Salah Jellali
email: s.jellali@squ.edu.om

Ahmed Amine Azzaz
email: amine.azzaz@uha.fr

Mejdi Jeguirim
email: mejdi.jeguirim@uha.fr

Helmi Hamdi
email: hhamdi@qu.edu.qa

Ammar Mlayah
email: ammarmlayah17@gmail.com

Use of Lignite as a Low-Cost Material for Cadmium and Copper Removal from Aqueous Solutions: Assessment of Adsorption Characteristics and Exploration of Involved Mechanisms

Keywords: lignite, heavy metals, adsorption, batch, isotherm, mechanism

Lignite, as an available and low-cost material, was tested for cadmium (Cd) and copper (Cu) removal from aqueous solutions under various static experimental conditions. Experimental results showed that the removal efficiency of both metals was improved by increasing their initial concentrations, adsorbent dosage and aqueous pH values. The adsorption kinetic was very rapid for Cd since about 78% of the totally adsorbed amounts were removed after a contact time of only 1 min. For Cd and Cu, the kinetic and isothermal data were well fitted with pseudo-second order and Freundlich models, respectively, which suggests that Cd/Cu removal by lignite occurs heterogeneously on multilayers surfaces. The maximum Langmuir's adsorption capacities of Cd and Cu were assessed to 38.0 and 21.4 mg g -1 and are relatively important compared to some other lignites and raw natural materials. Results of proximate, scanning electron microscopy/energy dispersive X-ray spectroscopy (SEM/EDS), Fourier transform infrared spectroscopy (FTIR) and X-Ray diffraction (XRD) showed that the removal of these metals occurs most likely through a combination of cation exchange and complexation with specific functional groups. The relatively high adsorption capacity of the used lignite promotes its future use as a low cost material for Cd and Cu removal from effluents, and possibly for other heavy metals or groups of pollutants.

Introduction

The contamination of water resources by heavy metals contained in industrial effluents is an important worldwide concern due to their toxicity, low biodegradability and high accumulation capacity in water-living organisms [START_REF] Morais | Environmental Health-Emerging Issues and Practice[END_REF]. When transferred to humans through the food chain, these metals could be accumulated in various body organs and tissues causing life-threating illness and possible damages to vital systems [START_REF] Morais | Environmental Health-Emerging Issues and Practice[END_REF]. Cadmium and copper are among the most toxic heavy metals. Indeed, exposure to cadmium, mainly used in batteries and the coating and plating industry, could result in stomach pains, bone fracture, possible infertility as well as serious damages to the central nervous and immune systems [START_REF] Sharma | The Characteristics, Toxicity and Effects of Cadmium[END_REF]. High ingested amounts of copper could increase infection frequencies, cardiovascular risks, and alterations in cholesterol metabolism [START_REF] Araya | Copper in human health[END_REF]. The removal of these pollutants from industrial wastewater before discharge into the environment is, therefore, an urgent task to be appropriately achieved.

Many technologies have been developed for heavy metal removal from industrial effluents. They include chemical precipitation, reduction, electrodialysis, and membrane separation [START_REF] Abdullah | Recent trends of heavy metal removal from water/wastewater by membrane technologies[END_REF][START_REF] Liu | Treatment of raffinate generated via copper ore hydrometallurgical processing using a bipolar membrane electrodialysis system[END_REF][START_REF] Sun | Performance evaluation and optimization of flocculation process for removing heavy metal[END_REF][START_REF] Cui | Novel forward osmosis process to effectively remove heavy metal ions[END_REF]. These technologies have shown interesting removal efficiencies at laboratory scale. However, their upscaling to real cases has been hindered by several limitations such as high capital and exploitation costs, very sensitive operational conditions, the use of large amounts of chemicals, and the production of secondary sludge that has to be sustainably handled [START_REF] Cui | Novel forward osmosis process to effectively remove heavy metal ions[END_REF].

The adsorption technique has been pointed out as a promising, eco-friendly and low-cost method for heavy metal removal from aqueous solutions [START_REF] Duan | Removal of heavy metals from aqueous solution using carbon-based adsorbents: A review[END_REF][START_REF] Qin | The improved methods of heavy metals removal by biosorbents: A review[END_REF][START_REF] Shakoor | A review of biochar-based sorbents for separation of heavy metals from water[END_REF][START_REF] Baby | Greener Method for the Removal of Toxic Metal Ions from the Wastewater by Application of Agricultural Waste as an Adsorbent[END_REF][START_REF] Cataldo | Complex formation of copper(II) and cadmium(II) with pectin and polygalacturonic acid in aqueous solution. An ISE-H + and ISE-Me 2+ electrochemical study[END_REF]. In general, adsorbents with high pH, important specific surface area, well developed microporosity and that are rich in specific functional groups are used for the efficient removal of heavy metals form effluents. Once the adsorption process is completed, the loaded-adsorbents could be regenerated and heavy metals could be recovered for a further subsequent use with respect to the circular economy and sustainable development concepts [START_REF] Zhong-Kai | High-efficiency removal of lead from wastewater by biochar derived from anaerobic digestion sludge[END_REF][START_REF] Chouchene | Biosorption performance, combustion behavior, and leaching characteristics of olive solid waste during the removal of copper and nickel from aqueous solutions[END_REF]. Various mineral and organic materials have been tested for Cd(II) and Cu(II) removal from synthetic/real wastewaters. They include powdered marble wastes [START_REF] Mlayah | Study of continuous lead removal from aqueous solutions by marble wastes: Efficiencies and mechanisms[END_REF], clays [START_REF] Li | Adsorption characteristics of white pottery clay towards Pb(II), Cu(II), and Cd(II). Arab[END_REF], lignocellulosic biomasses [START_REF] Foroutan | Characteristics and performance of Cd, Ni, and Pb bio-adsorption using Callinectes sapidus biomass: Real wastewater treatment[END_REF], biochars [START_REF] Jellali | Lead removal from aqueous solutions by raw sawdust and magnesium pretreated biochar: Experimental investigations and numerical modelling[END_REF][START_REF] Puglla | Biochar from Agricultural By-Products for the Removal of Lead and Cadmium from Drinking Water[END_REF], activated carbons [START_REF] Egirani | Preparation and characterization of powdered and granular activated carbon from Palmae biomass for cadmium removal[END_REF][START_REF] Vo | Efficient Removal of Cr(VI) from Water by Biochar and Activated Carbon Prepared through Hydrothermal Carbonization and Pyrolysis: Adsorption-Coupled Reduction Mechanism[END_REF] etc.

Lignite, often referred as brown coal, is a sedimentary rock that is naturally formed from naturally compressed peat. It is a low cost material and available in various countries at high amounts [START_REF] Ediger | An assessment of mining efficiency in Turkish lignite industry[END_REF]. Besides its classical use for energy generation, raw lignite has been tested for the removal of various pollutants from aqueous solutions such as phenol and chlorophenol [START_REF] Kuśmierek | Raw lignite as an effective low-cost adsorbent to remove phenol and chlorophenols from aqueous solutions[END_REF], trichloroethylene [START_REF] Grajales-Mesa | Pilot-Scale Evaluation of a Permeable Reactive Barrier with Compost and Brown Coal to Treat Groundwater Contaminated with Trichloroethylene[END_REF], phosphorus [START_REF] Zhang | Evaluation of five filter media in column experiment on the removal of selected organic micropollutants and phosphorus from household wastewater[END_REF] and ammonium [START_REF] Nazari | Application of Victorian brown coal for removal of ammonium and organics from wastewater[END_REF]. A special focus has been dedicated to the use of these raw materials as efficient adsorbents for heavy metals form aqueous solutions [START_REF] Zherebtsov | Sorption of Metal Cations by Lignite and Humic Acids[END_REF][START_REF] Binabaj | Comparative study on adsorption of chromium(VI) from industrial wastewater onto nature-derived adsorbents (brown coal and zeolite)[END_REF][START_REF] Zhao | Mechanism study of Cr(III) immobilization in the process of Cr(VI) removal by Huolinhe lignite[END_REF][START_REF] Pentari | Sorption properties of low calorific value Greek lignites: Removal of lead, cadmium, zinc and copper ions from aqueous solutions[END_REF]. As such, these materials have relatively high contents of functional groups such as carboxyl, alcoholic and carbonyl groups that could complex with metal ions [START_REF] Zherebtsov | Sorption of Metal Cations by Lignite and Humic Acids[END_REF][START_REF] Uçurum | A study of removal of Pb heavy metal ions from aqueous solution using lignite and a new cheap adsorbent (lignite washing plant tailings)[END_REF]. Furthermore, they have a large cation exchange capacity allowing cationic pollutants to be adsorbed [START_REF] Pehlivan | Removal of metal ions using lignite in aqueous solution-Low cost biosorbents[END_REF].

