Estimation and Optimization of Potentiometric Sensor Response ...

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SENSOR LETTERS Vol. 9, 1–8, 2011

Estimation and Optimization of Potentiometric Sensor Response Parameters from Surfactant Titration Data Using Microsoft Excel Solver and Mathematica ˇ ci´c2 , Nikola Sakaˇc1 , Milan Sak-Bosnar1
∗ , Dubravka Maduni´c-Caˇ Mirela Samardži´c1 , and Želimir Kurtanjek3 1

Department of Chemistry, Josip Juraj Strossmayer University of Osijek, F. Kuhaèa 20, HR-31000 Osijek, Croatia 2 Saponia, Chemical, Pharmaceutical and Foodstuff Industry, M. Gupca 2, HR-31000 Osijek, Croatia 3 Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, HR-10000 Zagreb, Croatia (Received: 21 May 2010. Accepted: 27 July 2010)

Keywords: Surfactant Sensor, Surfactants, Potentiometric Titration, Modeling, Solver, Mathematica.

1. INTRODUCTION Surfactants are a large group of synthetic substances that possess both a hydrophilic group, which determines its application, and a hydrophobic group, usually a long alkyl chain. Based on the hydrophilic group, surfactants can be divided into four types: anionic, cationic, amphoteric and non-ionic. The wide use of surfactants demands the development of modern methodologies for their accurate determination.1 The most widely used technique for the determination of ionic surfactant is the two-phase titration method based on the formation of ion-pair compounds between cationic dye methylene blue and the anionic surfactants, according to Epton,2 or with a mixed indicator, according to Reid, Longman and Heinerth.3 However, these methods suffer from many drawbacks, such as limited application to ∗

Corresponding author; E-mail: [email protected]

Sensor Lett. 2011, Vol. 9, No. 2

strongly colored and turbid samples, toxicity of the organic chlorinated solvent, emulsion formation during the titration that can disturb visual endpoint detection, and numerous matrix interferences. Potentiometry with ion-selective electrodes (ISE) used as sensors is a simple, fast and inexpensive method for determining surfactants and avoids most of these mentioned limitations.4 Since surfactant sensitive electrodes were developed by Birch and Clarke in 1974,5 they have been improved and used extensively for endpoint detection in surfactant analysis.6 Many of the analytical methods for determination of polyethoxylated nonionic surfactants are based on the formation of tetraphenylborate (TPB) salts of pseudocationic complexes of nonionic surfactants with barium cations, a reaction reported for the first time in 1961 by Seher.7 Levins and Ikeda were among the first who exploited this precipitation reaction for the potentiometric titration of nonionic surfactants.8 The Vytras9
10 and Moody and Thomas11–15 groups are among the pioneers in this area

1546-198X/2011/9/001/008

doi:10.1166/sl.2011.1503

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Theoretical models for potentiometric titration curves of some ionic and nonionic surfactants are proposed in this paper. The experimental data were compared with appropriate theoretical curves in which the sensor response parameters and analyte properties (sensor slope, constant potential term, solubility product constant, analyte concentration) were estimated and optimized with Solver (Excel) and Wolfram Research Mathematica using least-squares criterion to fit theoretical curves to an experimental data set. The initial parameter settings were defined, and no constraints on the variables were applied. The macro SolvStat provided the regression statistics of the Solver, and these statistics were confirmed by the NonlinearModelFit macro of Wolfram Research Mathematica by calculation of the standard deviations of the parameters, correlation coefficients and standard errors of the y estimate SE(y
. The two software tools gave almost the same estimates of the parameter models and the corresponding statistics. The theoretical models fit satisfactorily to the experimental values for all of the investigated surfactants.

