	6.645	3.056	5.712	3.013	4.954	2.971	4.338	2.934
	0.7413	6.417	2.303	5.547	2.277	4.844	2.253	4.265	2.230	3.785	2.209
	0.8184	5.439	1.582	4.767	1.570	4.217	1.560	3.755	1.549	3.369	1.539
	0.8990	4.668	0.860	4.137	0.856	3.698	0.853	3.326	0.850	3.010	0.846
	0.9488	4.295	0.434	3.827	0.431	3.439	0.430	3.108	0.428	2.827	0.427
	a Expanded uncertainties (k=2) U(P)=0.3 kPa, U(T)=0.01 K, U(=2.2 10 -5 K -1 , U(x)=0.002, U r ()=0.01., U(G E =0.002 J.mol -1

Table 8 : Continued

 8

	x 1	/mPa.s	/J.mol -1	/mPa.s	/J.mol -1	/mPa.s	/J.mol -1	/mPa.s	/J.mol -1	/mPa.s	/J.mol -1
		T=308.15 K	T= 313.15 K	T=318.15 K	T=323.15 K	T=328.15 K
	0.0501	2.068	2.871	1.831	2.852	1.636	2.841	1.472	2.833	1.336	2.835
	0.1000	3.164	4.022	2.752	3.974	2.418	3.937	2.139	3.900	1.908	3.872
	0.1801	4.736	5.026	4.040	4.941	3.482	4.867	3.032	4.800	2.662	4.741
	0.2598	5.646	5.343	4.788	5.246	4.093	5.152	3.540	5.069	3.092	4.997
	0.3390	5.823	5.221	4.939	5.119	4.237	5.028	3.673	4.947	3.214	4.875
	0.4201	5.468	4.799	4.684	4.712	4.053	4.635	3.540	4.564	3.119	4.502
	0.4952	4.951	4.265	4.290	4.196	3.752	4.135	3.308	4.078	2.939	4.028
	0.5807	4.351	3.580	3.827	3.534	3.387	3.491	3.021	3.451	2.712	3.417
	0.6594	3.830	2.900	3.407	2.869	3.052	2.842	2.750	2.817	2.489	2.792
	0.7413	3.383	2.191	3.042	2.172	2.750	2.155	2.501	2.140	2.288	2.130
	0.8184	3.041	1.531	2.761	1.523	2.519	1.516	2.309	1.509	2.127	1.505
	0.8990	2.740	0.843	2.506	0.839	2.303	0.837	2.126	0.835	1.970	0.833
	0.9488	2.585	0.427	2.374	0.424	2.191	0.425	2.031	0.425	1.888	0.424
	a Expanded uncertainties (k=2) U(P)=0.3 kPa, U(T)=0.01 K, U(=2.2 10 -5 K -1 , U(x)=0.002, U r ()=0.01., U(G E =0.002 J.mol -1

Table 8 : Continued

 8

	x 1	/mPa.s	/J.mol -1	/mPa.s	/J.mol -1	/mPa.s	/J.mol -1
		T=333.15 K		T=338.15 K		T=343.15 K
	0.0501	1.219	2.842	1.118	2.848	1.029	2.844
	0.1000	1.714	3.850	1.548	3.828	1.409	3.800
	0.1801	2.355	4.689	2.101	4.643	1.887	4.577
	0.2598	2.723	4.934	2.417	4.875	2.161	4.786
	0.3390	2.837	4.813	2.521	4.753	2.257	4.653
	0.4201	2.768	4.445	2.474	4.393	2.225	4.286
	0.4952	2.629	3.984	2.366	3.940	2.144	3.836
	0.5807	2.450	3.386	2.225	3.356	2.033	3.251
	0.6594	2.266	2.769	2.073	2.747	1.906	2.638
	0.7413	2.103	2.121	1.940	2.109	1.797	1.997
	0.8184	1.967	1.500	1.826	1.493	1.700	1.373
	0.8990	1.832	0.830	1.709	0.823	1.603	0.701
	0.9488	1.762	0.421	1.652	0.421	1.558	0.300
	a Expanded uncertainties (k=2) U(P)=0.3 kPa, U(T)=0.01 K, U(=2.2 10 -5 K -1 , U(x)=0.002, U r ()=0.01., U(G E =0.002 J.mol -1

Table 9 : Excess Gibbs energy of flow: Redlich-Kister parameters and deviation for DMI + water binary system.