Laboratory investigations regarding heavy metal removal by raw lignite have demonstrated that it could be considered as an interesting candidate for single metal removal form aqueous solutions [START_REF] Zherebtsov | Sorption of Metal Cations by Lignite and Humic Acids[END_REF][START_REF] Pentari | Sorption properties of low calorific value Greek lignites: Removal of lead, cadmium, zinc and copper ions from aqueous solutions[END_REF][START_REF] Pehlivan | Removal of metal ions using lignite in aqueous solution-Low cost biosorbents[END_REF]. However, when in multicomponent systems, the competition phenomenon decreases their maximal adsorption capacities. In both systems, results are sometimes contradictory and highlight that the ability of heavy metal retention depends on their physico-chemical characteristics as well as the lignite properties [START_REF] Doskočil | Removal of metal ions from multi-component mixture using natural lignite[END_REF][START_REF] Pehlivan | Electrochemical Investigation of Binding of Heavy Metal Ions to Turkish Lignites[END_REF]. For instance, Pehlivan et al. [START_REF] Pehlivan | Electrochemical Investigation of Binding of Heavy Metal Ions to Turkish Lignites[END_REF] studied the adsorption of Pb, Cd, Cu, Ni, and Zn on several Turkish lignite materials. They showed that the order of metal adsorption ability depends on the lignite type. However, Pb and Cu were the most adsorbed components in contrast to Ni and Zn. Furthermore, Pentari et al. [START_REF] Pentari | Sorption properties of low calorific value Greek lignites: Removal of lead, cadmium, zinc and copper ions from aqueous solutions[END_REF] studied the removal of Pb, Cd, Zn, and Cu from aqueous solutions by a lignite from Greece. They found that in single mode, the ability adsorption order was as follows: Pb > Cd > Cu > Zn. In a multicomponent system, they showed that Cu sorption was the most affected, while Zn adsorption capacity remained quasi-constant. For the same metals in a multicomponent system, Doskočil and Pekař [START_REF] Doskočil | Removal of metal ions from multi-component mixture using natural lignite[END_REF] found a different order: Pb > Cu > Zn > Cd. On the other hand, the involved mechanisms during heavy metal adsorption onto raw lignite are still not well identified. An in-depth kinetic and isothermal modeling study combined with lignite analysis before and after metal adsorption by using advanced techniques could reduce this gap.

As a consequence, the principal objectives of this work were: (i) to assess the adsorption characteristics of Cd and Cu onto a Tunisian lignite under various experimental conditions including contact time, initial concentration, pH, adsorbent dosage, and competition with other metals, (ii) to compare the efficiency of this lignite with raw and modified lignite available in the literature, (iii) to better understand the involved adsorption mechanisms of Cd and Cu through the combination of an in-depth modeling study and various physico-chemical analyses.

Materials and Methods

Adsorbent Preparation and Characterization

Raw lignite was collected from the Cap Bon region (northeastern part of Tunisia). This lignite was used in its natural state after a drying step at 60 • C for 24 h followed by manual grinding in ceramic grinder. The fraction with diameter size lower than 63 µm was selected for the adsorption tests. The preliminary characterization of the used lignite included the determination of its: (i) mineral composition by X-ray fluorescence spectrophotometer (Philips, Eindhoven, The Netherlands), (ii) Brunauer-Emmett-Teller BET specific surface area through N 2 gas adsorption method using a gas adsorption analyzer (Quantachrom Autosorb 1 sorptiometr), and (iii) pH of zero-point-charge value (pH ZPC ) [START_REF] Azzaz | Chemical treatment of orange tree sawdust for a cationic dye enhancement removal from aqueous solutions: Kinetic, equilibrium and thermodynamic studies[END_REF].

Furthermore, advanced analyses of the lignite before and after metals adsorption were performed for a better identification of the involved mechanisms. They included the assessment of: (i) the morphology and qualitative composition through scanning electron microscopy (SEM) coupled with energy dispersive X-ray (EDX) (Philips model FEI model Quanta 400 apparatus, Amsterdam, The Netherlands), (ii) proximate analysis using a TGA/DSC3+ device (Mettler-Toledo, Greifense, Switzerland), and (iii) the existing functional groups through a Fourier transform infrared (FTIR) analysis using an Equinox 55 spectrometer (Brucker, Billerica, MA, USA). The FTIR spectra were assessed between 4000 and 400 cm -1 with a resolution of 1 cm -1 . Experimental protocols for the above-cited analyses have already been detailed in a previous paper [START_REF] Azzaz | Olive mill wastewater: From a pollutant to green fuels, agricultural and water source and bio-fertilizer-Hydrothermal carbonization[END_REF].

Synthetic Heavy Metals Solutions Preparation and Analysis

Cadmium nitrate (Cd(NO 3 ) 2 ), copper nitrate (Cu(NO 3 ) 2 ), lead nitrate (Pb(NO 3 ) 2 ), and zinc nitrate (Zn(NO 3 ) 2 ) were used for the preparation of four stock solutions at a concentration of 1 g L -1 each (Fisher Scientific, Waltham, MA, USA). These solutions were used throughout this study for the preparation of adsorption solutions at precise concentrations. Metal concentrations were measured thorough an atomic absorption spectrometer (AAS) with an air-acetylene flame (Perkin Elmer Analyst 200, Waltham, MA, USA). The wavelengths used for the analysis of the Cd, Cu, Pb and Zn were 228.8, 324.8, 283.3, and 213.9 nm, respectively. The initial pH values of the solutions were adjusted by using dilute sodium hydroxide or nitric acid.

Batch Adsorption Investigation

Cd and Cu removal efficiency from aqueous solutions by raw lignite was carried under static conditions (batch mode). It consists in shaking, at room temperature (20 ± 2 • C), a given mass of lignite in 50 mL of aqueous solution containing the metal at a fixed concentration for a desired contact time at 400 rpm by using a Variomag-poly15 magnetic stirrer. Then, the suspension was filtrated through 0.45 µm cellulose acetate filter before analysis with AAS. During this study, the effect of the following experimental conditions on Cd and Cu removal efficiency were assessed: (i) particle size distribution for four lignite granulometries: <63 µm; between 63 and 500 µm; 500-1000 µm, and 1000-2000 µm; (ii) the contact time of 1; 5; 10; 20; 40; 60, and 90 min; (iii) the initial aqueous pH for 2.0; 3.0; 4.0 and 5.0, and (iv) lignite dosages for 0.4; 1.0; 1.6; 2.0; 2.4; 3; 3.5 and 4.0 g L -1 . During these assays, the default following parameters were used: a lignite size fraction lower than 63 µm, a contact time of 90 min, an initial Cd or Cu concentration of 100 mg L -1 , an initial pH of 5 and a lignite dosage equal to 2 g L -1 . Finally, the competing effect was determined for a multicomponent solution containing Cd, Cu, Pb and Zn at constant concentrations of 30 mg L -1 . All these experiments were carried out in triplicate and mean values are reported in this work. The standard deviation for all assays was lower than 5%.

The adsorbed metal amount at a given moment 't', (qt) and the related removal yield (Yt) were determined as follows [START_REF] Haddad | Mill Wastewater: From a Pollutant to Green Fuels, Agricultural Water Source and Biofertilizer[END_REF]:

q t = (C 0 -C t ) D (1)
Y e (%) = (C 0 -Ct) C 0 × 100 (2)
where C 0 and C t (mg L -1 ) are initial metal concentrations and at a given time 't', respectively, and D is the used adsorbent dose (g L -1 ). Cadmium and copper adsorption kinetics were fitted to three standard models namely, pseudo-first order (PFO), pseudo-second order (PSO), and intraparticle and film diffusion. Moreover, the experimental isothermal data were fitted to Langmuir, Freundlich and Dubinin-Radushkevich (D-R) isotherm models. The agreement between the experimental and theoretical adsorbed amounts was assessed through the determination of the average percentage errors (APE kinetic and APE isotherm) as follows:

APE kinetic (%) = ∑ q t,exp -q t,theo /q t,exp N × 100 (3)
APE isotherm (%) = ∑ q e,expq e,theo /q e,exp N × 100

where q t,exp and q t,theo (mg g -1 ) are the experimental and the theoretical adsorbed amounts at a given time 't'; q e,exp and q e,theo (mg g -1 ) are the experimental and the theoretical adsorbed masses at equilibrium.

Results

Lignite Characterization

The XRF analysis of the lignite showed that it contains various minerals. Silica, Sulfur, iron, and aluminum exist at relatively high contents of 6.2%, 5.1%, 2.6%, and 2.5% (dry mass), respectively. Lower contents of 0.06%, 0.46%, 1.21%, and 0.21% were observed for Na, K, Ca, and Mg, respectively. Some of these elements could be exchanged with Pb(II) during its adsorption by lignite. On the other hand, the used material has higher concentrations of Si, Al, Fe and Na and lower Ca and Mg content than a commercial and natural lignite from Czech Republic [START_REF] Havelcová | Sorption of metal ions on lignite and the derived humic substances[END_REF][START_REF] Huber | Evaluation of Factors Influencing Lab-Scale Studies to Determine Heavy Metal Removal by Six Sorbents for Stormwater Treatment[END_REF]. The BET surface area of the lignite was assessed to 11.2 m 2 g -1 , which is in the range of BET values reported for Greek lignites (between 3.6 and 23.8 m 2 g -1 ) [START_REF] Pentari | Sorption properties of low calorific value Greek lignites: Removal of lead, cadmium, zinc and copper ions from aqueous solutions[END_REF]. For instance, it is about 12-fold higher than a lignite from Poland (0.91 m 2 g -1 ) [START_REF] Kuśmierek | Raw lignite as an effective low-cost adsorbent to remove phenol and chlorophenols from aqueous solutions[END_REF], and about two-fold higher than a Hungarian lignite (5.3 m 2 g -1 ) [START_REF] Zhang | Evaluation of five filter media in column experiment on the removal of selected organic micropollutants and phosphorus from household wastewater[END_REF]. This relatively higher surface area suggests that the used lignite could exhibit more metal adsorption capacities compared to other lignites reported in the literature. On the other hand, its pH ZPC was estimated to 3.6. It is lower than that of a Czech lignite (5.0) [START_REF] Havelcová | Sorption of metal ions on lignite and the derived humic substances[END_REF], and a Polish lignite (6.2) [START_REF] Kuśmierek | Raw lignite as an effective low-cost adsorbent to remove phenol and chlorophenols from aqueous solutions[END_REF]. In contrast, it is higher than pH ZPC of a Hungarian lignite (2.6) [START_REF] Zhang | Evaluation of five filter media in column experiment on the removal of selected organic micropollutants and phosphorus from household wastewater[END_REF]. Since the net charge surface of the adsorbent becomes negative for aqueous pH values higher than pH ZPC , the removal of positively charged metals by the current lignite through electrostatic attraction will be favored for a wide pH range.

