Dr Cécile Danel
email: cecile.danel@univ-lille2.fr

Patricia Melnyk
email: patricia.melnyk@univ-lille2.fr

Nathalie Azaroual
email: nathalie.azaroual@univ-lille2.fr

Paul-Emmanuel Larchanché
email: paul-emmanuel.larchanche@univ-lille2.fr

Jean-Francois Goossens
email: jean-francois.goossens@univ-lille2.fr

Claude Vaccher
email: claude.vaccher@univ-lille2.fr

Jean- François Goossens

Evaluation of three neutral capillary coatings for the determination of analyte-cyclodextrin binding constants by affinity capillary electrophoresis. Application to N,N'-disubstituted piperazine derivatives

Keywords: Affinity capillary electrophoresis, Cyclodextrin, Neutral coating, Hydroxypropyl cellulose, Polyethylene oxide, Poly(N,N-dimethylacrylamide) Highlights

teaching and research institutions in France or abroad, or from public or private research centers.

Evaluation of three neutral capillary coatings for the determination of analyte-cyclodextrin binding constants

by affinity capillary electrophoresis. Application to N,N'-disubstituted piperazine derivatives.

Introduction

Cyclodextrins (CDs) are macrolytic compounds with several D-glucopyranose units linked by -1,4-glycosidic bonds. The common -CD, -CD and CD are composed of 6, 7 and 8 glucose units, respectively. The shape of CDs is a truncated cone with a central cavity due to the chair conformation of the glucopyranose units. Their exterior surface is hydrophilic thanks to the presence of hydroxyl groups whereas the central cavity is lined by skeletal carbons and ethereal oxygens of the glucose residues which give it a relatively lipophilic character [START_REF] Loftsson | Cyclodextrins in topical drug formulations: theory and practice[END_REF][START_REF] Szejtli | Introduction and general overview of cyclodextrin chemistry[END_REF]. These properties make them first interest complexing agents since they are able to form inclusion complexes with a great variety of molecules of appropriate polarity and size [START_REF] Duchêne | Cyclodextrins and their industrial uses[END_REF]. In pharmaceutical formulations, CDs are generally used to enhance the solubility, bioavailability and stability of drugs [START_REF] Carrier | The utility of cyclodextrins for enhancing oral bioavailability[END_REF][START_REF] Irie | Pharmaceutical Applications of Cyclodextrins. III. Toxicological Issues and Safety Evaluation[END_REF][6]. CDs can be found in at least 35 pharmaceutical products: the -CD, the hydroxypropylated HP--CD, the methylated Me--CD and the sulfobutyl ether SBE--CD are the most commonly used CDs [7].

Quantification of the interactions between the drug and the cyclodextrins based systems is a recurrent question in the literature. Many techniques may be used to assess the apparent stability constants of drug-CD complexes. Among them, the capillary electrophoresis has been widely used [START_REF] Chankvetadze | Capillary Electrophoresis in Chiral Analysis[END_REF] because of its automatisation, miniaturization, efficiency, rapidity and small amount of sample required.

Affinity capillary electrophoresis (ACE) is especially adapted to investigate the ligand-cyclodextrin interactions. It consists in studying the electrophoretic mobility variations of the ligand according to the various CD concentrations in the background electrolyte (BGE) [START_REF] Chankvetadze | Capillary Electrophoresis in Chiral Analysis[END_REF].

Alzheimer's disease (AD) is the most prevalent cause of dementia. It combines two pathophysiological mechanisms: amyloid pathology and Tau pathology [START_REF] Dubois | Revising the definition of Alzheimer's disease : a new lexicon[END_REF]. Itis characterized by a progressive decrease in acetylcholinesterase (AChE) enzymatic activity accompanied by a cholinergic neuron degenerescence. Only symptomatic treatments are currently available. Designing multitarget drugs is a recent and promising approach against multifactorial illnesses as neurodegenerative diseases such as Alzheimer's disease. We have designed and developed a novel family of N,N'-disubstituted piperazine [START_REF] Melnyk | Use of 1,4-bis(3-aminoalkylyl)piperazine derivatives in the treatment of neurodegenerative diseases[END_REF][START_REF] Melnyk | Chloroquine and chloroquine related compounds as a model for the design of anti-Alzheimer compounds[END_REF] that modulate APP metabolism by reducing the release of selective species of Aβ peptides, especially the long and toxic Aβ species, and by increasing several APP metabolites including the gene regulatory fragment of APP that is the amyloid intracellular domain (compounds 2, 3 and 4-Figure 1) [START_REF] Melnyk | Chloroquine and chloroquine related compounds as a model for the design of anti-Alzheimer compounds[END_REF]. We recently designed multifunctional compounds derived from N,N'-disubstitued piperazine and available AChE inhibitors. Compound 1 provided high potential as a drug candidate for AD as it inhibited hAChE very efficiently (IC50 < 1nM), and was able to decrease both amyloid and Tau pathology in vivo by oral route [START_REF] Melnyk | Novel 1,4-bis(3-aminopropyl)piperazine derivative and its use[END_REF][START_REF] Sergeant | Multi-target drugs prevents neurofibrillary degeneration, amyloid deposition and improves cognitive functions in animal models of Alzheimer's disease[END_REF].

Since the compound 1 possesses a tacrine moiety, particular attention must be paid to its eventual metabolites. Pool et al. [START_REF] Pool | Identification of a 3hydroxylated tacrine metabolite in rat and man : metabolic profiling implications and pharmacology[END_REF] have described the metabolism of tacrine by single or multiple hydroxylations of its aliphatic ring. These metabolites have been described to be responsible for hepatotoxicity. In order to protect its aliphatic ring against this metabolism, we have chosen to investigate the interest of CDs in the formulation of compound 1. Before envisaging a microsomal study of the metabolism of 1 under free and complexed forms, a study of the 1-CDs complexes must be achieved in order to select the more appropriate CD with a sufficient binding constant and a favourable structure.

Early CE analysis at pH 7.4 of compound 1, which contains a N,N'-disubstituted piperazine moiety (polycationic character), have shown its adsorption on the naked capillary. This adsorption may be relied to electrostatic interactions with the anionic surface of the capillary due to the deprotonation of the silanol groups at pH 7.4. We used coated capillaries in order to avoid this problem and to allow an accurate determination of the binding constants of the 1-CDs complexes.. To the best of our knowledge, no study regarding the determination of analyte-CD binding constants by ACE using coated capillaries is reported in literature. Only some chiral separation methods using such CDs are developed using coated capillaries to improve the enantioresolution and to reduce peak tailing resulting from solute interactions with silanol groups of capillary wall (for methods developed at neutral or basic pH) [START_REF] Assi | Enantiomeric purity determination of propranolol by capillary electrophoresis usind dual cyclodextrins and a polyacrylamide-coated capillary[END_REF][START_REF] Rudaz | Use of cyclodextrins in capillary electrophoresis: resolution of tramadol enantiomers[END_REF][START_REF] Lin | Capillary electrophoretic separation of tricyclic antidepressants using a polymer-coated capillary and -cyclodextrin as an electrolyte additive[END_REF].

As described in a recent review [START_REF] Huhn | Relevance and use of capillary coatings in capillary electrophoresis-mass spectrometry[END_REF], the coating of the inner wall of the capillary may be achieved in dynamic or static fashions. The dynamically coated capillaries require a regular regeneration and the addition of the coating agent into the BGE which is necessary to maintain the coating. Static coatings are the most popular capillary coatings. Two kinds of static coatings exist: the static-covalent coatings whose preparation are quite laborious but present a long-term stability and the static-adsorbed coatings which can be more easily regenerated.

In this paper, the performances of three static coatings obtained using hydroxypropylcellulose (HPC), polyethyleneoxide (PEO) or poly(N,N-dimethylacrylamide (PDMA) are evaluated: ability to cover the inner surface of the capillary and ability to strictly reduce the adsorption of the polycationic compounds 1-4 while avoiding the adsorption of the CDs. Then, the binding constants of the complexes formed between 1 and four CDs of pharmaceutical interest (-CD, HP--CD, Me--CD and SBE--CD) is determined by ACE to select the more appropriate CD. Last, the structure of the complex displaying the greater binding constant is investigated through a 2D ROESY NMR study and structure-binding constant relationships.

Material and methods

Chemicals

The original derivatives 1-4 (Figure 1) were designed and synthesized by some of us according to previously described protocols [START_REF] Melnyk | Chloroquine and chloroquine related compounds as a model for the design of anti-Alzheimer compounds[END_REF][START_REF] Melnyk | Novel 1,4-bis(3-aminopropyl)piperazine derivative and its use[END_REF][START_REF] Sergeant | Multi-target drugs prevents neurofibrillary degeneration, amyloid deposition and improves cognitive functions in animal models of Alzheimer's disease[END_REF][START_REF] Ryckebusch | Synthesis and in Vitro and in Vivo Antimalarial Activity of N1-(7-Chloro-4-quinolyl)-1,4bis(3-aminopropyl)piperazine Derivatives[END_REF]. Tacrine, ibuprofen, carbamazepine and poly(ethylene oxide) (PEO, average Mw 200,000) were purchased from Sigma-Aldrich (Saint-Quentin Fallavier, France). Hydroxypropylcellulose (HPC, average Mw 80,000) was purchased from Ashland Aqualon (Wilmington, SA). Poly(N,N-dimethylacrylamide) (PDMA, average Mw 2,400) was purchased from Polymer Source (Dorval, Quebec, Canada). The structure of these polymers are displayed in Figure 1.

