Ruming Pan

Marcio Ferreira Martins

Gérald Debenest
email: gerald.debenest@toulouse-inp.fr

Pyrolysis of waste polyethylene in a semi-batch reactor to produce liquid fuel: Optimization of operating conditions

Keywords:

This study investigated the interactive effects of temperature, residence time, and carrier gas flow rate on the liquid fuel production through the pyrolysis of waste polyethylene (WPE) in a bench-scale semi-batch reactor. To enhance the liquid fuel production, fifteen experiments were conducted based on a central composite design. The adaptive neural fuzzy model was adopted to establish the relationship between liquid fuel production and operating conditions. The R-squared value of the experimental and adaptive neural fuzzy model predicted that liquid fuel production was 0.9934. Four additional experimental results verified the adaptive neural fuzzy model's applicability. Subsequently, the genetic algorithm (GA) was adopted to optimize operating conditions to maximize liquid fuel production. The GA optimized operating conditions (temperature, residence time, and carrier gas flow rate) were: 488 • C, 20 min, and 20 mL/min. The liquid fuel under the optimal operating conditions was analyzed by Fourier-transform infrared spectroscopy (FTIR) and gas chromatography-mass spectrometry (GC-MS). The liquid fuel had similar main functional groups as diesel. The components of the liquid fuel were mainly 1-alkenes and n-alkanes ranging from C7 to C36. The effects of operating conditions on liquid fuel fractions and mean molecular weight were also investigated.

Introduction

Municipal solid waste (MSW) is accumulating rapidly due to the enormous resource consumption and inefficient recycling worldwide. It is estimated that within 34 years (from 2016 to 2050), MSW will increase from 2.01 billion tons to 3.40 billion tons [START_REF] Kaza | What a Waste 2.0: A Global Snapshot of Solid Waste Management to[END_REF]. Plastic waste accounts for a large part of Municipal Solid Waste (MSW) due to its wide range of uses [START_REF] Quesada | Production of an alternative fuel by pyrolysis of plastic wastes mixtures[END_REF]. Moreover, the waste polyethylene (WPE) accounts for 40% of plastics in MSW [START_REF] Onwudili | Composition of products from the pyrolysis of polyethylene and polystyrene in a closed batch reactor: Effects of temperature and residence time[END_REF]. Also, about 70% of the total produced plastics have been directly discarded in the environment [START_REF] Geyer | Production, use and fate of all plastics ever made[END_REF]. These combinations of mismanagement have aggravated environmental pollution and endangered human health [START_REF] Crippa | A circular economy for plastics -Insights from research and innovation to inform policy and funding decisions[END_REF]. Therefore, the recycling of waste plastics, especially the WPE, needs to be further promoted.

It is reported that pyrolysis is an efficient way to recycle the WPE [START_REF] Sharuddin | A review on pyrolysis of plastic wastes[END_REF][START_REF] Pan | Investigating waste plastic pyrolysis kinetic parameters by genetic algorithm coupled with thermogravimetric analysis[END_REF]. The polymer is thermally decomposed into gas, liquid fuel, and char products in an oxygen-free atmosphere. Moreover, slow pyrolysis with low heating rates can enhance liquid fuel production [START_REF] Das | The effect of slow pyrolysis on the conversion of packaging waste plastics (PE and PP) into fuel[END_REF]. The thermal lag phenomenon can also be diminished by adopting low heating rates [START_REF] Ceamanos | Kinetics of pyrolysis of high density polyethylene. Comparison of isothermal and dynamic experiments[END_REF]. The uniform temperature distributions inside the reactor and the reactants can be achieved during slow pyrolysis [START_REF] Mazloum | Modelling plastic heating and melting in a semi-batch pyrolysis reactor[END_REF][START_REF] Antelava | A review on thermal and catalytic pyrolysis of plastic solid waste (PSW)[END_REF]. This is conducive to heat and mass transfer and liquid fuel production [START_REF] Das | The effect of slow pyrolysis on the conversion of packaging waste plastics (PE and PP) into fuel[END_REF][START_REF] Kunwar | Plastics to fuel: a review[END_REF]. The distribution of products is determined by the reactor type, the presence of catalysts, and operating conditions, such as temperature, residence time, and carrier gas flow rate [START_REF] Sharuddin | A review on pyrolysis of plastic wastes[END_REF]. Many studies on the thermal pyrolysis of polyethylene (PE) have been conducted in semibatch reactors. It has been reported that high liquid fuel yields can be obtained [START_REF] Manos | Catalytic degradation of high-density polyethylene on an ultrastable-Y Zeolite. Nature of initial polymer reactions, pattern of formation of gas and liquid products, and temperature effects[END_REF][START_REF] Manos | Tertiary recycling of polyethylene to hydrocarbon fuel by catalytic cracking over aluminum pillared clays[END_REF][START_REF] Quesada | Performance of different catalysts for the in situ cracking of the oil-waxes obtained by the pyrolysis of polyethylene film waste[END_REF][START_REF] Kassargy | Experimental study of catalytic pyrolysis of polyethylene and polypropylene over USY zeolite and separation to gasoline and diesel-like fuels[END_REF][START_REF] Sm | Thermal pyrolysis of high density polyethylene (HDPE) in a novel fixed bed reactor system for the production of high value gasoline range hydrocarbons (HC)[END_REF][START_REF] Yan | Thermal cracking of virgin and waste plastics of PP and LDPE in a semibatch reactor under atmospheric pressure[END_REF].

In terms of operating conditions, the temperature is the dominant parameter during plastics' pyrolysis [START_REF] Sharuddin | A review on pyrolysis of plastic wastes[END_REF]. Onwudili et al. [START_REF] Onwudili | Composition of products from the pyrolysis of polyethylene and polystyrene in a closed batch reactor: Effects of temperature and residence time[END_REF][START_REF] Insura | Catalytic pyrolysis of low-density polyethylene over alumina-supported noble metal catalysts[END_REF] reported that the virgin low-density PE was completely pyrolyzed at temperatures above 425 • C. The liquid fuel yield reached 89.5 wt% at 425 • C. They also found that liquid oil production dropped drastically as the temperature increased beyond 425 • C. Tiikma et al. [START_REF] Tiikma | Coprocessing of heavy shale oil with polyethylene waste[END_REF] stated that the optimal temperature for the liquid oil production from the pyrolysis of the WPE was 450 • C. Quesada et al. [START_REF] Quesada | Optimization of the pyrolysis process of a plastic waste to obtain a liquid fuel using different mathematical models[END_REF][START_REF] Quesada | Characterization of fuel produced by pyrolysis of plastic film obtained of municipal solid waste[END_REF] and Rodríguez-Luna et al. [START_REF] Rodríguez-Luna | Two-step pyrolysis for waste HDPE valorization[END_REF] investigated WPE and high-density PE thermal pyrolysis processes in a temperature range of 450-550 • C, respectively. They both reported that the optimal temperature for liquid fuel production was 500 • C. Sharuddin et al. [START_REF] Sharuddin | A review on pyrolysis of plastic wastes[END_REF] also concluded that temperatures below 500 • C were suitable for plastics' pyrolysis to produce liquid fuel.

According to [START_REF] Onwudili | Composition of products from the pyrolysis of polyethylene and polystyrene in a closed batch reactor: Effects of temperature and residence time[END_REF], the residence time was related to the experiment's duration at the target temperature. It has been reported that the residence time is also a critical factor in determining the composition of pyrolysis products [START_REF] Onwudili | Composition of products from the pyrolysis of polyethylene and polystyrene in a closed batch reactor: Effects of temperature and residence time[END_REF][START_REF] Quesada | Optimization of the pyrolysis process of a plastic waste to obtain a liquid fuel using different mathematical models[END_REF]. Quesada et al. [START_REF] Quesada | Optimization of the pyrolysis process of a plastic waste to obtain a liquid fuel using different mathematical models[END_REF][START_REF] Quesada | Characterization of fuel produced by pyrolysis of plastic film obtained of municipal solid waste[END_REF] found that a long residence time could enhance the liquid fuel yield. Muhammad et al. [START_REF] Muhammad | Catalytic degradation of linear low-density polyethylene over HY-zeolite via pre-degradation method[END_REF] investigated the effect of the carrier gas flow rate on PE's pyrolysis in a 200 mL bench-scale semi-batch reactor with nitrogen as the carrier gas. They investigated the production changes of liquid fuel and char with gas flow rates in the range of 0-60 mL/min. Also, they found that the carrier gas flow rate could also determine the distribution of thermal pyrolysis products of PE. A higher carrier gas flow rate would increase the liquid fuel yield and reduce the char yield.

In short, on the one hand, many research works have been done to investigate the effects of the operating conditions (temperature, residence time, and carrier gas flow rate) on the liquid fuel yield through the pyrolysis of PE; On the other, most of them investigated such effects by varying operating conditions one-by-one. However, according to [START_REF] Quesada | Optimization of the pyrolysis process of a plastic waste to obtain a liquid fuel using different mathematical models[END_REF], the operating conditions have complex interactive effects on liquid fuel production, which requires establishing a multiparameter mathematical expression for evaluating the yield of liquid fuel. For these reasons, oneby-one relationship interaction makes the optimal operating conditions determined in such a way not particularly convincing.

The adaptive neural fuzzy model [START_REF] Karaboga | Adaptive network based fuzzy inference system (ANFIS) training approaches: A comprehensive survey[END_REF][START_REF] Jang | ANFIS: adaptive-network-based fuzzy inference system[END_REF] is a suitable mathematical method that can set up relations between attributive variables and multiple arguments. Paramasivam [START_REF] Paramasivam | Fuzzy prediction and RSM optimization of CI engine performance analysis: aegle marmelos non-edible seed cake pyrolysis oil as a diesel alternative[END_REF] adopted the adaptive neural fuzzy model to analyze the CI engine performance through engine load and fuel mixture ratio. High accuracy was obtained between the experimental, and the adaptive neural fuzzy model predicted results. Pan et al. [START_REF] Pan | Application of a neural fuzzy model combined with simulated annealing algorithm to predict optimal conditions for polyethylene waste non-isothermal pyrolysis[END_REF] established the mathematical relationship between the PE pyrolysis rate and the operating conditions (temperature and heating rate). The high R-squared value (>0.999) between the experimental and predicted values exhibited the adaptive neural fuzzy model's reliability. Dubdub et al. [START_REF] Al-Yaari | Application of artificial neural networks to predict the catalytic pyrolysis of HDPE using non-isothermal TGA data[END_REF] utilized the adaptive neural fuzzy model to conduct the thermogravimetric modeling of high-density PE catalytic pyrolysis. A good agreement between the experimental and predicted data was also obtained. Quesada et al. [START_REF] Quesada | Optimization of the pyrolysis process of a plastic waste to obtain a liquid fuel using different mathematical models[END_REF] adopted different mathematical models to establish the relationships between the yields of plastic waste pyrolysis products and operating conditions. They concluded that the adaptive neural fuzzy model predicted results were more accurate and reliable.