Estimation and Optimization of Potentiometric Sensor Response Parameters from Surfactant Titration Data

and the stoichiometry of these complexes has been extensively studied.16–18 One special feature of potentiometry is its applicability in physicochemical research on surfactant solutions, because direct measurements of surfactant activity are possible. In this paper, theoretical models for potentiometric titration curves of some ionic and nonionic surfactants are proposed and their estimations of response parameters for a few surfactant sensors are reported. Previously described liquid membrane surfactant-sensing electrodes19–21 were used to generate potentiometric experimental data, which were then incorporated into calculations using mass and charge balance equations.22–26 The models were optimized using Solver (Microsoft Excel) and Wolfram Research Mathematica v. 7.1.27

2. EXPERIMENTAL DETAILS

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2.1. Reagents and Materials Sodium tetraphenylborate (NaTPB), c = 5 mM, 1,3-didecyl-2-methylimidazolium chloride (DMIC), c = 4 mM, sodium dodecylsulfate (SDS), c = 4 mM and Hyamine 1622 (diisobutylphenoxyethoxyethyldimethylbenzylammonium chloride, benzethonium chloride), c = 4 mM were used as the titrants. The sodium tetraphenylborate solution was buffered with borate buffer at pH 10. The solutions of the following ionic surfactants were titrated: SDS, sodium dodecylbenzensulfonate (SDBS), cetyl trimethylammonium bromide (CTAB) and DMIC. All of the above-mentioned chemicals were of reagentgrade quality and supplied by Fluka, Switzerland, except for Hyamine, which was obtained from Merck, Germany. Diluted solutions of barium chloride (Riedel-de Haën, Germany), c = 02 M and Triton X-100 [Octylphenolpoly (ethyleneglycolether)10 , (Merck, Germany)], c = 50 mM were used for pseudo-ionic complex formation. 2.2. Sensors Poly(vinyl chloride) (PVC) liquid membrane surfactantsensing electrodes were used for all investigations. The DMI-TPB sensor contained a 1,3-didecyl-2-methyl-imidazolium-tetraphenylborate ion-exchange complex, which was prepared by adding a sodium tetraphenylborate solution to a 1,3-didecyl-2-methylimidazolium chloride solution. The resultant white precipitate was extracted with dichloromethane and dried with anhydrous sodium sulfate. After evaporation and recrystallization from a mixture of diethyl ether:methanol (1:1), the isolated ion-exchange complex was used for membrane preparation. A mixture of sodium chloride (c = 01 M) and sodium dodecylsulfate (c = 1 mM) was employed as the internal filling solution. Detailed preparation and characterization of the DMI-TPB sensor have been described previously.19
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The TA-DS sensor contained a tetrahexadecylammonium-dodecylsulfate ion-exchange complex, prepared by adding a hot sodium dodecylsulfate aqueous solution to a hot ethanol-water mixture (volume ratio = 2:1). The reaction mixture was heated up to 80  C, when the white precipitate was formed, and was stirred at 80  C until all of the ethanol had evaporated. After cooling, the precipitate was filtered off, washed with water and dried at 80  C. The isolated ion-exchange complex was used for PVC liquid membrane electrode preparation with o-NPOE as the plasticizer. A mixture of 5 mL of a sodium dodecylsulfate solution (c = 3 M) and 0.2 mL of a sodium chloride solution (c = 3 M) was employed as the internal filling solution. Detailed preparation and characterization of the TA-DS sensor have been described previously.20
29 The BaLx (TPB)2 sensor contained a pseudocationic tetraphenylborate ion-exchange complex, which was prepared by dropwise addition of a sodium tetraphenylborate solution to a solution of a pseudocationic barium-nonionic surfactant complex. The pseudocationic barium-nonionic surfactant complex has been prepared previously by addition of a barium chloride solution to a solution of tallow fatty alcohol polyglycol ether with 80 ethoxy (EO) groups (Genapol T 800, supplied by Clariant, Germany). The precipitate formed was extracted with dichloromethane, washed with water and dried with anhydrous sodium sulfate. After evaporation and recrystallization from a mixture of diethyl ether:methanol (1:1), the isolated ionexchange complex was used for the preparation of the PVC membrane with bis(2-ethylhexyl)phthalate (DEHP) as the plasticizer. A lithium chloride solution (c = 3 M) was employed as the internal filling solution. The detailed preparation and characterization of this sensor used for titrations of non-ionic surfactants (NS sensor) have been described elsewhere.21 2.3. Apparatus The all-purpose titrator 808 Titrando, combined with Metrohm 806 Exchange units, was employed for the performance of potentiometric titrations. The solutions during all measurements and titrations were magnetically stirred using the 727 Ti Stand. The above-mentioned apparatus was supplied by Metrohm, Switzerland. 2.4. Procedure 2.4.1. Titration Conditions The anionic surfactants were titrated with DMIC and Hyamine 1622 (both of 4 mM concentration), whereas the cationic surfactants were titrated with SDS (c = 4 mM) and NaTPB (c = 5 mM). The total volume of the solutions used for the titrations was 40 mL. The titrator was programmed to work in MET (Monotonic Equivalence Point Titration) mode with increments of 0.1 mL Sensor Letters 9, 1–8, 2011

