Ruming Pan

João Vitor

Ferreira Duque

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

Investigating Waste Plastic Pyrolysis Kinetic Parameters by Genetic Algorithm Coupled with Thermogravimetric Analysis

Keywords: Waste plastic, Reaction models, Genetic algorithm, Thermogravimetric analysis, Pyrolysis

Pyrolysis of waste plastic (WP) is a promising method to solve the plastic pollution issue. WP is mainly composed of polyethylene (PE). Moreover, the products of waste polyethylene (WPE) pyrolysis could serve as high quality fuels and the feedstocks of petrochemicals. Therefore, it is essential to investigate the WPE and WP pyrolysis process. This study evaluates pure PE, WPE and WP pyrolysis kinetic parameters by the use of genetic algorithm (GA) and isoconversional methods coupled with thermogravimetric analysis (TGA), respectively. Additionally, three representative reaction models, i.e. reaction-order, extended Prout-Tompkins and Sestak-Berggren models, are investigated for obtaining the most suitable model, which could describe the PE, WPE and WP pyrolysis process more accurately. Consequently, the reaction-order model turns out to be the optimal method for appropriately describing PE, WPE and WP pyrolysis processes. Hence, the pyrolysis parameters optimized by GA are proven to be accurate and reliable, in comparison of calculated values of activation energy by isoconversional methods and experimental data. Moreover, it might be applicable of GA coupled with TGA with reaction-order model to the future industrial WPE and WP pyrolysis circumstances that have variable heating rates.

that GA is a promising method to evaluate the pyrolysis kinetic parameters of polymers.

For the purpose of instructing the further application of chemical recycling of WPE and WP, this study adopts GA coupled with TGA to study the pyrolysis processes of WPE and WP at heating rates varied from 5 to 20 K/min in argon. Moreover, in order to obtain a better understanding of the pyrolysis processes, the pure PE from commercial corporation is utilized as a comparison. Three isoconversional methods are adopted to calculate the activation energy values. The results provide a reference to the GA calculated kinetic parameters. Additionally, three representative reaction models are investigated to determine the most favorable one which could describe the WPE and WP pyrolysis processes.

Experimental and Kinetic Modeling

Experimental

Materials

Pure PE, WPE and WP utilized in this study were offered by Lukplast Ind. (ES-Brazil), as illustrated in Fig. 1. WPE was recycled from waste polyethylene products with the certain selection, such as plastic bags, plastic films, milk buckets and et al. While WP was recycled plastic without selection, which composed of a majority of polyethylene (PE) and a minority of polystyrene (PS) and polyethylene terephthalate (PET).

Thermogravimetric Tests

The thermogravimetric tests were using a NETZSCH STA 449F3 thermal analyzer with 60 mL/min gas flow rate of argon. Approximately 13.0 mg powder test samples were heated from room temperature to 973.15 K at four representative heating rates (i.e., 5, 10, 15, 20 K/min).

Kinetic Modeling

Pyrolysis Reaction Description

According to the literature [START_REF] Jiang | Application of genetic algorithm to pyrolysis of typical polymers[END_REF][START_REF] Chen | Kinetic study on pyrolysis of waste phenolic fibre-reinforced plastic[END_REF], the pyrolysis of plastic could be described by: where ν represents reaction stoichiometry coefficient.

Based on the aforementioned Eq. ( 1), this study adopted Arrhenius Law to calculate mass loss rate (MLR) during plastic pyrolysis process. Hence, MLR could be expressed as follow:

where A, α, f(α), E, R and T represent the pre-exponential factor (1/s), the conversion rate, the dependence on the conversion rate, the activation energy (kJ/mol), the universal gas constant (equals to 8.314 J/(mol K)) and the temperature (K), respectively.

Accordingly, the mass fraction of test sample could use the following equation:

where y and t represent the mass fraction and the pyrolysis time (s), respectively.

Reaction Models

According to the literature [START_REF] Vyazovkin | ICTAC Kinetics Committee recommendations for performing kinetic computations on thermal analysis data[END_REF], three representative reaction models, namely reaction-order model, the extended Prout-Tompkins model, and the Sestak-Berggren model 43, are possible candidates for the pyrolysis description. The expressions are listed in Table 1.

(1) plastic → residue + (1 -)gas

(2) MLR = (1 -)Af ( )exp(-E∕(RT)) (3) y = 1 - t ∫ 0 MLRdt

Isoconversional Methods

In this study, three representative isoconversional methods, as listed in Table 2, were adopted for investigating the activation energy of plastic pyrolysis, namely one integral method (KAS) [START_REF] Akahira | Method of determining activation deterioration constant of electrical insulating materials[END_REF], one differential method (Friedman) [START_REF] Friedman | Kinetics of thermal degradation of char-forming plastics from thermogravimetry. Applications to a phenol plastic[END_REF] and AIC method [START_REF] Vyazovkin | ICTAC Kinetics Committee recommendations for performing kinetic computations on thermal analysis data[END_REF].

Genetic Algorithm

The genetic algorithm adopted in this study is described in Fig. 2. In each generation, there are N individuals with specific genes, such as the activation energy E, the pre-exponential factor A, the reaction order n and et al. These individuals serve as parents. Subsequently, N parents could generate N children. This procedure is known as reproduction, also accompanied by mutation. The arithmetic expressions were illustrated in Fig. 1, in which i and j denote the number of individual and gene; r i j is a random number varied from 0 to 1; ν mut is the possibility of mutation, which equals to 0.05 41; and s is a random number varied from -0.5 to 0.5 42. The next procedure is to calculate the fitness of each individual. The fitness is calculated by experimental and predicted MLR and mass fraction. The fitness is served as the target function. Additionally, φ denotes the weight coefficient, which equals to 0.5 [START_REF] Chen | Kinetic study on pyrolysis of waste phenolic fibre-reinforced plastic[END_REF]. Furthermore, the best individual, which had the maximum fitness value, was selected.

( ) = (1 -) n Extended Prout-Tompkins model f ( ) = m ⋅ (1 -) n Sestak-Berggren model f ( ) = m ⋅ (1 -) n ⋅ [-ln(1 -)] p
Table 2 Isoconversional methods adopted in this study β, E, T and R represent the heating rate (K/min), the activation energy (kJ/mol), temperature (K) and universal gas constant (equals to 8.314 J/(mol K)), respectively Method Equations KAS ln

T 2 = -E RT + const Friedman ln d dT = -E RT + const AIC Φ(E) = ∑ n i=1 ∑ n j≠i I(E,T i )∕ i I(E,T j )∕ j with I E, T i = ∫ T i 0 exp -E RT dT Fig. 2 GA schematic diagram
Lastly, offspring is undergoing selection based on the value of fitness. Individuals, of which fitness values lower than half of the maximum value fitness, would be replaced by the best individual in this generation. This process is called replacement. Therewith, the next inheritance will repeat the above processes. In this study, the number of individuals and generations are 500 and 200, respectively. The GA was coded in Matlab. In addition, ordinary differential equation ode23s was adopted to calculate the predicted mass loss rates (MLR) and mass fractions.

Results and Discussions

Thermogravimetric Analysis

Figure 3 illustrates the experimental MLR and TG (relative mass loss) curves of pure PE, WPE and WP at different heating rates during the pyrolysis process under argon atmosphere. The test samples were performed under the same experimental treatment, which heated from room temperature to 973.15 K. In general, the shape of MLR and TG curves remained unchanged regardless of the variation of the heating rate. Moreover, the peak values of MLR of PE, WPE and WP increase as the heating rate increases. For instance, as depicted in Fig. 3a, the peak value of MLR of PE increased from 0.2296 to 0.8751 wt%/s when the heating rate increased from 5 to 20 K/min. Additionally, a significant lateral shift to higher temperature occurred when the heating rate increased in the MLR and TG curves. It may be caused by the pyrolysis mechanism transformation when the heating rate increased [START_REF] Das | Thermal degradation kinetics of plastics and model selection[END_REF][START_REF] Kple | Kinetic study of low density polyethylene using thermogravimetric analysis, Part 2: Isothermal Study[END_REF].