Risperidone was a gift from Janssen-Cilag (Issy-les-Moulineaux, France). Sodium hydroxide, chlorhydric acid 37%, sodium dihydrogenophosphate monohydrate and di-sodium hydrogenophosphate were purchased from Merck (Nogent-sur-Marne, France). Deionized water was obtained from Milli-Q system (Millipore, Saint-en-Yvelines, France). Deuterium oxide (100%) was purchased from Euriso-top (Gif sur Yvette, France).

Cyclodextrins



-CD, HP--CD and Me--CD were kindly supplied by Roquette Laboratories (Lestrem, France).

The HP--CD and Me--CD represent multicomponent mixtures with molar substitution (MS) of 0.61 and 0.57 per glucose unit, respectively. Sulfobutylether--CD (SBE--CD, Captisol®, average degree of substitution: 6.2-6.9) was kindly supplied by Cydex Pharmaceutical (La Jolla, California, USA).

Averaged molecular weight was taken into account to calculate the molar concentration of the substituted CD.

Nuclear Magnetic Resonance

The NMR experiments were realized on a Bruker AVANCE 500 with a TXI probe operating at 500.13 MHz. To study the structure of the complexes by 2D ROESY experiments (homonuclear dipolar correlation 1 H-1 H), experiment spectra were recorded with a mixing time of 500 ms. Five hundred L of solutions were introduced into standard 5 mm NMR tubes and the experiments were performed at 298 K.

The 67 mM phosphate buffer pH 7.4 was prepared by mixing appropriate amounts of both sodium salts in D2O. The concentrations of 1 and the CD (-CD or SBE--CD) results from a compromise between a relative high analyte concentration (for better signal/noise ratios) and a good separation of analyte resonance signals (if possible). The optimal condition was obtained for an equimolar mixture (4 mM for 1 and 4 mM for theCD).

Capillary electrophoresis

CE experiments were performed on a Beckman P/ACE MDQ Capillary Electrophoresis system, including an on-column diode-array UV-detector driven by the 32 Karat software (Beckman Coulter, Villepinte, France) package for system control, data collection and analysis. It was equipped with For the complexation study, the capillary was mounted on a cartridge thermostated at 298 K ± 0.1 K. The 67 mM phosphate buffer pH 7.4 (BGE) was prepared by mixing appropriate amounts of NaH2PO4 and Na2HPO4. The applied voltage was 15 kV, the highest voltage that prevents significant Joule heating in our conditions. The concentration of the analytes was 0.1 mM (or 0.01 mM for ibuprofen) in water with 0.02% DMSO as EOF marker. To make the model valid, the analyte concentration must be at least ten times lower than the CD concentrations used [START_REF] Bose | Guidelines in selecting ligand concentrations for the determination of binding constants by affinity capillary electrophoresis[END_REF]. Hydrodynamic injections were made with a 5 s injection time at 1.0 psi pressure. Analytes were detected using the diode-array detector between 190 and 290 nm.

HPC coating

The capillary was coated with hydroxypropylcellulose (HPC) as described by Shen et al. [START_REF] Shen | High-resolution capillary isoelectric focusing of proteins using highly hydrophilic-substituted cellulose-coated capillaries[END_REF].

The fused-silica capillary was filled with a 5% w/v hydroxypropyl cellulose solution (in water) using a syringe pump (Harvard apparatus, Holliston, MA, USA). An amount of 20 µL was percolated at a flow rate of 0.5 µL/min and incubated for 2 h at room temperature. The capillary was purged under nitrogen pressure (20 psi) for 10 min, then heated from 60 to 140°C for 30 min, in a GC oven (Varian, Les Ulis, France).

PEO coating

The coating based on poly(ethylene oxide) (PEO) was prepared as described by Thuy Tran et al. [START_REF] Thuy Tran | Poly(ethylene oxide) facilitates the characterization of an affinity between strongly basic proteins with DNA by affinity capillary electrophoresis[END_REF]. The solution of PEO at 0.2% in 0.1 M HCl was prepared as follows : 20 mg of PEO were dissolved in 9 mL of water at 85°C and stored at room temperature. Then 1 mL of 1 M HCl was added to the PEO solution just before use. The coating procedure consists in successive flushes with water (20 psi, 2 min), HCl 1 M (20 psi, 5 min), PEO 0.2% (20 psi, 5 min) and BGE (20 psi, 2 min). It was necessary to regenerate the capillary daily and between each run with this rinse procedure to maintain the EOF constant.

PDMA coating

The solution of poly(N,N-dimethylacrylamide) (PDMA) was prepared in HCl 0.1 M at a concentration of 0.1% [START_REF] Lin | Capillary electrophoretic separation of tricyclic antidepressants using a polymer-coated capillary and -cyclodextrin as an electrolyte additive[END_REF]. The coating procedure consists in successive flushes with water (20 psi, 2 min), HCl 1 M (20 psi, 5 min), PDMA 0.1% (20 psi, 5 min) and BGE (20 psi, 2 min). It was necessary to regenerate the capillary daily and between each run with this rinse procedure to maintain the EOF constant.

EOF measurements

For uncoated capillaries, the EOF was measured using the migration time of a neutral marker (DMSO 0.02%). For neutral coated capillaries, for which EOF is too low to be classically evaluated, the residual EOF was measured according to the method developed by Williams and Vigh [START_REF] Williams | Fast accurate mobility determination method for capillary electrophoresis[END_REF].

Briefly, a neutral marker (DMSO 0,02%) was injected (1 psi, 5 s) in the capillary previously rinsed with the BGE (band N1). A pressure (0.5 psi, 30 s) was applied to shift the band into the capillary. A similar second band (N2) of DMSO was injected and a pressure (0.5 psi, 30 s) was applied to shift both bands N1 and N2 into the capillary. A voltage (V) of 15 kV was applied during 5 minutes (t voltage). During this time, both bands migrated towards with a mobility equal to EOF. A third band of DMSO was injected (N3) and a final pressure of 0.5 psi was applied during 5 minutes to allow the passage of the three bands past the UV detector. The times required for the bands N1, N2 and N3 to be pushed past the detector permit the calculation of the EOF.

Mesurement of the viscosity

The determination of analyte-CD binding constants by ACE requires correction to negate the changes in mobility caused by variation of the buffer viscosity induced by the addition of CD. The viscosities of the BGE without CD (0) or with CD from 1 to 10 mM () were evaluated using the Hagen-Poiseuille law as described by François et al. [START_REF] François | New integrated measurement protocol using capillary electrophoresis instrumentation for the determination of viscosity, conductivity and absorbance of ionic liquid-molecular solvent mixtures[END_REF]. All mobilities were corrected by the ratio 0/. These ratios varied from 0.99 to 0.97 for the -CD, from 0.99 to 0.96 for the Me--CD, from 0.99 to 0.95 for the HP--CD and from 0.99 to 0.93 for the SBE--CD.

Results and discussion

Preliminary experiments

The first analyses of 1 performed using a fused-silica capillary and a phosphate buffer pH 7.4

display an awful repeatability of the migration time and EOF (Fig. 3). .Seven successive analyses of 1 were performed and the migration times progressively increase as the EOF decreases which supposes a retention mechanism through an adsorption phenomenon. The severe peak broadening observed and the lack of efficiency is a second indication of the adsorption of 1 onto the negatively charged capillary surface. Similar observations were made with compounds 2, 3 and 4 (Figure A, Supplementary material). As the four compounds present a similar bis-aminoalkylpiperazine chain, their adsorption may be attributed to their polycationic character. Then, our strategy to analyze by CE these polycationic compounds was based on the use of coated capillaries. Numerous static coatings exist and are described in various reviews; most of them are developed to reduce the protein adsorption [START_REF] Huhn | Relevance and use of capillary coatings in capillary electrophoresis-mass spectrometry[END_REF][START_REF] Lucy | Non-covalent capillary coatings for protein separation in capillary electrophoresis[END_REF][START_REF] Stutz | Protein attachment onto silica surfacesa survey of molecular fundamentals, resulting effects and novel preventive strategies in CE[END_REF].

Additionally, the selected coatings must be based both on the the analyte and the cyclodextrin electrical charges: the compounds of pharmaceutical interest are polycationic, then the CDs selected will be neutral or anionic. Therefore, we chose to study the interactions between both partners potentially presenting opposite charges by focusing on neutral coatings. The compound 1 was used as a model to investigate and to reduce this adsorption phenomenon. It is worth mentioning that neutral coatings drastically reduce or eliminate the EOF [START_REF] Lucy | Non-covalent capillary coatings for protein separation in capillary electrophoresis[END_REF].

