A Factorial Design and Simplex Optimization of a Bismuth Film Glassy Carbon Electrode for Cd(II) and Pb(II) Determination

Abstract

This work presents the use of a fractional two-level factorial design to determine the influence of different factors involved in the development of the square-wave anodic stripping voltammetry method. A fractional factorial design of eight different electrodes using a bismuth film glassy carbon electrode (BiFGCE) was formulated, and the impact of the factors on the analytical performance of the electrodes was determined. The impact of six factors (deposition potential, deposition time, potential step, amplitude, frequency, and the concentration of Bi(III)) was tested for Cd(II) and Pb(II) determination. Next, simplex optimization was carried out to improve the electroanalytical performance of the BiFGCE. The objective of simplex optimization was to simultaneously obtain a low limit of quantification, a wide linear concentration range, high sensitivity, high accuracy, and good precision for the new BiFGCE developed. By employing the latter approach, an optimization criterion was improved by several orders of magnitude. Moreover, the possible interference effect of different species was tested using two optimized BiFGCE. Furthermore, a real sample analysis of tap water with two optimized BiFGCEs was also carried out.

1. Introduction

Environmental pollution with toxic heavy metals is a problem that harms human health. Various analytical methods, such as spectroscopic and electrochemical methods, are used to detect heavy metals [1,2,3,4,5,6,7,8]. The standard techniques for trace heavy metal determination are inductively coupled plasma mass spectrometry, inductively coupled plasma atomic emission spectrometry, and atomic absorption spectrometry. In the field of electrochemical methods, the most effective technique to determine trace heavy metals is the stripping voltammetric technique. However, there is always room for improvement, where the aim is to develop an analytical method with low costs and short analysis time. One such method, which uses portable devices, is square-wave anodic stripping voltammetry (SWASV). It has been shown previously that the modification of a bare glassy carbon electrode (GCE) to design a bismuth film electrode (BiFGCE) leads to improved electroanalytical properties [9,10]. Using a BiFGCE, a low limit of detection (LOD), a low limit of quantification (LOQ), a wide linear concentration range, high sensitivity, good precision, and accuracy have been reported [2,5,11,12,13,14,15,16,17,18]. Cd(II) and Pb(II) can be determined simultaneously by means of a BiFGCE. Cd(II) and Pb(II) are already toxic to human health at low concentrations [14,15,19,20,21]. Thus, an electrode with low LODs and LOQs is needed.

For SWASV, different factors are involved in conducting an experiment, such as the deposition potential (E_dep), the deposition time (t_dep), the amplitude, the potential step (ΔE), and the frequency. All these affect analytical performance. To design a BiFGCE, the concentration of Bi(III) present in the solution is also an essential factor affecting the analysis. A supporting electrolyte frequently used in combination with BiFGCE is 0.1 M acetate buffer solution [9,16,22,23,24,25,26,27,28,29], and the BiFGCE is often prepared in situ [9,24,30].

A systematic evaluation of the above-mentioned factors that affect the final electroanalytical methods is usually not carried out. Research is mainly oriented towards designing new BiFGCEs empirically by means of trial-and-error experiments. For such, much effort could be devoted to an approach that does not guarantee an improvement in the electroanalytical method. On the other hand, factorial design (frequently also called experimental design) can be a means to avoid a trial-and-error approach and to determine the impact of a particular factor on the performance of the electroanalytical method [31,32].

Moreover, a one-by-one optimization is performed, which means that the value of one factor is changed while the other factors are held constant. The best analytical signal obtained is then taken as the optimized one for that factor. This procedure is then repeated for the other factors, where the previous optimized factor (using one-by-one optimization) is held constant at its optimized value. This optimization procedure involves numerous (sometimes unnecessary) experiments and leads to local improvement, i.e., rarely to the local maximum and even less often to the global maximum. A significantly better approach to achieving the optimum factors is to use an optimization method where all factors are considered simultaneously in the optimization procedure, e.g., simplex optimization [33,34].

In this work, different electroanalytical factors, i.e., E_dep, t_dep, amplitude, ΔE, and frequency, were tested using a fractional two-level factorial design. The impact of these factors on the analytical performance of the BiFGCE was determined by employing a differently designed BiFGCE. Next, the optimization of these factors in order to perform a SWASV experiment and of the Bi(III) concentration in order to design the BiFGCE was performed using simplex optimization. Factor optimization was performed simultaneously and not by using a one-by-one optimization approach. As a response, two different optimization criteria for simplex optimization were used. Possible interferences were checked, and the applicability of the optimized BiFGCEs for real sample analysis was demonstrated.

2. Experimental

2.1. Solution Preparation

A 0.1 M acetate buffer solution was used as a supporting electrolyte for all SWASV measurements. Standard stock solutions (1000 mg/L, supplied by Merck, Darmstadt, Germany) of Bi(III), Cd(II), Pb(II), Zn(II), Cu(II), As(III), Sn(II), and Sb(III) were used to prepare the solutions of standards. NaCl, KCl, KNO₃, and (NH₄)₂Fe(SO₄)₂$·$6H₂O were supplied by Sigma Aldrich (St. Louis, MO, USA). CaCl₂ and MgCl₂ were supplied by Acros Organics (Fair Lawn, NJ, USA). Na₂SO₄ was supplied by Honeywell Fluka (Charlotte, NC, USA), and CH₃COONa$·$3H₂O was supplied by Fisher Chemical (Pittsburgh, PA, USA). All chemicals were p.a. grade. Ultrapure water with a resistivity of 18.2 MΩ cm (ELGA, Lane End, UK) was used to prepare the solutions as required.

2.2. Electrochemical Measurements

Electrochemical measurements were performed at room temperature using a PalmSens4 (Houten, The Netherlands) potentiostat/galvanostat, controlled with PSTrace 5.9 software. A GCE (Cat. No. 6.1204.300, disc with a diameter of 3.0 mm) was used as the working electrode, and a platinum wire was used as the counter electrode. The reference electrode was a Ag/AgCl (saturated KCl) electrode. The volume of the solution for the electrochemical analyses was approximately 5.0 mL. The cell and the electrodes were supplied by Metrohm (Herisau, Switzerland).