 9

	T/K

a Expanded uncertainties (k=2) U(p)=0.3 kPa, U(T)=0.01 K, U(=1.0 kg.m -3 , U(x)=0.002, U(V E =0.011 cm 3 .mol -1 , U(=2.2 10 -5 K -1 , U( E =4 10 -5 K -1

/K -1 x 1


Arthur Charguéraud

Begatim Bytyqi

Damien Rouhling

Yann Barsamian

OptiTrust: an Interactive Optimization Framework

We present OptiTrust, an interactive framework for optimizing general-purpose C code via series of programmer-guided, source-to-source transformations. Optimization steps are described in transformation scripts, expressed as OCaml programs. At every step, the programmer may interactively visualize the effect of the transformation as the difference between two pieces of human-readable C code. OptiTrust has been previously employed to optimize numerical simulation code. In this work, we showcase how to use OptiTrust to optimize matrix multiplication. We compare against TVM, which also relies on programmer guidance, but which restricts the input language and lacks easily readable feedback.

Motivation

Achieving high-performance on modern, parallel and heterogeneous hardware is a challenging task that is critical in many domains, e.g., numerical simulation, image processing and machine learning. It remains common industrial practice to optimize code by hand in languages such as C/C++ [START_REF] Memeti | Benchmarking OpenCL, OpenACC, OpenMP, and CUDA: programming productivity, performance, and energy consumption[END_REF][START_REF] Tousimojarad | Comparison of three popular parallel programming models on the Intel Xeon Phi[END_REF] or Fortran [START_REF] Vanderbauwhede | Domain-specific acceleration and auto-parallelization of legacy scientific code in FOR-TRAN 77 using source-to-source compilation[END_REF].

Manual code rewriting is highly time consuming, and must be repeated numerous times to reach high performance. Several optimization paths must be empirically explored because code performance is hard to assess without benchmarking. After this process, hand optimized code is generally hard to read, maintain, adapt to other hardware, and-worst of all-may contain bugs introduced in the process.

In order to reuse optimization efforts, optimized libraries such as BLAS or MKL for linear algebra were created. However, their use remains suboptimal as state-of-the-art generalpurpose compilers (e.g. Clang, GCC, ICX) fail to apply relevant optimizations across library calls. This limitation has motivated the development of specialized optimizing compilers, able to apply global optimizations on high-level algorithms written in Domain-Specific Languages (DSLs).

Examples of specialized compilers include Halide [START_REF] Ragan-Kelley | Halide: A Language and Compiler for Optimizing Parallelism, Locality, and Recomputation in Image Processing Pipelines[END_REF] and TVM [START_REF] Chen | TVM: An Automated End-to-End Optimizing Compiler for Deep Learning[END_REF], which have proved successful in industry for image processing and machine learning, respectively. With slightly larger application domain are compilers for array programming languages [START_REF] Manuel Mt Chakravarty | Accelerating Haskell array codes with multicore GPUs[END_REF][START_REF] Grelck | SAC-a functional array language for efficient multi-threaded execution[END_REF][START_REF] Henriksen | Futhark: purely functional GPU-programming with nested parallelism and in-place array updates[END_REF][START_REF] Steuwer | Generating performance portable code using rewrite rules: from high-level functional expressions to high-performance OpenCL code[END_REF], exploiting the fact that multi-dimensional arrays are a key abstraction for many performance-demanding applications. Although the success of specialized optimizing compilers is undeniable, they cannot optimize programs that fall outside of their domain. Morevover, extending these specialized compilers has proved to be challenging [START_REF] Barham | Machine learning systems are stuck in a rut[END_REF].