Figure 4 demonstrates the onset temperature T o , the end temperature T e and the maximum degradation temperature T m in the PE MLR curve at 5 K/min under argon atmosphere. T o , T e and T m were proposed for evaluating the plastic pyrolysis process [START_REF] Das | Thermal degradation kinetics of plastics and model selection[END_REF]. As listed in Table 3, due to the pyrolysis mechanisms transformation [START_REF] Das | Thermal degradation kinetics of plastics and model selection[END_REF][START_REF] Kple | Kinetic study of low density polyethylene using thermogravimetric analysis, Part 2: Isothermal Study[END_REF], T o , T e and T m of PE, WPE and WP increased with the heating rate. For instance, the onset temperature T o of PE, WPE and WP increased from 699 to 719 K, 700 to 725 K and 697 to 723 K gradually when the heating rate increased from 5 to 20 K/ min. By comparison, WP has the highest T e and T m , while PE has the lowest ones. As regards to T o , there is no obvious regularities among PE, WPE and WP at different heating rates. However, the differences value between T o and T e are nearly constant, around 60 K, despite of the variation of the heating rate.

In consideration of the MLR maximum value, the pyrolysis processes of PE, WPE and WP are relatively fast under argon atmosphere. For example, the maximum MLR of WPE could come up to 0.9142 wt%/s at 20 K/min. Besides, the maximum values of MLR of PE, WPE and WP varied from 0.2296-0.8751 wt%/s, 0.2233-0.9142 wt%/s and 0.1912-0.7784 wt%/s at the heating rates 5-20 K/min, respectively. It could be regarded that WPE underwent a relatively faster pyrolysis process, while WP underwent a relatively slower pyrolysis process under argon atmosphere.

The TG curves of PE, WPE and WP, which depicted in Fig. 3, indicated that they were decomposed to gaseous products (under the experimental environment) during the pyrolysis process. After PE, WPE and WP were heated from room temperature to 973.15 K, the mass fractions of the residue at different heating rates are listed in Table 4. Plastic pyrolysis products could be divided into char, wax/ oil and gases [START_REF] Sharuddin | A review on pyrolysis of plastic wastes[END_REF]. Moreover, the distributions of the products are affected by the heating rates [START_REF] Encinar | Pyrolysis of synthetic polymers and plastic wastes. Kinetic study[END_REF]. Therefore, the residues were different at different heating rates of pure PE, WPE and WP. When the pyrolysis processes finished, there were no residue of pure PE remained when the heating rates were 10 and 15 K/min. However, it existed 0.03 and 0.90 wt% residue of pure PE at 5 and 20 K/min, respectively. The values are negligible compared to the original weight. Therefore, it could be considered that pure PE can be decomposed thoroughly after the pyrolysis process under argon atmosphere [START_REF] Das | Thermal degradation kinetics of plastics and model selection[END_REF][START_REF] Aboulkas | Thermal degradation behaviors of polyethylene and polypropylene. Part I: Pyrolysis kinetics and mechanisms[END_REF]. In the case of WPE, a minor amount of residue remains after the pyrolysis process. However, a larger amount residue remained after WP pyrolysis process. As demonstrated in Table 4, it still remained around 2.15 wt% residue at the end of WP thermal degradation process under argon atmosphere.

Degradation Kinetics by the Isoconversional Methods

Three isoconversional methods were adopted for calculating the activation energy E of pure PE, WPE and WP by using four sets of TG data at different heating rates. As exhibited in Table 5, the activation energy values of pure PE, which were calculated by KAS, Friedman and AIC methods, are varied with the different conversion rates. However, the differences of the activation energy values are almost negligible particularly when the conversion rate varied from 0.3 to 0.9.

In addition, the activation energy is relatively lowered when the conversion rate is under 0.2. This is because that the initial degradation is relatively facilitated due to the presence of weak link sites intrinsic of PE [START_REF] Vyazovkin | Isoconversional kinetic analysis of thermally stimulated processes in polymers[END_REF][START_REF] Peterson | Kinetics of the thermal and thermo-oxidative degradation of polystyrene, polyethylene and poly(propylene)[END_REF]. The activation energy of WPE and WP were calculated by the same three isoconversional methods as its of pure PE. The results were listed in Table 6 and Table 7 separately. Hence, the calculated activation energy of pure PE, WPE and WP at different conversion rates by KAS, Friedman and AIC methods were plotted in Fig. 5. It is noticeable that the differences of activation energy, between the KAS and AIC methods, could be nearly neglected. It might be noticed that the activation energy of pure PE is larger than the one of WPE and WP. It indicates that the WPE and WP are more Fig. 5 The calculated activation energy of pure PE, WPE and WP by different isoconversional methods ignitable, whereas pure PE is relatively less ignitable [START_REF] Wang | Thermal degradation properties of LDPE insulation for new and aged fine wires[END_REF]. Once again, we can notice the decrease of the value differences of activation energy of pure PE, WPE and WP with the conversion rate.

Favorable Reaction Model Selection

In order to determine which reaction model could describe the PE pyrolysis process with accuracy, three representative reaction models were adopted in this study, i.e. reactionorder, extended Prout-Tompkins and Sestak-Berggren models. Hence, the different reaction models coupled with GA were investigated separately for obtaining the optimal pyrolysis kinetic parameters of pure PE. Consequently, the optimized values of PE pyrolysis kinetic parameters by adopting reaction-order, extended Prout-Tompkins and Sestak-Berggren models are listed in Table 8. The optimized values of activation energy were 241.55, 221.21 and 217.86 kJ/mol, which were calculated with reaction-order, extended Prout-Tompkins and Sestak-Berggren models, respectively. In comparison, the activation energy values of PE calculated by KAS, Friedman and AIC methods were 261.55, 252.68 and 261.86 kJ/mol, respectively. The GA optimized activation energy of reaction-order model is close to the value determined by the isoconversional methods.

Accordingly, the predicted MLR and TG curves were plotted in Fig. 6, which with the GA optimized values of PE by adopting reaction-order, extended Prout-Tompkins and Sestak-Berggren models at the different heating rates. It could be perceived that the predicted MLR and TG curves by using of three reaction models coupled with GA were promisingly consistent with the experimental ones.

Moreover, the R-squared values between experimental and predicted data of PE by adopting different reaction models were listed in Table 9. The predicted mass fraction was more precise than the predicted MLR in all three reaction models. The average R-squared value of mass fraction and MLR were ~ 0.998 and ~ 0.971, respectively. The overall average R-squared value gradually increased in reaction-order, extended Prout-Tompkins and Sestak-Berggren models. However, the increased value was almost negligible. The R-squared values of MLR and mass fraction had no obvious differences among different reaction models. To be more exact, the reaction-order and the extended Prout-Tompkins models were more accurate in the prediction of mass fraction. Whereas the Sestak-Berggren model could describe the MLR with a greater precision. In consideration of the activation energy, the GA optical value with the reactionorder was in agreement with the value calculated by using isoconversional methods. Although the activation energy calculated by isoconversional methods has no mechanistic significance and could not be utilized for extensive academic conclusions [START_REF] Simon | Isoconversional methods[END_REF]. The activation parameters are possible to obtain using isoconversional methods and could provide a guidance in plastic recycling industry [START_REF] Aboulkas | Thermal degradation behaviors of polyethylene and polypropylene. Part I: Pyrolysis kinetics and mechanisms[END_REF]. So we decide to use that as a base value. Therefore, the reaction-order model coupled with GA was selected to conduct the polyethylene pyrolysis kinetic modelling. Hence, GA coupled with the reaction-order model was adopted to investigate the pyrolysis processes of WPE and WP.

Kinetic Parameters Calculated by GA

According to the previous subsection discussion, GA coupled with the reaction-order model was adopted for calculating the optimal pyrolysis kinetic parameters, i.e., the pre-exponential factor A, the activation energy E and the reaction order n of pure PE, WPE and WP. As demonstrated in Table 10, the optimized values of pyrolysis kinetic parameters lnA, E and n are 34.52 ln(s -1 ), 241.55 kJ/mol and 0.58 of PE; 33.84 ln(s -1 ), 239.82 kJ/mol and 0.51 of WPE; and 33.06 ln(s -1 ), 234.51 kJ/mol and 0.71 of WP, respectively. The value of calculated activation energy of PE is the largest. Moreover, the value of calculated activation energy of WP is the smallest. It is consistence with the results calculated by the KAS and AIC methods. Furthermore, the differences of calculated E by using of GA and isoconversional methods are relatively small. It indicates the reliability of Figure 7 illustrates the experimental and predicted MLR and TG curves of PE, WPE and WP at different heating rates under argon atmosphere. The predicted MLR and mass fraction were calculated by Eq. ( 2) and (3), respectively. Due to the kinetic parameters were all calculated by GA, the variables were just the temperature and the pyrolysis time. The temperature was only determined by the heating rate within the specific pyrolysis time. Therefore, the MLR and mass fraction at different heating rates could be calculated subsequently. It indicated that a high degree of consistency between experimental and predicted data was obtained. Additionally, the R-squared values between experimental and predicted data at different heating rates were listed in Table 11. The R-squared values of WPE mass fraction are greater than 0.999 at each heating rate. However, the R-squared values of MLR of WPE are relatively lower, which the average value is around 0.980. Moreover, in the cases of PE and WP show the same tendency as WPE. In summary, the predicted PE, WPE and WP pyrolysis kinetic parameters are relatively reliable and accurate. Furthermore, GA coupled with TGA could be applied to the more practical pyrolysis circumstances, in which have variable heating rates. This will be discussed in the future work.