The GCE was prepared before every electrochemical measurement. First, the GCE surface was polished with 0.05 µm Al₂O₃ powder (supplied by Buehler, IL, USA), then it was rinsed with ultrapure water and cleaned by ultrasound in ultrapure water for 1 min. Finally, the electrode was chemically/electrochemically cleaned in 15 wt.% HCl for 15 min by applying a potential of 0.6 V. The GCE was then thoroughly rinsed with ultrapure water and gently wiped with a paper tissue without touching the active GCE surface. To check the properties of the GCE, cyclic voltammetry of the hexacyanoferrate system with different scan rates was performed, as described in the Supplementary Material.

SWASV measurements were performed at different E_dep, t_dep, amplitude, ΔE, and frequency, as explained below. The equilibration time for all SWASV measurements was 15 s. After the SWASV measurement, a potential of 0.6 V was applied for 60 s to remove possible residual metals from the GCE surface. During the accumulation and cleaning step, the solution was stirred, whereas stirring was turned off during the equilibration step and the SWASV voltammogram measurements.

2.3. Determination of the Optimization Criterium

When optimizing an analytical method, it is important to obtain the best validation parameters, which means the lowest limit of detection (LOD), the lowest limit of quantification (LOQ), the highest sensitivity (in the current case, the highest slope of the linear calibration curve), the highest precision (the lowest relative standard deviation, RSD), the highest accuracy (the best recovery, Re, which is the closest to 100.0%), and the lowest value of the lower limit of the linear calibration range (LLCR) [35].

In this work, BiFGCE optimization was performed based on the optimization criterion (OC). BiFGCE was prepared in situ. The higher the OC, the better the BiFGCE is for Cd(II) and Pb(II) determination. The OC was determined by taking into account the sensitivity (the calibration curve’s slope), LLCR, the LOQ (not the LOD, in order to not have a similar analytical property involved twice), the RSD, and the absolute value of 100% − Re [33]. The Re can be lower or higher than 100.0%, thus the deviation from 100.0% was considered to be significant and that is why the |100.0% − Re| term has been employed. The LLCR and not the width of the linear concentration range was considered, as the determination of low concentrations of analytes is most important when using electroanalysis. Based on these validation parameters, Equation (1) was used to calculate the OC for the Cd(II) and Pb(II) analytes (OC_an, where subscript an stands for the analyte, i.e., OC_Cd(II) and OC_Pb(II)). As the highest sensitivity is desired, this validation parameter is in the numerator. On the other hand, as the lowest LLCR, the lowest LOQ, the lowest RSD, and the lowest |100.0% − Re| are desired, these validation parameters are in the denominator.

$${\mathrm{OC}}_{\mathrm{an}}=\frac{{\mathrm{sensitivity}}_{\mathrm{an}}}{{\mathrm{LLCR}}_{\mathrm{an}}·{\mathrm{LOQ}}_{\mathrm{an}}·{\mathrm{RSD}}_{\mathrm{an}}·{\left|100.0\%-Re\right|}_{\mathrm{an}}}$$

(1)

In order to determine the value of every validation parameter, six repetitions were made.

Cd(II) and Pb(II) were determined simultaneously using all BiFGCEs presented in this work. The RSD and Re were calculated based on concentration determination using the multiple standard addition method.

On that basis, for both analytes, the OC_an was determined individually (OC_Cd(II) and OC_Pb(II)). The OC of the BiFGCE was defined as the combined OC (OC_comb) by including both OC_an. In this work, the OC_comb was calculated as the product of individual OC_an (OC_prod, Equation (2)) or as a sum of individual OC_an (OC_sum, Equation (3)). Thus, two different approaches were used to determine the response of the BiFGCEs. For simplicity, only numerical values without units were reported for the OCs.

OC_prod = OC_Cd(II) ∙ OC_Pb(II)

(2)

OC_sum = OC_Cd(II) + OC_Pb(II)

(3)

Figure 1 shows the measured SWASV voltammograms, linear calibration curves, and the shift of the given analyte’s stripping peak potential with increasing concentration (three out of six repetitions are shown). The results for other tested BiFGCEs are given in the Supplementary Material (Figures S1–S19).

3. Results and Discussion

3.1. Fractional Factorial Design

The factors for SWASV that can be varied are E_dep, t_dep, ΔE, amplitude, and frequency. As BiFGCEs were being investigated, the mass concentration of Bi(III) (γ_Bi(III)) must also be taken into consideration for optimization. On that basis, the impact of the above-mentioned six factors on BiFGCE performance was investigated by a fractional two-level factorial design, [33] as shown in Table 1 and

Table 2. Fractional two-level factorial design with six factors (the values of the factors at their high and low levels).

Electrode No.	Edep [V]	tdep [s]	ΔE [mV]	Amplitude [mV]	Frequency [Hz]	γBi(III) [mg/L]
1	−1.0	120	6	50	50	1.0
2	−1.0	120	3	50	25	0.2
3	−1.0	30	6	25	50	0.2
4	−1.0	30	3	25	25	1.0
5	−1.4	120	6	25	25	1.0
6	−1.4	120	3	25	50	0.2
7	−1.4	30	6	50	25	0.2
8	−1.4	30	3	50	50	1.0

. A fractional factorial design was employed to decrease the number of experiments compared to a full factorial design [36]. As the starting point in the fractional factorial design, the high (+ in Table 1) and low levels (− in Table 1) of the factors under investigation must be considered. The designation of the levels and experimental values for every factor at their low and high levels are shown in Table 1 and Table 2, respectively. These values were selected based on the previously reported work [33]. The E_dep value of −1.4 V was chosen because hydrogen evolution is significant at more negative potentials, while E_dep value of −1.0 V was selected as it is at still more negative potentials required to perform the deposition step of analytes. The reported Bi(III) concentration for the formation of the in situ film electrodes is frequently 0.5 or 1.0 mg/L [3,9,22,37,38,39,40,41]. To test the significance of an even lower Bi(III) concentration, 0.2 mg/L was set as the lower limit and 1.0 mg/L as the upper limit (the most commonly reported highest concentration for in situ film electrode formation). In addition, t_dep is often 60 s, so 30 s and 120 s were tested to verify the limits of t_dep. When selecting ΔE, amplitude, and frequency, care was taken to ensure that the SWASV voltammogram was still well pronounced and had no significant noise. These values are lower or higher than the most commonly used ΔE, amplitude, and frequency for performing SWASV [3,9,22,37,38,39,40,41].