OptiTrust is an optimization framework that aims to support general-purpose code, avoiding domain restriction. Op-tiTrust operates at the level of C code, by supporting sourceto-source transformations. The programmer interactively develops a transformation script. At every transformation step, the programmer may visualize the corresponding diff between pieces of human-readable C code.

Prior work [START_REF] Charguéraud | OptiTrust: an Interactive Framework for Source-to-Source Transformations[END_REF] has demonstrated the ability of OptiTrust to reproduce a state-of-the-art optimized implementation of a numerical Particle-In-Cell simulation. In the present work, we demonstrate the ability of OptiTrust to reproduce a reference implementation of matrix multiplication generated by the specialized compiler TVM.

The 5 Key Components of OptiTrust

1. An internal, simplified abstract syntax tree (AST) from which readable C code can be recovered throughout transformations. A similar idea of a simplified AST has been employed in the Cetus project [START_REF] Dave | Cetus: A Source-to-Source Compiler Infrastructure for Multicores[END_REF]. 2. A system for targeting program points, somewhat similar to XPath [START_REF] Clark | XML path language (XPath)[END_REF] for XML, but specialized for an AST. This system allows describing, in a concise and robust manner, one or several program points to transform. 3. A library of general-purpose transformations, including: core data layout transformations [START_REF] Sung | Data Layout Transformation Exploiting Memory-level Parallelism in Structured Grid Many-core Applications[END_REF], instruction-level transformations [START_REF] Allen | Optimizing Compilers for Modern Architectures[END_REF], control flow transformations [START_REF] Wolfe | High performance compilers for parallel computing[END_REF]. 4. A scripting language, embedded in OCaml, for describing transformations. Transformation scripts are a classic technique for programmer-guided optimization frameworks [START_REF] Bagnères | Opening Polyhedral Compiler's Black Box[END_REF][START_REF] Bispo | Clava: C/C++ source-to-source compilation using LARA[END_REF][START_REF] Bondhugula | A practical automatic polyhedral parallelizer and locality optimizer[END_REF][START_REF] Chen | CHiLL: A Framework for Composing High-Level Loop Transformations[END_REF][START_REF] Kedar | Loopy: Programmable and Formally Verified Loop Transformations[END_REF][START_REF] Rudy | A Programming Language Interface to Describe Transformations and Code Generation[END_REF][START_REF] Yi | Exploring the Optimization Space of Dense Linear Algebra Kernels[END_REF][START_REF] Yi | Specializing Compiler Optimizations through Programmable Composition for Dense Matrix Computations[END_REF][START_REF] Zinenko | Declarative Transformations in the Polyhedral Model[END_REF]. Transformation scripts provide fine-grained control, and they allow chaining large numbers of transformation steps. Other approaches include use of pragmas [START_REF] Chandra | Parallel programming in OpenMP[END_REF][START_REF] Kruse | A Proposal for Loop-Transformation Pragmas[END_REF][START_REF] Krzikalla | Scout: A Source-to-Source Transformator for SIMD-Optimizations[END_REF], schedules [START_REF] Baghdadi | Tiramisu: A polyhedral compiler for expressing fast and portable code[END_REF][START_REF] Chen | TVM: An Automated End-to-End Optimizing Compiler for Deep Learning[END_REF][START_REF] Hagedorn | Fireiron: a data-movement-aware scheduling language for GPUs[END_REF][START_REF] Ragan-Kelley | Halide: A Language and Compiler for Optimizing Parallelism, Locality, and Recomputation in Image Processing Pipelines[END_REF], or rewriting strategies [START_REF] Bagge | Design of the CodeBoost transformation system for domain-specific optimisation of C++ programs[END_REF][START_REF] Bravenboer | [END_REF][START_REF] Hagedorn | Achieving highperformance the functional way: a functional pearl on expressing high-performance optimizations as rewrite strategies[END_REF][START_REF] Kirchner | Rewriting strategies and strategic rewrite programs[END_REF]. 5. An interactive interface allowing to visualize code diffs associated with the transformation at a given line of the script, via a key shortcut in the code editor.