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Estimation and Optimization of Potentiometric Sensor Response Parameters from Surfactant Titration Data

and equilibrium times of 20 and 40 s for a signal drift of 10 mV/min. The wait time before the start of titration was 30 s. A pseudoionic complex of barium and Triton X-100 was prepared by mixing the appropriate amounts of barium chloride and Triton X-100, and was then titrated with sodium tetraphenylborate, c = 5 mM. The total volume of the solution (mixture of barium ion and Triton X-100) used for the titration was 40 mL. The titrator was programmed to work in MET mode with dosing increments of 0.1 mL and equilibrium times of 30 and 60 s for a signal drift of 5 mV/min. The wait time before the start of titration was 120 s. All measurements and titrations were performed at room temperature using a magnetic stirrer, without ionic strength or pH adjustment. 2.4.2. Optimization Strategy Using Solver

Sensor Letters 9, 1–8, 2011

The macro SolvStat23 provided the regression statistics for Solver by calculation of the standard deviations of the parameters, correlation coefficients and standard errors of the y estimate SE(y. 2.4.3. Optimization Strategy Using Wolfram Research Mathematica Wolfram Research Mathematica v. 7.0127 is a modular software system that excels in mathematical algorithms for symbolic calculations, graphical capabilities, numerical precision and computational efficiency. It provides a consortium of reliable and “intelligent” optimization algorithms and statistical packages for modeling and parameter estimation of nonlinear models, such as in chemical reaction systems. Here, we applied the NonlinearModelFit module to minimize the variance between the model predictions and the experimental data. (a) Experimental data stored in the Excel table were transferred to a text data file (.dat), which was read by the Import function to a Mathematica notebook cell in the form of a matrix. (b) The model nonlinear function was defined and evaluated with adjustable and very high WorkingPrecision (for example, from 15 to 50 digits). It was minimized under the assumption that the experimental data were independently and normally distributed with common standard deviation (i.e., all of the data had the same statistical weight set to 1). (c) An unconstrained optimization was applied with initially evaluated symbolic derivatives and numerically evaluated gradients during the course of the iteration. Mathematica minimization algorithm “intelligently” and automatically selected and adapted the best method during the iterations. In most cases, the optimization started with the Levenberg-Marquardt procedure when the initial estimates were far from the minima and gradually switched to the QuasiNewton method when the parameters became close to the solution. (d) The global solution was assured by a repetitive iteration procedure with different starting estimates or by use of the Mathematica global optimization suite NMinimize, which includes the main search algorithms, such as DifferentialEvolution, NelderMead, RandomSearch or SimulatedAnnealing. (e) The obtained model was validated by numerous statistics, such as AdjustedRSquared, ANOVA table, ParameterPValues, ParameterTstatistics, ParameterConfidenceRegion, and many others. 3