Conclusion

This study investigated the pyrolysis processes of waste polyethylene (WPE), waste plastic (WP), and pure polyethylene (PE) as a reference. Based on the pure PE, WPE and WP pyrolysis thermogravimetric experimental data at heating rates from 5 to 20 K/min under argon atmosphere, this study investigated the activation energy by using of three isoconversional methods, which are KAS, Friedman and AIC methods. Three representative reaction models, i.e. reaction-order, extended Prout-Tompkins and Sestak-Berggren models, were also investigated for determining the proper model which could describe the pyrolysis process with accuracy. Accordingly, genetic algorithm (GA) coupled with the reaction-order model was adopted for calculating the optimal pyrolysis kinetic parameters of pure PE, WPE and WP. The GA predicted optimal activation energy values are 241.55, 239.82 and 234.51 kJ/ mol of pure PE, WPE and WP, respectively. It was found that the calculated activation energy values by using of GA were in good agreement with that by using of isoconversional methods. Furthermore, it compared the predicted mass fraction and mass loss rate (MLR), calculated by the GA optimized parameters, with the experimental ones. According to the results, high consistencies between experimental and GA predicted data were obtained in pure PE, WPE and WP pyrolysis processes at heating rates of 5, 10, 15 and 20 K/min. Hence, it indicates that the predicted PE, WPE and WP pyrolysis kinetic parameters by using of GA are relatively reliable and accurate. Moreover, it might be applicable of genetic algorithm coupled with thermogravimetry analysis to the future industrial WPE and WP pyrolysis circumstances that have variable heating rates.

Fig. 1

 1 Fig. 1 Pictures of a pure PE, b WPE and c WP used in experiments

Fig. 3

 3 Fig. 3 Experimental MLR and TG curves at 5, 10, 15 and 20 K/min in argon: a MLR curves of PE; b) TG curves of PE; c MLR curves of WPE; d TG curves of WPE; e MLR curves of WP; f TG curves of WP

  Furthermore, the values of PE activation energy calculated by KAS and AIC were very close. While Friedman method calculated PE activation energy values were very different from KAS and AIC. This is consistent with the results in the work of Das et al.[START_REF] Das | Thermal degradation kinetics of plastics and model selection[END_REF]. And it indicated that the activation energy values calculated by the KAS and AIC were more convincible to a certain extent. The low-density PE activation energy value calculated by Encinar et al.[START_REF] Encinar | Pyrolysis of synthetic polymers and plastic wastes. Kinetic study[END_REF] was 285 kJ/mol. It is close to the calculated average value of pure PE activation energy in this work. However, in the works of Das et al.[START_REF] Das | Thermal degradation kinetics of plastics and model selection[END_REF], Xu et al.[START_REF] Xu | Thermal degradation of typical plastics under high heating rate conditions by TG-FTIR: Pyrolysis behaviors and kinetic analysis[END_REF] and Wang et al.[START_REF] Wang | Thermal degradation properties of LDPE insulation for new and aged fine wires[END_REF], the low-density PE activation energy values calculated by KAS were 162-242, 174.46 and 130.04-193.10 kJ/mol, respectively. The calculated results of activation energy in different literatures varied to a large extent. It may be caused by the different sources of the raw material and the varying processes of manufacturing[START_REF] Xu | Thermal degradation of typical plastics under high heating rate conditions by TG-FTIR: Pyrolysis behaviors and kinetic analysis[END_REF].

Fig. 6

 6 Fig. 6 Experimental and predicted MLR and TG curves at 5, 10, 15 and 20 K/min of pure PE by adopting reaction-order, extended Prout-Tompkins and Sestak-Berggren models in argon

Fig. 7

 7 Fig. 7 Experimental and predicted MLR and TG curves at 5, 10, 15 and 20 K/min of pure PE, WPE and WP in argon

Fig. 7

 7 Fig. 7 (continued)

Table 1

 1 Reaction models investigated in this study

	Reaction model	Equations
	Reaction-order model	f
	α, f(α) and n represent the conversion rate, the dependence on the
	conversion rate and the reaction order. m and p are the exponents
	which applied in different reaction models, respectively

Table 3

 3 Onset, end and maximum degradation temperatures and maximum MLR of PE, WPE and WP in argon

	Material Heating rate (K/ min) T PE 5 699	754	733	0.2296
		10	699	764	739	0.4451
		15	707	773	750	0.5664
		20	719	777	753	0.8751
	WPE	5	700	756	735	0.2233
		10	711	768	747	0.4617
		15	711	776	750	0.5820
		20	725	780	757	0.9142
	WP	5	697	757	735	0.1912
		10	714	770	748	0.4442
		15	716	775	753	0.5538
		20	723	781	756	0.7784

o (K) T e (K) T m (K) MLR max (wt%/s)

Table 4

 4 The mass fractions of the residue at different heating rates of pure PE, WPE and WP

	Residue (wt%)	Heating rate (K/min)
		5	10	15	20
	PE	0.02	0	0	0.9
	WPE	1.15	0.31	0.19	0.49
	WP	1.08	3.16	2.1	2.26

Table 5

 5 Calculation results of activation energy E of pure PE by KAS, Friedman and AIC methods Conversion KAS (kJ/mol) Friedman (kJ/mol) AIC (kJ/mol)

	0.1	236.67	239.97	237.00
	0.2	250.12	266.66	250.43
	0.3	260.39	260.24	260.70
	0.4	266.85	249.27	267.15
	0.5	265.02	255.90	265.32
	0.6	268.30	257.03	268.61
	0.7	270.58	271.33	270.88
	0.8	272.91	276.17	273.21
	0.9	263.12	197.59	263.43
	Average value 261.55	252.68	261.86

Table 6

 6 Calculation results of activation energy E of WPE by KAS, Friedman and AIC methods

	0.1	177.68	208.35	178.04
	0.2	215.85	245.45	216.17
	0.3	234.80	264.89	235.11
	0.4	240.37	241.46	240.67
	0.5	254.52	257.79	254.81
	0.6	257.34	271.78	257.62
	0.7	258.96	254.77	259.23
	0.8	260.94	258.39	261.22
	0.9	257.31	217.88	257.59
	Average value 239.75	246.75	240.05

Conversion

KAS (kJ/mol) Friedman (kJ/mol) AIC (kJ/mol)

Table 7

 7 Calculation results of activation energy E of WP by KAS,

	Friedman and AIC methods
	Conversion	KAS (kJ/mol) Friedman (kJ/mol) AIC (kJ/mol)
	0.1	126.75	162.39	127.20
	0.2	174.64	215.20	175.02
	0.3	213.99	285.69	214.32
	0.4	228.82	285.04	229.14
	0.5	241.26	267.45	241.57
	0.6	244.28	257.55	244.60
	0.7	251.43	244.72	251.74
	0.8	254.55	274.89	254.86
	0.9	258.38	254.81	258.70
	Average value 221.57	249.75	221.91

Table 8

 8 Optimized values of PE pyrolysis kinetic parameters by adopting reaction-order, extended Prout-Tompkins and Sestak-Berggren models

	Parameters	Optimized values
		Reaction-order Extended	Sestak-Berggren
			Prout-Tomp-
			kins
	lnA (ln(s -1 ))	34.52	31.78	31.23
	E (kJ/mol)	241.55	221.21	217.86
	n	0.58	1.08	1.12
	m	-	0.30	0.29
	p	-	-	0.07