Batch Adsorption Results

Effect of Lignite Granulometry

The impact of lignite granulometry on Cd and Cu removal from the synthetic solution was carried out for initial metal concentrations of 100 mg L -1 each, initial pH of 5 and a dosage of 2 g L -1 . The experimental results (Figure 1) showed that coarser is the lignite fraction, lower is Cd or Cu removal efficiency. Accordingly, the highest adsorbed amounts of Cd (26.1 mg g -1 ) and Cu (18.6 mg g -1 ) were observed for the finest granulometry (<63 µm). These adsorbed Cd and Cu amounts decreased by about 25% and more than 65%, respectively when lignite particles of 1-2 mm size were used. This outcome could be attributed to the lower microporosity and surface area generally observed for coarser media [START_REF] Kumar | Effect of pore size distribution and particle size of porous metal oxides on phosphate adsorption capacity and kinetics[END_REF]. Similar behavior was observed by PN and CP [START_REF] Pn | Overview on the Effect of Particle Size on the Performance of Wood Based Adsorbent[END_REF] when investigating an oily effluent treatment by hard wood based adsorbents. They reported a decrease of the removal efficiency by about 30% when the average size particles were increased from 0.8 to 3.5 mm.

Effect of Contact Time-Kinetic Study

The removal of Cd and Cu by the used lignite is clearly a time-dependent process as shown in Figure 2. Indeed, their removal was very rapid at the beginning of the adsorption (especially for Cd) since about 78% and 44% of the totally removed amounts were adsorbed after only 1 min for Cd and Cu, respectively. This finding suggests that Cd removal occurs mainly through surface reactions [START_REF] Baby | Greener Method for the Removal of Toxic Metal Ions from the Wastewater by Application of Agricultural Waste as an Adsorbent[END_REF]. The high reactivity of the used lignite could result in an important energy saving when such process is scaled up for field investigations. After this contact period, the adsorbed amounts continue to increase but at much slower rate. This behavior is linked to an intraparticle diffusion inside the pores of the lignite and adsorption by functional groups through complexation process [START_REF] Jellali | Biosorption characteristics of ammonium from aqueous solutions onto Posidonia oceanica (L.) fibers[END_REF]. The equilibrium state which corresponds to quasi-constant adsorbed amounts was reached after approximately 60 min for both metals. This time is 6-fold lower than the one reported by Havelcova et al. [START_REF] Havelcová | Sorption of metal ions on lignite and the derived humic substances[END_REF] when investigating Cd, Cu and Zn removal by a local lignite from south Moravian coal field (Czech republic) and 12-fold lower than the duration observed by Pentari et al. [START_REF] Pentari | Cadmium Removal from Aqueous Solutions Using Nano-Iron Doped Lignite[END_REF] for Cd removal by a raw and iron doped Greek lignite. However, relatively similar equilibrium contact times were observed for Cd and Cu removal by various Greek lignites [START_REF] Pentari | Sorption properties of low calorific value Greek lignites: Removal of lead, cadmium, zinc and copper ions from aqueous solutions[END_REF]; Cu, Pb and Ni removal by two Turkish lignites [START_REF] Pehlivan | Removal of metal ions using lignite in aqueous solution-Low cost biosorbents[END_REF]; Zn removal by a commercial lignite from South Korea [START_REF] Shrestha | Kinetic Parameters and Mechanism of Zn (II) Adsorption on Lignite and Coconut Shell-Based Activated Carbon Fiber[END_REF] and Cr(VI) adsorption onto a raw Iranian lignite [START_REF] Binabaj | Comparative study on adsorption of chromium(VI) from industrial wastewater onto nature-derived adsorbents (brown coal and zeolite)[END_REF]. For economic reasons, low contact times, ensuring percentages removal of more than 80% of the totally adsorbed Cd or Cu amounts could be used in real case application. These durations correspond to only 5 and 20 min for Cd and Cu, respectively.

At equilibrium, the lignite sample used in this study removed Cd better than Cu. Indeed, the adsorbed Cd amount by was assessed to 26.1 mg g -1 which is about 41% higher than Cu (Figure 2). A Similar trend was observed by various studies dealing with heavy metal adsorption onto lignite [START_REF] Pentari | Sorption properties of low calorific value Greek lignites: Removal of lead, cadmium, zinc and copper ions from aqueous solutions[END_REF][START_REF] Havelcová | Sorption of metal ions on lignite and the derived humic substances[END_REF]. This behavior is mainly imputed to their physico-chemical properties, especially electronegativity, ionic potential and ionic radius [START_REF] Pardo-Botello | Adsorption kinetics of zinc in multicomponent ionic systems[END_REF]. The parameters of the three theoretical models: PFO, PSO and diffusion models are given in Table 1. Based on these calculated parameters, it can be clearly deduced that the PFO model does not appropriately fit to the experimental data. In fact, the corresponding determination coefficients were low: 0.553 and 0.883 for Cd and Cu, respectively. The calculated APE between the measured and calculated adsorbed masses were high since they are about 28% for Cd and 22% for Cu. Figure 2 confirms this finding since the dashed lines are far from the measured data.

Table 1. Kinetic parameters of Cd and Cu removal by the used lignite (C 0 = 100 mg L -1 , D = 2 g L -1 ; t = 60 min; pH = 5; T = 20 ± 2 • C).

Cadmium

Copper q e,exp (mg g In contrast, the PSO model fits well the experimental data for both metals. The related determination coefficients (>0.89) were much higher than those for the PFO model. Furthermore, the APE for Cd and Cu were determined to only 7% and 12% for Cd and Cu, respectively and therefore were lower than those obtained for the PSO model. The theoretical adsorbed amounts of Cd and Cu at equilibrium (q e,theo ), were very close to the experimental ones (Figure 2 and Table 1), with different percentages of about 1% and 3%, respectively. Therefore, under the used experimental conditions, the PSO model is more suitable in fitting the Cd and Cu removal onto lignite. This model suggests that the rate limiting step might be chemical adsorption involving valency forces through sharing or exchange of electrons between these two metals and lignite [START_REF] Jellali | Adsorption characteristics of phosphorus from aqueous solutions onto phosphate mine wastes[END_REF].

The analysis of the adsorption of Cd and Cu onto the used raw lignite through the application of film and intraparticle diffusion models indicated clearly that that the adsorption process proceeds by surface interactions at earlier stages (short times, less than 5 min) and by intraparticle diffusion at later stages (Figure 2). For Cd, the film diffusion coefficient (through boundary layer) is about two times higher than the one of the intraparticle diffusion. This explains the very high adsorption percentage observed after only 1 min (Figure 2). This finding confirms that intraparticle diffusion process controls significantly the rate of Cd adsorption onto the lignite. Similar observations were reported by Pehlivan et al. [START_REF] Pehlivan | Removal of metal ions using lignite in aqueous solution-Low cost biosorbents[END_REF] when studying Pb, Cu, Ni, and Zn removal by Turkish lignites. For Cu, the film and intraparticle diffusion coefficients are quite similar (Table 1) indicating that both processes (boundary layer and intraparticle diffusion) control its removal by the raw lignite.

Effect of Initial Aqueous pH

The effect of the initial aqueous pH on Cd and Cu removal efficiency was performed under the experimental conditions presented in Section 2.3. The experimental results showed that for both metals, the removed amounts increased with the increase of the aqueous pH (Figure 3). As such, for an initial pH of 2, the adsorbed amounts of Cd and Cu were 10.9 and 5.4 mg g -1 , respectively. At a pH of 5, these quantities reached 26.1 and 18.6 mg g -1 which are about 2.4 and 3.4 times higher than those registered at an initial pH of 2. This outcome could be explained by the lignite surface charge that is positive at aqueous pH lower than the pH ZPC (3.6), which will repulse the two cationic metals. Furthermore, at this pH range, abundant H + ions in the solution will compete with Cd and Cu over the adsorption sites. However, when the used aqueous pH value is higher than the pH ZPC , the lignite surface will carry more negative charges and will consequently favor Cd and Cu adsorption through electrostatic reactions. At this pH range, H + and other exchangeable cations such as Ca 2+ , Mg 2+ , Na + , and K + could be exchanged with these metals and released in the aqueous solutions. Similar trends were reported by several studies dealing with heavy metals or cationic pollutant adsorption onto raw or modified lignites [START_REF] Pehlivan | Removal of metal ions using lignite in aqueous solution-Low cost biosorbents[END_REF][START_REF] Havelcová | Sorption of metal ions on lignite and the derived humic substances[END_REF] and other adsorbents [START_REF] Azzaz | Chemical treatment of orange tree sawdust for a cationic dye enhancement removal from aqueous solutions: Kinetic, equilibrium and thermodynamic studies[END_REF][START_REF] Jellali | Biosorption characteristics of ammonium from aqueous solutions onto Posidonia oceanica (L.) fibers[END_REF][START_REF] Haddad | Raw and treated marble wastes reuse as low cost materials for phosphorus removal from aqueous solutions: Efficiencies and mechanisms[END_REF][START_REF] Haddad | Investigations on phosphorus recovery from aqueous solutions by biochars derived from magnesium-pretreated cypress sawdust[END_REF]. For instance, Pehlivan et al. [START_REF] Pehlivan | Removal of metal ions using lignite in aqueous solution-Low cost biosorbents[END_REF] showed that increasing pH form 2 to 6 increased Cu removal efficiency yield by a raw Turkish lignite from about 8% to more than 88%.