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Solver is a spreadsheet optimization modeling system incorporated into Microsoft Excel for Windows. It can be used to solve a wide spectrum of linear, nonlinear or integer problems. When applied to the same data set, it can yield the same results as expensive commercial software packages. Solver was activated by choosing Add ins   in the Tools menu. It was used to compare an array of data predicted by the model with an initial set of parameter values over a range of dependent variable values with a set of experimental data. Then, the sum of squared residuals (SSR) between the two arrays was calculated by varying the parameter values according to an iterative search algorithm to minimize the error (SSR) between the two data sets. The following steps in the Solver optimization procedure were used: (a) A worksheet was generated containing the data, with an independent variable E (electrode potential in mV) and dependent variable V (volume of titrant in mL) to be fit. (b) A column was added containing Vcalc , which was calculated by means of an equation describing the titration course and involving the E values and appropriate number of parameters to be varied (Changing Cells): sample concentration Cs , constant potential term E 0 , sensor response slope S, and solubility product constant Ks . The initial settings were: Cs = 4 × 10−3 M, E 0 = +/ − 300 for the titration of anionic or cationic surfactants respectively, S = +/ − 50 for the titration of anionic or cationic surfactants respectively, and Ks = 1 × 10−10 . The automatic scaling option was used due to large differences in magnitude between the Target Cell and the Changing Cells. Instead of extremely low Ks values (below 1 × 10−10 , the corresponding pK s values should be used. Providing different sets of initial conditions ensured that Solver found a global minimum. (c) A column was added to calculate the squares of the residuals, E − Ecalc , for each data point.

(d) The sum of the squares of the residuals (Target Cell) was calculated. (e) Solver was used to minimize the sum of the squares of the residuals (Target Cell) by changing the selected parameters of the titration equation (Changing Cell). No constraints were applied on the variables.

Estimation and Optimization of Potentiometric Sensor Response Parameters from Surfactant Titration Data

3. RESULTS AND DISCUSSION

The electromotive force of the membrane sensor assembly dipped in the solution of anionic surfactant (AS) is given by the Nernst equation:

3.1. Ionic Surfactants Titration Ionic surfactants are usually titrated with an ion of opposite charge, thereby forming ion associates according to the following scheme: Cat + + An−
CatAn

(1)

where Cat+ is the cation of the cationic surfactant and An− is the anion of the anionic surfactant. Ion associates are minimally soluble in water and can be extracted into non-polar solvents. When the surfactant concentration is increased above the c.m.c. values, solubilization of the ion associate in the micelle often occurs. The solubility product of the ion associate from Eq. (1) is: 
(2) KS
CatAn = Cat + × An−  3.1.1. Anionic Surfactants Anionic surfactants (An−  are usually titrated using a standard solution of a cationic surfactant (Cat+  as the titrant, according to Eq. (1). The mass balance for the analyte (An− : Vs An  + CatAn + PCatAn = Cs × Vs + Vt 

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−

(3)

where Vs is the sample volume containing an analyte concentration of Cs . The mass balance for the titrant (Cat+ :


 Cat+ + CatAn + PCatAn = Ct ×

Vt Vs + Vt 

(4)

where Vt is the titrant volume with a concentration of Ct , CatAn is the equilibrium concentration of CatAn in the solution, and PCatAn is the concentration of analyte (An−  or titrant (Cat+  removed by the precipitation of CatAn. By subtracting (4) from (3):  C × Vs − Ct × Vt 
An−  − Cat + = s Vs + Vt 

(5)

After rearrangement, the following equation is obtained: Vt = Vs ×

Cs + Cat +  − An−  Ct − Cat +  + An− 

(6)

K

S
CatAn By substituting An−  = Cat from Eq. (2) into + Eq. (6), the following expression is obtained:   Cs + Cat +  − KS
CatAn /Cat +   Vt = Vs ×  (7) Ct − Cat +  + KS
CatAn /Cat + 

Equation (7) describes the course of the titration, i.e., the volume of titrant Vt as a function of its concentration change Cat +  during the titration procedure. 4

Sak-Bosnar et al.