Evaluation of neutral coatings

Two kinds of static neutral coating capillaries are intensively used: the covalently-linked and the physically-adsorbed neutral polymers. Among the neutral polymers, four have proven their high efficiency to coat the capillary and then to avoid protein adsorption: the cellulose derivatives, the polyvinylalcohol (PVA), the polyethylene oxide (PEO) and the acrylamide-based polymers [START_REF] Stutz | Protein attachment onto silica surfacesa survey of molecular fundamentals, resulting effects and novel preventive strategies in CE[END_REF][START_REF] Zou | Neutral coatings for the study of polycation/multicharged anion interactions by capillary electrophoresis : application to dendrigraft poly-L-lysines with negatively multicharged molecule[END_REF] or copolymers [START_REF] Zhang | Ploy(N,N-dimethylacrylamide)grafted polyacrylamide: a self-coating copolymer for sieving separation of native proteins by CE[END_REF]. We chose to study the performance of a covalently-linked HPC capillary, previously used in our laboratory [START_REF] Lecoeur | A multivariate approach for the determination of isoelectric point of human carbonic anhydrase isoforms by capillary isoelectric focusing[END_REF], and two physically adsorbed coated capillaries using PEO and PDMA. These polymers were selected since they possess a high affinity for the capillary wall and then form stable coating. Moreover, even if these coatings are not recommended for proteins (proteins possess hydrophobic patches that interact with these more hydrophobic polymers) [START_REF] Chiari | New adsorbed coatings for capillary electrophoresis[END_REF], it seems to be good candidates for the study of interaction between a cationic compound and various CD. Effectively, a moderately hydrophobic capillary coating must limit the adsorption of the CDs which present hydrophilic surfaces.

Four factors were considered to evaluate the coatings: (i) the measurement of the residual EOF, (ii) the repeatability of the migration time of the analytes, (iii) the separation efficiency N (plates per meter) and

(iv) the presence (or absence) of a baseline shift which is an indicator of an irreversible adsorption [START_REF] Lucy | Non-covalent capillary coatings for protein separation in capillary electrophoresis[END_REF].

According to the EOF values displayed in Table 1 and as expected the three coatings appear to be effective to drastically reduce the EOF. EOF mobilities using the PEO, HPC or PDMA coatings are around 350, 650 or 2700 times smaller than the EOF mobilities measured using the uncoated capillary, which prove the ability of these neutral polymers to mask the silanol groups of the capillary wall responsible of the analyte absorption. However, the RSD of the EOF appears to be higher than the one without coating (around 20-30%) but this result must be carefully discussed. Indeed, the EOF is so low that a variation of 0.01 minute on the migration times of the neutral marker, according to the Williams and Vigh's method, greatly impacts the very low EOF value. This result does not question the reliability of the coatings. The PDMA coating displays the lowest EOF (inferior to 1.10 -5 cm².V -1 .min -1 ) which is in accordance with the results published by Chiari et al. [START_REF] Chiari | New adsorbed coatings for capillary electrophoresis[END_REF]. Due to its more hydrophobic character, it is able to displace the solvent water molecules from the surface and then forms more stable coatings through the formation of a higher number of hydrogen bonds with the absorptive sites of the silicacapillary. Using the PDMA coating procedure, the silanol groups are more effectively shielded which provides the slowest EOF. Table 1 displays the three factors allowing to evaluate the performance of the coatings to avoid the adsorption of 1. The RSD of its migration time is very satisfactory whatever the nature of the coating (≤ 0.3%). These results are associated to higher peak efficiencies, the theoretical plates are around two times higher even if a slight tailing is still observed. Additionally, no baseline shift is observed with the use of the three coated capillaries. Similar results are obtained for the compounds 2-4 (Fig. B, supplementary material). Therefore, these results show that, whatever the nature of the polymer, the neutral coatings are efficient to strictly limit the adsorption of the compounds 1-4 on the capillary wall.

Determination of apparent binding constants

For analyte/CD complexes of assumed 1:1 stoichiometry, Benesi-Hildebrand's method [START_REF] Benesi | A Spectrophotometric investigation of the interaction of iodine with aromatic hydrocarbons[END_REF] and Scott's modified equation [START_REF] Scott | Some comments on the Benesi-Hildebrand equation[END_REF] allow the calculation of apparent binding constants from the three following linear equations obtained after mathematical rearrangements (y-reciprocal (1), doublereciprocal (2) and x-reciprocal (3)):

where K is the apparent binding constant, [CD] is the free CD concentration in the capillary, µi is the electrophoretic mobility observed and µf and µc are the electrophoretic mobilities of the analyte in their free and complexed forms, respectively. It is noteworthy that the free CD concentration is considered to be the total CD concentration since the complexed CD concentration is insignificant (the concentration of the injected analyte is at less ten times lower than the CD one). The electrophoretic mobilities are corrected for viscosity changes by multiplying the mobilities by the ratio 0/. The three linear equations are used since differences due to relative uncertainties of the variables before and after transformation

(1) (3)
(2)

) µ µ ( 1 ] CD [ 1 K ) µ µ ( 1 µ µ 1 f c f c f i      K ) µ µ ( 1 ] CD [ µ µ 1 µ µ ] CD [ f c f c f i      ) µ µ ( K ) µ µ ( K ] CD [ ) µ µ ( f c f i f i      
for plotting can occur [START_REF] Rundlett | Methods for the estimation of binding constants by capillary electrophoresis[END_REF][START_REF] Plätzer | Determination of formation constants of cyclodextrin inclusion complexes using affinity capillary electrophoresis[END_REF]. Moreover, deviations from linearity can be more easily observed using xor double-reciprocal plot than the y-reciprocal plot [START_REF] Rundlett | Methods for the determination of binding constants by capillary electrophoresis[END_REF]. However, no significant difference was obtained; all the reported results in this study are determined using the y-reciprocal equation. The binding constants are apparent because the concentrations are considered (instead of activities) and averaged when the complexation occurred with modified CDs of averaged degree of substitution. Since coated capillaries are employed, it is necessary to study the possible adsorption of the CD on the surface capillary. Indeed, if CD adsorption occurs, the CD concentration free to interact with the analyte is lower and this results to an undervaluation of the binding constant. Then, the determination of analyte-CD binding constants will be performed after proving the absence of CD adsorption.

Study of the CD adsorption on capillary wall:

The method classically used to study the adsorption on capillary wall consists in the evaluation of the recovery through the measurement of the peak areas, using a CE instrument with two on-capillary detectors [START_REF] Towns | Capillary electrophoretic separations of proteins using nonionic surfactant coatings[END_REF] or only one detector [START_REF] Staub | Use of organic solvent to prevent protein adsorption in CE-MS experiments[END_REF] (these methods were developed to study the protein adsorption).

In the case of non UV-absorbing substances such as CDs, these methods cannot be employed. If adsorption of CD onto the capillary wall occurs, the EOF may be altered. Thus, monitoring the EOF in absence and presence of CD should be informative. However, the addition of CDs in the BGE can generate other modifications such as the viscosity and the ionic strength of the BGE. Moreover, the interaction between the neutral marker and the CD can alter the measurement of the EOF. As previously described, the influence of the viscosity can be easily removed. The influence of ionic strength (which changes the -potential) is more problematic and can be significant using anionic SBE--CD. Müllerova et al. [START_REF] Müllerová | Determination of effective mobilities of EOF markers in BGE containing sulfated -cyclodextrin by a two-detector method[END_REF] have demonstrated that DMSO is a suitable neutral marker to determine the EOF in presence of CD in the BGE : the interaction between DMSO and CD is sufficiently weak to neglect the effect of their complexation on the mobility measurements. The EOF was evaluated for each coating in absence or presence of the neutral -CD at various concentrations in the BGE. Results are displayed in Figure 3. As illustrated, the errors associated to the electroosmotic mobilities are large and the results must be carefully discussed. For the PDMA coating, the EOF values are very low whatever the -CD concentration and no significant trend may be found. With the PEO and HPC coatings, a decrease of the EOF values using higher -CD concentration in the BGE is suggested. This may be in accordance with a CD adsorption on the capillary wall which may mask more efficiently the silanol groups. However, the lack of accurate conclusions about the adsorption of the CD by monitoring the EOF lead us to propose another methodology.

To the best of our knowledge, in literature, all the determinations of analyte-CD binding constants for neutral or anionic CD are evaluated using fused-silica capillaries and any adsorption phenomenon of these CD is reported. Moreover, various studies point out the good agreement between the binding constants obtained by ACE using uncoated capillaries and other static techniques as isothermal titration calorimetry (ITC) recognized as the reference method [START_REF] Danel | Complexation of triptolide and its succinate derivative with cyclodextrins: Affinity capillary electrophoresis, isothermal titration calorimetry and 1 H NMR studies[END_REF]. We have determined the binding constants using fused-silica capillary for the complexes ibuprofen--CD (K = 9 332 M -1 ) and ibuprofen-HP--CD (K = 5 178 M -1 ). Our results are in total agreement with the published results obtained by ITC (K = 9502 M -1 and K = 5146 M -1 , respectively) [START_REF] Bertaut | Improving ITC studies of cyclodextrin inclusion compounds by global analysis of conventional and non-conventional experiments[END_REF]. So, we have chosen to evaluate the possible adsorption of CDs by comparison of the binding constants determined for various analyte-CD complexes with the uncoated and coated capillaries (the reference being the result obtained with the uncoated capillary).The complexes must be carefully chosen, after will be assured that the analytes do not adsorb on the capillary wall.Several complexes were studied in order to generalize the study of the CD adsorption.