In this perspective, this study aims to comprehensively investigate the interactive effects of temperature, residence time, and carrier gas flow rate on the liquid fuel production through the slow pyrolysis (heating rate of 6 • C/min) [START_REF] Fakhrhoseini | Predicting pyrolysis products of PE, PP, and PET using NRTL activity coefficient model[END_REF] of the WPE in a bench-scale semi-batch reactor. The adaptive neural fuzzy model [START_REF] Karaboga | Adaptive network based fuzzy inference system (ANFIS) training approaches: A comprehensive survey[END_REF][START_REF] Jang | ANFIS: adaptive-network-based fuzzy inference system[END_REF] was adopted to determine the liquid oil production by the operating conditions (temperature, residence time, and carrier gas flow rate). The genetic algorithm (GA) is a promising method to determine the extremums of complicated functions [START_REF] Pan | Investigating waste plastic pyrolysis kinetic parameters by genetic algorithm coupled with thermogravimetric analysis[END_REF][START_REF] Ighose | Optimization of biodiesel production from Thevetia peruviana seed oil by adaptive neuro-fuzzy inference system coupled with genetic algorithm and response surface methodology[END_REF]. Due to the complex expressions established by the adaptive neural fuzzy model, GA was exploited to ascertain the maximum liquid fuel production's operating conditions. The adaptive neural fuzzy model was also used to investigate the effects of operating conditions on the gas yield through thermal pyrolysis of WPE. The pyrolysis liquid fuel under the optimal operating conditions was analyzed by Fourier-transform infrared spectroscopy (FTIR) and gas chromatography-mass spectrometry (GC-MS). Lastly, the effects of operating conditions on liquid fuel fractions and mean molecular weight were also investigated.

Material and methods

Material

WPE was provided from Wanbei Plastic Recycling Development Base in Anhui Province, China. It was recycled from MSW and cut into approximately 3 mm pellets.

Experimental setup

Fig. 1 illustrates the experimental schematic diagram of the pyrolysis of WPE for liquid fuel production. The reactor is a 200 mL bench-scale semi-batch reactor. The WPE weighing approximately 5 g was used in each experiment. The reactor was purged with nitrogen at a 100 mL/min flow rate for 30 min before each experiment to ensure an inert atmosphere, and then nitrogen flow was adjusted to the target flow rate. The internal pressure of the reactor was maintained at 0.1 MPa during the whole experiment. The reactor was heated from room temperature (20 • C) to the target temperature at a heating rate of 6 • C/min [START_REF] Fakhrhoseini | Predicting pyrolysis products of PE, PP, and PET using NRTL activity coefficient model[END_REF] in each experiment. Subsequently, the reactor was maintained at the target temperature for the specified duration (the residence time).

WPE was pyrolyzed into volatilized gas (liquid fuel and gas). The volatilized gas was purged out of the reactor by the carrier gas through the outlet pipe. The outer wall of the outlet pipe is equipped with metal cooling fins. The volatilized gas can be cooled to about 50 • C through the outlet pipe. Part of volatilized gas was condensed into liquid fuel during this process. The liquid fuel flowed through the rubber tube [START_REF] Muhammad | Catalytic degradation of linear low-density polyethylene over HY-zeolite via pre-degradation method[END_REF] into the glass bottle. The rest volatilized gas was condensed into a liquid by the ice-water mixture (0 • C) [START_REF] Rodríguez-Luna | Two-step pyrolysis for waste HDPE valorization[END_REF][START_REF] Muhammad | Catalytic degradation of linear low-density polyethylene over HY-zeolite via pre-degradation method[END_REF][START_REF] Das | Valorization of packaging plastic waste by slow pyrolysis[END_REF] and collected by two in-sequence

Results and discussion

Liquid fuel production

Accuracy of the adaptive neural fuzzy model

Fig. 3 illustrates the experimental and the adaptive neural fuzzy model predicted liquid fuel productions under different operating conditions. It is worth noting that liquid fuel's appearance resembles wax at room temperature [START_REF] Quesada | Optimization of the pyrolysis process of a plastic waste to obtain a liquid fuel using different mathematical models[END_REF][START_REF] Quesada | Characterization of fuel produced by pyrolysis of plastic film obtained of municipal solid waste[END_REF].

Fig. 3a shows that the adaptive neural fuzzy model predicted liquid fuel production results were close to the experimental ones (E1-E15). The absolute relative errors between the predicted and the experimental values were within 1.3%. Moreover, the R-squared value between the experimental and the adaptive neural fuzzy model predicted liquid fuel productions was 0.9934. Fig. 3b exhibits the adaptive neural fuzzy model's applicability through the four additional experiments (V1-V4). The absolute relative errors between the predicted and the experimental values were within 1.4%. It reveals that the adaptive neural fuzzy model predicted liquid fuel production is accurate and reliable. [3] obtained 89.5 wt% liquid fuel production at 425 • C in a batch reactor. However, Quesada et al. [START_REF] Quesada | Optimization of the pyrolysis process of a plastic waste to obtain a liquid fuel using different mathematical models[END_REF] obtained relatively lower liquid fuel production through the pyrolysis of WPE at 450 • C, which varied from 13.61 wt% to 48.38 wt%. This is because that they conducted the experiments under higher heating rates (20-50 • C/min).

Interactive effects of residence time and carrier gas flow rate on liquid fuel production

As depicted in Fig. 4a, a higher carrier gas flow rate could increase liquid fuel production under a shorter residence time at 425 • C. For instance, the liquid fuel production was increased from 65.32 wt% to 73.54 wt% when the carrier gas flow rate varied from 20 mL/min to 100 mL/min under residence time of 20 min. Muhammad et al. [START_REF] Muhammad | Catalytic degradation of linear low-density polyethylene over HY-zeolite via pre-degradation method[END_REF] investigated liquid fuel production by the thermal pyrolysis of linear lowdensity PE at 450-460 • C. The liquid fuel production increased from 45.0 wt% to 75.0 wt% when the carrier gas flow rate varied from 0 to 60 mL/min. This is consistent with the results of this study. The increase in liquid fuel production could be attributed to the higher carrier gas flow rate, which can quickly carry volatile products out of the reactor, thereby inhibiting the secondary reactions that consume liquid fuel [START_REF] Xu | Effects of operating parameters on products yield and volatiles composition during fast pyrolysis of References food waste in the presence of hydrogen[END_REF]. However, the liquid fuel production was decreased with the increasing carrier gas flow rate under a longer residence time at 425 • C. For example, the liquid fuel production was decreased from 82.43 wt% to 77.81 wt% when the carrier gas flow rate increased from 20 mL/min to 100 mL/min under residence time of 60 min. The reduction of liquid fuel production could be attributed to that the higher carrier gas flow rate inhibited polycondensation and repolymerization reactions of the pyrolysis gas for liquid fuel formation [START_REF] Heidari | Effect of process conditions on product yield and composition of fast pyrol ys is of Eucal yp tus grandis in fluidized bed reactor[END_REF]. As for the liquid fuel production at higher temperatures, as shown in Fig. 4b andc, increasing the carrier gas flow rate would inhibit the liquid fuel production under the residence time ranging from 20 min to 60 min. This is because higher temperatures can promote the liquid fuel's secondary cracking to generate the shorter-chain pyrolysis gas [START_REF] Wei | Nitrogen evolution during fast pyrolysis of sewage sludge wider inert and reductive abnospheres[END_REF].

Interactive effects of temperature and carrier gas flow rate on liquid fuel production

Fig. 5a-c show the interactive effects of the temperature and the carrier gas flow rate on liquid fuel production under residence times of 20 min, 40 min, and 60 min, respectively. The liquid fuel production had the same variation tendency under a higher carrier gas flow rate regardless of the residence time changes. The liquid fuel productions were increased when the temperature increased from 425 • C to 525 • C for all residence times. The liquid fuel production was increased from 73.54 wt% to 80.65 wt% (20 min), from 75.53 wt% to 81.02 wt% (40 min), and from 77.81 wt% to 81.45 wt% (60 min) when temperature increased from 425 • C to 525 • C, respectively. The increase in the liquid fuel production could be attributed to more intense random scission reactions of WPE at higher temperatures [START_REF] Antelava | A review on thermal and catalytic pyrolysis of plastic solid waste (PSW)[END_REF]. The temperature has a more complex influence on liquid fuel production under the lowest carrier gas flow rate (20 mL/min). This is due to the interaction between the random scission reactions of WPE and the liquid fuel's secondary cracking reactions [START_REF] Xu | Effects of operating parameters on products yield and volatiles composition during fast pyrolysis of References food waste in the presence of hydrogen[END_REF]. As illustrated in Fig. 5a, the liquid fuel production was firstly increased from 65.32 wt% to 83.63 wt% when temperature increased from 425 • C to 488 • C under residence time of 20 min. The increase in temperature had a greater impact on promoting random scission reactions of WPE in this temperature range. Therefore, the liquid fuel production increased with the increasing temperature. However, liquid fuel's secondary cracking reactions hold a dominant position in the higher temperature range. The liquid fuel production was then decreased from 83.63 wt% to 82.73 wt% when temperature increased from 488 • C to 525 • C under residence time of 20 min. While the changing trend of liquid fuel production under residence time of 60 min, as shown in Fig. 5c, was a reversal from the one under residence time of 20 min. It indicated that the dominance of the random scission reactions of WPE and the secondary cracking reactions of the liquid fuel had been reversed under the longest residence time (60 min).