E = E 0 − S × log aAS−

(8)

where E = constant potential term, S = sensor slope, and aAS− = activity of surfactant anion. The experimental data were compared with the appropriate theoretical curves in which the sensor response parameters (slope S, constant potential therm E 0  and unknown analyte properties (sample concentration Cs , solubility product constant Ks  were optimized. Solver (Excel) was used to find values for those variables that minimize the sum of the squares of the differences between the theoretical and experimental curves. In other words, the least-squares criterion fit the theoretical curve to the experimental data by using the entire data set. A part of the appropriate spreadsheet displaying the potentiometric titration data and model parameters for the titration of DMIC with SDS is shown in Table I. The potentiometric titration curve for sodium dodecylsulfate (SDS) after the optimization procedure using Solver and the sum of squared residual (SR) values are shown in Figure 1. 0

3.1.2. Cationic Surfactants Cationic surfactants (Cat+  can be titrated inversely using a standard solution of an anionic surfactant (An−  as the titrant. The electromotive force of the membrane sensor assembly when dipped in the solution of cationic surfactant (CS) of activity aCS+ is given by the Nernst equation: E = E 0 + S × log aCS+

(9)

The final equation that defines the course of the titration, as a function of the titrant volume Vt and its concentration change An−  during the titration, can be derived as in the previous section:   Cs + An−  − KS
CatAn /An−   Vt = Vs ×  (10) Ct − An−  + KS
CatAn /An−  The potentiometric titration curve for benzethonium chloride (Hyamine 1622) after the optimization procedure using Solver and the squared residual (SR) values are shown in Figure 2. 3.2. Nonionic Surfactants 3.2.1. Reaction Stoichiometry of the Formation of the Pseudoionic Complex of Barium and Ethoxylated Nonionic Surfactants Barium ions form pseudocationic complexes with ethoxylated nonionic surfactants (EONS), such as Triton X-100, according to the following scheme: Ba2+ + xEONS
BaEONSx 2+

(11)

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Estimation and Optimization of Potentiometric Sensor Response Parameters from Surfactant Titration Data

Table I. The potentiometric titration data (partial) and model parameters for the titration of DMIC with SDS. Potentiometric titration data and model parameter (titration of DMIC with SDS) Model parameter Vs /mL Ct /M

105 4.000E-03 CDS− = 10 expE − E 0 /S

Cs /M E0 S Ks pKs

0.00019141 −350.20503 −61.875342 5.7601E-13 12.2395736

Vt (mL)

E (mV)

E − E 0 /S

CDS − (M)

pDS

(Vt ) model. (mL)

res.

SR

0 0.23 0.43 0.63 0.83 1.03 1.23

176.9 175.9 174.7 173.5 172.2 170.9 169.5

−8.51882 −8.50266 −8.48327 −8.46387 −8.44286 −8.42185 −8.39923

3.028E-09 3.143E-09 3.286E-09 3.437E-09 3.607E-09 3.786E-09 3.988E-09

8.51882 8.50266 8.48327 8.46387 8.44286 8.42185 8.39923

0.02992124 0.20438116 0.40606144 0.59965861 0.80060901 0.99278257 1.19032704

−0.0299212 0.0256188 0.0239386 0.0303414 0.029391 0.0372174 0.039673

0.00089528 0.00065632 0.00057305 0.0009206 0.00086383 0.00138514 0.00157394

ssr

0.27984318

The “x” value varies depending on the number of ethoxy (EO) units in the surfactant molecule. One barium ion forms a complex with 10–12 EO groups. More simply, the above equation can be written as follows: Ba2+ + xL
BaLx 2+ (12)

BaLx 2+ + 2TPB−
BaLx TPB2

(13)

The stoichiometry of reactions (11)–(13) depends on the chain length of the oxyethylene part (hydrophilic) of the non-ionic surfactant and the nature of the rest of the surfactant molecule (hydrophobic part).