Due to the neutral character of the capillaries, at least one partner of the interaction must be ionized to allow the migration of the analyte. We excluded cationic CD because the CD of pharmaceutical interest are neutral or anionic. Complexes with identical charge for both partners (CD and analyte) were also excluded for reasons of electrical repulsion. The -CD and SBE--CD have been selected to study CDs presenting different charge and substitution. Four analytes, the carbamazepine, tacrine, risperidone and ibuprofen (which are neutral, cationic, cationic or anionic at pH 7.4, respectively) were selected after checking that they do not adsorb on the four capillary surfaces. by evaluation of their migration time repeatability (RSD under 1%), efficiency and absence of baseline shift (Table A, Supplementary material) Note that tacrine displays an excellent repeatability of its migration time on the fused-silica capillary (RSD = 0.5%) contrary to the compounds 1 derived from tacrine (32.5% RSD for tm of compound 1, Table 1). This result confirms that their adsorption may be due to the polycationic character of their piperazine chain. Lastly, four complexes (Table B, Supplementary material) were studied using the fused silica capillary and the three coatings in order to select the best one to avoid CD adsorption.

Choice of the coating

The apparent and averaged binding constants determined for the four complexes using fusedsilica capillary or the three coated capillaries are displayed in Table 2. Relative standard deviation and bias calculated taking as reference the result obtained with the uncoated capillary are enclosed. First, it appears that the binding constants determined using the PEO coating display poorer repeatabilities.

RSD values are higher than 5%, the generally admitted limit for binding constant determination by ACE [START_REF] Danel | Complexation of triptolide and its succinate derivative with cyclodextrins: Affinity capillary electrophoresis, isothermal titration calorimetry and 1 H NMR studies[END_REF][START_REF] Danel | Study of the complexation of risperidone and 9-hydroxyrisperidone with cyclodextrin hosts using affinity capillary electrophoresis and 1H NMR spectroscopy[END_REF]. Moreover, the binding constants measured with both PEO and HPC coatings are systematically lower than the reference value obtained using the uncoated capillary (bias around -10 to -20%). The undervaluation of the constants obtained with both more hydrophilic polymers can be explained by an adsorption of the CD through surface hydrophilic interactions, which reduce the CD concentration free to interact with the analyte. Only the PDMA coating permit to obtain binding constants values in agreement with those obtained using the uncoated capillary (bias around ± 5%).. Moreover, for the complex risperidone--CD, the apparent binding constant determined with the PDMA coating is 234 M - 1 , which is in accordance with our previously published results (K = 221 M -1 ) [START_REF] Danel | Study of the complexation of risperidone and 9-hydroxyrisperidone with cyclodextrin hosts using affinity capillary electrophoresis and 1H NMR spectroscopy[END_REF]. According to all these results, the PDMA coating is the best coating able to strictly limit the adsorption of the CDs and the adsorption of the compounds 1-4. Then, this coating was selected to investigate the complexation of the compound 1 with the four CDs of pharmaceutical interest.

Study of the (1)-CD complexes using the PDMA coating

First, the complexation of 1 with neutral CDs (-CD, HP--CD and Me--CD) has been investigated. Whereas the migration time shifts of 1 observed with additions of -CD and HP--CD in the BGE are too low to permit the binding constant determination, the additions of Me--CD involved sufficient migration time shifts to quantify the binding constant of the 1-Me--CD complex around 50 M - 1 . Due to the cationic character of 1 and so as to enhance the complexation, the complexing ability of the anionic SBE--CD, the only anionic CD used in marketed pharmaceutical products [7], was evaluated. It is worth mentioning than a previous work of Luppi et al. [START_REF] Luppi | Albumin nanoparticles carrying cyclodextrins for nasal delivery of the anti-Alzheimer drug tacrine[END_REF] about nanoparticles carrying cyclodextrins in order to load tacrine, has shown that SBE--CD significantly increases the drug loading whereas -CD and HP--CD did not affect it. Here, the apparent and averaged binding constant determined for the analyte 1-SBE--CD complex is more than 20 times greater than with the neutral Me--CD, around 1210 M -1 . This result can suggest a high contribution of electrostatic interactions in the complexation process between the sulfobutylether group of the CD and the cationic piperazine chain of 1 but this contribution would be only partially responsible of the interaction taking into account the results of Luppi et al [START_REF] Luppi | Albumin nanoparticles carrying cyclodextrins for nasal delivery of the anti-Alzheimer drug tacrine[END_REF]. Among the four CD, the SBE--CD appears to be the more appropriate CD to complex the compound 1.

Study of the structure of the 1-SBE--CD complex :

Before envisaging investigating the influence of the SBE--CD on the metabolism of 1, it is essential to study the structure of the complex. The ROESY is a powerful tool to investigate intra-and inter-molecular interactions. Presence of cross-peaks between the protons of two different species, which are generated by Nuclear Overhauser Effects (NOE) indicates that they are in a spatial contact within 3-5 Å [START_REF] Ribeiro | Multicomponent complex formation between vinpocetine, cyclodextrins, tartaric acid and water-soluble polymers monitored by NMR and solubility studies[END_REF]. The partial 1 H NMR 2D ROESY spectrum obtained for the complex formed between 1 and SBE--CD is displayed in Figure 4. One intramolecular interaction is observed between the protons a and e. Three cross-peaks may be attributed to intermolecular interactions between the analyte and the CD. However, the attribution of these interactions with the interior or exterior of the cavity is impossible due to overlapping of the various signals of the modified CD, The complex 1--CD was further studied since the 1 H signals of the -CD are well resolved and assigned, compared to the modified SBE--CD, and could provide more detailed information about the geometry of the complex. However, no cross peak was observed for intermolecular interactions and no information could be obtained due to low interactions between both partners even with an important accumulation of scans. A previous study on the complexation between tacrine and -CD by NMR and molecular dynamics [START_REF] Amato | NMR spectroscopic evidence and molecular dynamics studies on inclusion and non-inclusion phenomena between -cyclodextrin and new anti-Alzheimer's drugs tacrine (CI-970), velnacrine (HP-029) and suronacrine (HP-128)[END_REF] gave evidence of absolute lack of internal or external interaction between them. This result points that the structural modification of tacrine by substitution with the piperazine chain does not favour the interaction with the native -CD.

Since NMR does not allow to obtain information on the structure of the 1-SBE--CD complex, we have chosen to investigate structure-binding constant relationships by studying the complexes formed between this CD and the compounds presenting structural modifications on the tacrine moiety or on the piperazine chain. The apparent and averaged binding constants are displayed in Table 3. First, the comparison of the binding constants determined for tacrine and 1-4 points out that the interaction between the piperazine chain and the SBE--CD provide a high contribution to the binding constant. Indeed, the binding constant K obtained for tacrine is 3 to 9 times lower than those obtained for the substituted compounds 1-4.

Second, the comparison of the binding constants obtained for the various compounds 1-4 presenting structural modification on the tacrine moiety shows that the interaction is dependent of the nature of the ring. Only the replacement of the tacrine moiety by the benzimidazole moiety has low influence on the binding constant (1210 M -1 for 1 and 1066 M -1 for 2). On the contrary, the replacement by a quinoleine moiety in compound 4, has a great influence. The enhancement of the binding constant observed for 3 (1954 M -1 ) may be related to its chloro substitution and then its higher hydrophobicity which may favour its inclusion in the hydrophobic cavity of the CD. The comparison of the analogues 1 and 4 which present the tetrahydro-acridine ring or a quinoleine ring, respectively, without any other structural modification, is more interesting. Indeed, it appears that the aliphatic ring portion of the tacrine moiety greatly contributes to the interaction with the CD since the binding constant determined for the compound 1 is two times greater than for 4. This last result suggests that an inclusion of 1 in the SBE--CD cavity could occur through this ring.