Table 9

 9 The R-squared values between experimental and predicted data of pure PE by adopting reaction-order, extended Prout-Tompkins and Sestak-Berggren models

	Heating rate (K/min)	Reaction-order	Extended Prout-Tomp-	Sestak-Berggren
				kins
		Mass	MLR	Mass	MLR	Mass	MLR
	5	0.9948	0.9645	0.9949	0.9533	0.9966	0.9617
	10	0.9992	0.9696	0.9982	0.9713	0.9968	0.9648
	15	0.9995	0.9615	0.9997	0.9832	0.9992	0.9827
	20	0.9990	0.9871	0.9996	0.9759	0.9991	0.9760
	Average value	0.9981	0.9707	0.9981	0.9709	0.9979	0.9713
	Overall average value	0.9844		0.9845		0.9846

Table 10

 10 Optimized value of pure PE, WPE and WP pyrolysis kinetic parameters by adopting reaction-order model

	Parameters	Optimized values
		PE	WPE	WP
	lnA (ln(s -1 ))	34.52	33.84	33.06
	E (kJ/mol)	241.55	239.82	234.51
	n	0.58	0.51	0.71

Table 11

 11 The R-squared values between experimental and predicted data of pure PE, WPE and WP at different heating rates

	Heating rate (K/min)	PE		WPE		WP
		Mass	MLR	Mass	MLR	Mass	MLR
	5	0.9948	0.9645	0.9992	0.9945	0.9966	0.9956
	10	0.9992	0.9696	0.9993	0.9800	0.9966	0.9434
	15	0.9995	0.9615	0.9990	0.9543	0.9990	0.9752
	20	0.9990	0.9871	0.9992	0.9799	0.9992	0.9801
	Average value	0.9981	0.9707	0.9992	0.9772	0.9979	0.9736
	Overall average value	0.9844		0.9882		0.9857

Acknowledgments This work was supported by the China Scholarship Council (CSC) program. Thanks are given to Prof. Kinglston Soares for his assistance in the experimental work and Dr. H. Yao (Xiamen University) for providing the guidance of the ordinary differential equation calculation.

Compliance with Ethical Standards

Competing interest The authors declare no competing financial interest.

Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Affiliations


François Stoffelbach

Jerome Claverie

Rinaldo Poli
email: rinaldo.poli@u-bourgogne.fr

Atom Transfer Radical Polymerization of Styrene Controlled by Phosphine-Containing Coordination Compounds of Mo(III)/Mo(IV)

Keywords: Atom Transfer, Molybdenum, Polymerization, Radical, Styrene Molybdène, Polymérisation, Radicaux libres, Styrène, Transfert d'atomes

A controlled polymerization of styrene has been achieved under ATRP conditions by using a phosphine-containing Mo III /Mo IV system. Linearity of the Mn vs. conversion plot and low PDI's are observed in bulk, in a 30% (v/v) PhCl solution, and in a chain extension experiment when using MoCl3(PMe3)3/BIB. Controlled polymerization is also observed for a reverse ATRP experiment starting from MoCl4(PMe3)3/AIBN. The absence of chain transfer and termination is confirmed by NMR and MALDI-TOF analyses of polymers obtained by employing BEB and BIB as initiators.

Introduction

Controlled radical polymerization is becoming a heavily investigated new area of research, in view of the combined tolerance to polar functionalities and the ability to develop well-defined macromolecular architectures [START_REF] Matyjaszewski | Controlled Radical Polymerization[END_REF][START_REF] Patten | [END_REF]. After the concept of the persistent radical effect (PRE) [3,4] was fruitfully applied to the development of so-called Stable Free Radical Polymerization (SFRP) methodologies [5][6][7][8][9][10][11], the field of controlled radical polymerization has experienced tremendous advances following the independent reports by the groups of Sawamoto and Matyjaszewski [12,13] that the PRE can be realized by an atom transfer process whereby a halogen-capped dormant chain and a metal complex are in fast equilibrium with a reactive free radical and a metal halide complex. This process has been termed Atom Transfer Radical Polymerization (ATRP), see Scheme 1.

In a recent contribution [14], we have applied the analysis of homolytic Bond Dissociation Energies (BDE) to predict the ability of a metal complex to control radical polymerization of styrene. We have shown that, in agreement with the theoretical predictions, a family of Mo III /Mo IV cyclopentadienyl complexes controls the radical polymerization by both the SFRP and the ATRP protocols, while a system based on Mo II /Mo III produces a largely insufficient radical concentration to sustain the polymerization process. In addition, the polymerization with Mo III /Mo IV has been found to be polluted by a side reaction of hydride transfer. The use of a simple kinetic model has allowed to rationalize the variable degree of this Catalytic Chain Transfer (CCT) reaction as a function of reaction conditions (notably the Mo III /Mo IV ratio) when styrene polymerization is conducted with butadiene-containing cyclopentadienylmolybdenum complexes.

Since the variable of utmost importance for the regulation of the PRE, namely the BDE, appeared quite sensitive to the metal oxidation state but not so much to the nature of the ancillary ligands, [14] we wondered whether the controlled radical polymerization of styrene could also be achieved in the presence of cyclopentadienyl-free Mo III /Mo IV systems.

Numerous compounds of type MoCl3L3 and MoCl4Ln (L = ethers, thioethers, phosphines, etc.; n = =2 or 3) are known [START_REF] Wilkinson | Comprehensive Coordination Chemistry[END_REF] and are more easily accessible than the cyclopentadienylcontaining analogues [START_REF] Stoffelbach | [END_REF]. In this contribution, we report our initial studies indicating that, indeed, polystyrene is obtained by a controlled polymerization procedure under a variety of conditions, with control being exerted by phosphine-containing Mo III /Mo IV complexes.

Experimental Section

General Remarks

All manipulations were carried out under an atmosphere of dry and oxygen-free argon with standard Schlenk techniques. Styrene was washed with a aqueous NaOH solution (10%), neutralized with water, dried with MgSO4 and then distilled at 25°C under reduced pressure.

Chlorobenzene was bubbled with dry argon for more than 15 min immediately before use.

Molecular weights and molecular weight distributions were measured with a Gynkoteck P580 apparatus using THF as eluent (1mL/min) and being equipped with a refractometer and 2 columns B Jordi DVB (range 1000-1000000). MALDI-TOF mass spectrometric analyses were performed with a Perkin Elmer Voyager -DE STR. The matrix was the Dithranol and PS samples were cationized with silver salt (AgTFA). Compouds MoCl3(PMe3)3, 1 [17], and MoCl4(PMe3)3, 2 [START_REF] Stoffelbach | [END_REF]18], were obtained according to previously described synthetic procedures. (1-Bromoethyl)benzene (BEB) and ethyl-2-bromoisobutyrate (BIB) were purchased from Aldrich Chemical Co and degassed prior to use. Azobis(isobutyronitrile) (AIBN) was purchased from JANSSEN and recrystallized twice from MeOH before use.

ATRP polymerizations.

All ATRP polymerization reactions were performed following the same experimental procedure. The complex was added to a 50 mL Schlenk tube equipped with a stirring bar.

Styrene and the initiator (BEB or BIB) were added to the reaction flask by a syringe. The Schlenk tube was then immersed in an oil bath heated at 90°C. Aliquots were withdrawn periodically for a reaction monitoring. These were dried by room temperature evaporation followed by oven drying at 100°C for 8 h for the calculation of the conversion. The GPC analyses were carried out on the crude dried polymers after redissolution in THF.

Reverse ATRP polymerization.

Complex 2 and AIBN were added to a 50 mL Schlenk tube equipped with a stirring bar.

Chlorobenzene and styrene were added by a syringe. The Schlenk tube was immersed in an oil bath heated at 120°C for 15 min, then it transferred to another bath at 90°C. Aliquots were withdrawn periodically for a reaction monitoring by GPC, as described above.

Results and Discussion

Styrene is smoothly polymerized in the presence of compound 1 and PhCH(CH3)Br (BEB) in a 1:1 ratio at 90° in bulk. The progression of polydispersity and number average molecular weight as a function of conversion is shown in Table 1 and graphically in Figure 1.

