N,N-dimethylformamide (DMF) (≥99.9%) was purchased from Sacharlab. 1,4:3,6-dianhydro-d-glucitol (isosorbide) (IS) (≥99.9%) and bis(4-fluorophenyl) sulfone (DFDPS) (≥99.0%) were both purchased from Acros Organics. IS was employed after recrystallisation in acetone and dried over P₄O₁₀ in a desiccator. Acetic anhydride (AAn) (≥99.0%), 3-aminophenol (≥98.0%), maleic anhydride (MAn) (≥99. 0%), anhydrous potassium carbonate (K₂CO₃) (≥99.7%), anhydrous sodium acetate (NaOAC) (≥99.0%) and methanol (MeOH) (≥99.9%) were procured by Sigma Aldrich. These products were used as described. The REEs studied here, Cerium (Ce) and Lanthanum (La), were used without further purification.

The 300 MHz NMR spectra were recorded on a Bruker Advance spectrometer in 5 mm i.d. sample glass tubes using DMSO-d6 as solvent.

To obtain oligomer masses of 3000 g/mol, calculations of several polycondensation parameters were carried out using the following equations eq 1 and eq 2.

$M_n=M_0\times X_n+M_{\text{ext}}$ (eq 1)

with M_n = the number average molecular weight of oligomer;

M₀ = the molecular mass of repeating unit;

M_ext= the molecular mass of end-groups;

X_n = DP= Degrees of polymerization (the number of repeat units n).

Then, using the expression

$X_n=\left(1+r\right)/\left(1+r-2rp\right)$ (eq 2)

with r = stoichiometric ratio, p= degrees of advancement assuming 1 we can calculate the stoichiometric ratio $r=\left(X_n-1\right)/\left(X_n+1\right)=0.777$. Finally, we set the number of moles of excess monomer, to calculate the number of moles of default monomer by the relationship $r=n\left(\mathrm{default}reagent\right)/n\left(\text{excess reagent}\right)$.

A thermogravimetric analysis (TGA) was conducted using a TA Instruments TGA Q50 in a temperature range of 25˚C - 600˚C under nitrogen at a heating rate of 10˚C/min. Glass-transition temperatures (Tg) of the oligomers were determined using a Mettler-Toledo DSC822e differential scanning calorimeter at a heating rate of 10˚C/min under nitrogen atmosphere.

As illustrated in Figure 1 , a series of four oligo (ether sulfone)s with different end-groups (F, OH, NH₂ and maleimide) were prepared according to our published article [27] .

The polycondensation was performed by reacting 20.0 mmol of IS (1), 15.4 mmol of DFDPS (2), and 44 mmol of K₂CO₃ in 50 mL of DMF, into a 250-mL cylindrical two-neck glass reactor, equipped with a mechanical stirrer. 20 mL of toluene was dropwise added through a dropping funnel. The reaction mixture was heated until 140˚C so that a slow azeotropic distillation of the toluene was observable. The loss of toluene was replaced continuously by the dropping funnel. The temperature was then slowly increased to 160˚C for 24 h. The hydroxy end-group was controlled by ¹H NMR spectroscopy. Finally, the obtained viscous solution was precipitated into methanol. The precipitate was collected by filtration, washed thoroughly with cold methanol, and then dried in a vacuum. The oligomer P1 was isolated as a white powder with a 95% yield.

In a two-necked reactor equipped with a N₂ inlet and a mechanical stirrer, a mixture of 15.4 mmol of IS (1), 20.0 mmol of DFDPS (2), 33.88 mmol of K₂CO₃ and 50 mL of DMF was stirred and heated at 160˚C for 24 h. The fluoro end-group was controlled by proton NMR spectroscopy. The solution was then poured into methanol, filtrated and washed with methanol. After drying under vacuum, a white crystal was obtained with a 96% yield.