Table 1. Fractional two-level factorial design with six factors (+ stands for the factor when it is at the high level, and − stands for the factor when it is at the low level).

Electrode No.	Edep [V]	tdep [s]	ΔE [mV]	Amplitude [mV]	Frequency [Hz]	γBi(III) [mg/L]
1	+	+	+	+	+	+
2	+	+	−	+	−	−
3	+	−	+	−	+	−
4	+	−	−	−	−	+
5	−	+	+	−	−	+
6	−	+	−	−	+	−
7	−	−	+	+	−	−
8	−	−	−	+	+	+

Table 1. Fractional two-level factorial design with six factors (+ stands for the factor when it is at the high level, and − stands for the factor when it is at the low level).

Electrode No.	Edep [V]	tdep [s]	ΔE [mV]	Amplitude [mV]	Frequency [Hz]	γBi(III) [mg/L]
1	+	+	+	+	+	+
2	+	+	−	+	−	−
3	+	−	+	−	+	−
4	+	−	−	−	−	+
5	−	+	+	−	−	+
6	−	+	−	−	+	−
7	−	−	+	+	−	−
8	−	−	−	+	+	+

Eight BiFGCEs, named Electrode Nos. 1–8 in Table 2, for Cd(II) and Pb(II) determination, were validated (the values for the six factors used for these BiFGCEs are also given in Table 2). LOD, LOQ, linear concentration range, sensitivity, accuracy, and precision were determined. Based on the obtained validation parameter results, OC_prod and OC_sum values were calculated. The OC_prod and OC_sum values served as a response for the factorial design. The factor’s impact (FI) on the analytical performance of the BiFGCE was calculated using Equation (4), where $\overline{{\mathrm{OC}}_{{\mathrm{comb}}^{+}}}$ and $\overline{{\mathrm{OC}}_{{\mathrm{comb}}^{-}}}$ are the average values of OC_prod (or OC_sum) when the factor is at the high and low level, respectively.

$$\mathrm{FI}=\overline{{\mathrm{OC}}_{{\mathrm{comb}}^{+}}}-\overline{{\mathrm{OC}}_{{\mathrm{comb}}^{-}}}$$

(4)

The FI is significant when the $\left|\overline{{\mathrm{OC}}_{{\mathrm{comb}}^{+}}}-\overline{{\mathrm{OC}}_{{\mathrm{comb}}^{-}}}\right|$ is higher than the critical value. Critical values are calculated by Equation (5) and the pooled standard deviation (s_pool) by Equation (6) [33,36].

$$\mathrm{Critical}\mathrm{value}=t\left(0.05,{\mathrm{n}}_{{\mathrm{OC}}_{{\mathrm{comb}}^{+}}}+{\mathrm{n}}_{{\mathrm{OC}}_{{\mathrm{comb}}^{-}}}-2\right)·{\mathrm{s}}_{\mathrm{pool}}·\sqrt{\frac{1}{{\mathrm{n}}_{{\mathrm{OC}}_{{\mathrm{comb}}^{+}}}}+\frac{1}{{\mathrm{n}}_{{\mathrm{OC}}_{{\mathrm{comb}}^{-}}}}}$$

(5)

$${\mathrm{s}}_{\mathrm{pool}}=\sqrt{\frac{\left({\mathrm{n}}_{{\mathrm{OC}}_{{\mathrm{comb}}^{+}}}-1\right)·s_{{\mathrm{OC}}_{{\mathrm{comb}}^{+}}}^2+\left({\mathrm{n}}_{{\mathrm{OC}}_{{\mathrm{comb}}^{-}}}-1\right)·s_{{\mathrm{OC}}_{{\mathrm{comb}}^{-}}}^2}{{\mathrm{n}}_{{\mathrm{OC}}_{{\mathrm{comb}}^{+}}}+{\mathrm{n}}_{{\mathrm{OC}}_{{\mathrm{comb}}^{-}}}-2}}$$

(6)

The Student’s t value at 95% confidence is $t(0.05, {\mathrm{n}}_{{\mathrm{OC}}_{{\mathrm{comb}}^{+}}}+{\mathrm{n}}_{{\mathrm{OC}}_{{\mathrm{comb}}^{-}}}-2$) and has six degrees of freedom in the present case. In Equation (6), $s_{{\mathrm{OC}}_{{\mathrm{comb}}^{+}}}^2$ is the variance of the OC_comb when the factor is at a high level and vice versa for $s_{{\mathrm{OC}}_{{\mathrm{comb}}^{-}}}^2$. The terms ${\mathrm{n}}_{{\mathrm{OC}}_{{\mathrm{comb}}^{+}}}$ and ${\mathrm{n}}_{{\mathrm{OC}}_{{\mathrm{comb}}^{-}}}$ are the number of cases when the factor is at a high or low level, respectively [33,36]. Both ${\mathrm{n}}_{{\mathrm{OC}}_{{\mathrm{comb}}^{+}}}$ and ${\mathrm{n}}_{{\mathrm{OC}}_{{\mathrm{comb}}^{-}}}$ are four in the present case.

Moreover, an evaluation of the significance of a single factor on one validation parameter was performed. In this case, the response in the fractional factorial design was the product (or sum) of the single validation parameter for both analytes simultaneously (i.e., for the product: slope_prod, LOQ_prod, RSD_prod, |100% − Re|_prod, and the lower limit of the linear concentration range_prod, and for the sum: slope_sum, LOQ_sum, RSD_sum, |100% − Re|_sum, and the lower limit of the linear concentration range_sum) when the factor is at its high or low level. The response was also considered for individual OC_an, i.e., OC_Cd(II) and OC_Pb(II), or an individual validation parameter.

The FIs are shown in

Table 3. The FI on individual validation parameters (“No” or “Yes” is the answer if the factor has a significant impact on the given response).