Optimizing Matrix-Multiply in OptiTrust

We investigate how OptiTrust compares with the specialized compiler TVM for producing an optimized implementation of matrix multiplication-a standard benchmark. The function mm1024 specializes input sizes to 1024.

We start from the C code presented in Listing 1: a naive, unoptimized implementation of matrix multiplication. The aim is to produce similar code as the reference TVM schedule that was handwritten by an expert targeting Intel CPUs. 1We write an OptiTrust script in OCaml, as shown in Listing 2. It consists of 9 transformation steps, each one calling a function from the OptiTrust library, always providing a target describing where to apply the transformation as last argument. OptiTrust scripts are developed interactively: with the cursor on a given line of the script, we can press (e.g.) "F6" in the VSCode editor to visualize the diff associated with the transformation on that line. All intermediate versions of the code consist of human-readable, executable C code.

The final code is shown in Listing 3. It mirrors the structural optimizations described in the TVM reference schedule, including: improved data locality, parallelism, and specialization to 1024 sizes. A major difference is that we produce C code with OpenMP pragmas, whereas TVM targets directly LLVM IR. To check that the code produced using OptiTrust performs similarly to that produced by TVM, we benchmarked both codes on a 4-core Intel i7-8665U CPU with AVX2 support. Both codes have 90th percentile runtime of 9.4ms over 200 benchmark runs, corresponding to a speedup of 150× compared to the 90th percentile of the naive code. 2In summary, our case study shows that the OptiTrust general-purpose optimization framework can be used to interactively develop a code competitive with that produced by a state-of-the-art specialized compiler. Unlike specialized compilers, OptiTrust takes as input C code, and produces human-readable C code at every step.

Contents of the Talk

During the ARRAY workshop, we will present the OptiTrust matrix multiplication case study, and compare the script to the reference TVM schedule. We will give an overview of the transformations currently supported by OptiTrust, and describe ongoing work on leveraging Separation Logic annotations [START_REF] Charguéraud | Separation Logic for Sequential Programs (Functional Pearl)[END_REF][START_REF] Sammler | RefinedC: Automating the foundational verification of C code with refined ownership types[END_REF] to validate transformation correctness Listing 2. OptiTrust script for optimizing mm1024. Listing 3. Optimized C code produced by the OptiTrust script for mm1024. This code has similar structure and achieves similar performance as the reference output of TVM. beyond the limit of static analyses commonly used in generalpurpose compilers.

We hope to engage with the ARRAY community by presenting an approach that tackles the limitations of specialized langages and compilers. Suggestions of challenging optimizations, representative benchmarks, and compilation techniques to investigate will be especially welcomed.

Listing 1 .

 1 void mm(float* C, float* A, float* B, int m, int n, int p) { for (int i = 0; i < m; i++) { for (int j = 0; j < n; j++) { float sum = 0.0f; for (int k = 0; k < p; k++) sum += A[i][k] * B[k][j]; C[i][j] = sum; }}} void mm1024(float* C, float* A, float* B) { mm(C, A, B, 1024, 1024, 1024); } Unoptimized matrix multiplication. The function mm multiplies the matrices A and B and stores the result in C.

Figure 1 .

 1 Figure 1. Interactive diff for the Matrix.stack_copy step.

https://tvm.apache.org/docs/how_to/optimize_operators/opt_gemm.html

For the TVM code, the median is 2% faster than the 90th percentile; for the OptiTrust code, it is 8%. The manually optimized library Intel MKL delivers a 204× speedup compared to the 90th percentile of the naive code.