Conclusions

Among the three neutral coatings, the best performances were obtained for the PDMA coating. It proves to be more efficient to mask the silanol groups of the capillary wall and then to strictly reduce the adsorption of the polycationic compounds 1-4. Moreover, it permits to correctly determine the analyte-CD binding constants avoiding the CD adsorption. The results obtained by ACE using the PDMA coating and isothermal titration calorimetry (recognized as the reference method) are in good agreement. This coating would be further used for analysis of various polycationic molecules in presence of CD in the BGE (for chiral analysis or determination of binding constants). The SBE--CD appears to be the best pharmaceutical CD to complex our compounds. The investigation of the structure of the 1-SBE--CD complex shows that the inclusion of 1 in the CD cavity seems to occur through the aliphatic ring portion of the tacrine moiety. This structure may be favorable to a modification of the metabolism since it has been demonstrated by Pool [14] that metabolism of tacrine occurs by single or multiple hydroxylation of this aliphatic ring. Further study will consist to study the metabolism of 1 in the uncomplexed form or in association with the SBE--CD. Table 1. Evaluation of the EOF mobilities using the fused-silica capillary and the three coatings.Repeatability of the migration time, separation efficiency and presence of a baseline shift obtained for the compound 1 using the various capillaries EOF (cm².min -1 .V -

30 . 2 cm ( 20

 30220 or 10 cm effective length) x 50 µm I.D. capillaries (Composite Metal Services, Ilkley, UK). Each new fused-silica capillary was conditioned before use by successive flushes with NaOH 1 N (20 psi, 5 min), NaOH 0.1 N (20 psi, 5 min) and deionized water (20 psi, 5 min). For use of an uncoated capillary, the rinses between each run consist of flushes with deionized water (1 min, 20 psi) and BGE (3 min, 20 psi).

Figure 1 .Figure 2 .Figure 3 :Figure 4 .

 1234 Figure 1. Structure of the studied compounds and structure of the polymers used for the coatings

FigureFigure B. Electropherograms of 1 - 4 1 PEO - 1 PDMA - 1 HPC - 2 PEO - 2 PDMA - 2 HPC - 3 PEO - 3 PDMA - 3 HPC - 4 PEO - 4 PDMA - 4

 14111222333444 Figure 1

Table 2 .

 2 Apparent and averaged binding constants determined for the four complexes using the three coatings

			1 ) ( RSD)	RSD tm	N	Baseline shift
	Fused-silica				1793	Yes
		1.2 10 -2	(2%)	19%
	capillary
	HPC coating	1.8 10 -5	(32%)	0.3%	3251	No
	PEO coating	3.4 10 -5	(22%)	0.3%	3802	No
	PDMA coating	4.4 10 -6	(18%)	0.2%	3666	No
	mean values calculated from seven determinations

Table 3 .

 3 Apparent and averaged binding constants K determined by ACE for the complexes formed between Tacrine or compounds 1-4 and the SBE--CD.

	K ( M -1 )	%RSD
	Tacrine	215	5%
	1	1210	3%
	2	1066	1%
	3	1954	5%
	4	673	4%
	Mean values calculated from three experiments.

Table A .

 A RSD of the migration times and efficiencies measured for the four model analytes using uncoated and coated capillaries : lower efficiencies obtained for the neutral carbamazepine was expected since its signal is joined with the DMSO one. n.d. : not determined since the neutral carbamazepine does not migrate without charged CD in the BGE with these three neutral capillaries Conditions : capillary 30.2 cm (effective length, 20 cm) 50 mm i.d. ; BGE, 67 mM phosphate buffer, pH 7.4; anodic injection (1 psi, 5 s); temperature, 298 K; voltage, 15 kV

			%RSD tm ; N
	Analyte	Fused-silica capillary	HPC coating	PEO coating	PDMA coating
	Ibuprofen	0.2% ; 42 542	1.1% ; 14 438	0.9% ; 17 098	0.7 % ; 23 587
	Risperidone	0.4 % ; 24 929	3.1% ; 4 810	2.4% ; 9 964	0.3% ; 10 474
	Carbamazepine	0.7% ; 5 513*	n.d.	n.d	n.d
	Tacrine	0.5% ; 21 256	0.9% ; 5 604	2.4% ; 6 983	1.1% ; 10 478
	mean values calculated from three determination

*

Acknowledgments

The authors are grateful to Roquette Laboratories for the gift of -CD, HP--CD and Me--CD and to Cydex Pharmaceutical for the gift of Captisol®. This work was supported by the University of Lille 2.

The 500 MHz and 300 MHz NMR facilities were funded by the Région Nord-Pas de Calais (France), the Ministère de la Jeunesse, de l'Education Nationale et de la Recherche (MJENR) and the Fonds Européens de Développement Régional (FEDER).


Matthieu Petit
email: matthieu.petit@univ-amu.fr

Amine Boussadi

Vasile Heresanu

Alain Ranguis

Lisa Michez

Step flow growth of Mn 5 Ge 3 films on Ge(111) at room temperature

Keywords:

MBE) Critical nucleation volume Mn

Introduction

Since about 20 years, spintronics has brought significant advances in electronics speed, data storage and power consumption [START_REF] Duine | Spintronics: an alternating alternative[END_REF][START_REF] Ohno | Advancements in spintronics[END_REF]. However a key point is still restraining the full blossoming of spintronics: the lack of ferromagnetic materials fully compatible with the mainstream silicon technology, even though a great deal of efforts has been devoted to the development of semiconductors exhibiting magnetic properties. Two families of magnetic semiconductors exist. The first one is the concentrated magnetic semiconductors (CMS), based on rare-Earth chalcogenides. But the development of these compounds is hindered by their relatively low Curie temperature (T C ) (the highest value reported is 117 K for EuO) and notorious difficulty in materials synthesis [START_REF] Ahn | Relationship between stoichiometry and properties of EuO films[END_REF][START_REF] Von Molnár | New materials for semiconductor spin-electronics[END_REF]. The second family is the diluted magnetic semiconductors (DMS) one. These materials are extensively studied, in particular the (Ga,Mn)As alloys with T C remaining below 200 K and the Ge 1-x Mn x with T C around 150 K [START_REF] Ohno | Making nonmagnetic semiconductors ferromagnetic[END_REF][START_REF] Morresi | Structural, magnetic and electronic transport properties of Mn x Ge 1x /Ge(001) films grown by MBE at 350C[END_REF][START_REF] Arras | Phase diagram, structure, and magnetic properties of the Ge-Mn system: a first-principles study[END_REF][START_REF] Chen | Enhancing the curie temperature of ferromagnetic semiconductor (Ga,Mn)As to 200 K via nanostructure engineering[END_REF][START_REF] Aouassa | Mn-doping effects on structural and magnetic properties of Ge nanocrystals on insulator[END_REF].

In this context, Mn 5 Ge 3 has attracted a great deal of interest, since the demonstration of its epitaxial growth as a thin film on Ge(111) substrates with an atomically sharp interface, although the lattice mismatch is 3.7% [START_REF] Zeng | Epitaxial ferromagnetic Mn 5 Ge 3 on Ge(111)[END_REF]. That was the sign of the potential of Mn 5 Ge 3 as a ferromagnetic electrode suitable for the Si/Ge technology, in particular since the interfaces play a key role in spintronics [START_REF] Hellman | Interface-induced phenomena in magnetism[END_REF]. Mn 5 Ge 3 is a ferromagnetic metal with a T C of 296 K and a hexagonal crystal structure (space group P6 3 /mcm) [START_REF] Tawara | On the magnetic anisotropy of single crystal of Mn 5 Ge 3[END_REF]. The T C can be increased up to 450 K by introducing carbon atoms in the octahedral interstitial site of the lattice [START_REF] Gajdzik | Strongly enhanced Curie temperature in carbon-doped Mn 5 Ge 3 films[END_REF][START_REF] Slipukhina | Simulation of the enhanced Curie temperature in Mn 5 Ge 3 C x compounds[END_REF]. The spin polarization was measured at P = 15 ± 5% and the magnetic and electrical properties of the Mn 5 Ge 3 and Mn 5 Ge 3 C x alloys have been extensively investigated [START_REF] Picozzi | First-principles characterization of ferromagnetic Mn 5 Ge 3 for spintronic applications[END_REF][START_REF] Dedkov | Spin-resolved photoemission of a ferromagnetic Mn 5 Ge 3 (0001) epilayer on Ge(111)[END_REF][START_REF] Michez | Magnetic reversal in Mn 5 Ge 3 thin films: an extensive study[END_REF].