Exper. Model SR

and for titrant TPB− : TPB−  + BaLx TPB2  + PBaLx TPB2 =

Ct × Vt Vs + Vt 

(16)

150

0.016

Exper. Model SR

50

0.012

0.008 –50

SR/Arbitrary units

0.01

150

E/mV

0.02

250

SR/Arbitrary units

0.03

350

E/mV


C × Vs (15) BaLx 2+ + BaLx TPB2  + PBaLx TPB2 = s Vs + Vt 

where Vs is the sample volume of concentrations CBaLx 2+ and CBaLx TPB+ and Vt is the titrant volume of concentration Ct . BaLx TPB2  is the equilibrium concentration of

0.04

450

The formation of precipitate in the reaction described by Eq. (13) can be used to model the potentiometric titration. The solubility product of the precipitate is: 
(14) Ks = BaLx 2+ TPB− 2 
The mass balance equation for analyte BaLx 2+ :

0.004 50

0

0

2

4

6

8

Vt/mL

0

–150 0

2

4

6

8

Vt/mL Fig. 1. Comparison of experimental and modeled potentiometric titration curves of SDS with DMIC. Here and in the other figures, the values of the squared residuals (SR) are presented on the secondary Y-axis.

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Fig. 2. Comparison of experimental and modeled potentiometric titration curves of Hyamine 1622 with SDS.

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where L = EONS. The quantity of complex formed can be determined by titration with a standard sodium tetraphenylborate solution:

3.2.2. Modeling of the Potentiometric Titration of the Pseudoionic Complex of Triton X-100 and Barium with Sodium Tetraphenylborate

Estimation and Optimization of Potentiometric Sensor Response Parameters from Surfactant Titration Data

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titration is obtained:   Cs + TPB−  − Ks /TPB− 2 Vt = Vs   Ct − TPB−  + Ks /TPB− 2

(19)

Tetraphenylborate is the component of the ion-exchange complex used as the sensing material in the electrode membrane. The response of the nonionic surfactant sensor toward tetraphenylborate is described by the following equation: 0 − ETPB− = ETPB − + STPB− log TPB 

Fig. 3. Comparison of experimental and modeled potentiometric titration curves of pseudoionic complex Triton X-100 and barium with sodium tetraphenylborate.

BaLx TPB2 in the
solution,  and PBaLx TPB2 is the concentration of analyte BaLx 2+ or titrant TPB−  removed by the precipitation of BaLx TPB2 . By subtracting Eq. (16) from Eq. (15):

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C × Vs − Ct × Vt  BaLx 2+ − TPB−  = s Vs + Vt 

(17)

from which, after some rearrangement, the progress of the titration can be obtained:
 Cs + TPB−  − BaLx 2+ Vt = (18) Vs Ct − TPB−  + BaLx 2+ 
 By inserting BaLx 2+ = Ks /TPB− 2 from Eq. (14) into Eq. (18), the following expression for the progress of the

(20)

0 where ETPB − = constant potential term, STPB− = slope, aTPB− = activity of tetraphenylborate anion and TPB−  = concentration of tetraphenylborate anion. Following from Eq. (20), 0 (21) TPB−  = 10ETPB− − ETPB− /STPB−

which, after substitution into Eq. (19), gives: Cs + 10ETPB− −ETPB− /STPB− − Ks /10ETPB− −ETPB− /STPB−  0

Vt = Vs

0

Ct − 10ETPB− −ETPB− /STPB− + Ks /10ETPB− −ETPB− /STPB−  0

0

(22) The potentiometric titration curve of the pseudoionic complex Triton X-100 and barium with sodium tetraphenylborate after the optimization procedure using Solver is shown in Figure 3. The same titration data were also optimized using Wolfram Research Mathematica v. 7.01.27 The two software tools gave almost the same estimates of the parameter models and the corresponding statistics, as shown in Table II.