Interactive effects of temperature and residence time on liquid fuel production

Fig. 6a-c display the interactive effects of the temperature and the residence time on liquid fuel production under carrier gas flow rates of 20 mL/min, 60 mL/min, and 100 mL/min, respectively. The temperature has a more significant impact on liquid fuel production when the residence time is shorter under all carries gas flow rates. The residence time has less impact at higher temperatures [START_REF] Quesada | Optimization of the pyrolysis process of a plastic waste to obtain a liquid fuel using different mathematical models[END_REF]. As depicted in Fig. 6c, the liquid fuel production was increased with the increasing residence time regardless of the temperature. WPE could be fully decomposed into short-chain hydrocarbons with longer residence time through random scission reactions in the reactor [START_REF] Rodríguez-Luna | Two-step pyrolysis for waste HDPE valorization[END_REF]. Therefore, compared with the shorter residence time, the liquid fuel production was higher under longer residence time. The thermochemical conversion is an endothermic reaction, thereby increasing temperature is conducive to the thermal pyrolysis of WPE [START_REF] Morato-Godino | Pyrolysis of Cynara cardunculus L samples -effect of operating conditions and bed stage on the evolution of the conversion[END_REF]. Thus, a higher temperature could improve liquid fuel production. It is noteworthy that liquid fuel production was decreased with the temperature above 500 • C under carries gas flow rate of 20 mL/min in the lower range of the residence time. Sharuddin et al. [START_REF] Sharuddin | A review on pyrolysis of plastic wastes[END_REF] also suggested that temperatures below 500 • C are more conducive to liquid fuel production. It could be attributed to that the liquid fuel was further decomposed into low-molecular pyrolysis gas through β cleavage reactions at temperatures over 500 • C [START_REF] Cai | Pyrolysis-catal y--..is of different waste plastics over Fe/ Al203 catal ys e High-value hydrogen, liquid fuels, carbon nanotubes and possible reaction mechanisms[END_REF][START_REF] Vollmer | Beyond mechanical recycling: Giving new life to plastic waste[END_REF].

Gas production 3.2.1. Accuracy of the adaptive neural fuzzy model

Fig. 7 depicts the experimental and the adaptive neural fuzzy model predicted gas production results under different operating conditions. As illustrated in Fig. 7a, the gas productions predicted by the adaptive neural fuzzy model were all close to the experimental ones (E1-E15). The absolute relative errors between the predicted and the experimental values were within 4.6%. Besides, the R-squared value between the experimental and the adaptive neural fuzzy model predicted gas production was 0.9719. The predicted gas production was relatively less accurate than the predicted liquid fuel production. This is because the gas production was calculated from the difference between the initial WPE mass and the masses of liquid fuel and residue. Errors in gas production were accumulated, and thereby gas production became more inaccurate [START_REF] Quesada | Optimization of the pyrolysis process of a plastic waste to obtain a liquid fuel using different mathematical models[END_REF]. Fig. 7b shows the applicability of the adaptive neural fuzzy model predicted gas production. The absolute relative errors between the predicted and the experimental values were within 6.8%. Quesada et al. [START_REF] Quesada | Optimization of the pyrolysis process of a plastic waste to obtain a liquid fuel using different mathematical models[END_REF] adopted the fuzzy neural model to predict the WPE thermal pyrolysis's energy efficiency. The maximum absolute relative error was approximately 6.4%. The errors of the predicted gas production were in a reasonable range in this study. Therefore, the adaptive neural fuzzy model is qualified to predict gas production. • C, respectively. These results were similar to the ones reported by [START_REF] Das | The effect of slow pyrolysis on the conversion of packaging waste plastics (PE and PP) into fuel[END_REF][START_REF] Das | Valorization of packaging plastic waste by slow pyrolysis[END_REF]. Gas productions of 16.58-22.53 wt% and 17.80-27.52 wt% were obtained through thermal pyrolysis of virgin low-density PE and high-density PE within the temperature ranging from 350 • C to 400 • C, respectively [START_REF] Das | Valorization of packaging plastic waste by slow pyrolysis[END_REF]. Also, approximately 17 wt% of gas was obtained at a temperature of 500 • C [START_REF] Das | The effect of slow pyrolysis on the conversion of packaging waste plastics (PE and PP) into fuel[END_REF].

Interactive effects of residence time and carrier gas flow rate on gas production

As demonstrated in Fig. 8a, a longer residence time would increase the gas production under the lowest (20 mL/min) and the highest (100 mL/min) carrier gas flow rates at 425 • C. When the residence time was increased from 20 min to 60 min, the gas production was increased from 11.50 to 14.17 wt% and 14.14-18.75 wt% under the carrier gas flow rates of 20 mL/min and 100 mL/min, respectively. Onwudili et al. [START_REF] Onwudili | Composition of products from the pyrolysis of polyethylene and polystyrene in a closed batch reactor: Effects of temperature and residence time[END_REF] also found that gas production was increased from 8.70 wt% to 16.30 wt % at 450 • C when the residence time increased from 0 min to 30 min. This is because the random scission reactions of WPE were not intense to produce pyrolysis gas at low temperatures [START_REF] Xu | Effects of operating parameters on products yield and volatiles composition during fast pyrolysis of References food waste in the presence of hydrogen[END_REF]. Extending the residence time could make the random scission reactions of WPE proceed more thoroughly to obtain higher gas yields. As shown in Fig. 8b, the carrier gas flow rate had a stronger impact on gas production under the lowest residence time (20 min) at 475 • C. The gas production was first increased from 12.40 wt% to 15.82 wt% when the carrier gas flow rate increased from 20 mL/min to 64 mL/min. The increase in gas production could be attributed to the inhibition of polycondensation and repolymerization reactions of the pyrolysis gas at low carrier gas flow rates [START_REF] Heidari | Effect of process conditions on product yield and composition of fast pyrol ys is of Eucal yp tus grandis in fluidized bed reactor[END_REF]. While the excessive carrier gas flow rate would reduce the heat transfer efficiency inside the reactor, thereby inhibiting the gas production [START_REF] Xu | Effects of operating parameters on products yield and volatiles composition during fast pyrolysis of References food waste in the presence of hydrogen[END_REF]. Thus, gas production decreased from 15.82 wt% to 14.72 wt% when the carrier gas flow rate varied from 64 mL/min to 100 mL/min. The thermal pyrolysis of linear low-density PE at 450-460 • C has a similar phenomenon [START_REF] Muhammad | Catalytic degradation of linear low-density polyethylene over HY-zeolite via pre-degradation method[END_REF]. The gas production was first increased from 15.5 wt% to 20.0 wt% (by difference) when the carrier gas flow rate increased from 0 mL/min to 30 mL/min. Then, the gas production decreased from 20.0 wt% to 17.5 wt% when the carrier gas flow rate varied from 30 mL/min to 60 mL/min. At 525 • C, as depicted in Fig. 8c, a longer residence time would increase gas production under the lowest carrier gas flow rate (20 mL/min). The gas production was increased from 13.54 wt% to 14.49 wt% when the residence time increased from 20 min to 60 min. This is because that longer residence time enhanced the possibility of β cleavage reactions for pyrolysis gas formation [START_REF] Cai | Pyrolysis-catal y--..is of different waste plastics over Fe/ Al203 catal ys e High-value hydrogen, liquid fuels, carbon nanotubes and possible reaction mechanisms[END_REF].

Interactive effects of temperature and carrier gas flow rate on gas production

Fig. 9a-c present the interactive effects of the temperature and the carrier gas flow rate on gas production under residence times of 20 min, 40 min, and 60 min, respectively. The carrier gas flow rate had a stronger influence on gas production at the lowest temperature (425 • C) regardless of residence time changes. As illustrated in Fig. 9a, the temperature had a similar influence on gas production under the lowest (20 mL/min) and the highest (100 mL/min) carrier gas flow rates. The gas productions were both increased with the increasing temperature. The liquid fuel's secondary cracking reactions were more intense to generate pyrolysis gas at higher temperatures [START_REF] Xu | Effects of operating parameters on products yield and volatiles composition during fast pyrolysis of References food waste in the presence of hydrogen[END_REF]. However, the temperature had an opposite impact on the gas production under the medium (60 mL/min) carrier gas flow rate. The gas production was decreased from 16.17 wt% to 15.30 wt% when the temperature increased from 425 • C to 525 • C. The gas production was also decreased from 55.46 wt% to 32.63 wt% when the temperature varied from 450 • C to 550 • C in [START_REF] Quesada | Optimization of the pyrolysis process of a plastic waste to obtain a liquid fuel using different mathematical models[END_REF]. The increase in temperature promoted polycondensation and repolymerization reactions of the pyrolysis gas, resulting in a reduction in gas production. It also suggested that the polycondensation and repolymerization reactions of the pyrolysis gas played a dominant role, compared to the secondary cracking reactions of the liquid fuel, under the medium carrier gas flow rate (60 mL/min). As depicted in Fig. 9b andc, the temperature and the carrier gas flow rate had the same influences on gas production under the residence times of 40 min and 60 min. The lowest gas productions were obtained under the same operating conditions (temperature of 425 • C and carrier gas flow rate of 20 mL/min). While the highest gas productions were both obtained under 425 • C and 100 mL/min. It indicated that a higher carrier gas flow rate could suppress the polycondensation and repolymerization reactions for the pyrolysis gas consumption under the residence times of 40 min and 60 min.

Interactive effects of temperature and residence time on gas production

Fig. 10a-c demonstrate the interactive effects of the temperature and the residence time on gas production under carrier gas flow rates of 20 mL/min, 60 mL/min, and 100 mL/min, respectively. As shown in Fig. 10a, the temperature could enhance gas production under the carrier gas flow rate of 20 mL/min regardless of the residence time changes. As depicted in Fig. 10b andc, the interactive effects of the temperature and the residence time became more complicated on gas production under the carrier gas flow rates of 60 mL/min and 100 mL/min. For example, as demonstrated in Fig. 10b, the gas production was decreased from 16.17 wt% to 14.71 wt% when the residence time increased from 20 min to 60 min at 425 • C and 60 mL/min. The reason for the decrease in gas production was the promotion of the pyrolysis gas polymerization reactions at a longer residence time. Concurrently, the gas production was increased from 15.30 wt% to 16.35 wt% when the residence time varied from 20 min to 60 min at 525 • C and 60 mL/min. The longer residence time enhanced the possibility of char gasification and β cleavage reactions for pyrolysis gas formation [START_REF] Xu | Effects of operating parameters on products yield and volatiles composition during fast pyrolysis of References food waste in the presence of hydrogen[END_REF]. As depicted in Fig. 10c, the residence time's influence on gas production was opposite under 100 mL/min, compared with the results under 60 mL/min. Longer residence time would enhance the gas production at 425 • C; while it would inhibit the gas production at 525 • C.