Effect of Lignite Dosage

The impact of lignite dose on Cd and Cu removal from aqueous solutions at an initial concentration of 100 mg L -1 , an initial pH of 5 and 60 min of contact time is given in Figure 4. Their removal yields increase with the increase of the lignite dose. As such, rising the dose from 0.4 to 3 g L -1 , Cd and Cu removal efficiencies increased from about 40.5% and 19.8% to 61.9% and 46.7%, respectively. This important increase is linked to the presence of more available adsorption sites that could interact with Cd or Cu when using higher doses. Starting from a used dose of 3.5 g L -1 , Cd or Cu removal efficiency remains approximately constant due to the saturation of lignite particles. Such high removal efficiencies observed for relatively low lignite doses is a real asset for larger applications at industrial levels.

A similar trend was reported by Binabaj and Ramezanian [START_REF] Binabaj | Comparative study on adsorption of chromium(VI) from industrial wastewater onto nature-derived adsorbents (brown coal and zeolite)[END_REF] and Gurses et al. [START_REF] Gürses | Removal of methylene blue from aqueous solution using by untreated lignite as potential low-cost adsorbent: Kinetic, thermodynamic and equilibrium approach[END_REF] when studying chromium(VI) and methylene blue removal by raw lignites, respectively.

Competing Effect

Industrial effluents generally contain a complex mixture of organic and non-organic pollutants. The effect of the presence of Pb and Zn on the adsorption of Cd and Cu by the raw lignite was performed according to the experimental conditions described in Section 2.3. The experimental results (Figure 5) showed that for single mode, as reported by Pentari et al. [START_REF] Pentari | Sorption properties of low calorific value Greek lignites: Removal of lead, cadmium, zinc and copper ions from aqueous solutions[END_REF], the adsorption efficiency of the studied metals was as follows: Pb > Cd > Cu > Zn. The corresponding removal percentages were estimated at 100%, 74.6%, 71.7% and 44.8%, respectively. Similar observations were also reported by Pehlivan et al. [START_REF] Pehlivan | Electrochemical Investigation of Binding of Heavy Metal Ions to Turkish Lignites[END_REF] who studied the adsorption of Pb, Cd, Cu, Ni, and Zn on several Turkish lignites. They showed that for the majority of the studied lignites, Pb, Cu and Cd were the most adsorbed metals whereas Ni and Zn were the less adsorbed ones. In the multicomponent system, all the removal efficiencies of the four heavy metals significantly decreased due to competition. Accordingly, the highest removal decrease was observed for Cd (37.1%) and Zn (29.2%) and the adsorption pattern shifted to: Pb > Cu > Cd > Zn with removal efficiencies of 91.7%, 56.8%, 37.5% and 15.6%, respectively. Consequently, when treating real wastewater containing a mixture of heavy metals, Pb and Cu will be favorably retained by the current lignite compared to other metals. Similar trends were reported by Allen and Brown [START_REF] Allen | Isotherm analyses for single component and multi-component metal sorption onto lignite[END_REF] when investigating the removal of Cd, Cu and Zn by an Irish lignite in a single and multicomponent systems. Their experimental results showed that in single metal sorption mode, the adsorption ability decreases in the order Cd > Cu > Zn. However, in the multicomponent assays, this order changed to: Cu > Cd > Zn. This result is in a contradiction with the one reported by Pentari et al. [START_REF] Pentari | Sorption properties of low calorific value Greek lignites: Removal of lead, cadmium, zinc and copper ions from aqueous solutions[END_REF] who pointed out that in multicomponent system containing Pb, Cd, Cu, and Zn, copper adsorption was the most significantly influenced by the presence of the other elements with a total yield reduction of about 5%. This behavior should not be only attributed to metal characteristics including size, electronegativity, availability and hydration energy [START_REF] Mlayah | Study of continuous lead removal from aqueous solutions by marble wastes: Efficiencies and mechanisms[END_REF] but also to the physico-chemical properties of the lignite such as the donor atoms abundance (oxygen, nitrogen, sulphur [START_REF] Karabulut | Batch removal of copper(II) and zinc(II) from aqueous solutions with low-rank Turkish coals[END_REF]).

Isotherm Adsorption

For the experimental conditions cited in Section 2.3, the data of Cd and Cu adsorption isotherms in comparison with the predicted ones using Freundlich, Langmuir and D-R models are given in Figure 6. Table 2 gives the parameters of these models as well as their fitness to the experimental data. It can be clearly demonstrated from Figure 6 and Table 2 that the three applied models fit very well with the experimental data. Nevertheless, the highest determination coefficients (R 2 ) (0.990 for Cd and for Cu) and the lowest APE (2.3% for Cd and 1.5% for Cu) were observed for D-R model. This model predicts, however, very high adsorption capacities for both Cd and Cu with respective values of 124.8 and 40.5 mg g -1 , respectively. These values are unrealistic considering the isotherm curves shape (Figure 6) and are mainly imputed to this model's used assumptions especially the one related to the uniformity and homogeneity of the adsorbent's microporous structure [START_REF] Azzaz | Chemical treatment of orange tree sawdust for a cationic dye enhancement removal from aqueous solutions: Kinetic, equilibrium and thermodynamic studies[END_REF]. The calculated free energy (E = 1/ √ 2β) was assessed to 10.7 and 13.4 kJ mol -1 for Cd and Cu, respectively. Both of them are higher than 8 kJ mol -1 , indicating that Cd and Cu removal by the used lignite was mainly chemical [START_REF] Gürses | Removal of methylene blue from aqueous solution using by untreated lignite as potential low-cost adsorbent: Kinetic, thermodynamic and equilibrium approach[END_REF]. It is worth mentioning that compared to Langmuir model, The Freundlich one fits better the experimental data with higher R 2 coefficients and lower APE values (Table 2). This outcome indicates that both Cd and Cu adsorption by the used lignite occurs heterogeneously on multilayer surfaces through chemical processes [START_REF] Binabaj | Comparative study on adsorption of chromium(VI) from industrial wastewater onto nature-derived adsorbents (brown coal and zeolite)[END_REF].

On the other hand, the Freundlich constant 'n' values were 2.3 and 3.9 for Cd and Cu, respectively. They are in the range of 1-10, which indicates that the adsorption of these two heavy metals by the used lignite is a favorable process. Values in the same range were determined by Pentari et al. [START_REF] Pentari | Sorption properties of low calorific value Greek lignites: Removal of lead, cadmium, zinc and copper ions from aqueous solutions[END_REF] when studying Cd and Cu removal by a raw Greek lignite. Moreover, the highest Langmuir's parameter values (obtained for the lowest used initial concentrations) 'R L = 1 1+K L * C 0 ' were estimated to 0.41 and 0.24 for Cd and Cu, respectively. They are lower than 1, suggesting that lignite could be considered as a favorable material for Cd and Cu retention from aqueous solutions. The Langmuir's maximum adsorption capacities of Cd and Cu were determined to 38.0 and 21.4 mg g -1 , respectively. A comparison of the used lignite efficiency in removing Cd and Cu with other lignites (Table 3) clearly shows that it can be considered as a promising medium for the removal of heavy metals form wastewater. As a matter of fact, its Cd adsorption capacity was about 5.6, 1.7 and 1.5 times higher than a lignite activated carbon from China [START_REF] Sun | Nitric acid-anionic surfactant modified activated carbon to enhance cadmium(II) removal from wastewater: Preparation conditions and physicochemical properties[END_REF], a HNO 3 treated lignite activated carbon [START_REF] Sun | Nitric acid-anionic surfactant modified activated carbon to enhance cadmium(II) removal from wastewater: Preparation conditions and physicochemical properties[END_REF], and a raw lignite form Greece [START_REF] Pentari | Cadmium Removal from Aqueous Solutions Using Nano-Iron Doped Lignite[END_REF]. Its Cu adsorption capacity was higher than a Beypazari lignite [START_REF] Karabulut | Batch removal of copper(II) and zinc(II) from aqueous solutions with low-rank Turkish coals[END_REF], Ilgin lignite and Beysehir lignite [START_REF] Pehlivan | Removal of metal ions using lignite in aqueous solution-Low cost biosorbents[END_REF]. However, other tested lignites exhibited higher adsorption capacities [START_REF] Zherebtsov | Sorption of Metal Cations by Lignite and Humic Acids[END_REF][START_REF] Pentari | Sorption properties of low calorific value Greek lignites: Removal of lead, cadmium, zinc and copper ions from aqueous solutions[END_REF][START_REF] Pekař | Comparison of Copper Sorption on Lignite and on Soils of Different Types and Their Humic Acids[END_REF]. It is important to underline that the observed lignite adsorption capacities of Cd and Cu are also higher than various other agricultural, animal and industrial wastes [START_REF] Singh | Current advancement and future prospect of biosorbents for bioremediation[END_REF][START_REF] Chakraborty | Adsorption of heavy metal ions by various low-cost adsorbents: A review[END_REF]. As compared to natural agricultural wastes, Cd adsorption capacity of the studied lignite was about 7.5 and 2.3 higher than corncob [START_REF] Leyva-Ramos | Adsorption of cadmium(II) from aqueous solution on natural and oxidized corncob[END_REF] and rice husk [START_REF] Krishnani | Biosorption mechanism of nine different heavy metals onto biomatrix from rice husk[END_REF], respectively. Moreover, Cu removal was 2.8 and 2.1 times more important than Banana peels [START_REF] Demessie | Cleaning Water Contaminated With Heavy Metal Ions Using Pyrolyzed Biochar Adsorbents[END_REF] and grape stalks [START_REF] Villaescusa | Removal of copper and nickel ions from aqueous solutions by grape stalks wastes[END_REF], respectively. As illustrated in Table 3, Cd or Cu adsorption efficiency by lignites depend not only on their specific surface area (case of the activated lignites [START_REF] Sun | Nitric acid-anionic surfactant modified activated carbon to enhance cadmium(II) removal from wastewater: Preparation conditions and physicochemical properties[END_REF]) but also on their functional group types and richness and the experimental conditions. The same trend was reported by Puglla et al. [START_REF] Puglla | Biochar from Agricultural By-Products for the Removal of Lead and Cadmium from Drinking Water[END_REF] who found relatively low adsorption capacities of Cd onto biochars derived from peanut shell (1.038 mg g -1 ), chonta pulp (0.655 mg g -1 ) and corncob (0.857 mg g -1 ) even if they have interesting textural properties. It is important to underline that the structural and textural properties of the lignite could be improved through specific physical/chemical/thermal modifications [START_REF] Sun | Nitric acid-anionic surfactant modified activated carbon to enhance cadmium(II) removal from wastewater: Preparation conditions and physicochemical properties[END_REF][START_REF] Zhang | Factors controlling adsorption of recalcitrant organic contaminant from bio-treated coking wastewater using lignite activated coke and coal tar-derived activated carbon[END_REF][START_REF] Zhang | One-step pyrolysis of lignin and polyvinyl chloride for synthesis of porous carbon and its application for toluene sorption[END_REF]. For instance, Sun et al. [START_REF] Sun | Nitric acid-anionic surfactant modified activated carbon to enhance cadmium(II) removal from wastewater: Preparation conditions and physicochemical properties[END_REF] found that the chemical/thermal modification of lignite by a HNO 3 solution simultaneously increased its surface area (from 158.1 to 185.1 m 2 g -1 ) and its surface carboxylic and lactone functional groups (from 0.55 to 1.26 mmol g -1 ). Consequently, such modification increased Cd removal capacity from 6.8 to 26.6 mg g -1 .