Regarding the growth processes, two methods can be employed to grow Mn 5 Ge 3 thin films on Ge(111) substrates with very good structural properties, especially without threading dislocations despite the significant lattice mismatch. The solid phase epitaxy (SPE) was the first method used by Zeng et al., and it is the commonly used one [START_REF] Zeng | Epitaxial ferromagnetic Mn 5 Ge 3 on Ge(111)[END_REF]. It consists on the deposition of manganese on Ge(111) at room temperature, followed by an annealing at 700-720 K. The Mn 5 Ge 3 phase is formed thanks to the diffusion process and the epitaxial stabilization offered by the three order symmetry of the Ge(111) surface. The main drawback of this growth process is the annealing. This step is an impairment in the control of the interface Mn 5 Ge 3 /Ge. On one hand, it entails a diffusion profile of Mn at the Mn 5 Ge 3 /Ge interface, affecting the electrical properties which is detrimental to electrical spin injection. On the other hand, such an annealing step prevents the tailoring of the Mn 5 Ge 3 /Ge Schottky contact by, for instance, the insertion of heavy doped Ge layers at the interface, needed to maximize the spin injection through the Schottky barrier existing at the Mn 5 Ge 3 /Ge contact [START_REF] Albrecht | Spin-polarized electron transport at ferromagnet-semiconductor Schottky contacts[END_REF]. To overcome these issues, our team developed a non-diffusive growth method by molecular beam epitaxy (MBE): it consists on the formation of the Mn 5 Ge 3 thin films by stoichiometric co-deposition of both Mn and Ge on Ge(111) at room temperature. The formation of a Mn 5 Ge 3 film is favourable thanks to the low interface energy between Ge(111) and Mn 5 Ge 3 , and to its low enthalpy of formation compare to the other https://doi.org/10.1016/j.apsusc.2019.01.164 Received 27 September 2018; Received in revised form 11 January 2019; Accepted 20 January 2019 phases of the Mn-Ge phase diagram [START_REF] Arras | Phase diagram, structure, and magnetic properties of the Ge-Mn system: a first-principles study[END_REF][START_REF] Arras | Étude théorique de la structure et de la stabilité des alliages GeMn dans le cadre de la spintronique. Un prototype de semiconducteur magnétique confronté aux résultats expérimentaux[END_REF][START_REF] Berche | Thermodynamic modeling of the germaniummanganese system[END_REF]. The growth at room temperature allows to create well defined interfaces on a structural point of view but especially on an electrical account [START_REF] Petit | Very low-temperature epitaxial growth of Mn 5 Ge 3 and Mn 5 Ge 3 C 0.2 films on Ge(111) using molecular beam epitaxy[END_REF][START_REF] Petit | Mn 5 Ge 3 C 0.6 /Ge(111) Schottky contacts tuned by an n-type ultra-shallow doping layer[END_REF].

Using the co-deposition process, we focused on the first step of the Mn 5 Ge 3 thin film growth on Ge(111) and on the interface between these two materials. In-situ RHEED intensity monitoring and ex-situ atomic force microscopy (AFM) images, taken at different deposition times and for two different ratios of the respective Mn and Ge fluxes, are combined to describe the formation and the morphology of the first nanometres of the Mn 5 Ge 3 films. Transmission electron microscopy (TEM) pictures of the interface revealed the atomic organization. We compared these experimental results and a numerical calculation of the critical nucleation volume V C to determine whether the accommodation of the lattice mismatch involves a pseudomorphic growth or interfacial dislocations.

Experimental details

The growth method of the Mn 5 Ge 3 thin films was described in an earlier paper [START_REF] Petit | Very low-temperature epitaxial growth of Mn 5 Ge 3 and Mn 5 Ge 3 C 0.2 films on Ge(111) using molecular beam epitaxy[END_REF]. The samples are synthesised by MBE in a ultra-vacuum chamber with a base pressure of 2.7 × 10 -8 Pa. On-axis oriented Ge(111) substrates are used and are chemically cleaned before introduction into the MBE chamber. The substrate RMS roughness measured by AFM prior to the growth is 0.52 nm, on 5 × 5 μm -2 field. The Ge and Mn flows of the Knudsen cells were carefully calibrated and lead to a Mn 5 Ge 3 growth rate of 1 nm⋅min -1 . The growth can be monitored in real time by RHEED (acceleration voltage: 30 kV) and a CCD camera fixed above the fluorescent screen allows to record the evolution of the RHEED patterns versus time of deposition. Indeed, the Mn 5 Ge 3 films exhibit an easily identifiable ( ×

3)R30°reconstruction RHEED pattern (see for instance inset of Fig. 2 or Fig. 1e)-f) of ref. [START_REF] Olive-Mendez | Epitaxial growth of Mn 5 Ge 3 /Ge(111) heterostructures for spin injection[END_REF]). For some of the samples, the growth was interrupted in order to image by AFM or TEM the surface of the samples at the corresponding time (i.e at an equivalent deposited Mn 5 Ge 3 thickness d of 0.4, 3.6 and 20 nm for AFM and 20 nm for TEM) or to perform XRD analysis (at d = 20 nm). For the AFM images, we used a Nanoscope IIIA Multimode (Digital instruments) equipped with a 10 × 10 × 2.5 μm scanner. The images were recorded in tapping mode at room temperature using silicon probes (HQ:NSC15/AL BS, Mikromasch). The curvature radius of the silicon tips was about 8 nm (supplier specifications) and the vertical resolution is 0.4 Å. The AFM images were analyzed using the Gwyddion software [START_REF] Nečas | Gwyddion: an open-source software for SPM data analysis[END_REF]. TEM investigations were performed at an accelerating voltage of 300 kV on a JEOL JEM-3010 instrument with a spatial resolution of 1.7 Å. Cross-section samples were prepared with a precision ion polishing system (PIPS). XRD measurements were done on a θ-θ diffractometer, Panalytical X'Pert Pro MPD (for quantitative analysis) and a on a rotating anode Rigaku RU200BH equipped with a Mar345 detector (mainly for qualitative observations). In both cases the Cu (Kα1 + Kα2) radiation was used (λ = 0.1542 nm). X'Pert HighScore software was used for peak analysis.

Calculation

The aim of the following calculation is to obtain a trend whether the lattice mismatch between Mn 5 Ge 3 and Ge(111) gives rise to a pseudomorphic growth or to an array of interfacial dislocations, since no other accommodation mechanisms are observed. The first stages of an epitaxial growth can be assimilated to a condensation process provided that the chemical potential of the gas phase is higher than the one of the solid phase. Thermodynamics foretells a critical volume V C of the nuclei deposited on the surface above which they are stable. V C is calculated by maximizing the nucleation free enthalpy ΔG(V) of a deposited material on a substrate [START_REF] Matthews | Defects in epitaxial multilayers: I. Misfit dislocations[END_REF][START_REF] Matthews | Defects associated with the accommodation of misfit between crystals[END_REF][START_REF] Markov | Recent theoretical developments in epitaxy[END_REF][START_REF] Saint-Girons | Accommodation at the interface of highly dissimilar semiconductor/oxide epitaxial systems[END_REF]. ΔG(V) can be written as follows:

= - + - + G V E μ V γ γ S γ S Δ ( ) ( Δ ) ( ) el interf sub interf dep dep (1)
E el is the elastic energy per unit of volume stored in the deposit, Δμ is the chemical potential difference per unit of volume of the deposit, between the gas phase and the solid phase. γ sub , γ dep , γ interf are the surface energies of the substrate, the deposit and the interface between the substrate and the deposit, respectively. S dep is the free surface of the deposit and S interf , the area of the interface between the substrate and the deposit. From our previous work, the growth of Mn 5 Ge 3 films is two dimensional, hence we limited the calculation of the critical nucleation volume V C to the bidimensional cases [START_REF] Petit | Very low-temperature epitaxial growth of Mn 5 Ge 3 and Mn 5 Ge 3 C 0.2 films on Ge(111) using molecular beam epitaxy[END_REF]. To simplify the problem, the nuclei are described by a cylindrical shape with a diameter d and a height h equal to one monolayer. Thus ΔG(V) can be rewritten:

⎜ ⎟ ⎜ ⎟ = ⎛ ⎝ -⎞ ⎠ + ⎛ ⎝ + + ⎞ ⎠ G V μ E μ πd h γ μ πdh πd a Δ ( ) Δ Δ 1 4 Δ 4 ( 1
)
el dep 2 2 1 ( 2
)
where a 1 = (γ interf -γ sub )/γ dep ). ΔG(V)/Δμ exhibits a maximum for a critical diameter d c :

= - - + ( ) d γ μ a 2 /Δ 1 ( 1 ) c dep E μ γ h μ Δ Δ 1 el dep (3) corresponding to a critical volume = V h c πd 4 c 2
. Two cases are foreseeable: a pseudomorphic growth mode of the deposit on the substrate meaning that the deposit is elastically strained on the substrate (corresponding critical volume V c el ) or a growth mode where the lattice mismatch is accommodated by an array of interfacial dislocations (corresponding critical volume V c dis ). In the first case, E el can be express as 1)

= - E Y ν Rm /(
el dep dep
2 , with Y dep and ν dep the Young modulus and the Poisson coefficient of the deposited materials. m is the lattice mismatch between the deposit and the substrate: m = (a depa sub )/a sub . R takes into account the fact that the nuclei are not of infinite size in the lateral dimensions, which entails a partial relaxation, through free surfaces, of the stored elastic energy: R = 1e -2kd/h . For a cylindrical shape, k = 0.073 and R can be sensibly approximated to 1 [START_REF] Tillmann | Critical dimensions for the formation of interfacial misfit dislocations of In 0.6 Ga 0.4 As islands on GaAs(001)[END_REF]. In the second case, E el is equal to zero and the energetic cost of the formation of the dislocations has to be introduced in Eq. ( 3). The element a 1 becomes ′ = + a a a dis

1 1
where [START_REF] Saint-Girons | Accommodation at the interface of highly dissimilar semiconductor/oxide epitaxial systems[END_REF]: el and V c dis can be compared in order to determine which mechanism is more favourable for the growth of Mn 5 Ge 3 on Ge(111).