In a 100 mL two round-bottom flask equipped with N₂ inlet, 4 g of the resulting di-fluorinated oligomer P2, 0.60 g of 3-aminophenol and 4.00 mmol of K₂CO₃ were dissolved in 20 mL of DMF. The mixture was then stirred for 48 h at 130˚C. The reaction solution was poured into methanol, filtered, and dried under vacuum at 40˚C, giving a brown powder with a yield of 87%.

A 100-mL two-necked, round-bottomed flask equipped with a magnetic stirrer bar, a nitrogen-gas inlet tube, and a calcium chloride drying tube, was charged with 1.80 g of resulting diamine P3, 115.0 mg of maleic anhydride (MAn) and 10 mL of DMF. The mixture was stirred at room temperature for 5 h. The formed bismaleamic acid was then converted into bismaleimide via a chemical-imidization process. The amide acid cyclization was carried out by adding 0.3 mL of acetic anhydride and 20 mg of anhydrous sodium acetate to the intermediate solution, while stirring it at 100˚C for 1 h. The mixture was then stirred overnight at room temperature. Finally, the oligomer solution was slowly poured into methanol to form a precipitate. The precipitate was then filtered, washed thoroughly with methanol, and dried overnight under a vacuum at 80˚C, a yield of 85% was obtained.

EIS measurements were carried out using a SP-200 potentiostat (Biologic) using EC-Lab V11.43 software. This device generates the sinusoidal signal with the desired amplitude and frequency and recovers the real and imaginary parts of the impedance system under study. In these tests, the initial potential was E = −0.15 V, the frequency range was set from 1 kHz to 100 mHz, and an amplitude of 10 mV superimposed on the DC voltage. All measurements were performed in an identical solution and at room temperature to minimize the variations of resistance. Real and imaginary impedances were calculated and a Nyquist diagram was traced. The impedance values were adjusted using Scribner Associates “Zview” software.

The electrochemical measurements were performed in a Pyrex glass cell using a polymer-coated platinum electrode as the working electrode, an Ag/AgCl reference electrode, and an auxiliary platinum electrode (99% purity), used to close the electrical circuit in the electrochemical cell. For each measurement, the position of the electrodes remained the same. The electrochemical cell was placed in a Faraday cage to reduce the influence of electromagnetic waves. All measurements were performed with the polymer-modified platinum electrode immersed in a 0.2 M potassium citrate solution (pH 4.0) containing the rare earth ions after 25 min of equilibration.

Platinum electrodes were 1.2 × 1.2 cm square plates. The surface was rinsed with ethanol for 10 min, rinsed with ultrapure water, and then dried under flowing nitrogen. 5 µL of 1 w/w% dimethylsulfoxyde solution of P1, P2, P3 and P4 polymer were mixed separately and drop-casted on the electrode surface and then dried under nitrogen flow for 48 h, at room temperature.

The dihydroxy and difluoro-ether sulfone oligomers P1 and P2 were prepared with a molecular ratio calculation of isosorbide (Is) and 4,4’-difluorodiphenylsulfone (DFDPS) monomers to get a M_n of 3000 g/mol. The monomer polycondensation was performed in DMF in the presence of potassium carbonate at 160˚C, as reported previously in the literature [27] . The condensation reactions were monitored by ¹H NMR spectroscopy to obtain the desired hydroxylated and fluorinated oligomers in satisfactory yields and almost quantitative functionalization. The oligoethersulfones P1 and P2 were isolated as a white powder after precipitation in methanol and dried under vacuum.

The repeat unit molecular weight (M₀) of each oligomer was equal to 361.07 g/mol, and the molecular mass of end groups was 146.06 g/mol and 254.02 g/mol for P1 and P2, respectively. Since the average molecular weight (M_n) of both target oligoethersulfones P1 and P2 was fixed at 3000 g/mol, the number of repeating units n in their principal chains, was found to be around 8.