Raw and Metal-Loaded Lignite Characterization and Adsorption Mechanism Exploration

Various techniques were used for the characterization of lignite before and after metal adsorption in order to get a better understanding of the probably involved mechanisms. Morphological properties of the lignite before and after adsorption of metals were investigated through imagery comparison using scanning electron microscopy and energy-dispersive X-ray spectroscopy (Figures 7 and8). It appears that the raw lignite (particle size lower than 63 µm) presents an irregular and heterogeneous surface. Moreover, similarly to the majority of natural carbonaceous materials, the used lignite does not present a concrete porosity (Figure 7a), which explain its measured specific surface area. SEM imaging reveals also the presence of crystalline structures on the surface. According to Zhang and Chen [START_REF] Zhang | Particle size effect on pore structure characteristics of lignite determined via low-temperature nitrogen adsorption[END_REF], raw uncrushed lignites could present some typical cellular plant structures such as fusinites and semifusinites, intergranular and plant tissue pores along with some crystalline impurities identified mainly as calcium carbonates and silicon dioxide as well as other trace elements such as aluminum and iron ions. This was confirmed by EDX analysis where the raw material presents high peak intensities of oxygen, silica and calcium (Figure 8a). After the adsorption process, SEM images of the metal-loaded lignite presented a clearer contrast on its surface for both metals, while the crystalline structure present in the raw material became less apparent (Figure 7b,c). This observation could be due to an adsorption reaction of metals on the lignite surface and at the same time to a possible inverse movement of some minerals from lignite to the liquid phase. The EDX analysis confirmed this hypothesis since the peaks related to oxygen, silica, potassium, and calcium decreased in intensity, while peaks attributed to copper and cadmium appeared, which confirms the adsorption process (Figure 8b,c).

The proximate analysis (Figure 9) indicates that the raw lignite is mainly composed of minerals with an ash content of about 39%. Its moisture, fixed carbon and volatile matter contents are respectively 7%, 21% and 33%. Similar trends were reported by Kanca [START_REF] Kanca | Investigation on pyrolysis and combustion characteristics of low quality lignite, cotton waste, and their blends by TGA-FTIR[END_REF] for a Turkish lignite, where ash content had the major proportion (48%) followed by volatile matter and fixed carbon (25% and 24%, respectively). After metal adsorption, a slight decrease of the fixed carbon and volatile matter contents occurred in favor of an ash content increase by about 3.7% and 5.8% for Cd-and Cu-loaded lignite, respectively. To align these results with SEM/EDS observations, it is possible that the adsorption of cadmium and copper were driven by three mechanisms: (i) adsorption of a small fraction of metal ions onto lignin decayed matrix which constitutes the small specific surface area of the feedstock yet characterized, (ii) ion-exchange reaction where ions such as potassium, calcium then silica at lower percentage were diffused from the solid matrix to the aqueous solution thus vacating the oxygenic functional groups, and (iii) chemical adsorption on free surface functional groups.

In order to confirm this hypothesis, FTIR analyses were performed on both raw and metal-loaded lignite and results are depicted in Figure 10 and Table 4. The lignite presented a heterogeneous surface with the co-presence of acidic and alkaline functional groups namely, hydroxyl (-OH; 3600-3200 cm -1 ), aliphatic (C-H; 3000-2700 cm -1 ), carbonyl and acetyl esters (C=O and C-O; 1650-1600 cm -1 and 1180-980 cm -1 , respectively), methyl and methylene aromatic (-CH 2 /-CH 3 ; 1465-1320 cm -1 ) and out-of-plane aromatic groups (C-H; 896-809 cm -1 ). After metal adsorption, few modifications were noticed for peak positions of some functional groups due to their involvement in the adsorption process (Table 4). For instance, a peak shift of about +8 and +4 cm -1 was noticed for hydroxyl groups after Cd and Cu adsorption, respectively. Similar observation for carboxylic group where a vibration was spotted by about -4 and -6 cm -1 as compared to raw lignite for the same exhausted adsorbents, respectively. Moreover, a significant change was observed in the-C=C aromatic structure where a peak appeared for both Cd-and Cu-related specters at 1382 and 1380 cm -1 , respectively (Table 4). This could be related to the uptake of nitrate ions (entering in the composition of copper and cadmium reagents) by the lignite matrix, causing a slight alteration in its surface functionalities [START_REF] Qi | Characterisation of lignite as an industrial adsorbent[END_REF]. On the other hand, the presence of positively charged metals (i.e., Cd 2+ and Cu 2+ ) caused a thermodynamic balance between solid and liquid phases. It is possible that a cation release phenomenon occurred leading to the release of protons (H + ) or other cations (Na + ; K + ; Ca 2+ , and Mg 2+ ) followed by the fixation of bivalent metals on the oxygenic functional groups as follows [START_REF] Puglla | Biochar from Agricultural By-Products for the Removal of Lead and Cadmium from Drinking Water[END_REF][START_REF] Azzaz | Investigations on a dye desorption from modified biomass by using a low-cost eluent: Hysteresis and mechanisms exploration[END_REF]:

2 A -C -O --H + + Me 2+ ↔ A -C -O --Me 2+ + 2H + (5) 2 A -O --H + + Me 2+ ↔ A -O --Me 2+ + 2H + (6)
A -Na + /K + /Ca 2+ /Mg 2+ + Me 2+ ↔ A -Me 2+ + 2Na + /2K + /Ca 2+ /Mg 2+ [START_REF] Cui | Novel forward osmosis process to effectively remove heavy metal ions[END_REF] where A stands for the aromatic structure of lignite and Me 2+ is either Cu 2+ or Cd 2+ . On the basis of FTIR investigation, we can deduce that Cd and Cu removal by lignite is governed not only by its textural properties (especially surface area and microporosity), but also by its richness in functional groups.

Conclusions

The current research study demonstrates that lignite, as a low cost and abundant material, can achieve very rapid and effective removal of cadmium and copper from aqueous effluents under wide experimental conditions. The removal efficiency seems not only dependent on the heavy metal properties but also on the textural and structural characteristics of the lignite. The heavy metal adsorption process occurs through a combination of several mechanisms, including mainly cation exchange and complexation with various functional groups. Lignite adsorption capacity could be further enhanced through physical, chemical, and thermal treatment methods. However, the cost and the environmental impact of such modifications should be accurately assessed. Moreover, further studies are required in order to assess the efficiency of raw/modified lignite for the removal of other heavy metals and recalcitrant organic pollutants under dynamic conditions.

Figure 1 .

 1 Figure 1. Impact of lignite particle size on Cd and Cu removal efficiency (C 0 = 100 mg L -1 , D = 2 g L -1 ; t = 60 min; pH = 5; T = 20 ± 2 • C).

Figure 2 .

 2 Figure 2. Kinetics of Cd and Cu removal by lignite and its fitting with PFO and PSO models (C 0 = 100 mg L -1 , D = 2 g L -1 ; t = 60 min; pH = 5; T = 20 ± 2 • C; Exp.: Experimental value).

Figure 3 .

 3 Figure 3. Impact of initial aqueous pH on Cd and Cu removal efficiency by lignite (C 0 = 100 mg L -1 , D = 2 g L -1 ; t = 60 min; T = 20 ± 2 • C).

Figure 4 .

 4 Figure 4. Impact of lignite dose on Cd and Cu removal yields (C 0 = 100 mg L -1 , t = 60 min; pH = 5; T = 20 ± 2 • C).

Figure 5 .

 5 Figure 5. Cd, Cu, Pb and Zn removal yield in single and multicomponent systems (C 0 = 30 mg L -1 for all metals; D = 2 g L -1 ; t = 60 min; pH = 5; T = 20 ± 2 • C).