Response	Edep [V]	tdep [s]	ΔE [mV]	Amplitude [mV]	Frequency [Hz]	γBi(III) [mg/L]
OCprod	No	No	No	No	No	Yes
OCsum	No	No	No	No	No	No
1 slopeprod	No	No	No	No	No	No
2 LOQprod	No	No	No	No	No	No
3 RSDprod	No	No	No	No	No	No
4 |100% − Re|prod	No	Yes	No	No	No	No
5 LLCRprod	No	Yes	No	No	No	No
6 slopesum	No	Yes	No	No	No	No
7 LOQsum	No	Yes	No	No	No	No
8 RSDsum	No	No	No	No	No	No
9 |100% − Re|sum	No	Yes	No	No	No	No
10 LLCRsum	No	No	No	No	No	No
slopeCd(II)	No	Yes	No	No	No	No
LOQCd(II)	No	Yes	No	No	No	No
RSDCd(II)	No	No	No	No	No	No
|100% − Re|Cd(II)	No	Yes	No	No	No	No
LLCRCd(II)	No	No	No	No	No	No
OCCd(II)	No	No	No	No	No	No
slopePb(II)	No	Yes	No	No	No	No
LOQPb(II)	No	Yes	No	No	No	No
RSDPb(II)	No	No	No	No	No	No
|100% − Re|Pb(II)	No	No	No	No	No	No
LLCRPb(II)	No	No	Yes	No	No	No
OCPb(II)	No	No	No	No	No	Yes

¹ slope_prod = slope_Cd(II) ∙ slope_Pb(II). ² LOQ_prod = LOQ_Cd(II) ∙ LOQ_Pb(II). ³ RSD_prod = RSD_Cd(II) ∙ RSD_Pb(II). ⁴ |100% − Re|_prod = |100% − Re|_Cd(II) ∙ |100% − Re|_Pb(II). ⁵ LLCR_prod = LLCR_Cd(II) ∙ LLCR_Pb(II). ⁶ slope_sum = slope_Cd(II) + slope_Pb(II). ⁷ LOQ_sum = LOQ_Cd(II) + LOQ_Pb(II). ⁸ RSD_sum = RSD_Cd(II) + RSD_Pb(II). ⁹ |100% − Re|_sum= |100% − Re|_Cd(II) + |100% − Re|_Pb(II). ¹⁰ LLCR_sum = LLCR_Cd(II) + LLCR_Pb(II).

. The term “No” means that the factor does not have a significant impact on the given response (FI is not significant when the $\left|\overline{{\mathrm{OC}}_{{\mathrm{comb}}^{+}}}-\overline{{\mathrm{OC}}_{{\mathrm{comb}}^{-}}}\right|$ is lower than the critical value), while “Yes” means that the factor has a significant impact on the given response (FI is significant when the $\left|\overline{{\mathrm{OC}}_{{\mathrm{comb}}^{+}}}-\overline{{\mathrm{OC}}_{{\mathrm{comb}}^{-}}}\right|$ is higher than the critical value). Table 3 presents different responses. The evaluation of the impact on one validation parameter is only given for comparison since the main purpose of this study is to improve validation parameters simultaneously by employing the OC_prod and OC_sum responses [33].

Table 3 shows that none of the factors for conducting SWASV measurements (the electrochemical setup factors, not the γ_Bi(III)) has a significant impact on the validation parameters simultaneously, i.e., when the response is OC_prod or OC_sum. On the other hand, γ_Bi(III) has a significant impact on all validation parameters simultaneously when the response is OC_prod. Moreover, t_dep has an impact on |100% − Re|_prod, the lower limit of the linear concentration range_prod, the calibration curve’s slope_sum, the LOQ_sum, the |100% − Re|_sum, the calibration curve’s slope_Cd(II), the LOQ_Cd(II), the |100% − Re|_Cd(II), the calibration curve’s slope_Pb(II), and the LOQ_Pb(II). Furthermore, the ΔE has an impact on the lower limit of the linear calibration range_Pb(II) and γ_Bi(III) has an impact on the OC_prod and OC_Pb(II).

3.2. Simplex Optimization Procedure

As six factors are optimized (E_dep, t_dep, ΔE, amplitude, frequency, and γ_Bi(III)), seven BiFGCEs with different factors are needed for simplex optimization as a starting point. The BiFGCEs that were tested with the fractional factorial design were also employed for simplex optimization. As eight BiFGCEs were tested with the fractional factorial design, the BiFGCE with the lowest value of OC_prod and OC_sum was not used (this was Electrode No. 8). Two different OCs were employed for simplex optimization, i.e., OC_prod and OC_sum. The higher the value of OC_prod or OC_sum, the better the BiFGCE is considered to be.

The simplex optimization procedure follows three steps: (i) first the BiFGCE with the lowest OC_prod or OC_sum is determined (designated by the worst point, Wi), (ii) then the centroid point (CEN) of other BiFGCEs for each factor is calculated (CEN is calculated as an average value), and (iii) reflection of the Wi through the CEN is performed (α is a multiplication factor and is taken as 1 in the present case [33] and is common in Simplex optimization [36]) according to Equation (7) [33]. By means of simplex optimization, a new BiFGCE is suggested, and named Bi (B for best point and i for the number of simplex reflections).

Bi = (1 + α) ∙ CEN − α ∙ Wi

(7)

When calculating the factors for a new BiFGCE, it can happen that impossible or values over a certain limit are calculated, thus boundary (limit) conditions are needed. For example, a negative value for γ_Bi(III) is impossible; that is why the lowest boundary is taken as γ_Bi(III) = 0.2 mg/L (the same boundary was proved successful in the previous study [33]). The latter was applied for B3 and B4 when OC_prod was used, and for B1, B3, and B6 when OC_sum was used as the OC.

Table 4. The simplex optimization procedure with designated Wi and Bi, when OC_prod is taken as the OC.