= + + + ⎛ ⎝ + ⎞ ⎠ a mb Y Y Y ν Y ν π ln h b (1 ) (1 )

Results and discussion

Numerical results on the critical nucleation volumes

The numerical values of the experimental parameters needed to perform the calculations of the critical nucleation volumes V c el and V c dis are summarised in Table 1. Δμ was fixed at 0.5 eV. This value was chosen using ref. [START_REF] Irisawa | Periodic changes in the structure of a surface growing under MBE conditions and RHEED oscillation[END_REF] where the authors demonstrate that a value of μ k T Δ B around 19 corresponds to the limit between a growth mode prone to kinetic roughening and a layer-by-layer one. Two parameters remain unknown: the surface energy γ dep of the Mn 5 Ge 3 film and the Poisson's ratio ν dep of Mn 5 Ge 3 . We chose to draw the maps of V c el and V c dis versus γ dep and ν dep . The range of values for the γ dep axis was determined knowing that the Mn 5 Ge 3 surface is Mn rich terminated [START_REF] Zeng | Initial stages of Mn adsorption on Ge(111)[END_REF]. According to the literature, the Mn surface energy exhibits values from 1.5 to 3.75 J⋅m -2 [START_REF] Keene | Review of data for the surface tension of pure metals[END_REF][START_REF] Aldén | Surface energy and magnetism of the 3d metals[END_REF][START_REF] Vitos | The surface energy of metals[END_REF][START_REF] Tran | Surface energies of elemental crystals[END_REF]. Regarding the abscissa axis, the range [0.20-0.50] corresponds to typical Poisson's coefficients observed for several families of materials [START_REF] Greaves | Poisson's ratio over two centuries: challenging hypotheses[END_REF]. The two maps are presented on Fig. 1.

The two maps are very similar in terms of values reached by the critical volumes and in terms of distribution of these values as a function of γ dep and ν dep . From an epitaxial growth point of view, it means that the two mechanisms of accommodation of the lattice mismatch could be simultaneously observed during the growth of the Mn 5 Ge 3 film on Ge(111), as confirmed by the following experimental results.

Mismatch accommodation

The deposition of the Mn 5 Ge 3 thin films are followed in situ by RHEED. The intensity of the specular streak is recorded along with the distance between the 00 and 01 streaks, which is converted into the value of the lattice parameter. The evolution of the intensity is displayed on Fig. 2.

As soon as the co-deposition of Ge and Mn started, the intensity decreased significantly. Although the intensity dropped, the RHEED streak patterns were still detected, allowing the measurement of the lattice parameter all along the co-deposition. From 0 to a thickness of 4.0 nm, the lattice evolved from (0.400 ± 0.005) nm (Ge(111) surface) to (0.420 ± 0.050) nm. After a deposition of 4.0 nm, the lattice value stabilised around (0.418 ± 0.025) nm. This indicates that a slight tensile strain (0.7%) remains in the Mn 5 Ge 3 film since the lattice bulk value of Mn 5 Ge 3 is 0.415 nm. The fact that the RHEED pattern of Mn 5 Ge 3 was quickly obtained, is consistent with the small values of the critical volumes of nucleation calculated on Fig. 1. The crystallinity of the 20 nm thick Mn 5 Ge 3 /Ge(111) thin film was examined by XRD, in order to get information regarding the c-axis of the Mn 5 Ge 3 lattice. XRD patterns are displayed on Fig. 3.

The 2D-XRD image of Fig. 3a) is consistent with a single crystal Mn 5 Ge 3 layer, the position of the diffraction spots being in good agreement with the reflections (as indicated in the image) mentioned in the reference file of Mn 5 Ge 3 (ICSD-01-089-4887). On Fig. 3b), the θ-θ measurements show that only the diffraction peaks corresponding to the 002 and 004 reflections were detected. This indicates a monocrystalline Mn 5 Ge 3 thin film with the c-axis normal to the surface of the substrate, in agreement with the 2D-XRD data. Fitting these XRD data with the profile fitting software X'Pert HighScore, the interplanar distance d 001 is measured at (0.5035 ± 0.0001) nm. This value indicates a compressive strain of -(0.5 ± 0.1)% of the Mn 5 Ge 3 lattice, along the c-axis. The full width at half maximum (FWHM) of the diffraction peaks of Mn 5 Ge 3 is 0.51°(the instrumental width evaluated with the 111 reflection of the Ge substrate is 0.03°). Considering that the broadening of the diffraction peaks arose from the low thickness of the film, we apply the Scherrer equation to obtain a minimal thickness value of the Mn 5 Ge 3 film of 17 nm, which is consistent with the 20 nm thick film planned with the deposition fluxes [START_REF] D'agostino | Determination of thin metal film thickness by X-ray diffractometry using the Scherrer equation, atomic absorption analysis and transmission/reflection visible spectroscopy[END_REF]. The values of strain obtained with the RHEED and XRD characterizations have to be compared to the 3.7% lattice mismatch between Ge(111) and the bulk Mn 5 Ge 3 : the Mn 5 Ge 3 thin film is almost relaxed. To go further in the description of the Mn 5 Ge 3 /Ge(111) interface, cross-sectional TEM images of the interface have been taken (Fig. 4).

A periodical intensity contrast can be observed on the top image, with an occurrence every 16 nm along the [010] direction of the Mn 5 Ge 3 film. It corresponds to an array of dislocations spaced of a distance corresponding to 24 unit cells. This arrangement entails a relaxation of the thin film with a slight remaining tensile strain of 0.65%, in good agreement with the RHEED measurements. The bottom image shows an enlargement focussed on a dislocation: a stacking fault can be seen at the centre of the white dotted ellipse. Interfacial dislocations along the Ge(111)/Mn 5 Ge 3 interface have also been observed in the case of the SPE growth mode by several teams, with the same periodicity [START_REF] De Padova | Mn 5 Ge 3 films grown on Ge(1 1 1)-c(2 × 8)[END_REF][START_REF] Spiesser | Large spin accumulation voltages in epitaxial Mn 5 Ge 3 contacts on Ge without an oxide tunnel barrier Aur[END_REF]. Bringing together the results from the RHEED and XRD patterns, and the TEM images, the lattice mismatch between Mn 5 Ge 3 and Ge(111) seems to be accommodated by two mechanisms: a slightly strained layer as shown by the residual stress calculated with the RHEED streaks shift and the XRD measurements, and the interfacial dislocations observed on the TEM images. The previous numerical calculations are consistent with these experimental observations.

Step-flow growth mode and surface morphology

The in situ monitoring of the specular RHEED intensity, as already shown on Fig. 2, gives information on the growth mode and the surface roughness of the Mn 5 Ge 3 thin films. Two monitoring have been carried out, according to the respective Mn and Ge fluxes: the first monitoring has been done with a co-deposition of Mn and Ge in the stoichiometric ratio, the second one in Mn-rich condition (Mn/Ge ratio equal to 1.9). The evolution of the specular intensities is the same for both growth conditions. The persistent streaky RHEED patterns along with the quickly damped intensities and one discernible oscillation are commonly attributed to a one-dimensional step-flow growth mode [START_REF] Neave | Reflection high-energy electron diffraction oscillations from vicinal surfaces -a new approach to surface diffusion measurements[END_REF][START_REF] Adelmann | Dynamically stable gallium surface coverages during plasma-assisted molecularbeam epitaxy of (0001) GaN[END_REF]. This implies a quite high surface mobility of the adatoms. The difference between the stoichiometric and Mn-rich conditions lies in the background intensity, which is higher in the case of the Mn-rich growth. This can be linked to a bigger surface roughness of the thin film, maybe due to the increase of Mn adatoms. To confirm these statements related to the RHEED observations, AFM images have been taken at different film thicknesses on as-grown samples. Panels a) and g) in Fig. 5 (resp. d) and h)) show the surface morphology of 0.4 and 20 nm thick Mn 5 Ge 3 films grown with stoichiometric (resp. Mn-rich) condition.

On Fig. 5a) and g), corresponding to thin films grown under stoichiometric fluxes, terraces with non-uniform length are clearly visible, and the surface morphology exhibits a slight fingers like pattern. This confirms the step-flow growth mode and points out a meandering instability which may arise from different surface diffusion lengths and an asymmetric attachment to the steps of the Ge and Mn adatoms as described in refs. [START_REF] Kallunki | Breakdown of step flow growth in unstable homoepitaxy[END_REF] and [START_REF] Misbah | Crystal surfaces in and out of equilibrium: a modern view[END_REF] and observed in the case of plasma-assisted MBE growth of GaN layers on different kinds of GaN substrates [START_REF] Chèze | Step-flow growth mode instability of N-polar GaN under N-excess[END_REF]. The height and the maximum length of these terraces have been measured on a 2 × 2 μm 2 scans and were evaluated to (0.40 ± 0.08) nm and (140 ± 10) nm, respectively (cf. AFM profile Fig. 5b)). The terrace height is closed to the thickness of one monolayer of Mn 5 Ge 3 along the c-axis of the lattice: 0.5 nm [START_REF] Hirvonen Grytzelius | Mn 5 Ge 3 film formation on Ge(111)c(2 × 8)[END_REF]. The RMS roughness for both films thicknesses is 0.76 nm. Hexagonal pits are present for both thicknesses (No pits are observed on the Ge(111) substrates surfaces before the growth). The hexagonal shape is attributed to the hexagonal lattice of Mn 5 Ge 3 . The pits can be classified in two groups (labelled 1 and 2 on the AFM images) according to their location on the surface and their profile (Fig. 6).