The continuous decrease of PDI as the conversion increases, reaching values under 1.5 at 50% conversion, is as expected in the presence of a PRE. The Mn are quite close to the theoretical values. A similar behavior is observed when the polymerization is conducted in chlorobenzene, see Table 2 and Figure 2. The polymers obtained in solution have slightly lower PDI than those obtained in bulk. The rate of polymerization, on the other hand, is greater for the bulk experiment (see Figure 3), as expected from the increased monomer and radical concentrations. <Figure 1 and Table 1> <Figure 2 and Table 2> <Figure 3>

The absence of chain transfer in this polymerization process has been confirmed by the NMR and MS analyses of the polymer. The 1 H NMR spectrum on the polymers obtained at small conversions shows clearly the presence of brominated chain-ends (peak at 4.3-4.6 ppm) [19] and the absence of chains terminated by vinyl groups (for which a resonance would be expected in the 6.0-6.4 ppm range) [20].

A MALDI-TOF-MS analysis of the polymer provides supporting evidence for the presence of Br-terminated chains, but shows also three additional families of polymer chains (see Figure 4a). The position and relative intensities of the four series of peaks are identical with those of the polymer obtained by using the CpMoCl2(PMe3)2/BEB system and analyzed under the same instrumental conditions [14]. The fours peaks at m/z = 3321.7, 3337.9, 3353.5, and 3379.6 in Figure 4a CH2CHPh and n = 28. The presence of Cl-terminated chains is not surprising, since the radical forming equilibrium (equation 1) can be reversed in principle by abstraction of either a bromine or a chloride atom. However, the Mo-Cl BDE is stronger than the Mo-Br BDE according to our calculations [14], consequently favoring the bromine atom transfer. Indeed, the Cl-terminated chains are less abundant than the Br-terminated chains. The most intense peaks, on the other hand, are those related to vinyl-terminated chains with either Ag + or Na + being provided by the matrix. Sodium contamination is a common phenomenon for MALDI-TOF analyses of polymers [21,22]. The related Na peaks of the Br-or Cl-capped polymer chains are either masked by other peaks or too small to be observed.

MoCl3(PMe3)3 + R-Br MoCl3Br(PMe3)3 + R • (1) <Figure 4>
The question arises as to the origin of these vinyl-terminated chains.

Dehydrobromination processes are known to occur readily for Br-terminated polymers upon contact with the silver salt and the Cl-terminated chains could analogously be dehydrochlorinated. This possibility seems also confirmed by the inability to observe vinylterminated end group resonances in the NMR. However, the alternative of a CCT process remains a possibility, since after the transfer process the resulting metal hydride would, upon insertion of the first styrene monomer, yield the same CH3CH(Ph)-end group provided by the BEB initiator. In order to settle this point, the polymerization was repeated with a different initiator, BIB, which differs from BEB in having 10 additional mass units. The result of the MALDI-TOF-MS analysis of the resulting polymer (see Figure 4b) shows quite clearly that all four families of polymer chains are shifted by exactly 10 mass units and consequently demonstrates that they are all initiated by BIB. In other words, there is no CCT in this polymerization process.

A common test for a living polymerization process is also the so-called "stop-and-go" experiment (chain extension). In a genuine "living" polymerization, after the chain growth process is stopped by lowering the temperature or by the complete monomer consumption, resuming the polymerization by warming or by adding new monomer restores the growth of the previously formed chains without initiating any new ones. The results shown in Table 3 and Figure 5 for a polymerization of styrene with 1/BEB clearly show that (i) the polymerization process takes place with the same rate before and after being stopped for ca.

14 h, indicating that the total number of chains is conserved; (ii) the Mn of the polymers before and after the stop fall on the same line and are close to the theoretical masses; (iii) the PDI continues to decrease after resumption of the polymerization. <Figure 5 and Table 3> A final test for the involvement of Equation 1 in determining the PRE for the living polymerization process has been the establishment of a controlled polymerization under "reverse" conditions. That is, equilibrium 1 can in principle be equally established starting from the left hand side, as shown above (regular ATRP process), or from the right hand side by exposing the preformed Mo(IV) complex to a radical (reverse ATRP). Complex MoCl3Br(PMe3)3 does not exist to the best of our knowledge and out attempts to synthesize it by Br2 oxidation of 1 have only yielded analytically impure products. However, the related complex MoCl4(PMe3)4, 2, is known [18] and we have recently improved its preparation procedure [START_REF] Stoffelbach | [END_REF]. A reverse ATRP experiment, the initial radical being provided by the thermal decomposition of AIBN, confirms that the polymerization process is controlled under these conditions, see Table 4 and Figure 6. The control of the radical concentration would be established by Equilibrium 2. The experimental molecular weight is higher than theoretical molecular weight, indicative that the initiator efficiency (AIBN) is less than 1. In addition, PDI decreases with conversion, as expected. The successful controlled polymerization under "reverse" ATRP conditions demonstrates that the PRE is established by use of the Mo III /Mo IV couple.

MoCl3(PMe3)3 + R-Cl MoCl4(PMe3)3 + R • (2)
<Figure 6 and Table 4> Complex MoCl3(THF)3, i.e. the synthetic precursor of complex 1, does not exert any control in the polymerization of styrene. In fact, a faster polymerization process having all the characteristics of cationic polymerization is observed when using this complex. We presume that the Lewis acidity of the Mo(III) center in this coordination compound may favor the opening process of the THF ligand to afford a Mo IV -OCH2CH2CH2CH2 + moiety faster than the atom transfer process. The zwitterionic species formed would then trigger the cationic polymerization mechanism.

No degree of control is witnessed in the radical polymerization of MMA with 1. The PMMA obtained at low conversions is characterized by a high Mn and a broad PDI. Our interpretation of this result is that Equilibrium 1 is shifted to a greater extent toward the right end side when R • = (Polymer-CH2)(CH3)C • -COOCH3, in agreement with a lower BDE of the corresponding C-Br bond and an uncontrolled radical polymerization caused by a large radical concentration.

Conclusions and Perspectives

In conclusion, we have reported here for the first time that the ATRP polymerization of styrene can be accomplished by simple coordination compounds of molybdenum, the PRE being established by a redox couple containing Mo III and Mo IV species. A controlled radical polymerization of styrene has been accomplished by using the lithium molybdate(V)

complexes [LiMo(NAr)é(C-N)R] (C-N = C6H4(CH2NMe2)-2; Ar = C6H3Pr i 2-2,6; R = C-N,
Me, CH2SiMe3, p-Tol) and benzyl chloride as initiator [23]. The direct implication of the Mo complex in the polymerization control mechanism was not unambiguously proven in that case, however. Current studies are aimed at the modification of the metal coordination sphere for achieving control in the radical polymerization of other monomers. a The polymerization was stopped after 740 min and resumed after 1580 min (see Figure 5a).

  , for instance, are attributed to the formulas MeCH(Ph)-(Sty)n-CH2CH(Ph)Br+Ag + , MeCH(Ph)-(Sty)n+1-CH=CH(Ph)+Ag + , MeCH(Ph)-(Sty)n+2-CH=CH(Ph)+Na + , and MeCH(Ph)-(Sty)n+1-CH2CH(Ph)Cl+Ag + , respectively, with Sty =

Figure 1 .

 1 Figure 1. Plot of Mn (black squares) and PDI (open triangles) vs. conversion for the ATRP

Figure 2 .

 2 Figure 2. Plot of Mn (black squares) and PDI (open triangles) vs. conversion for the ATRP

Figure 3 .

 3 Figure 3. Plots of ln([M]0/[M]) vs. time for the bulk and solution (30% v/v in PhCl) ATRP

Figure 4 .

 4 Figure 4. Details of the MALDI-TOF spectra for polystyrene obtained using BEB (a) or BIB

Figure 5 .

 5 Figure 5. Plots of (a) conversion vs. time and (b) Mn (black squares) and PDI (open

Figure 6 .

 6 Figure 6. Plot of Mn (black squares) and PDI (open triangles) vs. conversion for the reverse

  Figure 1

Figure

  Figure 4

  Figure 5

Table 1 .

 1 Results of the ATRP of styrene in bulk with MoCl3(PMe3)3 and BEB. T

	= 90°C, [styrene]/[BEB]/[Mo] = 196/1/1.
	Time (min) Conversion (%)	Mn (exp.)	Mn (theoret.)	PDI
	262	17,0	3752	3460	1,6
	580	24.3	4867	4968	1,5
	1025	36,8	5873	7505	1,5
	1317	47,5	7538	9698	1,5
	1625	55,4	9197	11301	1,4

Table 2 .