Electrode No.	Edep [V]	tdep [s]	ΔE [mV]	Amplitude [mV]	Frequency [Hz]	γBi(III) [mg/L]	OCprod
1	−1.0	120	6.0	50.0	50.0	1.0	1.15 ∙ 10−10	W5
2	−1.0	120	3.0	50.0	25.0	0.2	3.58 ∙ 10−9
3	−1.0	30	6.0	25.0	50.0	0.2	1.32 ∙ 10−11	W1
4	−1.0	30	3.0	25.0	25.0	1.0	6.53 ∙ 10−11	W4
5	−1.4	120	6.0	25.0	25.0	1.0	3.68 ∙ 10−11	W3
6	−1.4	120	3.0	25.0	50.0	0.2	3.46 ∙ 10−9
7	−1.4	30	6.0	50.0	25.0	0.2	1.01 ∙ 10−9
8	−1.4	30	3.0	50.0	50.0	1.0	6.64 ∙ 10−12
Simplex Optimization
Electrode Designation	Edep[V]	tdep[s]	ΔE [mV]	Amplitude [mV]	Frequency [Hz]	γBi(III)[mg/L]	OCprod
B1prod	−1.4	150	3.0	50.0	16.7	1.0	6.96 ∙ 10−12	W2
B2prod	−1.0	30	6.0	25.0	50.0	0.2	1.27 ∙ 10−10	W6
B3prod	−0.9	30	3.0	50.0	50.0	0.2	1.43 ∙ 10−10
B4prod	−1.2	120	6.0	58.3	58.3	0.2	2.91 ∙ 10−9
B5prod	−1.3	30	3.0	36.1	36.1	0.2	4.83 ∙ 10−8
B6prod	−1.4	120	2.0	64.8	31.5	0.2	1.92 ∙ 10−6

and

Table 5. The simplex optimization procedure with designated Wi and Bi, when OC_sum is taken as the OC.

Electrode No.	Edep [V]	tdep [s]	ΔE [mV]	Amplitude [mV]	Frequency [Hz]	γBi(III) [mg/L]	OCsum
1	−1.0	120	6.0	50.0	50.0	1.0	2.25 ∙ 10−5	W3
2	−1.0	120	3.0	50.0	25.0	0.2	1.32 ∙ 10−4
3	−1.0	30	6.0	25.0	50.0	0.2	1.82 ∙ 10−5	W2
4	−1.0	30	3.0	25.0	25.0	1.0	7.72 ∙ 10−5	W6
5	−1.4	120	6.0	25.0	25.0	1.0	1.41 ∙ 10−5	W1
6	−1.4	120	3.0	25.0	50.0	0.2	1.73 ∙ 10−4
7	−1.4	30	6.0	50.0	25.0	0.2	8.12 ∙ 10−5
8	−1.4	30	3.0	50.0	50.0	1.0	5.16 ∙ 10−6
Simplex Optimization
Electrode Designation	Edep[V]	tdep[s]	ΔE [mV]	Amplitude [mV]	Frequency [Hz]	γBi(III)[mg/L]	OCsum
B1sum	−0.9	30	3.0	50.0	50.0	0.2	4.35 ∙ 10−5	W5
B2sum	−1.2	120	2.0	58.3	25.0	0.7	1.68 ∙ 10−4
B3sum	−1.3	30	0.7	36.1	16.7	0.2	4.98 ∙ 10−6	W4
B4sum	−1.0	120	6.0	50.0	50.0	0.6	7.71 ∙ 10−4
B5sum	−1.4	150	4.7	36.1	16.7	0.8	1.99 ∙ 10−4
B6sum	−1.5	190	5.2	64.8	38.9	0.2	3.97 ∙ 10−3

show new BiFGCEs using simplex optimization when OC_prod and OC_sum are taken as the OC, respectively. Six simplex reflections are reported for each OC. Bi_prod and Bi_sum represent the Bi, when OC_prod and OC_sum were applied as the OC, respectively. The best BiFGCE (with the highest OC) was B6_prod, when the OC_prod was the OC. The OC_prod for B6_prod is significantly higher than for B5_prod and Electrode No. 2 (with the best OC_prod in the fractional factorial design reported above). When the OC is OC_sum, the best BiFGCE is B6_sum. The OC_sum for B6_sum is significantly higher than for B5_sum and Electrode No. 6 (with the best OC_sum in the fractional factorial design reported above).

3.3. Method Validation

For all tested BiFGCEs, the stripping peaks for Cd(II), Pb(II), and Bi(III) were well separated from each other; therefore, all BiFGCEs showed good selectivity for Cd(II) and Pb(II) determination (Figure 1e and Figures S1–S19e in the Supplementary Material). In order to accept a linear concentration range, the square of the correlation coefficient (R²) should be larger than 0.995 and the quality coefficient (QC) should be lower than 5.00% [36,42]. Figure 2 shows the linear concentration ranges for Cd(II) and Pb(II) determination using the different BiFGCEs tested using the factors in Table 4 and Table 5 (in the fractional factorial design and simplex optimization).

For Pb(II), the linear concentration ranges determined were wider than the linear concentration ranges for Cd(II), apart from Electrodes No. 1, 5, 8, B6_prod, and B2_sum. The widest linear concentration range for Cd(II) was determined for Electrode No. 4, and the widest linear concentration range for Pb(II) was determined for Electrode No. 7.

The sensitivity was evaluated based on the slope of the linear calibration curve (Figure 3). Most of the BiFGCEs show higher sensitivity to Cd(II) compared with Pb(II) (except for Electrode B1_prod, Figure 3b) when the OC was OC_prod. However, when the OC was OC_sum, the BiFGCEs had a similar or higher sensitivity to Pb(II) compared with Cd(II) (Figure 3c). The highest sensitivity to Cd(II) was seen for Electrode B4_prod, and regarding Pb(II) for Electrode B6_sum.

The LOD is the concentration where the signal-to-noise ratio is equal to or higher than 3 (it should be close to 3 and lower than 10). The LOQ is the concentration where the signal-to-noise ratio is equal to or higher than 10 (but close to 10) [43]. The methodology for LOD and LOQ determination is described in [33].

Table 6. The LOD and LOQ values determined during the course of the fractional factorial design testing and simplex optimization (with two different optimization criteria, i.e., OC_prod and OC_sum).

