Anal. Chem. 1983, 55, 855-861
855
Determination with Ion Selective Electrodes in the Low-Level, Non-Nernstian Response Region Jack W. Frazer,"' Davld J. Balaban, Hal R. Brand, Gaythla A. Robinson, and Stanley M. Lanning Lawrence Liverniore National Laboratoty, P.O. Box 808 L-3 1 1, Livermore, California 94550
Ion Selective ellectrodes can be utlllzed to obtaln quantltatlve analytical data even when the concentratlons to be determined result in EMF outputs In the non-Nernstlan response region. We describe an experlmental electrochemical analytlcal cell deslgned to obtain such data from tltratlons using multlple ISEs and reference electrodes. A mlnlcomputer Is used for data aicqulsltlon and the control of a Mettler buret. Two pattern mirtchlng methods are used for comparlng data sets obtalned from the sample wlth data sets obtained from callbratlon runs. Callbratlons are run In a manner ldentlcal wlth sample analysis but wlth a blank solution. Finally, an artlflclal lntelllgsnce method (speclflcally an expert system) Is used to analyze the resultant data sets obtained from the comparison of a sample wlth callbratlon data. Quantltatlve analytlcal resullts can be obtalned from the non-Nernstlan response regloin down to the ISE noise level.
Traditionally, ion selective electrodes (ISEs)have been used for analytical determinations in the range where the electrode output can be represented by the Nernst equation. In this range, straight-line calibration curves can easily be constructed using semilog p,aper. However, in general no serious attempt has been made to determine ions whose concentrations fall in a range so low that the ISE response is non-Nernstian, i.e., nonlinear on semilog paper. The reasons for avoiding the low-level, non-Nernstian ISE response region are lack of reproducibility of the cell-electrode system response to changes in ion concentraltion and the difficulty in determining the ion concentration even when there is reasonably good cell-electrode stability. However, the development of new experimental techniques and instrumentation utilizing microprocessors provides the computational and control capabilities necessary for the development of analytical procedures which will extend ISE determinations to concentrations below the traditional linear-response region. For purposes of environmental monitoring or control of impurity concentrations in processing systems, there is often a need to measure ions at very low concentrations. In many applications ISEs would be the ideal instrument choice, provided the sensitivity could be extended below what is commonly called the linear response region. The difficulty in operating outside the linear response region can be overcome by using nonlinear pattern matching programs to compare the titration curve obtained from an unknown concentration with a similar titration curve obtained by using known concentration of the same ion. For practical applications a microprocessor could conceivably be incorporated into a bench-top instrument and, upon the completion of titrations, provide an immediate estimate of the ion concentration. However, non-Nernstian ISE response is not the only problem. The lack of ISE output stability at any given ion concentration in the non-Nernstian response region causes far more difficulties. 'Present address: Keithley Instruments, Inc., 28775 Aurora Road, Cleveland, Ohio 44139. 0003-2700/83/0355-0855$0 1.50/0
Solid-state electrodes, such as the bromide and thiocyanate electrodes used in these experiments, may or may not provide a reproducible EMF at fiied ion concentrations. The response appears to be a function of the physical state of the electrode surface and the chemical environment in which the electrode was previously operated and is currently operating. The physical state of a freshly prepared electrode will have multiple surface defects spanning some energy range. The distribution and magnitude of these defect energies depend on the device surface preparation. They can affect the absorption-desorption properties of the electrode interface. It has been postulated (1, 2) that the absorption of material on the electrode surface can result in shifts in EMF. Thus, given an electrode surface with a fixed number of defects, the environments within which the ISE has been functioning can result in small changes of EMF with respect to a fiied concentration of the ion for which it is most selective. Obviously, if ions other than the primary ion can be concentrated a t the surface defects, they will also influence the cell EMF. Thus, minor variations in EMF, at fixed low concentrations of the primary ion, are to be anticipated when ISEs are used for normal routine analysis. These minor variations in E M F can result in relatively large percentages of the signal when working at low concentrations, i.e., in the non-Nernstian response region. In this paper, we shall discuss experimental and calculational techniques that provide the basis for low-level determinations. In addition, the artificial intelligence techniques required for the evaluation of the large data sets obtained from the simultaneous utilization of multiple electrodes will be discussed. Finally, results will be presented to illustrate the quality of determinations and the problems encountered with regard to cell-electrode signal reproducibility.