Optimization of operating conditions by GA

As discussed in Section 3.1, the operating conditions (temperature, residence time, and carrier gas flow rate) had very complex interactive effects on liquid fuel production. To obtain the maximum liquid fuel production, GA was adopted to determine the optimal operating conditions. Fig. 11a exhibits the variations of the average and the optimal values of 1000 individuals' fitness in 1000 iterations. In this study, fitness was liquid fuel production. The optimal value reached stability after 100 iterations, while the average value was stabled after 800 iterations.

Fig. 11b shows the maximum liquid fuel production under the optimal operating conditions. The GA optimized operating conditions were 488 • C, 20 min, and 20 mL/min. The optimal liquid fuel production was 83.63 wt%. Rodríguez-Luna et al. [START_REF] Rodríguez-Luna | Two-step pyrolysis for waste HDPE valorization[END_REF] reported 500 • C was the most suitable temperature for pyrolysis of high-density PE to produce liquid fuel in a semi-batch reactor. Quesada et al. [START_REF] Quesada | Optimization of the pyrolysis process of a plastic waste to obtain a liquid fuel using different mathematical models[END_REF] concluded that the optimal operating conditions for liquid fuel production were 500 • C and 120 min. The experiments were conducted in a horizontal tubular reactor. The optimized temperature was close to the one in this study. While the optimized residence time was much longer than the one in this study. This is because the experiments in [START_REF] Quesada | Optimization of the pyrolysis process of a plastic waste to obtain a liquid fuel using different mathematical models[END_REF] were conducted under higher heating rates (20-50 • C/min) and a faster carrier gas flow rate (833 mL/min). Sharuddin et al. [START_REF] Sharuddin | A review on pyrolysis of plastic wastes[END_REF] also concluded that temperatures below 500 • C were suitable for liquid fuel production.

The experiment under GA optimized operating conditions (488 • C, 20 min, and 20 mL/min) was conducted to verify the GA predicted results. The experimental liquid fuel production was 83.50 ± 0.59 wt%. The absolute relative error between the predicted and the experimental values was within 0.16%. It suggests that the GA optimized results were accurate and reliable.

FTIR analysis

Fig. 12 shows the FTIR results of liquid fuels from this study and Quesada et al. [START_REF] Quesada | Characterization of fuel produced by pyrolysis of plastic film obtained of municipal solid waste[END_REF]. The main functional groups of liquid fuels do not change under different operating conditions [START_REF] Quesada | Characterization of fuel produced by pyrolysis of plastic film obtained of municipal solid waste[END_REF]. Therefore, the liquid fuel under the optimal operating conditions (488 • C, 20 min, and 20 mL/ min) was chosen to conduct the FTIR analysis. As illustrated in Fig. 12, the following functional groups in liquid fuel of this study were determined: C-H stretch at 2916-2848 cm -1 [START_REF] Ghenai | Design of hybrid solar photovoltaics/shrouded wind turbine power systein for thermal pyrolysis of plastic waste[END_REF]; C C stretching at 1642 cm -1 and 1462 cm -1 ; C-H scissor and bend at 1377 cm -1 ; C-H out of the plane bend at 909 cm -1 ; and C-H bend at 719 cm -1 . The linear alkanes were generated by intermolecular hydrogen transfer reactions [START_REF] Mccaffrey | Tertiary recycling of pol ye thy!ene: mechanism of liquid production from pol ye thylene by thermol y--..is/reactive distillation[END_REF]. While the β cleavage coupled with the intramolecular hydrogen transfer reactions are responsible for the alkenes yields during the pyrolysis of WPE [START_REF] Rodríguez-Luna | Two-step pyrolysis for waste HDPE valorization[END_REF].

Fig. 12 also demonstrates the FTIR results of liquid fuel from Quesada et al. [START_REF] Quesada | Characterization of fuel produced by pyrolysis of plastic film obtained of municipal solid waste[END_REF] and diesel. The liquid fuel sample in [START_REF] Quesada | Characterization of fuel produced by pyrolysis of plastic film obtained of municipal solid waste[END_REF] was obtained under 500 • C, 80 min, and 833 mL/min. The liquid fuel from this study had the same characteristic peaks like the one in [START_REF] Quesada | Characterization of fuel produced by pyrolysis of plastic film obtained of municipal solid waste[END_REF]. It also suggests that the operating conditions do not change the main functional groups of WPE thermal pyrolysis liquid fuels. Besides, the liquid fuel from this study had similar characteristic peaks as diesel.

GC-MS analysis

The GC-MS analysis was conducted to determine the specific components of the liquid fuel. Fig. 13 demonstrates the liquid fuel chromatogram under the optimal operating conditions (488 • C, 20 min, and 20 mL/min). The identified compounds present in the liquid fuel under the optimal operating conditions are tabulated in Table 2. The components of the liquid fuel were mainly 1-alkenes and n-alkanes ranging from C7 to C36. The liquid fuel's mean molecular weight was 291 g/mol. respectively. The liquid fuels E1-15 had the same component types as the liquid fuel under the optimal operating conditions (mostly 1-alkenes and n-alkanes ranging from C7 to C36). However, the specific component proportions of the liquid fuels were different. This indicated that the operating conditions had effects on the composition of the liquid fuel produced by the thermal pyrolysis of waste polyethylene. The liquid fuel was classified into light (C 7 -C 11 ), middle (C 12 -C 20 ) and heavy (C 21 -C 36 ) fractions [START_REF] Das | The effect of slow pyrolysis on the conversion of packaging waste plastics (PE and PP) into fuel[END_REF]. Fig. 14 shows the liquid fuel fractions and mean molecular weight under different operating conditions. The light, middle and heavy fractions varied from 3.61 to 6.79%, 28.65-42.54% and 50.72-66.83%, respectively. Moreover, the liquid fuel's mean molecular weight was varied from 291.00 g/mol to 325.23 g/mol. Fig. 15 demonstrates the effects of operating conditions on liquid fuel fractions and mean molecular weight. Samples of E3, E8 and E13 were taken into consideration to analyze the effect of temperature on the liquid fuel composition. As shown in Fig. 15a, the light and middle fractions were decreased from 6.74 to 4.25% and 42.54-32.22% when the temperature was increased from 425 • C to 525 • C, respectively. In comparison, the heavy fraction was increased from 50.72% to 63.53% when the temperature was increased from 425 • C to 525 • C. Besides, the liquid fuel's mean molecular weight was also increased from 291.78 g/ mol to 313.96 g/mol. This indicates that high temperature is conducive to the formation of the heavy fraction in the liquid fuel. It can be ascribed to that higher temperature would enhance the possibility of secondary reactions of liquid fuel's light and middle fractions for pyrolysis gas formation [START_REF] Das | The effect of slow pyrolysis on the conversion of packaging waste plastics (PE and PP) into fuel[END_REF].

Fig. 15b illustrates the effect of residence time on liquid fuel fractions and mean molecular weight (E6, E8 and E10). The middle fraction was decreased from 38.18% to 32.02%, whereas the heavy fraction was increased from 56.80% to 63.73% when residence time was increased from 20 min to 60 min. The light fraction was firstly increased from 5.02% to 6.73% when residence time was increased from 20 min to 40 min. The light fraction was then decreased from 6.73% to 4.25% when residence time varied from 40 min to 60 min. The longer residence time enhanced the possibility of β cleavage reactions of liquid fuel's light and middle fractions [START_REF] Das | The effect of slow pyrolysis on the conversion of packaging waste plastics (PE and PP) into fuel[END_REF][START_REF] Xu | Effects of operating parameters on products yield and volatiles composition during fast pyrolysis of References food waste in the presence of hydrogen[END_REF]. Therefore, the liquid fuel's mean molecular weight was increased from 301.54 g/mol to 319.73 g/mol when residence time was increased from 20 min to 60 min. Fig. 15c shows the effect of carrier gas flow rate on liquid fuel fractions and mean molecular weight (E7, E8 and E9). The light, middle and heavy fractions varied from 3.61 to 6.73%, 29.57-33.42% and 59.86-66.82%, respectively. Moreover, the liquid fuel's mean molecular weight was increased from 309.02 g/mol to 325.23 g/mol when the carrier gas flow rate increased from 20 mL/min to 100 mL/min. The increase in liquid fuel's mean molecular weight could be attributed to the higher carrier gas flow rate, which can quickly carry volatile products out of the reactor, thereby inhibiting β cleavage reactions of liquid fuel's heavy fraction [START_REF] Xu | Effects of operating parameters on products yield and volatiles composition during fast pyrolysis of References food waste in the presence of hydrogen[END_REF] and polycondensation and repolymerization reactions of the pyrolysis gas [START_REF] Heidari | Effect of process conditions on product yield and composition of fast pyrol ys is of Eucal yp tus grandis in fluidized bed reactor[END_REF] for the light and middle fractions' liquid fuel formation.

Conclusion

This study aims to obtain the optimal operating conditions for liquid fuel production through thermal pyrolysis of waste polyethylene in a bench-scale semi-batch reactor. Three operating conditions were considered, i.e., the temperature, the residence time, and the carrier gas flow rate. The adaptive neural fuzzy model comprehensively described the interactive effects of operating conditions on liquid fuel production. The R-squared value between the experimental and the adaptive neural fuzzy model predicted liquid fuel production was 0.9934. It revealed that the adaptive neural fuzzy model predicted liquid fuel production was accurate and reliable.