The chemical structures of these oligoethersulfones were characterized by ¹H NMR. The proton NMR spectra of (OH)-oligomer P1, presented in Figure 2 , exhibit the characteristic protons of isosorbide diol (H1 - H6) and DFDPS (H7 - H8). Moreover, the spectrum reveals the presence of low-intensity peaks corresponding to protons at the end groups of isosorbide (H1* - H6*) confirming the formation of hydroxyl-terminated oligomer. The proton NMR spectra of oligomer P2 clearly show the total protons characteristic of the repetitive unit with the presence of low-intensity peaks in the aromatic zone (H7* and H8*).

Fluorine atoms at oligomer P2 chain ends were successfully substituted with 3-aminophenol in the presence of K₂CO₃ in DMF at 130˚C for 48 h to afford the diamine-oligoethersulfone P3 ( Figure 1 ). The nucleophilic displacement reaction of F-atoms by OH-PhNH₂ was confirmed by the appearance of the characteristic signal of amine protons H9 located in ¹H NMR spectra at 5.30 ppm ( Figure 2 ), as well as the disappearance of the fluorine chain ends signals.

Oligomer containing maleimides end-groups P4 was finally synthesized with a good yield of 85%, according to a two-step procedure, which includes the condensation of the diamine P3 with maleic anhydride followed by the maleamic acid cyclization, as outlined in Figure 1 . The first reaction was performed at room temperature in DMF. This fast and exothermic reaction leads to the intermediate bismaleamic acid. The chemical imidization process was then performed with acetic anhydride and a basic catalyst like sodium acetate giving the bismaleimide P4. The ¹H NMR spectrum of the representative oligomer ( Figure 2 ), presents the total protons related to bismaleimide groups (H14 - H19) at the end of chains which are consistent with the proposed chemical structure for this oligomer.

The solubility of the resulting oligo (ether sulfone)s was tested by dissolving 10 mg of oligomers in 10 ml of solvent; in the case of insolubility, we assayed 1 mg in 1 ml of solvent. The results are summarized in Table 1 . All the obtained oligomers (P1 - P4) exhibited excellent solubility in aprotic polar solvents such as DMSO, DMF, and NMP. They are insoluble in heptane, methanol, ethyl acetate, diethyl ether, acetonitrile and water, and partially soluble in acetone and THF, in particular for the P4 oligomer, which is soluble.

The thermal properties of functional oligo (ether sulfone)s P1-P4 were determined by Thermogravimetric analysis (TGA) and Differential Scanning Calorimetry (DSC). DSC analysis revealed the amorphous behaviour of the oligomers with high glass transition temperature values (Tg) ranging from 160 to 176˚C, depending on the nature of the functional group at the end of oligomer chains. The Tg values of diamine and dihydroxyoligoethersulfones (P3, 176˚C) and (P1, 171˚C) are higher than the Tg of difluoro-oligomer (P2, 160˚C) ( Table 2 ). This result can be explained by the formation of inter- and intra-chain hydrogen bonds in P1 and P3 oligomers, leading to a decrease in chain mobility and flexibility with an increase in structural rigidity and glass transition temperature (Tg) [28] .

The TGA experiments ( Table 2 ) shows that all oligo (ether sulfone)s P1-P4 were thermally stable with decomposition temperatures for 5% weight losses (T_d5) higher than 300˚C, depending on the structure of the polymers. The highest thermal stability is observed for bismaleimide oligomer (P4) with a structure rich in aromatic rings (Td5 = 447˚C).

^a) After precipitation into MeOH. ^b) Determined by DSC under nitrogen at a heating rate of 10˚C/min. ^c) Temperature at 5 wt % loss.