Electrode No.	Cd(II)		Pb(II)
	LODCd(II) [µg/L]	LOQCd(II) [µg/L]	LODPb(II) [µg/L]	LOQPb(II) [µg/L]
1	1.2	2.7	0.2	2.0
2	2.3	3.8	1.6	3.1
3	7.4	10.7	3.8	5.7
4	10.7	16.7	2.7	7.4
5	2.7	5.7	0.8	2.3
6	2.0	3.5	0.4	0.8
7	2.3	5.7	1.6	3.5
8	2.7	5.7	1.6	3.8
Electrode Designation
B1prod	0.4	3.1	0.8	2.0
B2prod	2.3	5.7	1.2	3.1
B3prod	1.6	3.8	0.8	2.3
B4prod	0.2	1.6	0.2	0.4
B5prod	1.6	3.8	0.8	3.1
B6prod	0.4	3.1	0.4	0.8
B1sum	3.8	10.7	1.2	3.1
B2sum	0.8	3.1	0.4	1.6
B3sum	3.1	13.8	1.6	7.4
B4sum	0.8	2.3	0.2	0.4
B5sum	0.4	3.1	0.4	0.8
B6sum	0.4	2.3	0.2	0.4

shows the LODs and LOQs for all the BiFGCEs tested. In general, the LODs and LOQs determined for Pb(II) were lower than for Cd(II), apart from Electrode B1_prod.

The precision of the system and the precision of the method were also determined. The precision of the system (the precision at the repeatability level) was determined by calculating the RSD using the stripping peak height values of twelve consecutive measurements of the Cd(II) and Pb(II) stripping peaks in the same solution. The precision of the method was calculated from the determined concentrations of Cd(II) and Pb(II) using the multiple standard addition method. For the latter, at least six repetitions of a solution (every measurement was performed using a freshly prepared solution) with the same concentrations of Cd(II) and Pb(II) were measured and the RSD was calculated from the determined concentrations. The precision of the method was determined at the lowest possible concentration of Cd(II) and Pb(II), where both of the analytes were within their linear concentration range. A method was deemed precise (for both the precision of the system and the precision of the method), when the RSD was lower or equal to 20.0% [44,45].

Table 7. RSD values, representing the precision of the system.

Electrode No.	γCd(II) [µg/L]	RSDCd(II) [%]	γPb(II) [µg/L]	RSDPb(II) [%]
1	19.4	44.4	19.4	21.1
2	10.7	15.2	10.7	12.1
3	19.4	9.4	19.4	5.7
4	19.4	7.0	19.4	6.1
5	16.7	43.4	16.7	8.6
6	13.8	17.0	13.8	6.0
7	35.7	1.8	35.7	2.3
8	16.7	25.1	16.7	9.5
Electrode Designation
B1prod	19.4	22.7	19.4	5.9
B2prod	10.7	10.0	10.7	7.1
B3prod	7.4	11.6	7.4	5.1
B4prod	21.9	25.4	21.9	5.9
B5prod	13.8	17.6	13.8	8.2
B6prod	10.7	26.7	10.7	5.9
B1sum	10.7	9.4	10.7	4.3
B2sum	10.7	19.9	10.7	3.1
B3sum	19.4	19.1	19.4	11.6
B4sum	13.8	24.6	13.8	9.7
B5sum	10.7	17.0	10.7	9.0
B6sum	3.8	39.7	3.8	5.1

presents the precision of the system and

Table 8. RSD values, representing the precision of the method.

Electrode No.	γCd(II) [µg/L]	RSDCd(II) [%]	γPb(II) [µg/L]	RSDPb(II) [%]
1	19.4	89.1	19.4	23.5
2	10.7	24.6	10.7	11.6
3	19.4	35.3	19.4	41.4
4	19.4	31.1	19.4	23.2
5	16.7	76.6	16.7	16.1
6	13.8	9.3	13.8	19.8
7	35.7	14.1	35.7	16.1
8	16.7	28.1	16.7	21.5
Electrode Designation
B1prod	19.4	95.2	19.4	14.5
B2prod	10.7	22.5	10.7	17.1
B3prod	7.4	26.1	7.4	38.7
B4prod	21.9	137.8	21.9	24.5
B5prod	13.8	7.3	13.8	5.0
B6prod	10.7	20.5	10.7	7.0
B1sum	10.7	24.3	10.7	16.3
B2sum	10.7	21.2	10.7	13.7
B3sum	19.4	43.9	19.4	9.7
B4sum	13.8	34.7	13.8	26.0
B5sum	10.7	28.7	10.7	37.2
B6sum	3.8	59.9	3.8	19.6

presents the precision of the method using different BiFGCEs. For Cd(II) determination, Electrodes No. 1, 5, and 8, and Electrodes B1_prod, B4_prod, B6_prod, B4_sum and B6_sum showed RSDs higher than 20.0, which exceeded the criterium for the precision of the system. On the other hand, all of the electrodes for Pb(II) determination were precise, apart from Electrode No. 1 (Table 7). Moreover, it terms of the precision of the method, only few of the electrodes for Cd(II) determination were precise (as they had an RSD lower than 20.0%), i.e., Electrodes No. 6, 7, and B5_prod. For Cd(II) determination, in general, the electrodes demonstrated the acceptable precision of the method (with an RSD lower than 20.0%), apart from Electrodes No. 1, 3, 4, and 8, and Electrodes B3_prod, B4_prod, B4_sum, and B5_sum (Table 8).

The Re was determined at the same concentration as reported above for the precision of the method (Table 8). If the Re was between 80.0% and 120.0%, the BiFGCE was deemed accurate [44,45].

Table 9. Re values for different BiFGCEs.

Electrode No.	γCd(II) [µg/L]	ReCd(II) [%]	γPb(II) [µg/L]	RePb(II) [%]
1	19.4	25.0	19.4	21.0
2	10.7	87.5	10.7	67.4
3	19.4	97.2	19.4	160.4
4	19.4	100.2	19.4	49.1
5	16.7	43.2	16.7	39.5
6	13.8	187.1	13.8	188.6
7	35.7	113.7	35.7	101.2
8	16.7	56.8	16.7	58.3
Electrode Designation
B1prod	19.4	331.6	19.4	64.3
B2prod	10.7	56.3	10.7	61.0
B3prod	7.4	63.0	7.4	51.3
B4prod	21.9	371.6	21.9	188.9
B5prod	13.8	94.4	13.8	97.8
B6prod	10.7	95.8	10.7	90.9
B1sum	10.7	31.1	10.7	56.2
B2sum	10.7	94.2	10.7	58.8
B3sum	19.4	384.6	19.4	108.7
B4sum	13.8	65.5	13.8	41.9
B5sum	10.7	82.4	10.7	3.8
B6sum	3.8	70.6	3.8	7.5

shows the determined Re values for different BiFGCEs. Electrodes No. 2, 3, 4, 7, and B5_prod, B6_prod, B2_sum, and B5_sum were deemed to be accurate for Cd(II) determination and Electrodes No. 7, B5_prod, B6_prod, and B3_sum were deemed to be accurate for Pb(II) determination.