Table II. Sensor and titration parameters for the various surfactants obtained by modeling their potentiometric titration data after the optimization procedure using Solver (S) and Mathematica (M). Anionic surfactants Parameter Sensor used Titrant

Cationic surfactants

Nonionic surfactant

DS

DBS

CTAB

DMIC

Triton X-100

DMI-TPB DMIC

TA-DS Hyamine

DMI-TPB NaTPB

DMI-TPB SDS

BaLx (TPB)2 NaTPB

Slope*/ (mV/decade)

S M

53.1 ± 0.5 52.7 ± 0.5

86.9 ± 1.0 88.6 ± 1.0

−53.6 ± 0.6 −53.5± 0.6

−61.9 ± 0.7 −61.6 ± 0.7

−89.0 ± 0.9 −88.8 ± 0.9

E 0 */mV

S M

651.9 ± 1.9 649.7 ± 1.8

557.5 ± 4.3 557.6 ± 4.3

−505.1 ± 2.0 −504.9 ± 1.9

−350.2 ± 2.8 −349.0 ± 2.9

−353.7 ± 3.9 −352.9 ± 3.9

pKs∗

S M

13.67 ± 0.07 13.72 ± 0.07

10.24 ± 0.03 10.20 ± 0.03

14.79 ± 0.09 14.71 ± 0.09

12.24 ± 0.05 12.26 ± 0.05

15.72 ± 0.03 15.73 ± 0.03

Concentration*/ (mol/L)

S M

(6.738 ± 0.014)×10−4 (6.742 ± 0.014)10−4

(2.666 ± 0.011)×10−4 (2.702 ± 0.010 × 10−4

(8.575 ± 0.025)×10−4 (8.634 ± 0.024 × 10−4

(1.914 ± 0.006)×10−4 (1.903 ± 0.006 × 10−4

(6.100 ± 0.027)×10−5 (6.102 ± 0.026 × 10−5

Correlation coefficient R2

S M

0.9995 0.9995

0.9988 0.9989

0.9993 0.9992

0.9992 0.9992

0.9994 0.9993

SE(y)

S M

0.0469 0.0359

0.0259 0.0257

0.0732 0.0733

0.0713 0.0689

0.0575 0.0575

∗

value ± standard deviation.

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20

Eo

S

Ks

Cs

18 16

Log(J)

14 12 10 8 6 4 2 0

DS

DBS

CTAB

DMIC

Triton X-100

Surfactants Fig. 4. Sensitivities of the model variance on the model parameters for all the sensors and surfactants investigated.

where SS is the sum of the squares of the model residuals, the p values are the model parameters, and p∗ values are their optimal estimates. The sensitivities were numerically approximated by the finite difference method with 1% perturbation of each parameter, and are expressed on a logarithmic scale due to several orders of magnitude differences among the values. The results presented in Figure 4 show that Ks is the dominant modeling parameter (four orders of magnitude) for all of the sensors and titration parameters applied.

4. CONCLUSIONS Theoretical models for the potentiometric titration curves of some ionic and nonionic surfactants have been proposed. Potentiometric titration data were used to estimate sensor response parameters and analyte properties (sensor slope, constant potential term, solubility product constant, and analyte concentration). The experimental data were compared with appropriate theoretical curves in which the unknown properties (sensor Sensor Letters 9, 1–8, 2011

Acknowledgment: The authors gratefully acknowledge financial support from the Croatian Ministry of Science, Education and Sports given to project No. 291-05800000169.