 2 Results of the ATRP of styrene in PhCl solution (30% v/v) with MoCl3(PMe3)3 and

	BEB. T = 90°C, [styrene]/[BEB]/[Mo] = 235/1/1.
	Time (min)	Conversion (%)	Mn (exp.)	Mn (theoret.)	PDI
	385	18,9	6388	4617	1,5
	1072	24.3	7396	7108	1,5
	1332	34,5	7473	8442	1,5
	1702	37,3	8103	9129	1,4
	3181	49	10095	12000	1,3

Table 3 .

 3 Results of the stop-and-go ATRP of styrene in bulk with MoCl3(PMe3)3 and BEB. T

		= 90°C, [styrene]/[BEB]/[Mo] = 239/1/1. a
	Time (min) Conversion (%)	Mn (exp.)	Mn (theoret.)	PDI
	150	11.7	4097	2903	2
	303	24.3	4693	3666	1.7
	425	17.6	4747	4365	1.8
	565	20.1	5682	4994	1.7
	740	23	6574	5717	1.6
	1580	22.9	5719	5696	1.7
	1775	26.9	7119	6680	1.6
	1985	30.3	7981	7526	1.5
	2105	31.9	7706	7938	1.5

Table 4 .

 4 Results of the reverse ATRP of styrene in bulk with MoCl4(PMe3)3 and AIBN. T

		= 120°C for 15 min, then T = 90°C; [styrene]/[AIBN]/[Mo] = 240/0.5/1.
	Time (min) Conversion (%)	Mn (exp.)	Mn (theoret.)	PDI
	488	18.2	7895	4147	2.3
	1373	29.6	9350	7382	2.0
	2123	38.1	11206	9507	1.9
	5643	55.2	15080	13764	1.6

  4

	% Intensity	20 40 60 80 100	3337.9 3353.5 3321.7	3379.6	3442.0 3457.8 3425.5	3483.0	3546.2 3530.0	3562.2	3588.6	3650.3 3666.5 3634.0	3684.0	3754.5 3770.5 3738.1
		0	3300		3400	3500			3600	3700	3800
										Mass (m/z)
	% Intensity	20 40 60 80 100	3347.5 3364.2 3331.1	3387.9	3451.5 3468.4 3435.4	3494.1	3555.7 3572.4 3539.5	3597.2	3659.8 3676.4 3643.7	3703.0	3764.1 3747.7
		0	3300		3400		3500		3600	3700	3800
										Mass (m/z)

Acknowledgments

We are grateful to the CNRS for support this work (Programme Catalyse pour l'Industrie et l'Environnement) and for a BDI Ph.D. fellowship to F.S. Support from the Ministery of

Research and COST D17 (Working group on Transition Metal Mediated Living Radical

Polymerization) is also gratefully acknowledged.

Scheme 1


Gabriel Boitel-Aullen

Pr Laure F Fillaud

François Huet

Dr Iwona Nierengarten
email: nierengarten@unistra.fr

Béatrice Delavaux- Nicot

Jean-François Nierengarten

Emmanuel Maisonhaute
email: emmanuel.maisonhaute@sorbonne-universite.fr

Professor J.-M Savéant

Dr Mr G Boitel-Aullen

Pr E Huet

Maisonhaute

Dr J.-F Nierengarten

Dr B Delavaux-Nicot

Electron Transfer Inside a Decaferrocenylated Rotaxane Analyzed by Fast Scan Cyclic Voltammetry and Impedance Spectroscopy

Keywords: cyclic voltammetry, impedance, self-assembled monolayers, rotaxane, electron transfer

The electrochemical behavior of a giant rotaxane bearing 10 electroactive ferrocene centers has been analyzed by both fast scan cyclic voltammetry and impedance measurements. The measurements support the hypothesis that all redox centers are not equivalent and display different electron transfer rate constants, which would arise from several possible conformations of the adsorbed molecule. Effectively using such systems in molecular devices thus necessitates to minimize or at least control those dispersions.

Introduction

Approaches such as those developed by Jean-Michel Savéant and his school greatly contributed to the understanding of electron transfer mechanisms inside complex systems. Those may be revealed by following specific observables such as peak potential or current in cyclic voltammetry (CV). [START_REF] Savéant | Elements of Molecular and Biomolecular Electrochemistry: An Electrochemical Approach to Electron Transfer Chemistry 2 nd edition[END_REF] Deviations from this general framework often reveal interactions or heterogeneities of the system, as may be observed with modified electrodes. For example, it has been recognized from a long time that even if selfassembly of thiolated systems onto gold surfaces produces very rapidly (within minutes) a large coverage, obtaining reproducible results necessitates long immersion times (usually one night). [START_REF] Nuzzo | [END_REF][3][4][5] Molecule/surface interactions and ordering indeed resort to many dynamic processes that cover multiple timescales, and reaching equilibrium may take up to a few days. [3][4][5][6] For ideal isolated redox centers that transfer a single electron, the CV peak should present a symmetric bell-shape with a full width at half maximum of 91 mV at room temperature. [START_REF] Savéant | Elements of Molecular and Biomolecular Electrochemistry: An Electrochemical Approach to Electron Transfer Chemistry 2 nd edition[END_REF]7,8] This behavior is however seldom observed even within self-assembled monolayers (SAMs) of rather simple systems because redox centers may interact together or with their matrix. Many interesting models have been proposed to take these interactions into account and also sometimes include kinetic aspects. For example, Laviron considered Frumkin isotherms [9] while Smith et al. calculated the influence of electrostatic interactions. [10] More recently, Levillain et al. proposed a generalized interaction model that can take into account random or non-random distribution of electroactive centers. [11][12][13][14][15] In all cases, the response tends towards ideality when the surfacic concentration decreases. Nevertheless, despite the care in electrode preparation, monolayers are sometimes not uniform, and present defects such as pinholes [16] or distribution of different environments or distances from the electrode for the redox centers. [17] Those may translate in distributions of apparent standard potentials E 0 and kinetic rate constants k ET 0 as examined initially by Albery and then by Murray for self-assembed monolayers. [18,19] Disentangling all effects benefits from analyzing the systems with a range of techniques. In this paper, we have examined in details the kinetic dispersion of the decaferrocenylated rotaxane 1 presented in Figure 1 adsorbed onto a gold ultramicroelectrode. Rotaxanes represent a new avenue in supramolecular chemistry since they offer the possibility to tailor either the axle or the macrocyclic ring with many functions, in particular redox centers. [20] A wide field of potential applications are concerned, such as molecular electronics, [21] sensing [22] or artificial muscles [23][24][25] for example. Those systems allow to propagate an electrochemical perturbation over long distances through electron hopping from the ferrocene layer close to the electrode towards centers positioned at larger distances from the electrode. Practical applications require fast signal transduction, which translates into the necessity to develop adequate instrumentation and methodologies. [26][27][START_REF] Bard | Electrochemical Methods: Fundamentals and Applications 2 nd edition[END_REF][START_REF] Baranski | [END_REF][30][31][32] The synthesis, surface modification procedure and CV characterization of molecule 1 were already examined earlier, [33] It was demonstrated with fast scan CV that hopping within this structure is extremely fast and then transparent in the measurements. Here we will focus on the complementarity of CV and impedance measurements using high frequencies to analyze slight kinetic deviations from ideality. Figure 2 presents a typical CV at 2000 Vs -1 acquired in acetonitrile + 0.50 M tetrabutylammonium hexafluorophosphate for 1 adsorbed onto a 10400 µm 2 gold ultramicroelectrode during 48 hours. In this low scan rate regime, the peaks present a bellshape with a mean peak potential value of +0.239 ± 0.003 V vs Pt. At 20000 Vs -1 , the oscillations at the potential inversions are due to ohmic drop compensation that is necessary to obtain reliable data. The molecular coverage is estimated by integration of oxidation and reduction peaks at Γ 0 = 98 ± 3 nmolm -2 . Thus each molecule occupies in average an area of 17.0 ± 0.4 nm 2 , confirming formation of a monolayer. The peak to peak potential difference at low scan rate is only 8 mV, very close to the theoretical one of 0 mV for ideal Nernstian systems. [START_REF] Savéant | Elements of Molecular and Biomolecular Electrochemistry: An Electrochemical Approach to Electron Transfer Chemistry 2 nd edition[END_REF]9] At 20000 Vs -1 , the peaks shift from equilibrium position because the characteristic time scale of the voltammetry becomes comparable to the electron transfer one. CVs in the range 10-40000 Vs -1 are provided in Figure S1, and the corresponding peak potentials are reported in Figure 2b in the so-called "trumpet plots". [9] Their fit with Butler-Volmer theory provides an average rate constant k ET 0 = 65000  5000 s -1 at E = E 0 . The apparent disagreement at scan rates above 20000 Vs -1 is purely instrumental and caused by the bandwidth limitation of the electrochemical equipment. Nevertheless, simulation of the oxidation and reduction peaks as displayed in Figure 2 (dashed lines) reveals that the peak width is larger than the theoretical one, and that this deviation is more apparent at large scan rates. Indeed, at 20000 Vs -1 , a full-width at half maximum of 178 ± 4 mV was found, whereas theoretical values expected from Butler-Volmer (BV) and Marcus-Hush-Chidsey (MHC) theories are of 120 mV and 129 mV, respectively. In fact, MHC theory is more accurate and physically significant than Butler-Volmer one. [27,[34][35][36] In this framework, at large overpotentials k ET departs from its intrinsic exponential dependence with the electrode potential to potentially level out. This translates as broader CV shapes. [36][37][38][39] However, here overpotentials of less than 100 mV are concerned until 20000 Vs - 1 , so that the larger full width at half maximum cannot be attributed to Marcus-Hush-Chidsey effects. These effects may only explain the small "tail" in the peak as confirmed by the numerical simulations presented in Figure S2.