EXPERIMENTAL SECTION Apparatus. The unstable EMF outputs from chemical cells operating in the non-Nernstian response region dictated a nontraditional approach for obtaining useful information. First,, it was supposed that the widely varying EMF results were inot stochastic, Le., they were not random but rather erratic due to varying physical properties of the ISE and its surface. Second, it was thought that if enough unique measurements (estimates of ion concentration) could be obtained, artificial intelligence techniques could be used to select the quality results from data sets containing a high percentage of erratic results. With these two suppositions in mind, a titration cell containing three ISEs and three reference electrodes was constructed and interfaced to a PDP 11/45 computer (3, 4). In the cell shown in Figure 1, the electrodes, buret tip, and stirring rod are held firmly in a Teflon cap that fits into the 1,op of a 150-mL beaker. The rims of the titration beakers have been sawed off at exactly the same distance from the bottom of each beaker. The Teflon cap (Figure 1) holds the electrodes firnnly in a fixed geometry in the titration beaker. (In order to assure reproducible cell configuration and resultant EMF outputs.) Figure 2 is a schematic showing the computer, buret, multiplexer, analog-to-digitalconverters, amplifiers, thermometer, and titration cell. The design provides for the acquisition of data from all nine combinations of ion selective and reference electrodes as well as from the digital thermometer. Analog-to-digital conversion is performed by a 128-channelADC with 14 bits plus sign resolution and a 40 kHz throughput rate. 0 1983 American Chemical Society
856
ANALYTICAL CHEMISTRY, VOL. 55, NO. 6, MAY 1983 Buret tip 7
Temperature probe Ion selective electrode 3
Raference electrode 1
Reference electrode 3
Ion selective electrode 1
electrode 2
electrode 2
150 ml cut to 2.5 inch height
Figure 1. The titration cell consisting of a modified 150-mL beaker and a Teflon cap to position and hold, three ISEs, three reference electrodes, stirring rod, buret tip, and temperature probe.
Digital Thermometer
Figure 2. A schematic diagram of the computerized apparatus showing the processing and control components.
The buret is a Mettler DVlO which is controlled by the PDP 11/45. Reproducible and uniform stirring of the titration cell were found to improve the quality of the analytical estimates. Satisfactory speed control was obtained with a Barber-Coleman Co. Type FYQM 63260-15 speed-regulated DC motor and the CYMQ-568-64 speed control circuit board. The amplifier circuitry was designed and constructed at LLNL. (Schematics can be obtained upon request.) Determinations utilized two different temperature control schemes. When good temperature control was desired, the sample cell was immersed in a water recirculating bath maintained at 19.0 & 0.1 "C. However, many determinations were made at ambient temperatures with control to a 1 "C range for an individual data set of four determinations plus five calibrations. Reagents. Ultrapure analytical reagent grade KBr (Alfa) was used for preparation of standard and titrant Br- solutions. A
0.8131 M KBr stock solution was prepared from which all other solutions were prepared by means of serial dilution with deionized water. Three different titrant solutions containing 0.1 M KNOB (EM Suprapure) for ionic strength adjustment were used, i.e., 3.662 X M Br- and 3.749 X lob M Br- for all determinations except the one determination at the 2.9 ppb level which used a titrant containing 2.747 X lo4 M Br-. The "unknown" solutions were prepared from the same stock solution and also contained 0.1 M KNOa for ionic strength adjustment. The thiocyanate solutions were prepared from analytical reagent grade NaSCN (Mallinckrodt). All standards and titrants were prepared by serial dilution of a stock solution with deionized water. Ionic strength adjustment was accomplished by making the solutions 0.1 M in KN03. Adjustment of solution pH to 4.0 was done by using ultrapure HN03 (Ventron). The titrants used were 3.179 X M SCN-, 3.002 X M SCN-, and 2.252 X M SCN-. Experimental Design. The method requires that a calibration run be made with each determination of an unknown. Ideally, calibrations and unknown determinations are made alternately, thereby providing two calibrations for each determination of an unknown, i.e., a calibration immediately preceding and following each determination. Calibrations and determinations both used the standard titrant solution. For a calibration, 3 mL of the titrant solution was added in 300 equal size steps (0.01 mL each) to a 50-mL blank sample solution, Le., deionized water with 0.1 N KN03 ionic strength adjustor. An EMF measurement was made for each of the nine ISE-reference electrode pairs just prior to each titrant addition. The total titration time was 216 s during which time 2700 EMF measurements were made. For the determination of unknowns, the sample volume was adjusted to 50 mL at a KNOBconcentration of 0.1 M. A total of 1.5 mL of titrant was then added in 300 increments of 0.005 mL each. An EMF measurement was made for each of the nine ISE-reference electrode pairs prior to each titrant addition. Total titration time for an unknown sample was 108s. Thus the titration time for a calibration-unknown pair was 324 s. Shorter titration times could have been used if desired. However, no effort was made to determine what, if any, effect