References and Notes 1. T. Schmitt, Analysis of Surfactants, M. Dekker, New York, Basel (2001). 2. S. R. Epton, Nature 160, 795 (1947). 3. V. W. Reid, G. F. Longman, and E. Heinerth, Tenside 4, 292 (1967). 4. W. S. Selig, Fresenius Z. Anal. Chem. 329, 486 (1987). 5. B. J. Birch, D. E. Clarke, R. S. Lee, and J. Oakes, Anal. Chim. Acta 70, 417 (1974). 6. K. Vytras, Ion-Selective Electrodes Rev. 7, 77 (1985). 7. A. Seher, Fette Seifen Anstrichm. 63, 617 (1961). 8. R. J. Levins and R. M. Ikeda, Anal. Chem. 37, 671 (1965). 9. K. Vytras, V. Dvorakova, and I. Zeman, Analyst 114, 1435 (1989). 10. K. Vytras, I. Varmuzova, and J. Kalous, Electrochim. Acta 40, 3015 (1995). 11. D. L. Jones, G. J. Moody, and J. D. R. Thomas, Analyst 106, 439 (1981). 12. D. L. Jones, G. J. Moody, J. D. R. Thomas, and B. J. Birch, Analyst 106, 974 (1981). 13. P. H. V. Alexander, G. J. Moody, and J. D. R. Thomas, Analyst 112, 113 (1987). 14. J. Gwilym, G. J. Moody, and J. D. R. Thomas, Nonionic Surfactants (Chemical Analysis), edited by J. Cross, M. Dekker, New York (1987). 15. G. J. Moody, J. D. R. Thomas, J. L. F. C. Lima, and A. S. C. Machado, Analyst 113, 1023 (1988). 16. P. G. Delduca, A. M. Y. Jaber, G. J. Moody, and J. D. R. Thomas, J. Inorg. Nucl. Chem. 40, 187 (1978). 17. R. D. Gallegos, Analyst 118, 1137 (1993). 18. T. Okada, Analyst 118, 959 (1993). ˇ ci´c, M. Sak-Bosnar, R. Mateši´c-Puaˇc, and 19. D. Maduni´c-Caˇ Z. Grabari´c, Sensor Lett. 6, 339 (2008). 20. R. Matesic-Puac, M. Sak-Bosnar, M. Bilic, and B. S. Grabaric, Sens. Actuators, B 106, 221 (2005). 21. M. Sak-Bosnar, D. Madunic-Cacic, R. Matesic-Puac, and Z. Grabaric, Anal. Chim. Acta 581, 355 (2007). 22. D. Diamond and V. C. A. Hanratty, Spreadsheet Applications in Chemistry Using Microsoft Excel, Wiley, New York (1997).

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An excellent fit of the theoretical models to the experimental values was obtained in all cases for the tested sensors and the titration parameters. By using the above methodologies, the values for the four model parameters (slope S, constant potential term E 0 , sample concentration expressed as recovery, solubility product constant Ks  were calculated (Table II). The obtained estimates of the parameter standard errors, standard error of the output variable, and correlation coefficients support the validity of the model. In order to discern the key model parameters, the local sensitivities of the model variances at the optimal estimates were evaluated. Evaluation of the sensitivities J was based on the following expression:    SS p∗     (23) J p = Log  p 

slope, constant potential term, solubility product constant, and analyte concentration) were optimized. Solver (Excel) was used to find the values for those variables that minimize the sum of the squares of the differences between the theoretical and experimental curves using the least-squares criterion to fit the theoretical curve to the experimental data set. The initial parameter settings were defined, and no constraints were applied on the variables. The regression statistics for Solver were performed by calculating the standard deviations of the parameters, correlation coefficients and standard error of the y estimate SE(y with the macro SolvStat. Wolfram Research Mathematica was used to evaluate the local sensitivities of the model variances at the optimal estimates. The theoretical models fit satisfactorily to the experimental values for all of the investigated surfactants.

Estimation and Optimization of Potentiometric Sensor Response Parameters from Surfactant Titration Data

26. S. Walsh and D. Diamond, Talanta 42, 561 (1995). 27. Mathematica, v.7.1., Wolfram Research, Champaign, Illinois (2009). ˇ ci´c, M. Sak-Bosnar, O. Galovi´c, N. Sakaˇc, and 28. D. Maduni´c-Caˇ R. Mateši´c-Puaˇc, Talanta 76, 259 (2008). ˇ ci´c, and 29. M. Sak-Bosnar, R. Mateši´c-Puaˇc, D. Maduni´c-Caˇ Z. Grabari´c, Tenside Surfact. Det. 43, 82 (2006).

RESEARCH ARTICLE

23. E. J. Billo, Excel for Chemists, Wiley, New York (2001). 24. S. R. Crouch and F. J. Holler, Application of Microsoft Excel in Analytical Chemistry, Thomson Learning, Belmont (2004). 25. R. de Levie, How to Use Excel in Analytical Chemistry and General Scientific Data Analysis, Cambridge University Press, Cambridge (2001).

Sak-Bosnar et al.

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Sensor Letters 9, 1–8, 2011