3.4. A Comparison of Optimized BiFGCEs

Simplex optimization resulted in two best BiFGCEs, i.e., B6_prod and B6_sum.

Table 10. A comparison of optimized Electrodes B6_prod and B6_sum.

	Electrode B6prod
	Cd(II)	Pb(II)
LOD [µg/L]	0.4	0.4
LOQ [µg/L]	3.1	0.8
Linearity [µg/L]	10.7–54.9	0.8–24.2
Sensitivity [µA$·$L/µg]	0.51	0.44
RSD [%]	20.5 at 10.7 µg/L	7.0 at 10.7 µg/L
Re[%]	95.8 at 10.7 µg/L	90.9 at 10.7 µg/L
OCprod	1.92 ∙ 10−6
	Electrode B6sum
	Cd(II)	Pb(II)
LOD [µg/L]	0.4	0.2
LOQ [µg/L]	2.3	0.4
Linearity [µg/L]	3.8–13.8	2.3–28.6
Sensitivity [µA$·$L/µg]	0.84	6.52
RSD [%]	59.9 at 3.8 µg/L	19.6 at 3.8 µg/L
Re[%]	70.6 at 3.8 µg/L	7.5 at 3.8 µg/L
OCsum	3.97 ∙ 10−3

summarizes the validation parameters for these optimized BiFGCEs. Electrode B6_sum has a lower LOD and LOQ for Pb(II), and a lower LOQ for Cd(II), but the same LOD for Cd(II) as Electrode B6_prod. However, Electrode B6_prod has a wider linear calibration range for both analytes compared to Electrode B6_sum. The sensitivity of Electrode B6_sum is higher for Cd(II) and significantly higher for Pb(II) than the sensitivity of Electrode B6_prod. The RSD (for the precision of the method) and the Re for both BiFGCEs were determined at different concentrations, due to their different linear concentration ranges. The RSD is lower for Pb(II) determination using Electrode B6_sum than the RSD for Pb(II) determination using Electrode B6_prod. Both BiFGCEs produce precise results (with an RSD lower than 20.0%) for Pb(II) determination, but do not produce precise results for Cd(II) determination. Electrode B6_prod produces accurate results for both analytes (with the Re in an interval between 80.0 and 120.0%), but Electrode B6_sum does not produce accurate results for any of the analytes. It must be pointed out that by optimizing the BiFGCEs for heavy metal determination, an OC_comb was used as a criterion for optimization taking into account different validation parameters simultaneously and not individually, which resulted in these BiFGCEs not having the best results for individual validation parameter (such as the Re).

3.5. Interferences

A possible interference effect was tested for Electrodes B6_prod and B6_sum in 0.1 M acetate buffer solution, containing 10.0 µg/L Cd(II) and Pb(II). This concentration of Cd(II) and Pb(II) was used because this is the limit concentration in drinking water reported by the World Health Organization (WHO) [20,21,46,47]. The following species were tested as possible interferents: Na(I), K(I), Mg(II), Ca(II), Fe(II), As(III), Cu(II), Sn(II), Sb(III), Zn(II), Cl⁻, NO₃⁻, and SO₄²⁻. The influence of possible interferents was tested at three ratios relative to the mass concentration of the analytes Cd(II) and Pb(II), namely 1:1, 1:10, and 1:100.

An interference effect was determined using the stripping peak height for Cd(II) and Pb(II) with and without the interferent present. Three repetition measurements were performed for every solution (with and without the possible interferent present). The influence of the interferent at a certain analyte:interferent ratio was calculated as % = 100% (Δi_interferent − Δi_analyte)/Δi_analyte, where Δi_analyte is the average analyte’s stripping peak height and Δi_interferent is the average analyte’s stripping peak height in the presence of the interferent. The increase and decrease of the analytes stripping peak height or splitting to double stripping peaks, i.e., the stripping peak for Pb(II) at 1:1 and 1:10 ratios of Pb(II):Fe(II) using Electrode B6_prod, as shown in Figure S20e, in the presence of different interferents, are given in

Table 11. The possible interference effect of different species for Cd(II) and Pb(II) determination using Electrode B6_prod.

Interferent	Mass Concentration Ratio Cd(II):Interferent			Mass Concentration Ratio Pb(II):Interferent
	1:1	1:10	1:100	1:1	1:10	1:100
Na(I)	−24.4	−46.3	−87.9	−4.7	−13.2	−59.7
K(I)	−33.4	−50.1	−82.1	−7.4	−10.5	−47.5
Ca(II)	−9.5	−26.8	−79.9	−1.8	−7.7	−52.3
Mg(II)	−24.2	−44.8	−77.9	−4.1	−13.0	−49.8
Fe(II)	−58.7	−92.9	*	**	**	*
As(III)	16.4	10.5	−64.9	15.0	15.5	−45.4
Cu(II)	−89.9	*	*	−48.6	−87.6	*
Sn(II)	−21.8	−44.9	−93.8	−2.8	−10.9	−73.8
Sb(III)	−15.8	−17.2	−58.9	−3.1	−3.3	−62.0
Zn(II)	−24.0	−65.3	−97.9	−3.7	−31.3	−65.6
Cl−	−41.6	−62.1	−89.2	−8.1	−16.5	−57.2
NO3−	−22.3	−48.8	−86.4	−5.5	−14.7	−59.8
SO42−	−28.3	−47.9	−84.1	−6.2	−11.3	−53.8

* an analyte stripping peak did not develop. ** a double stripping peak for Pb(II) developed.

and

Table 12. The possible interference effect of different species for Cd(II) and Pb(II) determination using Electrode B6_sum.