Subsequently, the genetic algorithm (GA) was adopted to optimize the operating conditions to maximize liquid fuel production. The GA optimized liquid fuel production was 83.63 wt% under the operating conditions of 488 • C (temperature), 20 min (residence time) and 20 mL/ min (carrier gas flow rate). The experimental liquid fuel production was 83.50 wt% under these operating conditions, a value very close to the

Fig. 1 .

 1 Fig. 1. The experimental schematic diagram of the pyrolysis of WPE for liquid fuel production.

  Fig. 4a-c demonstrate the interactive effects of the residence time and the carrier gas flow rate on liquid fuel production at temperatures of 425 • C, 475 • C, and 525 • C, respectively. The liquid fuel production varied from 65.32 wt% to 82.43 wt% at 425 • C; from 79.05 wt% to 83.00 wt% at 475 • C; and from 80.65 wt% to 82.73 wt% at 525 • C, respectively. Das and Tiwari [32] obtained around 81.4 wt% liquid fuel through thermal pyrolysis of virgin PE under the operating conditions of 400 • C, 8 h and 200 mL/min in a 1000 mL semi-batch reactor. They also obtained around 82.7 wt% liquid fuel under the operating conditions of 500 • C, 30 min, and 100 mL/min in the same reactor [8]. Onwudili et al.

Fig. 3 .

 3 Fig. 3. The experimental and the adaptive neural fuzzy model predicted liquid fuel productions: (a) Training; (b) Testing.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Interactive effects of residence time and carrier gas flow rate on liquid fuel production at different temperatures: (a) 425 • C; (b) 475 • C; (c) 525 • C.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Interactive effects of temperature and residence time on liquid fuel production under different carrier gas flow rates: (a) 20 mL/min; (b) 60 mL/min; (c) 100 mL/min.

  Fig. 8a-c illustrate the interactive effects of the residence time and the carrier gas flow rate on gas production at temperatures of 425 • C, 475 • C, and 525 • C, respectively. The gas production varied from 11.50 wt% to 18.75 wt% at 425 • C; from 12.40 wt% to 16.93 wt% at 475 • C; and from 13.54 wt% to 16.35 wt% at 525 • C, respectively. Onwudili et al. [3] obtained 10 wt% and 25 wt% gas productions through thermal pyrolysis of low-density PE in a batch reactor at temperatures of 425 • C and 450

Fig. 8 .Fig. 9 .

 89 Fig. 8. Interactive effects of residence time and carrier gas flow rate on gas production at different temperatures: (a) 425 • C; (b) 475 • C; (c) 525 • C.

Fig. 10 .

 10 Fig. 10. Interactive effects of temperature and residence time on gas production under different carrier gas flow rates: (a) 20 mL/min; (b) 60 mL/min; (c) 100 mL/min.

Fig. 11 .

 11 Fig. 11. Illustrations of optimized operating conditions by GA: (a) Optimization process; (b) Liquid fuel production under optimal operating conditions.

Fig. 12 .

 12 Fig.12. FTIR analysis of liquid fuels from this study and Quesada et al.[START_REF] Quesada | Characterization of fuel produced by pyrolysis of plastic film obtained of municipal solid waste[END_REF], and diesel[START_REF] Quesada | Characterization of fuel produced by pyrolysis of plastic film obtained of municipal solid waste[END_REF].

Fig. A. 1 -

 1 [START_REF] Quesada | Performance of different catalysts for the in situ cracking of the oil-waxes obtained by the pyrolysis of polyethylene film waste[END_REF] illustrate the chromatograms of the liquid fuels E1-15,

Fig. 13 .

 13 Fig. 13. GC-MS analysis of the liquid fuel under the optimal operating conditions (488 • C, 20 min, and 20 mL/min).

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Liquid fuel fractions and mean molecular weight under different operating conditions.

Table 2

 2 Identified compounds present in the liquid fuel under the optimal operating conditions (488 • C, 20 min, and 20 mL/min).

	Peak	Time	Compound	Relative area	Molecular
		(min)		(%)	weight
	1	2.32	cyclopentane (C7)	0.71	98
	2	2.38	n-heptane (C7)	0.63	100
	3	2.76	1-octene (C8)	1.67	112
	4	2.98	n-octane (C8)	0.20	114
	5	3.31	1-nonene (C9)	1.41	126
	6	3.40	n-nonane (C9)	0.74	128
	7	4.66	1-decene (C10)	1.33	140
	8	4.79	n-decane (C10)	0.78	142
	9	6.19	1-undecene (C11)	1.51	154
	10	6.31	n-undecane (C11)	1.01	156
	11	7.70	1-dodecene (C12)	1.53	168
	12	7.82	n-dodecane (C12)	1.21	170
	13	9.15	1-tridecene (C13)	1.75	182
	14	9.26	n-tridecane (C13)	2.12	184
	15	10.50	1-tetradecene (C14)	2.09	196
	16	10.60	n-tetradecane (C14)	1.47	198
	17	11.78	1-pentadecene (C15)	2.18	210
	18	11.87	n-pentadecane (C15)	1.66	212
	19	12.99	1-hexadecene (C16)	2.45	224
	20	13.07	n-hexadecane (C16)	2.01	226
	21	14.13	1-heptadecene (C17)	2.62	238
	22	14.20	n-heptadecane (C17)	2.08	240
	23	15.21	1-octadecene (C18)	2.87	252
	24	15.28	n-octadecane (C18)	2.39	254
	25	16.24	1-nonadecene (C19)	2.74	266
	26	16.30	n-nonadecane (C19)	2.35	268
	27	17.21	1-eicosene (C20)	2.54	280
	28	17.27	n-eicosane (C20)	2.50	282
	29	18.15	1-heneicosene (C21)	2.66	294
	30	18.20	n-heneicosane (C21)	2.64	296
	31	19.04	1-docosene (C22)	2.62	308
	32	19.09	n-docosane (C22)	2.89	310
	33	19.90	1-tricosene (C23)	2.24	322
	34	19.94	n-tricosane (C23)	2.63	324
	35	20.79	1-tretacosene (C24)	2.10	336
	36	20.83	n-tretacosane (C24)	2.75	338
	37	21.83	1-pentacosene (C25)	1.93	350
	38	21.88	n-pentacosane (C25)	2.76	352
	39	23.11	1-hexacosene (C26)	1.66	364
	40	23.17	n-hexacosane (C26)	2.71	366
	41	24.71	1-heptacosene (C27)	1.22	378
	42	24.79	n-heptacosane (C27)	2.59	380
	43	26.75	1-octacosene (C28)	1.31	392
	44	26.84	n-octacosane (C28)	2.60	394
	45	29.36	1-nonacosene (29)	0.88	406
	46	29.47	n-nonacosane (29)	2.37	408
	47	31.56	1-triacontene (C30)	0.82	420
	48	31.60	n-triacontane (C30)	2.27	422
	49	33.60	n-hentriacontane (C31)	2.18	436
	50	34.64	n-dotriacontane (C32)	1.80	450
	51	35.81	n-tritriacontane (C33)	1.54	464
	52	37.16	n-tetratriacontane	1.00	478
			(C34)
	53	38.76	n-pentatriacontane	0.72	492
			(C35)
	54	40.67	n-hexatriacontane	0.58
			(C36)

Acknowledgements

This work was supported by the China Scholarship Council (CSC) program (No. 201906120036). Thanks are given to Dr. Yue ZAN from LCC-CNRS-UPS (Laboratoire de Chimie de Coordination du CNRS, Université Toulouse III -Paul Sabatier) for her assistance in the FTIR test and analysis.

GA predicted one. It indicated that the adaptive neural fuzzy model coupled with GA was qualified to optimize the operating conditions for liquid fuel production through thermal pyrolysis of waste polyethylene.

FTIR and GC-MS analyses were conducted to determine the liquid fuels' main functional groups and the chemical components. It was found that the operating conditions did not change the liquid fuels' main functional groups. The waste polyethylene thermal pyrolysis liquid fuels had similar characteristic peaks as commercial diesel. The liquid fuels had the same component types under different operating conditions (mostly 1-alkenes and n-alkanes ranging from C7 to C36). However, the specific component proportions of the liquid fuels varied with the operating conditions. It could be concluded that high temperature, long residence time and high carrier gas flow rate were conducive to formation of the liquid fuel's heavy fraction. While low temperature, short residence time and low carrier gas flow rate were beneficial to form light and middle fractions in liquid fuel.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi. org/10.1016/j.enconman.2021.114114.


Fabien Chirot

Pierre Feiden

Sébastien Zamith

Pierre Labastie

Jean-Marc L'hermite

A novel experimental method for the measurement of the caloric curves of clusters

Keywords: numbers: 36.40.-c, 34.10.+x, 82.60.Nh

A novel experimental scheme has been developed in order to measure the heat capacity of mass selected clusters. It is based on controlled sticking of atoms on clusters. This allows one to construct the caloric curve, thus determining the melting temperature and the latent heat of fusion in the case of first-order phase transitions. This method is model-free. It is transferable to many systems since the energy is brought to clusters through sticking collisions. As an example, it has been applied to N a + 90 and N a + 140 . Our results are in good agreement with previous measurements.

I. INTRODUCTION

Although melting is a universal property of bulk matter, no quantitative theory of this phenomenon is known. A possible way to tackle the problem is to study small systems, expecting that the number of degrees of freedom is low enough for the theory to be tractable. On the other hand, this makes comparison to experiments harder, since melting of very small particles cannot be seen directly. It has been shown however, that caloric curves could theoretically [1,2,3] and experimentally [4,5] be used to characterize the melting transition. Generally, the melting temperatures of clusters are much lower than in the bulk, but higher temperatures have been reported for tin [6] and for gallium [7]. The most thorough study on small clusters is that of Schmidt et al [4]. They found that the melting temperature of small sodium particles (in the range of a hundred atoms) is far from monotonous as a function of size. The latent heat per atom also varies considerably. Those variations originate from a complex interplay between electronic, geometric and entropic effects [5,8].

The general scheme for cluster calorimetry is the following. Clusters are produced at a given temperature T using a buffer gas as a heat bath. They are mass selected and then some energy E is added to the clusters in order to bring them to a known reference state. An increase δT of the temperature is compensated by a decrease δE of the energy needed to reach the same state. The heat capacity is then deduced as the ratio C(T ) = δE/δT . A peak in the curve C(T) is the signature of a phase transition. The maximum of this peak gives the melting temperature and its area is the latent heat [2].