The first group consists of pits (labelled 1 on AFM images from Fig. 5) with a depth of tens of nanometres. The second group contains pits with a depth of only a few nanometres and with a flat bottom. Regarding the films growth under a Mn-rich co-deposition, the surface morphology appears more disturbed with lot of deep hexagonal holes, and still exhibits a fingers like morphology but with smaller features than the stoichiometric co-deposition case. Examining carefully the AFM 2 × 2 μm scans, terraces are visible with a height of (0.45 ± 0.05) nm, which is still consistent with the step-flow growth detected by RHEED. The RMS roughness is around 2.2 nm. To understand the evolution of the surface morphology, we have analyzed the distribution (caption on next page) of the pits depth versus the thickness d of the thin films (Fig. 7).

The distributions of the depths evolve with d: at the very beginning of the deposit, the majority of the pits exhibits a depth in the sampling interval of [-0.5;-1.5] nm i.e in the range of the deposited thickness. Some holes already have a deeper depth around -10 nm. For d = 3.6 nm, the distribution is bimodal: one part of the holes sampling presents a depth centred in the interval [-3.5;-4.5] nm, and the second part consists of holes with a depth between -40 and -50 nm. As the deposition time increases, the depth distribution becomes once again mono-modal: the deepest pits disappear, and the majority of the depths is in the interval [-0.5;-1.0] nm for d = 20 nm: the pits may be filled during the growth. At this thickness, the pits density is 12.5 μm -2 compared to 0.3 μm -2 in the case of the stoichiometric codeposition.

Such pits were also observed in the case of GaN growth by MBE or plasma assisted MBE under Ga rich growth conditions [START_REF] Adelmann | Dynamically stable gallium surface coverages during plasma-assisted molecularbeam epitaxy of (0001) GaN[END_REF][START_REF] Chèze | Step-flow growth mode instability of N-polar GaN under N-excess[END_REF][START_REF] Tarsa | Homoepitaxial growth of GaN under Ga-stable and N-stable conditions by plasma-assisted molecular beam epitaxy[END_REF]. The pits of group 2 have been described as vacancy islands caused by the self-crossing of the meandering steps, and which are not filled by the growth thereafter. These pits pin the steps and entail the fingers like morphology of the surface. Regarding the deeper holes of group 1, it seems unlikely that they can be related to dislocations or impurities: no dislocations (except interfacial ones) are observed in the TEM images of the Mn 5 Ge 3 thin films, and the pits densities should be almost the same for both growth conditions if impurities were involved. So the group 1 may be also characteristic of the growth mode. This conclusion meets Fig. 5. 5 × 5 μm 2 and 2 × 2 μm 2 AFM images of Mn 5 Ge 3 thin films corresponding to a deposit d of a)-b) 0.4 nm in stoichiometric condition and d)-e) in Mn-rich conditions. g) and h) are images of d = 20 nm in stoichiometric and Mn excess conditions, respectively. c) and f) are the profiles along the white lines in the b) and e) AFM scans. Labels 1 and 2 refer to two kinds of hexagonal pits observed on the surface. Fig. 6. Profiles from the hexagonal pits labelled 1 and 2 on the AFM images displayed on Fig. 5. Profile 1 corresponds to deep pits seemingly coming from the growth process and profile 2 corresponds to vacancy islands caused by selfcrossing steps. Fig. 7. Distributions of the depth of the holes from the AFM scans performed at film thickness d of 0.4, 3.6, 6 and 20 nm (not shown here). The number of pits is expressed in percentage of all the counted holes on each AFM images. The RMS roughness is reported for each deposition thickness. the one drawn by Chèze et al. regarding the growth of GaN films by plasma assisted MBE [START_REF] Chèze | Step-flow growth mode instability of N-polar GaN under N-excess[END_REF]. Another argument in favour of a characteristic of the growth mode comes from the other growth process of the Mn 5 Ge 3 films, the SPE. Indeed, in the case of the SPE growth of manganese silicides and germanides, deep holes are also observed in the early stages of the thin films [START_REF] Hirvonen Grytzelius | Mn 5 Ge 3 film formation on Ge(111)c(2 × 8)[END_REF][START_REF] Kumar | Thin manganese films on Si(111)-(7 × 7): electronic structure and strain in silicide formation[END_REF]. Hirvonen Grytzelius et al. have shown that an almost fully covering Mn 5 Ge 3 film is obtained only after a deposition of 32 monolayers of Mn, corresponding to a Mn 5 Ge 3 thickness of 50 nm. The holes are thought to play two main roles: local areas of strain relaxation of the thin films and complementary Ge sources for the formation of the Mn 5 Ge 3 phase. This last point is consistent with our experiments where the pits density is higher in the case of a Mn-rich co-deposition. It means that to form the Mn 5 Ge 3 phase, additional Ge atoms are drawn off in the substrate, thus creating deep holes.

Conclusion

A study of the early stages of the non-diffusive growth of Mn 5 Ge 3 thin film on Ge(111) substrate has been carried out. This system exhibits a significant lattice mismatch of 3.7%. XRD measurements demonstrate that the thin film is monocrystalline and, combined with RHEED diffraction patterns analysis, that the Mn 5 Ge 3 lattice is slightly strained. TEM images show the presence of interfacial dislocations at the Mn 5 Ge 3 /Ge interface, every 16 nm along the [010] direction of the Mn 5 Ge 3 film. Thus, the lattice mismatch is accommodated by two phenomena: a pseudomorphic growth and interfacial dislocations. Numerical calculations support these observations: the critical nucleation volumes calculated for both the case of a pseudomorphic growth and the case of an accommodation of the lattice mismatch by an array of dislocations, are very similar in values. AFM images taken at different deposition times and the evolution of the specular RHEED intensity, both recorded in stoichiometric and Mn-rich co-deposition conditions, show that the Mn 5 Ge 3 films grow according to a step-flow mode, with a meandering instability.

Y

  sub and ν sub are the Young modulus and the Poisson coefficient of the substrate, and b is the Burger vector of the dislocations. The two different values of the critical nucleation volumes V c

Fig. 1 .

 1 Fig. 1. Maps of the critical volume of nucleation as a function of the surface energy and of the Poisson's ratio of the Mn 5 Ge 3 deposit. Top figure: accommodation of the lattice mismatch between the Mn 5 Ge 3 film and the Ge(111) substrate via an elastically strained film V ( ) c el . Bottom figure: accommodation

Fig. 2 .

 2 Fig. 2. Typical evolution of the intensity of the 00 streak of the RHEED pattern taken along the azimuth [210] of the typical Mn 5 Ge 3 (001) × ( 3 3)R30°s urface. The white dotted rectangle on the pattern indicates the area used to record the intensity. The white double arrow shows the distance between the streaks used to calculate the value of the lattice parameter of the Mn 5 Ge 3 (001) plan.

Fig. 3 .

 3 Fig. 3. a) 2D-XRD pattern of a 20 nm thick Mn 5 Ge 3 thin film. The signal of the Ge substrate has been subtracted. The angle between the incident X-ray beam and the sample surface (ω angle) varied from 15 to 20°during the acquisition. b) XRD pattern of the 20 nm thick Mn 5 Ge 3 thin film. The diffraction peaks correspond to the 002 and 004 reflections of Mn 5 Ge 3 . The insert displays the peaks corresponding to the 111 reflection of Ge substrate (CuK α1 , CuK α2 ), the Ge signal was attenuated by a factor 85, in order to avoid the damage of the detector.

Fig. 4 .

 4 Fig. 4. Top image: TEM image of a cross section of Mn 5 Ge 3 film. The periodical contrast of the intensity at the interface indicates the presence of interfacial dislocations every 16 nm. Bottom image: zoom on a dislocation from the top image.

Table 1

 1 Experimental parameters needed to perform the calculations of the critical nucleation volumes V c el and V c dis .

		Ge	Mn 5 Ge 3	Interface
	Young modulus Y (GPa)	103 [31,32,33]	110 [34]
	Poisson's ratio ν	0.26 [31,32,33]
	Surface/interface energy (J⋅m 2 )	1.06 [35]		0.53 [7,19]
	Burger's vector b (Å)		7.18 [36]
	Monolayer height h (Å)		5 [37]

Acknowledgments

The authors would like to thank D Chaudanson from the electron microscopes service of the CINaM for his support regarding the TEM images and C Coudreau for the technical support on the MBE growth set-up.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.apsusc.2019.01.164.