Interferent	Mass Concentration Ratio Cd(II):Interferent			Mass Concentration Ratio Pb(II):Interferent
	1:1	1:10	1:100	1:1	1:10	1:100
Na(I)	−7.9	−24.7	−79.6	3.9	0.3	−51.9
K(I)	−19.5	−40.0	−82.6	−2.9	−12.2	−56.5
Ca(II)	−12.0	−32.9	−86.2	−4.2	−10.9	−57.0
Mg(II)	−9.9	−34.1	−81.6	−2.7	−14.1	−61.9
Fe(II)	−81.3	−73.7	*	−53.5	*	*
As(III)	33.8	13.5	−57.2	24.6	20.1	−40.8
Cu(II)	−93.2	*	*	−55.3	−89.1	*
Sn(II)	15.6	−17.5	−92.5	3.7	−8.7	−80.8
Sb(III)	−2.0	−13.3	−49.1	−3.6	−18.0	−68.3
Zn(II)	−20.0	−71.8	*	7.6	−22.6	−77.2
Cl−	−8.9	−33.4	−81.3	−5.7	−20.8	−59.4
NO3−	−9.5	−30.4	−81.0	−6.8	−21.4	−68.3
SO42−	−8.6	−26.8	−79.1	−7.4	−15.3	−56.2

* an analyte stripping peak did not develop.

.

Cu(II) and Fe(II) have the greatest influence on Cd(II) and Pb(II) determination. In the presence of Cu(II) and Fe(II), the analyte’s stripping peak height significantly decreased, splitting into double stripping peaks, or did not develop (Figures S20e,g and S21e,g in the Supplementary Material). The stripping peak height for Cd(II) also significantly decreased at an analyte:interferent ratio of 1:100 for all interferents using Electrodes B6_prod and B6_sum. Moreover, when testing the interference effect with Electrode B6_sum at a ratio of γ_Cd(II):γ_Zn(II) = 1:100, the stripping peak for Cd(II) did not develop (Figure S21j).

3.6. Real Sample Analysis

The real sample analysis was performed with Electrodes B6_prod and B6_sum. The real sample was tap water obtained in our laboratory, with which a 0.1 M acetate buffer was prepared instead of using ultrapure water. First, the voltammogram of the blank solution was measured. The stripping peak for Pb(II) did not develop. Therefore, the concentration of Pb(II) was below the LOD or Pb(II) was not present in the tap water. As Zn(II) was present in the solution, it overlapped with the Cd(II) stripping peak and, therefore, it was not possible to analyze the Cd(II) (as reported above for the interference effect). The tap water was then spiked to 10.0 µg/L Pb(II) (the limit concentration for Pb(II) in drinking water allowed by the WHO [20,21,46,47]). For the spiked real sample analysis for determining the Pb(II) concentration, a multiple standard addition method was used. This determination was repeated three times (Figure 4). The average Re for Pb(II) was 91.5% and 107.7% for Electrodes B6_sum and B6_prod, respectively. The determined RSD for Pb(II) was 17.8% and 52.8% for Electrodes B6_sum and B6_prod, respectively. Therefore, only Electrode B6_sum was deemed to be precise and accurate for Pb(II) determination in the real sample.

4. Conclusions

In this work, an evaluation of the impact of different factors for square wave anodic stripping voltammetry on the performance of electrodes for the determination of Cd(II) and Pb(II) analytes was performed using a fractional two-level factorial design. The optimization of the electrodes for the determination of Cd(II) and Pb(II) was further performed using simplex optimization. A glassy carbon electrode, modified with Bi(III) (BiFGCE), was used as the working electrode. Eight different BiFGCEs were validated and employed in a fractional two-level factorial design to understand the significance of the impact of individual factors on the analytical performance of the BiFGCEs. A fractional two-level factorial design was used with six different factors: deposition potential (E_dep), deposition time (t_dep), amplitude, potential step (ΔE), frequency, and mass concentration of Bi(III) (γ_Bi(III)). The response in the factorial design was OC_an (OC_Cd(II) or OC_Pb(II)) or combined OC (OC_sum or OC_prod), which included sensitivity, the lower limit of the linear calibration range (LLCR), the LOQ, the RSD, and the absolute value of |100% − Re|.

It was found that γ_Bi(III) has a significant impact on the sensitivity, LLCR, LOQ, precision of the method, and accuracy simultaneously when the response was OC_prod. Moreover, t_dep has an impact on |100% − Re|_prod, the LLCR_prod, the calibration curve’s slope_sum, the LOQ_sum, |100% − Re|_sum, the calibration curve’s slope_Cd(II), the LOQ_Cd(II), |100% − Re|_Cd(II), the calibration curve’s slope_Pb(II), and the LOQ_Pb(II). Furthermore, ΔE has an impact on the LLCR_Pb(II) and γ_Bi(III) has an impact on OC_prod and OC_Pb(II).

Next, simplex optimization was performed. As the optimization criteria, the OC_prod and OC_sum were used. With both criteria, six new BiFGCEs were developed and validated. The optimized BiFGCEs determined were Electrodes B6_prod and B6_sum, with significantly better OC_prod and OC_sum compared with the BiFGCEs before the simplex optimization. Electrode B6_sum has a lower limit of detection (LOD) and LOQ for Pb(II), and a lower LOQ for Cd(II), but the same LOD for Cd(II) compared with Electrode B6_prod. However, Electrode B6_prod has a wider linear concentration range for both analytes than Electrode B6_sum. The sensitivity of Electrode B6_sum is higher for Cd(II) and significantly higher for Pb(II) than the sensitivity of Electrode B6_prod. The RSD and Re for both of these BiFGCEs were determined at different concentrations due to their different linear concentration ranges.

The possible interferent effect of different species (Na(I), K(I), Ca(II), Mg(II), Fe(II), As(III), Cu(II), Sn(II), Sb(III), Zn(II), Cl⁻, NO₃⁻, and SO₄²⁻) was studied. For Electrodes B6_prod and B6_sum, a significant interference effect on Cd(II) and Pb(II) determination was shown by Fe(II) and Cu(II), and Zn(II) when a 1:100 mass concentration ratio of the analyte relative to Zn(II) was studied. The applicability of Electrodes B6_prod and B6_sum was tested for a real sample (tap water), regarding which Electrode B6_sum showed precise and accurate results for Pb(II) determination.