The impedance spectra of platinum electrode/oligo (ether sulfone)s/electrolyte obtained in the presence of increasing concentrations of La ions in an acetate solution are shown on the Nyquist plot in Figure 3 , as solid points. An equivalent circuit modeled the electrical properties: Nyquist diagrams were compared with several electrical circuit models through Z-View software until the experimental data were fitted satisfactorily. The chosen model must fit the data with good precision and should not contain too many components. In our experiment, each spectrum shows the typical behavior of a semicircle associated with an equivalent circuit consisting of a resistor, R, in parallel with a constant phase element, CPE ( Figure 4 ). As the concentration of lanthanum increases, the diameter of the semicircles decreases. The right-hand (low-frequency) end of a semicircle corresponds to the value of the resistor. The values of the equivalent electrical circuit’s elements were obtained, corresponding to a correlation coefficient χ² equal to 10⁻³. A higher concentration of La ions decreases the polarization resistance, which generates a lower impedance. This result can be explained by the adsorption of La ions on the oligo (ether sulfone)s.

The calibration curve of the sensor was obtained by plotting the relative variation of the charge transfer resistance of each platinum electrode modified with P1, P2, P3, and P4 oligomer respectively as a function of the logarithmic concentration of La ion using the following equation: $\left|Rp0-Rp\right|/Rp0\left(\Delta R/R\right)$. This parameter was found to be linearly proportional to the logarithmic value of La concentrations (pM) ( Figure 5 ). To ascertain the limits of detection (LOD) and quantification (LOQ), the formulas Equation (3) and Equation (4), respectively, were used.

$\text{LOD}=\frac{3\times \text{Standard}\text{deviation}\text{of}\text{the}\text{intercept}}{\text{Slope}\text{of}\text{the}\text{calibration}\text{curve}}$ (eq 3)

$\text{LOQ}=\frac{10\times \text{Standard}\text{deviation}\text{of}\text{the}\text{intercept}}{\text{Slope}\text{of}\text{the}\text{calibration}\text{curve}}$ (eq 4)

As shown in Table 3 , the OH-oligo(ether-sulfone) (P1), presents a higher sensitivity value. The presence of the dihydroxyl groups at the end of P1 chains improves the hydrophilic properties of the polymer. Lanthanum ions present six coordination links with oxygen atoms of the isosorbide group, and the sulfone group as designed for 2-MNBBSH-Yterbium-complex ( Figure 6 ). Other trivalent ions were tested in this work and presented lower affinity for the 2-MNBBSH than the REE, Y³⁺, even at a 10 times higher concentration [29] .

The analytical performance of this impedimetric sensor for La detection was compared to that of previously published electrochemical sensors in Table 4 . A very low detection limit is obtained with EIS, compared to those obtained with potentiometric and voltammetric La sensors.

The obtained Nyquist plots in the complex plane are shown in Figure 7 . A significant decrease in impedance was observed after increasing cerium concentration. This observation confirms that the polarization resistance of the polymer/electrolyte interface decreases. This is attributed to chemical bonds between Ce ions and the oligo (ether sulfone) as described for La ions.

Calibration curves were obtained by plotting the relative variation of the polarization resistance as a function of Ce concentrations ( Figure 8 ). The relative change (ΔR/R) of the oligo (ether sulfone)s P1 - P4 modified electrodes is linearly proportional to the logarithmic value of Ce concentration. The detection sensitivity is defined as the slope of the ΔR/R calibration curves as a function of the logarithm of Ce concentration. Table 5 gathers the results obtained with the different oligo (ether sulfone)s, showing that the P1 oligomer presents a higher sensitivity. Due to hydrophilic properties, the presence of dihydroxyl groups in the structure of oligomer P1 improves the adsorption efficiency of Ce ions. The relative standard deviation on the relative variation of the polarization resistance is 5%. The repeatability between the three P1-based sensors was 10%.

The analytical performance of this impedimetric sensor for Ce detection was compared to that of previously published electrochemical sensors in Table 6 . A very low detection limit is obtained with EIS, compared to those obtained with voltammetric Ce sensors, except in [31] when an ion-imprinted polymer was used.