the rate of titration addition and subsequent EMF measurements had on the accuracy of the determination. For the results reported here, identical unknown samples were determined, each interleaved between two calibrations. Thus for each determination of unknown, there were two calibration runs each determination of an unknown resulted in, two data sets with each set consisting of nine separate sample titration curves and nine separate calibration curves for a total of 18 EMF curves of 300 measurements each. Thus, when an unknown determination was bracketed by two calibration titrations, there were a total of 18 calibration-unknown pairs of data sets which could be used to estimate the concentration in the unknown. Analyses were performed as follows: A calibration data set was obtained by taking a blank to which a total of 300 additions were made, each titrant addition being followed by an EMF measurement. An unknown was analyzed by obtaining a data set as above but with the unknown sample (same size as the blank) being used in place of the blank. The above was repeated for the samples to be analyzed. Finally one additional blank was run. Calculations, For the calculations discussed below, the entire calibration data set, ~ u with n a blank, was used together with the data points 31 through 155 of the data set taken for the determination of an unknown. The first 30 data points of the data set from the unknown were discarded, because, during the period these data pointa are taken, the system was coming to equilibrium and the quality of the data were not representative of that obtained later in the titration, i.e., the front portion of the data set was "hook" shaped. A typical calibration plot of potential vs. log concentration for a bromide ISE is shown in Figure 3. The plot in this figure has been divided into two sections by a dotted line. The section on the right is described by the Nernst equation E = E o+ S l n C (1) where S is the electrode "slope" (RTIBF), C is the concentration
ANALYTICAL CHEMISTRY, VOL. 55, NO. 6, MAY 1983
E Non-Nernstian region
I1 \ \
\ E
I I
.~
Log c
857
CALIBRATION , r -
CURVE
\
Figure 3. A calibratlon plot of the potential vs. log concentrations for a bromide ISE. The dotted line d l v i s the Nernstian region on the right and the non-Nernistian region on the left.
of bromide, and ,Eo is the potential at unit bromide concentration. Although most electrochemical analyses using ISEs have been performed on solutions having concentrations in this Nernstian region, for the work reported here the curved section to the left of the dotted lint?(non-Nernstian region) in Figure 3 is the region of interest,. In this non-Nernstian region, eq 1no longer describes the behavior of the electrode, and a new method of data analysis is required. No analytical model for the non-Nernstian behavior of the electrode will be assumed for the new method. Only empirical models will be used. The proposedl method requires taking the two data sets, a calibration data set and an unknown data set, and comparing the unknown data set to the calibration data set in order to obtain estimates of the concentrations of the unknown. The details of the method follow. A calibration data set is a set of measurements of potentials after specified volumes of standard have been added. This means that a calibratiori data set can be represented by a set of ordered pairs of numbers. This set of ordered pairs will be written as follows: {(El,VI), (E23 4, (EN,V N ) ) = ((Ei,ui), i = 1, 2, .**,N ) where E; is the potential at data point i, ui if the volume added up to point i, and N is the number of volume additions made. In a calibration run, a blank with zero (undetectable amount) concentration of the species to be determined is titrated with a standard solution. Thus the concentration of unknown initially in the titration cell and the concentration of standard to be added to the cell are both known. This means that, after each addition of standard, the concentration of standard in the vessel can be computed by using the following formula: -7
C=
voc, + u c s uo + u
where C is the concentration of standard as a function of volunie added, Vo is the initial volume in the vessel, C, is the initial concentrationof standard m the vessel, u is the volume of standard added, and Cs is the concentration of standard being added to the vessel. The parameters Vo and C, in eq 2 are always known. The parameter Co is known only for calibration runs. In fact, Co is the parameter which is to be determined by the electrochemical analytical method. Equation 2 then means that, if Co is known, C can be found as a function of u. This can be written in the following symbolic form:
c = f(u;Co)
(3) Equation 3 is a convenient way of writing eq 2. The reason for this more abstract notation will become clear later. With eq 2, a new set of ordered pairs can be computed. Each ordered pair in this set consists of a potential and a corresponding concentration of Btandard. This set is written as (IC&) such that i = 1, 2, ...,N (4 where E, ie, the potential a t point i and C, is the concentration at point i. A least-riquares spline function is then fit to this new set of ordered pairs (5, 6). This type of fit tends to follow the general
\
LOG C
Figure 4. The No-Shift method for matching unknown data anid a callbratlon curve. A sequence of estimates of the true Co.Estimate (2) would be selected as the best match. Value of Co increases in order 1-2-3.