In the existing experimental methods, the energy E is brought either by a laser [4] or by collisions [7]. The method of Ref. 4 is accurate but requires a laser excitation. If photoabsorbtion is very efficient for metallic clusters, this is not the case for all kind of nanoparticles: most of them often undergo direct photodissociation rather than heating. This drawback can be overcome by the collisional technique [7]. There is price to pay however because the determination of the amount of energy transferred by inelastic collisions relies on a model. Another method, which is not based on the determination of the caloric curve, has been used to extract melting temperatures [9]: one measures the mobility of clusters in a drift tube, from which collision cross sections are deduced. A variation of the cross section as a function of the temperature is the signature of the phase transition. This method is the only one able to detect a second-order phase transition. Nevertheless, it relies on a subtle effect that might be hard to detect. Since we can also measure collision cross sections in our experiment [10], we should also be able to detect the transition in a similar way, but preliminary measurements have not yet revealed any evidence of second order phase transition in sodium clusters.

This paper reports a new way to acquire caloric curves of mass selected clusters. Basically, as in previous experiments, we take advantage of the relation between evaporation rate and internal energy: the reference state, hence its internal energy, corresponds to a given evaporation rate. In our case the energy is brought by sticking collisions. In this way, unlike in Ref. 7, the energy is increased at each collision by a well defined quantity E c + D, where E c is the collision energy and D the dissociation energy. Moreover, as our method does not require laser excitation, it is easily transferable to many different systems. Furthermore, our method is parameter free: performing the experiment at two different collision energies allows us to get rid of the unknown dissociation energies in the final expression of the heat capacity. In this paper, we demonstrate the validity of our method using sodium clusters, in order to compare our results with those of Ref. 5. In the following, we first introduce the principle of the method used for extracting the caloric curves (Section II), then we briefly describe the experimental setup (Section III). The evolution of the number of sticking in the multicollision regime is discussed in section IV. Section V is devoted to a detailed description of the method whose validity and robustness is discussed with the help of Monte Carlo simulations in section VI. Finally, in section VII, we show the experimental caloric curves obtained for N a + 90 and N a + 140 and compare them to previous results.

II. PRINCIPLE OF THE METHOD

When performing calorimetry with nanoclusters, there is no way to measure the temperature after adding energy to the system, in contrast to macroscopic systems, where the temperature can be monitored as a function of added energy. Only the initial temperature can be fixed by a suitable preparation of the system. Another difficulty is related to the fact that measurements are performed on an assembly of clusters with some statistics involved. In order to illustrate the principle of the method, we neglect all statistical effects here. Further details are given in section V.

So we assume that all the clusters have initially a given internal energy E 0 (T ). The purpose of the experiment is to add a known amount of energy ∆E to the cluster such as to bring it to a well defined and recognizable state.

The energy E f = E 0 (T ) + ∆E of this state is not necessarily known. All we need is that E f does not depend on the partition between E 0 and ∆E. In our experiment, this final state is given by the condition that its lifetime is equal to the detection time in the experiment. Other experiments (see refs. 4, 7) use slight variations of this condition. The fact that this state is well defined is given by the theory of the evaporative ensemble [11]: since the lifetime with respect to evaporation depends exponentially on the energy, fixing the lifetime, even loosely, gives a well-defined value to the energy.

There are various ways to pour energy into the cluster. Haberland's group uses photons [4], Jarrold's uses inelastic collisions with a rare gas [7]. In our group, we use sticking collisions. In this case, the size n is increased by one at each collision and the added energy is D n+1 + E c , where D n is the dissociation energy of the cluster of size n and E c the collision energy in the center of mass frame (it is easily shown that the rotational energy can be neglected). Basically, the energy E f is reached when no more sticking collision is possible, that is, the maximum number of sticking collisions depends on the initial energy. Fig. 1 illustrates this schematically. The solid line represents the internal energy E 0 (T ) of the cluster as a function of its temperature. A first order phase transition shows up as a smooth jump in this curve (which gives a peak in the heat capacity). The vertical arrows in the figure represent the quantities D n + E c . E f is depicted as constant, although it depends slightly on the size reached after the stickings have occurred. As the initial temperature is increased, the cluster can stand less and less sticking. Five situations are depicted where the number of sticking evolves from 5 to 1. Before and after the phase transition, the number of sticking evolves slowly. But at the phase transition it evolves much more rapidly due to the rapid change in energy as the temperature increases.

Of course, the scheme presented on Fig. 1 is oversimplified. The main correction is that even if the initial energy is supposed to be well defined, the number of sticking collisions results from a statistical process, with Energy Temperature E f FIG. 1: (Color on line) Illustration of the principle of our experiment. The cluster energy is plotted as a function of the initial temperature. Each small vertical arrow represents the quantity Dn + Ec. E f is the energy such that the dissociation time equals the observation time in our experiment. As the initial temperature is increased, less and less sticking are observed. This decrease in the number of sticking evolves more rapidly as a phase transition occurs.

Poisson probability. Furthermore, it is impossible to rule out evaporation during the process. For the sake of simplicity however, let us assume for the present discussion that we can extract from the experiment a well defined "maximum number of sticking" n max for a given initial temperature. It is then theoretically possible to obtain the initial internal energy E 0 of a cluster of initial size j as:

E 0 = E f - nmax i=1 (D j+i + E c ) (1)
This equation cannot be used directly however, because neither E f nor the D n 's are known. We get rid of this problem by using a differential method. The basic idea is to measure the number of sticking n max (T, E c1 ) and n max (T, E c2 ) as a function of the initial temperature T for two different collision energies E c1 and E c2 , and to find two temperatures T 1 and T 2 such that n max (T 1 , E c1 ) = n max (T 2 , E c2 ) (see Fig. 2). Then, E f and the D n 's are the same in both processes. This can be written as:

E f = E 0 (T 1 ) + nmax i=1 (D j+i + E c1 ) (2) = E 0 (T 2 ) + nmax i=1 (D j+i + E c2 ) (3)
In a first approach, one can neglect the variation of the collisional energies with the number of sticking. By subtracting Eq. 2 to Eq. 3 and using a finite differential approximation, one can derive:

∂E 0 ∂T ≈ n max E c1 -E c2 T 2 -T 1 (4)
Let us recall that this equation is a rough approximation however. It is useful for the understanding of the principle of the differential method, but it has been established with a number of simplifying hypotheses. Specially, the quantity n max is not well defined. Instead, we will show in section IV that a well defined and relevant quantity is the average number of sticking. This quantity can be easily extracted from the experimental mass spectra after sticking.

III. EXPERIMENTAL SETUP

The experimental setup is depicted in Figure 3 and has already been described in details elsewhere [10]. The key points of the experiment are highlighted in the following.

The clusters are produced in a gas aggregation source. A crucible filled with sodium is heated up to 573 K. The source walls are cooled down with liquid nitrogen. A controlled flux of cold Helium (173 K) flows through the hot sodium vapor and condensation occurs. A hollow cathode discharge in the crucible ionizes the produced clusters.

The helium buffer gas brings the sodium clusters to the next stage of the experiment, the thermalization chamber. Its temperature can be set between 140 K and 500 K. Clusters thermalization occurs thanks to collision with the helium buffer gas (≈ 10 5 collisions). Out of the thermalization chamber flows a continuous beam of charged, temperature controlled, sodium clusters. The size distribution is controlled by varying the helium gas flow, the temperature of the crucible and the output diameter of the thermalization chamber. Typically, the center of the size distribution ranges from 30 to 300 atoms with a width of about 100.

We then apply pulsed voltages to electrodes in order to mass select, energy focus and slow down the clusters. The first acceleration stage consisting in 3 electrodes serves two purposes: first it allows a first raw mass selection, but more importantly, it is used to bring the clusters at the distance D f oc with a linear relation between their kinetic energy and their position. Clusters of different sizes will reach D f oc at different times. By applying a voltage on the energy focusing electrode at a specific time T f oc , we compensate the energy dispersion for a given size of clusters, giving all of them the same energy to a very good accuracy.

Mass selected clusters are then decelerated with a gradual potential barrier until they reach the end of the slowing down apparatus. The voltages applied on electrodes are then suddenly shut down.

At this point, clusters are mass selected, thermalized and slowed down to a known controlled kinetic energy. They enter a cylindrical collision cell (5 cm long, 2.5 cm radius, 5 mm entrance and exit holes). The vapor density ρ of sodium atoms in the cell is controlled by the temperature T cel . After colliding, clusters finally reach the second acceleration stage, which, used in combination with a reflectron, gives us the size distribution at the output of the cell.

Clusters can be routinely slowed down to E k = 10 eV. The full width at half maximum of the kinetic energy distribution is about 2 eV. The mean collision energy in the center of mass frame is given by

E c = E k n + 1 + 3nk B T cel 2(n + 1) . ( 5
)
where the second term is the contribution from the mean kinetic energy of the atoms in the vapor.

Our raw experimental data are the mass spectra recorded at the output of the cell, after parent clusters have undergone collisions. The mass spectra are recorded as a function of several parameters such as the initial temperature of the clusters, the density in the cell or the kinetic energy E k .

The pulsed voltages are operated at 200 Hz, and mass spectra are generally averaged over 6000 sweeps. The thermalizer or cell temperature is ramped (1 K/minute) and mass spectra are continuously acquired with the following sequence:

-Mass spectrum at E c1 -Background spectrum at E c1 -Mass spectrum at E c2 -Background spectrum at E c2 until the final temperature is reached. Background peaks originate from masses close to the selected one, which, although not ideally transmitted through the energy focusing device, can also cross the collision cell and be detected. These peaks are present in the absence of sodium vapor in the cell. Background spectra are obtained by slightly changing T f oc so that neither the chosen mass nor any other mass is energy focused.

In the following section we analyze the evolution of the mass distributions at the output of the cell as the experimental parameters are varied.