Figure 6 .

 6 Figure 6. Isotherm experimental and fitted data with Freundlich, Langmuir and Dubinin-Radushkevich (D-R) models for Cd (a) and Cu (b) removal by lignite (D = 2 g L -1 ; t = 60 min; pH = 5; T = 20 ± 2 • C; Exp.: Experimental value).

Figure 7 .

 7 Figure 7. SEM images of: (a) raw lignite; (b) after adsorption of cadmium; (c) after adsorption of copper (magnitude: ×2500).

Figure 8 .

 8 Figure 8. EDX analysis of: (a) raw lignite; (b) cadmium-loaded lignite; (c) copper-loaded lignite.

Figure 9 .

 9 Figure 9. Proximate analysis of lignite before and after copper and cadmium adsorption.

Figure 10 .

 10 Figure 10. FTIR analysis of lignite before and after Cu (II) and Cd (II) adsorption.

Table 2 .

 2 Adsorption isotherm parameters of Cd and Cu removal by lignite (D = 2 g L -1 ; t = 60 min; pH = 5; T = 20 ± 2 • C).

	Metal	Cd	Cu
	Freundlich model
	Freundlich constant: K F	4.7	6.4
	Freundlich constant: n	2.3	3.9
	Determination coefficient: R 2	0.991	0.983
	Average percentage error: APE (%)	2.8	2.3
	Langmuir model
	Langmuir's maximum adsorption capacity; q m (mg g -1 )	38.0	21.4
	Langmuir constant: K L Determination coefficient: R 2	0.048 0.979	0.104 0.957
	APE (%)	5.4	2.5
	D-R model
	DR's maximum adsorption capacity; q m (mg g -1 ) E (kJ mol -1 )	124.8 10.7	40.5 13.4
	Determination coefficient: R 2	0.990	0.990
	APE (%)	2.3	1.5

Table 3 .

 3 Comparison of the used lignite removal efficiency of Cd and Cu with other lignites from various regions.

	Material	BET Surface Area (m 2 g -1 )	Adsorption Conditions	Cd Langmuir (mg g -1 ) Adsorption Capacity	Cu Langmuir (mg g -1 ) Adsorption Capacity	Reference
	Lignite, Beypazarı, Turkey	2.56	C 0 = 0.00015-0.0015 M; pH = 4; D = 10 g L -1 ;	-	1.62	[51]
			t = 1 h ; T = 20 • C
	Ilgin lignite, Konya, Turkey Beysehir lignite, Konya, Turkey	2.06 2.96	C 0 = 0.0025-0.025 M; pH = 4.5; D = 4 g L -1 ; t = 120 min; T = 20 • C	--	17.8 18.9	[32]
	Tyul'gansk lignite, South Ural Basin Russia Tisul'sk lignite, Kansko-Achinsk	--	C 0 = 0.1 M; pH = 4.1; D = 40 g L -1 ; t = 5 days; T = 25 • C	--	27.3 27.3	[27]
	Basin, Russia
	Lignite, South Moravian deposit, Mikulcice,	-	C 0 = 0.001-0.2 M; pH = 4-5; D = 20 g L -1 ;	-	82.0	[53]
	Czech Republic		t = 24 h; T = 25 • C
	Lignite activated carbon (LAC), China	158.11		6.8	-
	Lignite activated carbon,					[52]
	treated with sodium dodecyl benzene	118.2		26.6	-
	sulfonate (SAC) HNO 3 -treated lignite activated carbon (NAC)	185.07	C 0 = 0.00015-0.0015 M; pH = 7.2; D = 1 g L -1 ; t = 24 h ; T = 25 • C	22.8	-
	HNO 3 -treated lignite
	activated carbon, post
	treated with sodium	131.4		44.2	-
	dodecyl benzene
	sulfonate (NSAC)
	Lignite, Drama, northern Greece Same lignite doped with Nano-scale-zero-valent-	154 109	C 0 = 0.00015-0.015 M; pH = 4.5; D = 10 g L -1 ; t = 12 h; T = 25 • C	25.5 34.7	--	[43]
	iron particles
	Lignite, Macedonia and
	Thrace, Greece

Table 4 .

 4 Peak location for each functional group detected in FTIR spectra for raw and metal-loaded lignite.

	Functional Group	-OH	C=O	C-H	-C=C	-C-O	-C-H
	Raw	3400	1705	1614	Absent	1105	794
	Cd 2+ -loaded lignite	3408	1701	1600	1382	1101	794
	Cu 2+ -loaded lignite	3404	1699	1593	1380	1103	794

Funding: This research was funded by The Tunisian Ministry of Higher Education and Scientific Research.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: Data available on request.

Acknowledgments: Authors would like to thank all technicians involved in this work. The Tunisian Ministry of Higher Education and Scientific Research is gratefully acknowledged for financing this research project.

Author Contributions: Conceptualization, S.J. and A.M.; methodology, S.J., A.M. and M.J.; validation, S.J., A.M. and A.A.A.; formal analysis, S.J., A.A.A., M.J., H.H. and A.M.; investigation, S.J. and A.M.; resources, S.J. and A.M.; writing-original draft preparation, S.J. and A.A.A.; writing-review and editing, A.M., H.H. and M.J.; visualization, S.J., A.M., A.A.A., H.H. and M.J.; supervision, S.J. and A.M.; project administration, A.M.; funding acquisition, A.M. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:

The authors declare no conflict of interest


Etienne Gravier
email: etienne.gravier@univ-lorraine.fr

Thomas Drouot

Maxime Lesur

Alejandro Guillevic

Guillaume Lo-Cascio

Jérôme Moritz

Dominique Escande

Giovanni Manfredi
email: giovanni.manfredi@ipcms.unistra.fr

Collision rates estimated from exact N -body simulations of a one-dimensional plasma

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

The most accurate way to simulate the many-body plasma dynamics is by solving the full Nbody system of Newton's equations. But for a typical fusion or astrophysical plasmas, it is obviously impossible to follow the trajectories of each one of the ≈ 10 23 particles that constitute the plasma.

Hence the need for kinetic models, which are mean-field models that neglect microscopic two-body collisions and only retain the macroscopic self-consistent electromagnetic fields. Kinetic models typically solve the evolution equation (Vlasov equation) of a one-particle probability distribution in the six-dimensional phase space, coupled to the Maxwell equations for the fields.

Rigourously, kinetic models can be derived from the full N -body dynamics through a BBGKY hierarchy truncated to the lowest order, which amounts to neglecting two-body and higher-order correlations between the particles. The smallness parameter that defines this hierarchy and justifies the truncation procedure is the so-called coupling or "graininess" parameter g, which is proportional to the ratio of the kinetic energy to the Coulomb energy. Ideal collisionless plasmas are characterized by g ≪ 1, while strongly coupled plasmas have g ≥ 1.

Even though fusion plasmas are not collision-dominated, a collision operator is often needed for simulating the plasma behavior on long timescales, for instance to take into account neoclassical transport in tokamaks, which can be dominant when a transport barrier is triggered, or to model the diffusion of high-Z impurities [1,2]. For this purpose, the relevant Vlasov equations are augmented by a collision term, often in the form of a simple Bhatnagar-Gross-Krook (BGK) operator [3]. The collision rate involved in this collision operator is then given by some Spitzer-like formula [4].

However, this simple procedure is conceptually shaky. The Vlasov equation is already an averaged equation, and the electromagnetic fields that appear there are mean macroscopic fields.

Hence, adding to this Vlasov equation a collision term based on the microscopic collision frequency may lead to an incorrect estimation of the effects of the collisions. Recent works [4][5][6], based on a weak-turbulence calculation starting from the Klimontovich equation and including all wave numbers, suggest a modification of the collision operator. Using this theory, it was estimated that the collisional damping of Langmuir waves is up to two orders of magnitude smaller than the one predicted by a BGK operator [4]. In other works [7,8] the authors solve either a Vlasov-like equation with ad-hoc collision terms (eq. 1 in [7]) or a Fokker-Planck equation (eq. 6 in [8]).

The present work aims at investigating a simple one-dimensional (1D) N -body model, in order to extract the exact collision rates and compare them to those given by the heuristic Spitzer formula.

We want to stress that our approach is based on the N -body Newton equations for the charged particles. Therefore, we have here a microscopic model that includes, at least potentially, all plasma effects, both collective (such as Landau damping) and individual (such as collisional damping). We will focus on two typical plasma phenomena, namely plasma Langmuir waves and the two-stream instability. As our N -body model is 1D, whereas collision rates are usually estimated in 3D, the comparison can only be approximate, and should be regarded as a first step aimed at clarifying the predictions put forward in Refs. [4,5].

In the past, several computer experiments were performed with 1D N -body models, investigating various phenomena such as Debye shielding and plasma oscillations [9,10]. The relaxation towards a Maxwellian distribution was also studied, revealing different scalings of the relaxation time for sub-populations of the particles and for the overall population [11]. The relaxation times were compared to the results of the Lenard-Balescu equation [12]. Other works investigated similar relaxation problems for the dynamics of self-gravitating systems [13].

The present paper is organized as follows. The N -body model and corresponding computational code are described in Sec. II. In Sec. III, we study the impact of collisions on the two-stream instability and estimate the collision frequency by subtracting the observed N -body instability rate from the theoretical collisionless instability rate. In Sec. IV, a similar analysis is performed for Langmuir waves. Comparison with a Vlasov kinetic code allowed us to study the coexistence of collisional damping and Landau damping. Finally, in Sec. V we summarize the preceding results and discuss the dependence of the collision rates on the plasma coupling parameter g. Conclusions are drawn in Sec. VI.