Results and Discussion

We underline that such deviations remain tiny and are frequently observed for adsorbed systems. Considering that the density of redox centers is low and that a large supporting electrolyte concentration is used, we will below neglect their interactions even if we are aware that this may be an oversimplification. [14] In the following, in order to examine more deeply the origin of this slight discrepancy, we complement the CV results with impedance ones performed onto exactly the same modified electrode and under the same conditions. Indeed, impedance provides the possibility to examine the behavior of a system at a given potential and is thus complementary to cyclic voltammetry. [START_REF] Orazem | Electrochemical Impedance Spectroscopy[END_REF] For that, it is necessary that the system does not degrade, which limits the potential excursion. To check the SAM integrity, a control cyclic voltammogram was performed after each impedance spectrum acquisition. We observed that the surface coverage did not decrease by more than 5% compared to the one measured at the beginning of the experiment. Electrochemical impedance spectroscopy has been less used than cyclic voltammetry in molecular electrochemistry though Savéant et al. punctually demonstrated its interest. [START_REF] Garreau | [END_REF]42] For selfassembled monolayers, Creager et al. proposed in 1998 impedance as an alternative method to assess electron transfer rates within self-assembled monolayers. [14,[43][44][45] Here, the electrode is polarized at a potential E m , onto which is added a small sinusoidal perturbation of amplitude E, so that:

 e (1)
with ω the angular frequency. The authors relied on the equivalent electrical model enclosed in the red box in Scheme 1a and we recall below the main formulations and conclusions. In this model, the complex impedance Z of the system may be expressed as:

° ° ° ° (2)
Where Γ 0 is the density of molecules having n redox sites transferring electrons at the rate constant k ET 0 onto the electrode of surface S ( is the charge transfer coefficient taken as 0.5 in all this study, R = 8.314 JK -1 mol -1 , F = 96485 C.mol -1 and T = 293 K). At low frequencies, faradaic processes are at equilibrium with the electrode potential and can thus be modeled with a potentialdependent capacitance proportional to the surface coverage:

° ° (3)
At very high frequencies, charge transfer kinetics interferes, and for k ET 0 << ω the system is purely resistive, the charge transfer resistance R CT being expressed as:

° ° (4)
One can notice that when the electrode is polarized at the standard potential of the redox process, that is, if E m = E°, equation (3) reduces to equation ( 5), and the rate constant k ET 0 can be determined with equation ( 6):

(5)

For real systems that display heterogeneities, capacitances are often replaced by a constant phase element (CPE) so that:

The parameter  (0 ≤  ≤ 1) reflects the heterogeneities. [START_REF] Orazem | Electrochemical Impedance Spectroscopy[END_REF] When  = 1, Q CPE is a pure capacitance, and conversely when  = 0, Z CPE is a pure resistance.

In practice, one has to handle also the solution resistance R sol and double layer capacitance C DL so that the global equivalent impedance for an ideal system is now expressed as: (8) This formulation demonstrates that at low frequencies or equivalently for Nernstian systems, the impedance is purely capacitive, being the sum of the adsorption and double layer capacitance. Hence impedance will be sensitive to adsorbed centers only provided that they are in sufficient number so that C ads is not negligible compared to C DL . At very high frequencies the response is dominated by the electrolyte resistance as usual in any impedance measurement. Therefore, the impedance is sensitive to the electron transfer kinetics only provided that the quantity R CT C ads is non negligible compared to R sol C ads and R sol C DL as detailed further in SI. These conclusions are reminiscent from those encountered in fast scan cyclic voltammetry for which one has to fulfill k ET 0  1/R sol C DL to observe a significant peak shift. One difference however is that in CV, Γ 0 only plays a role in the signal/noise resolution since the peak current is itself proportional to Γ 0 but does not appear directly in the kinetic measurements. Conversely, it plays a major role in impedance through the term C ads which is important also in signal/noise resolution but also for kinetic determination through the term R CT C ads . Whereas Creager et al. chose some frequencies and recorded the impedance in the so-called AC voltammetry configuration, we wished to avoid polarization of the electrode at oxidative potentials for a too long time to prevent SAM degradation. [14,43] We thus investigated a modified electrode by impedance spectroscopy in the frequency range 10 2 -10 7 Hz at fixed potentials. Figure 3 presents the results obtained at two potentials for the same electrode. We chose two complementary representations to interpret and fit our data. In the first one impedance modulus and phase are represented as a function of frequency in the classical Bode plots. In the second one, proposed by Bueno et al., [46][47][48] the so-called complex capacitance C * is calculated with equation ( 9): * j "

The real (C') and imaginary (C'' or -C") capacitances can thus be represented in semi-log plots as a function of frequency. Alternatively, -C" versus C' Nyquist plots are provided in SI. The advantage of such approach is to provide a convenient visualization of data over a large range of frequencies. At E << E 0 (cf. Figure 3, cyan dots, see also Figure S3-S6) C ads = 0. The response is dominated by the R sol and C DL components which is translated into a peak in the -C'' component at  ≃ R sol C DL (see SI for the analytical formulation). However, as observed for other self-assembled monolayers, the phase does not reach -90° for low frequencies as should be expected for a pure RC system (cf. Figure 3b). This is attributed to ion permeation into the layer, a potential-dependent artefact first described by Lennox et al. [49][50][51] . This can be accounted for by adding the additional high impedance circuits represented in yellow in Scheme 1 and further depicted in SI. Very large impedances are involved in those additional elements, so that they do not influence the behavior and fits in the high frequency region. They are better evidenced in the range 0.1-100 Hz (Figure S3-S6) that could be explored in this potential range. This behavior at low frequencies also explains the poor resolution of the faradaic peaks observed at low scan rates (i.e. 10 Vs -1 , see Figure S1). The equivalent circuit is potential dependent and simplifies to a single resistance R surf at E  E 0 . Otherwise, we note that the value of C DL (0.731 nF at -0.150 V vs Pt polarization) is in agreement with the one that can be assessed though with less precision by cyclic voltammetry (0.75 nF at -0.150 V vs Pt on the CV at 2000 Vs -1 ). It will not be discussed further below and we focus now onto the high frequency range (10 2 -10 7 Hz).

When E gets closer to E 0 (blue dots in Figure 3, see also Figures S7-S14), the phase shift transition occurs at lower frequencies, in agreement with apparition of the additional capacitance C ads . Impedance data were then analyzed in the framework presented above. In Figures 3a-d (green line), a single R CT C ads circuit is added. A discrepancy is observed in the Bode plot and emphasized in the C" fits. Therefore, we progressively added other R CTi C adsi elements to include faster and slower electron transfers as suggested by Creager et al. and displayed in Scheme 1. [44] The progressive improvement of the fit is presented in SI and in Figure 3. For 4 components, an excellent fit is obtained (see Scheme 1). The calculated k ETi 0 and corresponding Γ i 0 for one and four components are given in Table 1. In all fits, we note that  ≥ 0.890 for expected C DL and C ads capacitances, so that the proposed electrical equivalent schemes are physically relevant.