IV. MASS DISTRIBUTIONS IN THE MULTICOLLISION REGIME

When the thermalized, mass selected clusters fly through the cell, they undergo a number of collisions, depending on the density of atoms in the cell. If the collision energy is low enough, those collisions lead to the sticking of sodium atoms onto the clusters [12]. Fig. 4 shows the evolution of the size distribution at the exit of the cell when the atom density ρ is increased. These spectra are obtained for N a + 90 slowed down to E k = 20 eV and thermalized at a temperature T = 146 K. The top panel, for almost no density in the cell, demonstrates the ability of our setup to mass select the clusters N a + 90 . As can be seen, peaks appear after the selected mass. Particular care has been taken in order to eliminate these background peaks. Then, as the density is increased, bigger sizes appear. Since the collision energy is low (0.25 eV in the center of mass frame) compared to the dissociation energy (about 1 eV) and the clusters have a low temperature, several stickings are observed (up to 8 in the lower panel of Figure 4).

Starting from a cluster of size j, a convenient way to quantify the number of stickings, is to use the size distribution barycenter n. It is calculated as: where I(i) is the integrated signal of clusters of size i + j. Fig. 5 shows the evolution of n as a function of the vapor density in the cell for N a + 90 with E k = 20 eV and T = 146 K. The barycenter first increases until it reaches a plateau at n ≈ 4.5 for a density of 0.3×10 20 m -3 . It then decreases continuously and can even reach negative values. Two competitive reactions occur in the cell that govern the evolution of the size distribution, the sticking of an atom onto the cluster:

n = i i × I(i) i I(i) (6)
N a + n + N a → N a + n+1 (7) E = E 0 + D n+1 + E c (8)
and the evaporation:

N a + n → N a + n-1 + N a (9) E = E 0 -D n -ε (10)
where E and E 0 are the internal energies after and before reaction respectively, D n are the dissociation energies of cluster of size n, E c is the collision energy in the center of mass frame and ε the fragment kinetic energy. Three regimes can be distinguished:

1. The number of sticking increases linearly with the density. Each collision leads to a sticking and E remains low enough so that no evaporation occurs.

2. The number of sticking reaches a maximum (saturation). At each collision, the internal energy is increased by the dissociation energy D n plus the collision energy E c . When the internal energy gets high enough, evaporation sets in and counterbalances the sticking: the mean evaporation time is nearly equal to the mean time between two collisions.

3. After saturation is reached, each extra collision leads to evaporation. Sticking-evaporation cycles increase the internal energy of the clusters by E c -ε (ε ≈ k B T ). Since ε < E c , after a number of evaporation-sticking cycles, the clusters heat up and evaporation dominates.

The reliability of our method rests on the fact that the state of energy E f on Fig. 1 can be reached in the course of the flight through the cell. Thus, the density has to be high enough so that regime (2) or (3) occurs. This implies that evaporation occurs also during the process, so that the analysis leading to Eq. ( 4) has to be refined in order to account for evaporation. This is done in the next section.

V. EXTRACTION OF THE CALORIC CURVES A. Mathematical expression for the heat capacity

Let us consider a cluster of initial energy E 0 and size j when it enters the cell. Suppose now that it undergoes i collisions during its flight through the cell, and that the final observed number of stickings is n, with n < i because some evaporation occurs. Our basic hypothesis is that no matter the detailed history of these events, the final value of n depends only on the total energy E which has been brought to the cluster:

E = E 0 + iE c + f ( 11
)
where f is an unknown quantity, which depends on the various values of the dissociation energies but neither on E 0 nor on E c . There is no need in knowing f , as it does not appear in the final expression of the heat capacity. This hypothesis amounts to say that the number of evaporations is such as to bring back the cluster to the energy E f defined in section II, and thus depends only on the excess energy E -E f . Now, all three quantities E 0 , E c and i are statistically distributed. In this treatment, we suppose that the widths of the E 0 and E c distributions are small respective to their averages, so that E 0 can be replaced by its average E 0 (T ) and E c by its average Eq. ( 5). The main statistical effects come from the distribution of the number of collisions i, whose width is close to the average. The probability P (i) is a Poisson law :

P (i) = ı i e -ı i! (12)
with an average ı, which can be easily computed knowing the density ρ in the cell.

Computing the mean number of sticking, we obtain:

n = i P (i)n(E 0 (T ) + iE c ). (13)
By the implicit functions theorem, we have:

∂E 0 ∂E c n = - ∂n ∂E c ∂n ∂E 0 (14)
The two derivatives on the r.h.s. above are given by:

∂n ∂E 0 = i P (i) ∂n ∂E (E 0 (T ) + iE c ) (15)
∂n ∂E c = i iP (i) ∂n ∂E (E 0 (T ) + iE c ) (16)
We now use a specific property of the Poisson law:

iP (i) = ı P (i) + ∂P (i) ∂ı (17)
so that Eq. ( 16) can be rewritten as:

∂n ∂E c = ı ∂n ∂E 0 + ∂ 2 n ∂E 0 ∂ı (18)
So far, we neglected the change in cluster kinetic energy after each collision. When the number of collisions becomes large, this has to be accounted for through an effective value E * c , which is to a good approximation:

E * c = E c j + 1 ı 1 - j j + 1 ı (19)
We use now

C(T ) = dE 0 dT (20)
together with equations (14 and (18), and we eventually obtain the final expression for the heat capacity:

C(T ) = -ı 1 + ∂ 2 n ∂T ∂ı ∂n ∂T ∂E * c ∂T n (21)
We describe in the next section how the different terms in Equation ( 21) are deduced from the experiment.

B. Practical extraction of C(T ) from experimental curves

In order to use Eq. (21), we need three partial derivatives and the value of ı. All these quantities can be extracted from experiment.

The derivative ∂Ec ∂T n

The method can be summarized as follows (see Fig. 6 for an illustration):

1. We record the barycenter n of the size distribution as a function of the initial cluster temperature for two different kinetic energies of the clusters E c1 and E c2 . 2. We then chose a given value of the barycenter. For this value, we record the two corresponding temperatures T 1 and T 2 (as shown on Fig. 6).

3. We approximate:

∂E c ∂T n = E c2 -E c1 T 2 -T 1 (22)

Determination of ı

The average number of collisions ı can be considered to be the same for the two collision energies E c1 and E c2 , as long as the difference δE c remains small enough (in other words, that the method is indeed differential). For instance, for clusters as small as N a + 30 at E k = 15 eV and 20 eV (that correspond to 0.53 and 0.70 eV collision energies) and for a density ρ = 0.2 × 10 20 m -3 in the cell, Monte Carlo simulations (see next section) show that the average number of collisions are respectively 2.11 and 2.08. As one can see, the difference is small enough that it can be neglected. Note that as the cluster size increases it becomes even easier to fulfill this requirement.

The average number of collisions ı is deduced from saturation curves such as the ones presented in Figure 7. There is no need in knowing the value of the cross sections to determine this parameter. The main reason is that in regime (1), each collision leads to a sticking. The proportionality factor between the cell density ρ and ı is thus deduced from the initial slope of the curves. Actually collisions at high impact parameter might not lead to sticking. These collisions involve a very small energy exchange anyway, so that they can be neglected [12]. The principle is presented on Fig. 7 too. The slopes S 1 and S 2 are the derivatives ∂n ∂ı at temperatures T 1 and T 2 respectively. ∂ 2 n ∂T ∂ı is then estimated as (S 2 -S 1 )/(T 2 -T 1 ). Note that its value does not depend on the collision energy (which needs nevertheless to be the same for the two curves at T 1 and T 2 ).

Finally the derivative ∂n

∂T is evaluated using the curves n(T ) (see Fig. 6).

VI. MONTE CARLO SIMULATION: VALIDATION OF THE METHOD A. Description of the simulations

Realistic Monte Carlo simulations have been performed in order to check the validity of the method under real conditions where the above analytical ideal approach might be invalidated by a number of details that are not taken into account: initial internal and collisional energy distributions, spatial divergence due to collisions, variations of ∂ 2 n ∂T ∂ı , dissociation events between the output of the cell and the detector,... The initial internal energy of the clusters is randomly picked such that the average energy is equal to:

E = (3j -6)hν 0 e hν 0 k B T -1 ( 23
)
where j is the number of atoms in the cluster, h the Planck constant, ν 0 the vibration frequency. The frequency ν 0 is taken from 13 and is equal to 2.3×10 12 Hz.

The phase transition is taken into account by adding the latent heat L j (T ) to E in the following way assuming a gaussian peak in the heat capacity:

L j (T ) = L j0 (1 -erf(- (T -T jf ) ∆Tj )) 2 (24)
where T jf is the temperature of the phase transition, ∆T j is the width of the phase transition and erf is the error function. L j0 is the latent heat for a cluster of size j. This way we simulate the broadening of the phase transition for a finite system. The evaporation rate is estimated using a Weisskopf approximation:

Γ evap = 8πν 0 (3j -7)σ (1 -D j /E) 3j-8 E ( 25
)
where σ is the sticking cross section to the cluster of size j -1. The collision rate is given by:

Γ coll = ρσv rel . ( 26
)
where v rel is the relative velocity of the cluster and the sticking atom. The velocity distribution of atoms is assumed to follow a Boltzman distribution. At each collision, the change in speed of the clusters is taken into account. Since only clusters with the right velocity are detected, clusters that undergo a lot of collisions become too slow to be detected and are discarded in the simulation.

We first checked that our Monte Carlo simulations could reproduce the experimental results of Figs. 5 and8. The agreement between the simulations and the experimental data is good. In order to reproduce correctly the experimental results it is important to put in the correct latent heat and melting temperature for N a + 90 . We took T f = 205 K and L = 8 meV/atom for the melting temperature and the latent heat respectively. On the other hand, for the other sizes the latent heat and temperature of fusion can be chosen arbitrarily: there is no change in the curve n(ρ).

For the dissociation energies, only values of D n up to n=37 [START_REF] Bréchignac | [END_REF] are available. In the simulations presented here, the dissociation energies are all taken equal to 0.94 eV.