II. ONE-DIMENSIONAL N -BODY MODEL AND COMPUTATIONAL CODE

We consider a two-component plasma made of N electron sheets and N ion sheets with surface charge density ±σ. The plasma is confined between two reflecting walls, separated by a distance L, where L is the length of the box (see Fig. 1, top panel). There is no magnetic field. We consider infinite sheets in the direction perpendicular to the x axis, hence the electric field between the sheets is constant, and is equal to an integer multiple of σ/2ϵ 0 . Thus, the total electric field has a step-like profile and the motion of the sheet is uniformly accelerated as long as it does not cross one of its neighbors (Fig. 1, bottom panel). As the electric field is constant between two planes and using Newton's law, the equation of motion can be integrated exactly up to computer round-off errors.

In the remainder of the paper, lengths are normalized to the Debye length λ De , times are normalized to ω -1 pe , and velocities to the electron thermal velocity v T e . Starting from Newton's second law applied to one sheet labeled j, the normalized acceleration âj that the sheet experiences is:

âj = L 2N c j m e m j [2Q g j + c j ], (1)
where L is the length of the box normalized to the Debye length, 2N is the total number of sheets (ions and electrons), m e is the mass of one electron, m j is the mass of one particle that belongs to the sheet j (ion or electron), Q g j is the number of sheets located on the left side of the considered sheet j. The sign of the charge belonging to sheet j is c j , and is equal to c j = +1 for ion sheets and c j = -1 for electron sheets.

As âj is constant between two sheets, we can evolve the dynamical variables v j and x j of a single sheet by integrating the equations of motion, which can be done analytically in between two successive collisions, as long as the j-th sheet does not cross one of its two neighbors. After a crossing, the sheets experience a new constant field. Then the trajectories of the sheets in phase space are known without any approximation except for the round-off errors due to the finite number of digits to code real numbers the computer. We have checked that the total energy remains constant, with a relative error close to 10 -13 . In this code, the sheets pass freely through one another, interacting only via their coupling to the electric field, and, as particles interact via long-range forces, two-body collisions as well as collective effects are both taken into account by the model. We also consider that the sheets collide elastically with the walls so that they reverse their velocities whenever they hit one of the walls.

The algorithm starts by computing the collision time of each sheet with its neighbor to the right, or with the wall. Then the smallest crossing time is identified, and the system is evolved up to just before this event. Actually only the two crossing sheets are evolved, while the others are frozen. The time when a sheet last experienced a crossing is recorded in a table. Finally the replacement on the index numbers j and j + 1 of the two sheets is performed.

The most time consuming operation in the code is the search of the minimum collision time. In order to minimize this step, the collision times are ordered on a heap [14]. The crossing times in the heap obey the following heap condition: each element in a child node is greater than or equal to the elements in its parent node. The smallest time is located at the top of the heap (Fig. 2).

It is not necessary to update all the states of particles while processing the heap. For a large number of sheets, i.e. a large value of N , the operation count is dominated by the cost of reordering the heap after each crossing, which goes asymptotically as log N [14]. We introduce an indexing array, mapping the position in the heap to the position of the sheets in space. Therefore, this algorithm determines the smallest crossing time at a low numerical cost, but also maps the crossing times in the heap to the spatial positions of the sheets. In the end, the overall duration of a typical run scales as N log N .

III. TWO-STREAM INSTABILITY

As a first example, we consider the case of two cold oppositely-directed electron streams with velocities ±v 0 , and heavy ions (m i ≫ m e ) with zero velocity. The ions will hardly move on the electronic time scales relevant to this study, and only act as a homogeneous neutralizing background.

A random generator is used to distribute the positions of the sheets at t = 0. This two-stream configuration can then be unstable under certain conditions. The growth rate of the instability may be derived from the linearized Vlasov-Poisson equations for the electrons:

∂f e ∂t + v x ∂f e ∂x - eE m e ∂f e ∂v x = 0, (2)
∂E ∂x = e ε 0 n 0 - +∞ -∞ f e dv x , (3)
with n 0 the ion density, here assumed to be constant and equal to the electron plasma density at equilibrium, e = +1.6 × 10 -19 C, m e = 9.1 × 10 -31 kg, E the electric field, ε 0 is the dielectric constant in vacuum. The initial electron distribution function is taken as

f e (x, v x , t = 0) =f 0 (v) = 1 2 n 0 [δ(v x -v 0 ) + δ(v x + v 0 )],
where δ denotes the Dirac delta function, i.e. the streams are assumed to be cold. Linearizing the Vlasov-Poisson equations and expanding in plane waves ∼ e i(kx-ωt) , we obtain the instability growth rate γ as the imaginary part of the frequency ω:

γ = -2k 2 v 2 0 -ω 2 pe + ω 2 pe 1 + 8k 2 v 2 0 ω 2 pe 2 , ( 4
)
where k is the wave number and ω pe is the electron plasma frequency.

As the beams are initially cold, their thermal velocity is zero. However, for long times (t → ∞)

the electron population will eventually relax to an equilibrium Maxwellian distribution with thermal velocity v T e . At thermal equilibrium, all the initial drift kinetic energy of the streams ( 1 2 m e v 2 0 ) will be converted into thermal energy (k B T e = 1 2 m e v 2 T e , where k B is the Boltzmann constant and v T e the electron thermal speed), so that v T e = v 0 . The effective Debye length is then defined as

λ De = v T e /ω pe .
From Eq. ( 4), the maximum linear growth rate of the instability is γ max = 0.3536 ω pe , with k max λ De = 0.6124. The initial perturbation is provided by the intrinsic numerical noise. The length of the box is chosen to be L = 10.2606 λ De , so that the first mode is the most unstable one, i.e., k max = 2π/L. Hereafter this linear collisionless growth rate will be denoted γ no-coll = γ max , As the total potential energy of the two-stream system is expected to increase exponentially with time (exp(2γ N B t)) during the linear phase of the instability, γ N B can be easily extracted from the calculated data.

In Fig. 3 Here, our working hypothesis is that the growth and damping rates are additive. Therefore, the collision rate due to two-body Coulomb collisions, denoted γ coll N B , can be written as the difference between the theoretical collisionless growth rate (γ no-coll ) and the observed numerical growth rate of the instability (γ N B ):

γ coll N B = γ no-coll -γ N B . (5)
The dependence of γ coll N B on N is shown in Fig. 4.

The hypothesis of Eq. ( 5) is in agreement with the linear theory of the Vlasov equation augmented by a BGK collision operator (∂f e /∂t) coll = -ν(f e -f M ), where ν is the BGK collision rate and f M is the Maxwellian distribution [15], which predicts that the effective linear growth rate is lowered by the collision rate. This property was also verified by numerical simulations of the Vlasov-BGK equation using a semi-lagrangian Vlasov code [15,16].

The importance of collective versus individual two-body effects can be measured using the so-called graininess or coupling parameter g, which for our 1D model reads as [17]:

g = 1 nλ De = 1 N L λ De = L N , (6)
where n = N/L is the plasma density and L = L/λ De is the normalized length of the computational box ( L ≈ 10 in the present case). When the graininess parameter is of order unity or larger, the plasma is dominated by individual effects, i.e. two-body collisions. Again, we observe that the two-body collisions can be neglected when N is larger than 1000 (g < ∼ 10 -2 ). Finally, it should be mentioned that, on longer timescales, the two streams are destroyed by the instability and the system evolves towards thermal equilibrium. This occurs over a characteristic time that scales as g -2 ∼ (nλ De ) 2 , in agreement with earlier results [9,11,12].

IV. LANGMUIR WAVES

We now consider the phenomenon of plasma oscillations and Langmuir waves. If we shift the electrons with respect to the positive ions, the mutual attraction of the two species acts as a restoring force that causes the electrons to oscillate relative to the ions at a certain frequency ω.

The well-known Bohm-Gross dispersion relation [18] quantifies this effect:

ω 2 = ω 2 pe + 3k 2 v 2 T e (7)
If the wave number k is equal to zero (homogeneous perturbation), the electrons should oscillate at the plasma frequency. We shall first consider this case. At t = 0, the distribution of the electrons is a Maxwellian. In order to excite the plasma oscillations, we adopt the same procedure as described in [10]: we impart to all the electrons a positive velocity kick, meaning that each v j is instantaneously changed to v j +v kick . This procedure is equivalent to applying a large and almost instantaneous electric field to the electrons between t = 0 and t = τ , which induces a velocity change eEτ /m e = v kick , with E → ∞, τ → 0, but Eτ finite (Fig. 5).

In order to assess the plasma oscillations, we study the evolution of ve (t), the spatial average of the electron velocities. Indeed ve (t) is much less sensitive to thermal fluctuations than the density or the electric potential. As a first check, we measured the oscillation period for different values of N and different lengths L of the box. As expected, we found a period equal to 2π/ω pe with a relative discrepancy less than 5%. Next, we estimate the damping rate by plotting the amplitude of the oscillations on a semilogarithmic scale and determine its slope using a least squares method. As in this case there is neither instability nor Landau damping (k = 0), the observed damping rate directly represents the collision rate γ coll N B . The results are presented in Fig. 6. The collision rate is plotted for four values of the kick ranging from 0.3 v T e to 1.0 v T e , with L = 100 λ De . We observe that the collision rates depend little on the initial kick. Again, as observed in the case of the two-stream instability, collisional damping becomes significant for values of N below N ≈ 2000 (g > 5 × 10 -2 ).