Table 1. Values of kinetic parameters kETi 0 and corresponding molecular coverages Γi 0 obtained at E = E 0 . Fits for impedance spectroscopy and simulation parameters used for reconstruction for cyclic voltammetry with 1 and 4 populations. Similarly, we analyzed further the oxidation and reduction peaks of the CVs at 4 different scan rates by incorporating 4 different rate constants. The results are presented in Figure 4 and Table 1 for 20000 Vs -1 and in Figures S15-18 for lower scan rates. A very accurate reconstruction of both waves is observed, and the obtained values for rate constants and surface coverages are comparable to those obtained with impedance, confirming that the dispersion indeed originates from a distribution of kinetic rate constants, probably due to slightly different adsorption configurations onto the gold electrode.

In our previous study, we analyzed the kinetics for different coverages and found that electron transfer was faster for less dense monolayers. All k ET 0 were however found in the range 1.7-5.8x10 5 s -1 whereas coverages were larger. The discrepancy is thus significant. Albeit there is no apparent influence of counter ion limitation, previous experiments were performed with tetraethylammonium tetrafluoroborate instead of tetrabutylamonium hexafluorophosphate that was used here. One possibility is that PF 6 -and BF 4 -contribute in the global reorganization energy and thus may influence k ET 0 . We suggest alternatively that in the present case, the contributions from different populations are modified when electrode preparation, adsorption time and rinsing procedures are different. Surface roughness may also play a role that will be scrutinized in the future. This is probably involved by changes in the average distance between redox centers and the electrode. In the complex assemblies presented by the rotaxane, there are indeed many degrees of freedom. The ring may translate onto the axle, and the anchor is very flexible. Overall, the precise conformation of the adsorbed molecule is unknown and thus the distance between the ferrocene entities and the electrode, which regulates the electronic coupling and finally k ET 0 . For future integration of these giant supramolecular structures in devices, it will be essential to further tailor their interaction with the electrode to impose a single conformation. 1.

Conclusion

In this contribution, we demonstrated that impedance measurements performed in a large frequency range are complementary to voltammetric ones for investigating fast electron transfer within a complex supramolecular system. We showed with our combined approach that for the complex molecule 1, the kinetic behavior is complex and that a distribution of different populations with different rate constants on the electrode surface should be considered. We expect in the future to push further the methodology and include the interactions and possible recognition and conformation changes in the analysis. [20,52,53] In the example examined therein, CV and impedance data were consistent, but we expect that this may not be the case in the future for other systems, for example those that display molecular movements. The results obtained on molecule 1 inform that even if flexibility of molecular devices may be an advantage compared to those used in conventional electronics, this may also introduce kinetic dispersions deleterious for practical applications. Solving these issues may be addressed either by appropriate molecular design or by the development of new surface preparation procedures.

Experimental Section Electrode preparation and modification

Gold ball ultramicroelectrodes were made by sealing a 30µm gold wire inside a glass pipet. About 100 µm of the wire protruded and was them melted in a butane flame. The electrode area was estimated by performing cyclic voltammetry in a ferrocene solution as described by Compton et al. [54] Prior to each experiment the gold surface was cleaned by applying 3 cycles in the range -0.60 to 1.20 V vs Pt in a 0.50 M H2SO4 solution. The electrode surface was then reconstructed by flame annealing. We observed that this procedure provided the best reproducibility. Molecule 1 was then adsorbed onto the electrode by immersion into a 10 µM solution of 1 in dichloromethane for at least 40 h in a fridge. Then, the electrode was rinsed and transferred in pure dichloromethane for 30 mins to remove any physisorbed molecule. The electrode was then directly transferred to the electrolyte solution.

Electrochemical measurements

For each electrode, all CV and impedance measurements were performed the same day in acetonitrile (VWR) + 0.50 M tetrabutylammonium hexafluorophosphate (Alfa Aesar, 98%) as supporting electrolyte. A platinum wire was used as counter electrode, and another platinum wire was used as a reference electrode. We indeed observed artefacts in the measurements at high frequencies with a classical Saturated Calomel Electrode (SCE) as often observed. [55] The Pt reference was calibrated versus SCE prior each experiment (EPt = 0.203 V vs ESCE). Impedance measurements were performed with an Autolab potentiostat equipped with the ECI10M module that allows impedance acquisitions up to 10 MHz. Prior each impedance spectrum acquisition, a control CV at 2000 Vs -1 was performed to check the SAM integrity. 6 frequencies per decade with an excitation amplitude of 10 mV were first acquired in the range 0.1 Hz-10 MHz for 3 DC potentials below E 0 . Then, the same procedure was applied at 3 potentials E close to E 0 but in the range 10 Hz-10 MHz to avoid degrading the SAM at low frequencies. The validity of impedance measurements was verified with Kramers-Krönig relations. Deviations at frequencies larger than 1 MHz are ascribed to limitations of the potentiostat. Fast scan measurements were performed in the range 10 -40000 Vs -1 with a home-made potentiostat allowing ohmic drop compensation. [56] All experiments were performed at room temperature (293 K).

Figure 1 .

 1 Figure 1. Molecule 1 presents 10 ferrocene units that display ultrafast electron

Figure 2 .

 2 Figure 2. a) Cyclic voltammograms at 2000 (red) and 20000 Vs -1 (blue) of molecule 1 adsorbed on a gold microelectrode recorded in acetonitrile + 0.50 M tetrabutylammonium hexafluorophosphate. Dashed lines: fits with kET 0 = 65000 s -1 ,  = 0.5 using Butler-Volmer theory. b) Dots: Trumpet plots recording the forward (black) and backward (blue) peak potentials at different scan rates. Red lines: theoretical variations according to Butler-Volmer theory with a predominant rate constant of kET 0 = 65000 s -1 .

Scheme 1 .

 1 Scheme 1. Electrical equivalent schemes of the electrochemical cell. Rsol: solution resistance. CDL: double layer capacitance. a) At E  E 0 several populations i 0 of electron transfer rate constants kETi 0 may contribute to the signal through different RCTi and Cadsi. These elements are absent at E << E 0 in (b). Deviations at low frequencies are taken into account by additional high impedance elements represented in yellow (see text).

Figure 3 .

 3 Figure 3. Analysis of impedance data acquired at Em = -0.150 V vs Pt << E 0 (cyan dots) and Em = +0.240 V vs Pt = E 0 (blue dots) onto a gold ball ultramicroelectrode with an area of 10400 µm 2 . Pink line: simulation without faradaic process. Green line: simulations with a single faradaic process (RCT = 908 Ω ; Cads = 8.01 nF,  = 0.912) ; Rsol = 1190 Ω ; CDL = 1.27 nF, DL = 0.948). Red line: simulations with 4 populations, each having its own RCTi and Cadsi value (RCT1 = 77700 Ω ; Cads1 = 1.64 nF, 1 = 0.891 ; RCT2 = 12000 Ω ; Cads2 = 1.84 nF, 2 = 0.940 ; RCT3 = 1870 Ω ; Cads3 = 2.79 nF, 3 = 0.938 ; RCT4 = 758 Ω ; Cads4 = 1.69 nF, 4 = 0.916 ; Rsol = 1190 Ω ; CDL = 1.18 nF, DL = 0.947). a) Impedance b) Phase. c) Real capacitance C' d) Imaginary capacitance -C".

Figure 4 .

 4 Figure 4. Reconstruction of the CV oxidation and reduction peaks at 20000 Vs - 1 taking into account 4 populations. Γi 0 and kETi 0 values are given in Table1.

Acknowledgements

This worked was founded by (grant FastGiant ANR-17-CE07-0012-01)

Data analysis

Simulations of CVs were performed with a home-made Matlab program (Butler-Volmer) or with the Digielch® software (Marcus). For impedance data fitting and simulations, we used the 1.0.4 version of the Matlab (R2019a) program ZfitGUI developed by Jean-Luc Dellis (Laboratoire de la Physique de la Matière Condensée, UFR des Sciences, Amiens, France). [START_REF] Dellis | Laboratoire De La Physique De La Matière Condensée[END_REF] Mathematica® was used for establishing analytical expressions given in the supplementary information.

Entry for the Table of Contents

Fast scan cyclic voltammetry and high frequency electrochemical impedance spectroscopy are very complementary methods to analyze electron transfer processes in an electroactive giant rotaxane.