These simulations allow us to follow the evolution of the internal energy distribution of the clusters as a function of their size as the density in the cell is varied. In Figure 9 we present the energy distribution of the clusters for three different densities in the cell. In Figure 9(a) the density in the cell is 0.1 × 10 20 m -3 , corresponding to an average number of sticking n ≈ 3 (in this case evaporation is completely negligible so that each collision leads to a sticking). We see in the figures well separated peaks corresponding to the different masses obtained after the cell. The energy separation between the peaks correspond to the dissociation energy plus the collision energy. The width of the peaks originates mainly from the initial canonical energy distribution. This can be seen from the n = 90 energy distribution: indeed, in this example part of the parent clusters did not collide nor evaporate. Nevertheless, as the cluster size increases we observe an increase in the width. This comes from the width of the collision energy distribution which adds at each new collision.

In Figure 9(b), the density is 0.32 × 10 20 m -3 (ı ≈ 7). This corresponds to the saturated case (regime (2) of Figure 5). If we consider the sizes n = 93 and 94, the internal energy distribution becomes bimodal: the high energy part comes from the evaporation of bigger clusters. For the two biggest sizes, namely n = 95 and 96, one can see as well that the energy separation between the peaks gets smaller. In fact, there is a strong contribution coming from the evaporation of n = 96 clusters in the internal energy distribution of the n = 95 clusters. For the size n = 96, there is of course also contribution from the evaporation of bigger sizes that are not observed. Indeed due to their high evaporation rate, they evaporate before detection.

As the density is increased to 0.72 × 10 20 m -3 (Figure 9(c)), all clusters end up with almost the same internal energy. In this last case the average number of collisions is about 17, much more than the maximum number of sticking observed (which is about 6). The observed internal energy is the maximum energy clusters can bear without evaporating before detection. In this case the observed clusters have all undergone evaporation.

Despite the complicated evolution of the internal energy of the clusters as the number of collision increases, the size distribution, for a given collision energy and density in the cell, is still essentially governed by the internal energy of the incoming cluster. Furthermore, as shown below, the fact that clusters of a specific size don't have necessarily a well defined internal energy do not prevent the determination of the caloric curves.

B. Test of the method of section V

In order to test the method of section V, we start from a theoretical caloric curve for N a + 90 and we perform a set of simulations that mimics the experimental curves. Then, using these fake experimental curves, we apply the method and we extract a caloric curve, which is then compared to the theoretical one we started with.

First of all, two saturation curves at E k = 20 eV were simulated for two initial temperatures of the clusters, namely 146 K and 293 K. These curves are presented in roughly corresponds to the maximum of the saturation curve (see Fig 7). It would not be a very good choice in a real experiment since the curvature of ∂n ∂ı , thus the value of the second derivative in Eq. ( 21), is maximum. For ı = 10.6, the second derivative in Eq. ( 21) is much smaller because the saturation curves already become much more parallel (see Fig 7).

Figure 10 shows a comparison between the theoretical caloric curve introduced in the calculation and the curves deduced from the simulation of the experiment. Fig. 10 In order to evaluate the effect of the second derivative in Eq. ( 21), two curves are represented for each vapor density in each panel of Fig. 10, respectively with and without this term. For ı = 10.6, within the noise, the curves are virtually indistinguishable. On the other hand, for ı = 5.3, although the general shape of the caloric curve is correctly obtained with or without the second derivative, a quantitative agreement is obtained only by having it taken into account. The correction nicely improves the curve even in this unfavorable case. Nevertheless the smallest the correction the more accu-rate the result. Note that the noise is more important for ı = 5.3 than for ı = 10.6: since the mean energy difference is roughly ıδE c , the two curves n(T ) get closer for small ı values.

As already mentioned the simulations presented here are done with all clusters having the same dissociation energies. Nevertheless, we carefully checked that the introduction of magic numbers (i.e. higher dissociation energies) for arbitrary sizes does not influence our determination of the caloric curve.

Furthermore we performed simulations with randomly chosen temperature of fusion, latent heat and width of the transition for cluster sizes other than the initial one, and again this does not affect our determination of the latent heat and temperature of fusion of the initial cluster size.

From these simulations, we find our method particularly robust. The simulations demonstrate that the average number of sticking n is a well suited quantity to characterize the size distribution, even when evaporation dominates, provided Eq. ( 21) is used.

VII. EXAMPLES OF EXPERIMENTAL

RESULTS WITH N a + 90 AND N a + 140

We present in Figure 11 the experimental caloric curves obtained for N a + 90 and N a + 140 clusters. The melting temperature is obtained from these curves as the maximum of the peak. The latent heat is deduced by integrating the peak area, after removing the base line (grey filled area under the curves in figure 11). The caloric curve for N a + 90 was not measured previously. We find a melting temperature of 205 ±5 K and a latent heat per atom of 8 ±2 meV.

For the N a + 140 clusters, the melting temperature is 272 ±4 K and the latent heat is 14 ±2 meV/atom. This is consistent with previous measurements by Schmidt et al [5] (T f us = 262 K, L = 11 meV/atom). The results for N a + 90 are quite noisy: this is due to the relatively small latent heat of this cluster and the fact that the width of the transition is relatively large. Nevertheless we can still identify a peak in the caloric curve for this disfavoring case. We observe that the latent heat increases at high temperatures. Such an increase has already been observed, although less pronounced (see [4] for N a + 192 and [15] for N a + 139 ). It might be due to either a rising background due to a high Debye temperature or a melting occurring in several steps. However the noise level is such that we can not confirm the significance of this increase in our case.

As shown from Fig. 10 the exact choice of the vapor density in the cell is not critical. Nevertheless, an optimal choice exists. As already mentioned in Sec. VI, it can be chosen so that it minimizes the noise. In order to minimize the second derivative in Eq. ( 21), the vapor density ρ has to be chosen high enough such that the curves n(ρ) become almost parallel. On the other hand, if ρ be- comes too large, evaporation becomes too strong and the signal gets scattered over many mass peaks, which is unfavorable for the statistics. Furthermore the background pressure in the collision chamber gets too high and leads to unwanted collisions out of the cell. Finally, too large ı result in angular scattering of the clusters.

The use of a differential method has several advantages. First, we do not need to know the dissociation energies. Second, the presence of magic numbers is not a problem with this method. Indeed, any accident in the curves n(T ) is canceled out by the differentiation. These all in one make this method reliable and precise. On the other hand, noise is a drawback. Since we rely on the calculation of finite differences for the determination of the caloric curves, and that the experimental data is discrete, the numerical differentiation is inherently noisy.

VIII. CONCLUSION

We have developed a novel experimental method for the measurement of caloric curves of small nanoparticles based on the sticking of atoms onto clusters. We have mathematically established the method and shown with the help of simulations that it is extremely robust. It associates the accuracy of the photoexcitation methods and the large application field of collisional methods. Moreover, its principle makes it rigorously independent on preliminary assumptions about properties such as dissociation energies or collisional energy transfer.

We have experimentally demonstrated the use of the method on sodium clusters (namely N a + 90 and N a + 140 ).

FIG. 2 :

 2 FIG. 2: (Color on line) Schematic view of the average number of sticking as a function of the temperature. Two curves are shown corresponding to two energies of collision Ec 1 and Ec 2 , with Ec 2 > Ec 1 . When the collision energy is increased at a given temperature less sticking occurs.

FIG. 3 :

 3 FIG. 3: Schematic view of the experiment.

3 FIG. 4 :

 34 FIG. 4: Size distributions for N a + 90 clusters going through the cell with T = 146 K and E k = 20 eV. As the density in the cell increases (from top to bottom), the number of sticking onto N a + 90 increases.

FIG. 5 :

 5 FIG. 5: Evolution of the sticking distribution barycenter as a function of the density of sodium atoms in the cell for N a + 90 , T = 146 K and E k = 20 eV. Three regimes can be distinguished: (1) linear increase, (2) saturation and (3) decrease.

FIG. 6 :

 6 FIG. 6: (Color on line) Experimental data showing the barycenter n as a function of the initial temperature for N a + 140 clusters. The two curves correspond to collision energies of Ec 1 = 0.14 eV and Ec 2 = 0.21 eV

FIG. 8 :

 8 FIG. 8: (Color on line) Evolution of the size distribution barycenter as a function of the density of sodium atoms in the cell for N a + 90 for (a) three different initial temperatures (the kinetic energy is E k = 20 eV) and (b) two different kinetic energies (T = 213 K). Full lines are the results of Monte Carlo simulations (see Section VI).

FIG. 9 :

 9 FIG. 9: (Color online) Simulation of the internal energy distribution for three different densities in the cell: (a) 0.1, (b) 0.32 and (c) 0.72×10 20 m -3 . The initial temperature of the clusters is 146 K and E k =20 eV. Peaks are labeled by the size of the clusters.

FIG. 10 :

 10 FIG.10:(Color online) The thick solid line is the caloric curve introduced in the simulation (L = 8 meV/atom, T f =205 K and ∆T = 30 K). The dotted curves correspond to the caloric curves obtained by setting the second derivative in Eq. 21 to 0 whereas for the curves in thin solid line it is taken into account. The top panel correspond to ı = 5.3 and the bottom one to ı = 10.6

  (a) corresponds to ı = 5.3 and Fig. 10(b) to ı = 10.6.

FIG. 11 :

 11 FIG. 11: (Color online) Experimental caloric curves for (a) N a + 90 and (b) N a + 140 clusters. C(T ) is in unit of kB per degree of freedom. The full lines represent the latent heat used in the simulations (see Eq. 24) and are used to guide the eye. The grey filled area under the curves represent the latent heat.

  (see Section VI for details). The kinetic energy of the cluster is 20 eV. The bottom scale is in average number of collisions ı whereas the top scale is in vapor density. At low density, the number of sticking is equal to the number of collisions. S1 and S2 are the slopes used in the determination of ∂ 2 n ∂ı∂T .

					20		-3
					(10	m	)
			1	0.2	0.3		0.4	0.5	0.6
	barycentre						s 2 s 1
		0	2	4	6		8	10	12
			Average number of collisions
	FIG. 7:	Monte Carlo simulation of n(ρ) for two initial
	temperatures of N a + 90 clusters: 146 K (circles) and 293 K
	(squares)

Acknowledgments

This work has been partly funded by the Agence Nationale de la Recherche under grant ANR-05-BLAN-0145. We gratefully acknowledge M. Schmidt for fruitful discussions.