Finally, in order to investigate the influence of collisions on the total damping of Langmuir waves, we consider modes with k ̸ = 0. In order to excite a particular wave number, we modulate the initial velocity of each particle as follows:

v j = v j,M axwell + v 0 sin m 2π L x j ( 8
)
where v j,M axwell is the velocity of the sheet j, located at x = x j , before the kick, and m is the mode number. The wave number k is then equal to 2πm/L. The amplitude v 0 is chosen to be 0.1 v T e . Landau and collisional damping, a Vlasov code (COBBLES [16]) and a Particle-In-Cell (PIC) code [19], without any collision operators, were used to determine the Landau damping rates for the m = 5 and m = 6 modes. Both codes yield γ Landau = 0.028 for m = 5 and γ Landau = 0.061 for m = 6, with the same initial conditions as those used in the N -body code. In Fig. 7, the horizontal lines correspond to the Landau damping rates given by the Vlasov and PIC codes.

Again, for small values of g (g < 5×10 -2 ), i. e. large values of the number of sheets (N > 2000), the effect of Coulomb collisions can be neglected when compared to the Landau damping rate. For both modes m = 5 and m = 6 the transition is located at the same values of N (and therefore of g).

V. TWO-BODY COLLISION RATES

In the preceding sections, we found that -in all three cases considered here of plasma waves (k = 0), Langmuir waves (k ̸ = 0), and two-stream instability -collisional effects start playing a role for g > ∼ 10 -2 . The agreement of the three results suggests the method we used to determine the two-body collision rate can be trusted to deduce some general laws.

Fig. 8 summarizes the findings detailed in the preceding sections, augmented by some further simulations performed with different parameters, such as different stream velocities or box lengths.

The red and blue lines represent linear regression fits of the collision rates with the coupling parameter, γ coll N B ∝ g, with slopes equal to 0.326 (coefficient of determination, R 2 = 0.97) for the two-stream instability, and 0.135 (R 2 = 0.98) for the plasma oscillations. This linear dependence on g appears to be rather natural, as two-body collisions are a first-order effect (in g 1 ) in a BBGKY hierarchy, where the mean-field Vlasov approximation holds at the lowest order (g 0 ). We will address shortly the discrepancy between the collision rate observed for the two-stream instability and for plasma waves. The comparison with the standard collision rates published in the literature is not straightforward, because usual estimations are done for realistic 3D plasmas, whereas our model is 1D.

Nevertheless we attempt a comparison in order to fix the ideas and check that the 1D results are not too far from reality.

For an electron with a velocity v passing throughout many ions, the electron-ion collision rate is [START_REF] Nishikawa | Plasma Physics[END_REF]21]:

ν ei single = ne 4 4πϵ 2 0 m 1/2 e (k B T e ) 3/2 ln Λ, ( 9
)
where n is the plasma density, ln Λ = ln (λ De /λ L ) is the Coulomb logarithm, and λ L is the smallest impact parameter -i.e., two-body collisions that lead to large deflection angles have been neglected in the calculation. λ L is equal to 2e 2 /(4πϵ 0 m e v 2 T e ), and therefore ln Λ = ln(2π/g). Nevertheless, we stress that the notions of impact parameter and Coulomb logarithm have no counterpart for 1D collisions, as approaching particles will always cross each other, and the electric force is constant rather that decreasing as the square of the distance.

Normalizing to the electron plasma frequency yields:

γ single ω pe = ν ei single ω pe = g 4π ln 2π g . (10)
The above collision rate is plotted on Fig. 8 (green line), and reproduces quite well the results for the two-stream instability. This seems rather natural, as Eq. ( 9) applies to electrons all having the same velocity, which is indeed the case for our monochromatic streams. Nevertheless, it should be noted that the expression of ν ei single given in Eq. ( 10) takes into account the Coulomb logarithm and its g ln g dependence on the coupling parameter g, whereas the blue and red lines in Fig. 8 are simply proportional to g. Although the order of magnitude given by Eq. ( 10) is in agreement with the collision rates measured from the N -body simulations, we emphasize that the comparison is not straightforward as the Coulomb logarithm has no counterpart for 1D collisions.

For the case of plasma oscillations, the starting distribution function is a Maxwellian. In that case, the electrons with velocities larger than v T e experience fewer collisions with the ions (as ν ei is proportional to 1/v 3 ). Hence, the collision rate has to be averaged over the electron distribution function [21], which yields [22]:

⟨ν ei ⟩ = 2 π ν ei single ≈ 0.8 ν ei single . (11)
A further correction factor to ⟨ν ei ⟩ should be included to take into account the distortion of the electronic distribution that appears in the presence of an electric field [21,[START_REF] Kuritsyn | [END_REF][24][25]. Electrons of different velocities respond differently to the combined effect of an electric field and collisions with ions. The faster electrons are less collisional and therefore are more effectively accelerated by the field, which modifies the shape of the distribution function. When all these effects are included, the net result is the Spitzer collisionality, about twice as small as the simple Maxwellian-averaged estimate:

γ Spitzer ≈ 0.5 ⟨ν ei ⟩ ≈ 0.4 ν ei single . (12)
The Spitzer rate, which takes into account the Maxwellian distribution and other corrections, should be more relevant to the collisional damping of plasma waves. Indeed, the ratio between the Spitzer and the single-particle collision rates γ Spitzer /γ single ≈ 0.4 is similar to the ratio of the observed collision rates for the two-body instability (monochromatic velocity distribution) and for the plasma waves (Maxwellian distribution): 0.135/0.326 = 0.41 (see Fig. 8).

To test the above ideas, a further series of numerical experiments was performed for plasma oscillations with cold monochromatic electrons. The slope of the damping rate as a function of g was found to be equal to 0.364 (R 2 = 0.99), close to the one obtained for the two-stream instability (0.326). Therefore, the same factor ≈ 0.4 is observed between the slope of the damping rate of plasma oscillations for cold monochromatic electrons and the slope of the damping rate of the same plasma oscillations for a Maxwellian distribution. This is a further hint that the different observed slopes are indeed due to the different types of velocity distributions used in the simulations.

To conclude, we underline that the usual 3D collision rates published in the literature, such as Eq. ( 10), appear to compare rather well with the rates observed in our simulations. Although the substantial difference between 1D and 3D plasmas precludes a definite conclusion from these numerical experiments, these results seem in contradiction with a recent theoretical analysis [4,5] where both collective and individual effects were incorporated from first principles. Further work will be necessary to elucidate this discrepancy.

VI. CONCLUSION

The purpose of this work was to provide an estimation of the collision rates by performing computer experiments with a 1D N -body plasma model. The advantage of this model is that it contains both collective and collisional effects, and that the relevant equations of motion can be integrated exactly (up to round-off errors). Therefore it provides an ideal tool to compare theoretical estimates with measured collision rates.

We implemented such comparison for three relevant cases involving basic plasma phenomena:

the two-stream instability, simple plasma waves at the plasma frequency, and Langmuir waves obeying the Bohm-Gross dispersion relation. When collisional effects were competing with either an instability or with collisionless Landau damping, the collision rate was obtained by simply subtracting the collisionless rate from the observed one.

The collision rates observed in all three cases were of comparable magnitude, pointing at an intrinsic collisional origin, independent on the details of the phenomena at play. The collision rate for the case of plasma oscillations was found to be slightly smaller (by a factor 0.4) compared to the collision rate for the two-stream instability. This discrepancy was explained in terms of the initial conditions used: Maxwellian distribution for the former case and monochromatic beams for the latter. Using the corrective factors provided in the literature, we could justify this small difference.

The observed collision rates were also well approximated by the standard "Spitzer-like" [24,25] rates usually published in the literature, in contrast to what was suggested by recent theoretical results in 3D [4,5]. Further work remains to be done to extend the present results to a more realistic 3D scenario.

FIG. 1 :

 1 FIG. 1: Schematic view of the 1D planar geometry of the simulations. Top panel: System of N electron and N ion planar sheets, confined between two walls separated by a distance L. Bottom panel: The total electric field has a step-like profile.

FIG. 2 :

 2 FIG. 2: Heap based algorithm, following [14]: Starting from the 2 n = 2N crossing times at the bottom of the heap, the crossing times are compared by pairs and the smallest one ends up at the top of the heap.

  FIG. 3: Measured linear instability growth rate 2γ N B plotted against the number of sheets N . The horizontal blue line corresponds to the expected value when collisions are neglected.

FIG. 4 :

 4 FIG. 4: Collision rate γ coll N B obtained from the N -body code for the two-stream instability, as a function of the number of sheets N .

FIG. 5 :

 5 FIG. 5: Electron (red) and ion (blue) phase-space distribution function (left panel) and electron velocity distribution (right panel) for v kick = 0.5 v T e .

FIG. 6 :

 6 FIG. 6: Measured collision rate γ N B plotted against the number of sheets N , in the case of plasma oscillations, for four different values of the kick (0.3, 0.5, 0.75 and 1.0 v T e ). In this case γ N B = γ coll N B because the mode k = 0 is considered (plasma oscillations with no Landau damping).

FIG. 7 :

 7 FIG. 7: Measured damping rates γ N B plotted against the number of sheets N , for Langmuir waves. The horizontal lines correspond to the Landau damping rates given by the Vlasov code COBBLES [16] and PIC code [19]. We have γ N B = γ coll N B + γ Landau . For large values of N , the collision rate becomes negligible.

FIG. 8 :

 8 FIG.8: Measured collision rates γ coll N B as a function of the coupling parameter g, in the case of the two-stream instability (blue) and plasma oscillations (in red). The solid lines correspond to the linear fits (γ coll N B ∝ g), with a slope equal to 0.326 for the two-stream instability, and 0.135 for plasma oscillations. The solid green line corresponds to Eq.(10).

Acknowledgments

The authors would like to thank Jean-Louis Rouet, Pierre Bertrand, and Xavier Caron for fruitful discussions.