shape of the data while ignoring the noise. The purpose of this fit is to find an equation to represent the behavior of the electrode in the non-Nernstian region This new spline function becornes an empirical model for the non-Nernstian behavior of the ISE just as eq 1 models the Nernstian behavior. The relationship between potential and concentration which the spline function describes will be denoted by E = g(c) (5) Two things should be noted about eq 5. First, all the information which was in the set of ordered pairs, eq 4, is also in eq 5. In fact if one of the parameters in the least-squaresspline fitting process is chosen properly, then E, = g(c,) i = 1, 2, ...,N (6) This means that g goes through all the data points. Even if the parameter in the least-squares spline process is not chosen in this way, it will always by true that E, = g(C,) i = 1, 2, ...,N (7) The second thing to note is that g can be evaluated at any concentration value, not just at the C, values. In other words, 6’ is continuous and smooth. So far, only the calibration data have been discussed. The volume values are transformed and the transformed data ueed to make a least-squares spline fit. No such manipulations are necessary with the unknown data. These data will always be used in its original form of an ordered pair data set. (E,,u,) such that i = 1, 2, ..., N ((8) The Co of interest is the concentration of the ion of interest in the unknown solution. If a value of Co is guessed, eq 3 can be used to determine the unknown concentration values that would have resulted after each volume addition. These concentration values can then be used with eq 5 to determine the calibration curve potential values that would have corresponded to each volume addition. Thus, eq 3-5 allow potential values to be computed which are predictions of what the measured potential values would be. The results of choosing three values of Co and then predicting the measured potential value are shown in Figure 4. As the value of Co increases,the predicted unknown data sets slides to the riglit. All of these predicted data sets are parallel. Thus, there will be a value of Co which gives the best match between the calibration and unknown data. The criterion used to determine the best match is the standard least-squarescriterion. The sum of squares to be minimized is given in eq 9. N
SSQi = C ( E ,- g(f(u,;Co)))2 ,=I
(9)
This method is called the Non-Shift method for reasons to be discussed shortly. There are problems with this method. First,
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 6, MAY 1983
Table I. Rules Associated with the No-Shift/Shift Ratios No-Shift/Shift ratios
rule
class A 0.97 t o 1.035
identifies the data as a quality estimate of the ion concentration class B 1.035 to 1.8 and identifies data that may be a good estimate of the ion 0.97 t o 0.2 concentration, however, further analysis is required class C t o 1.8 and identifies erroneous data that 0.2 to 0.0 should be immediately discarded
-
E /
CRLIBRATION CURVE
I
/\
i
Figure 5. The Shift method for matching unknown data and a callbration curve. Posltion (2) is the approximate positlon where the concentration estimate would be made.
the method compares only absolute potentials between calibration and unknown data curves. If the two curves had radically different shapes, and nearly the same mean value, a match would be made anyway. Second, if the two data sets are offset from each other in potential, then this method must give the wrong answer. Third, this method is intuitively unsatisfying. A human could easily recognize that matching curves were very different even though they had the same mean. Therefore, this method cannot be used by itself to evaluate data sets. However, the results obtained by this method and the results from a more intuitive method can be combined to create more reliable results. A second more intuitive method called the Shift method was developed next. In order to take the shapes of the curves into account, another parameter, the Shift parameter, was introduced and varied during the matching process. Figure 5 shows the basic idea behind this new matching process. In this case, the measured data set slides along the calibration curve until the best fit is found. The motion of the measured data sets is downward and to the right as Co increases. This method is therefore insensitive to potential offsets between the calibration data and measured data. It is, however, very sensitive to the shapes of the curves. If the calibration and data curves have different shapes, then this method may give the wrong answer. The detailed implementation of this method was carried out as follows. A combination of linear and nonlinear least-squares methods was used. It assumed that the best match between unknown and calibration data occurred when N
C(Ej - g(f(u,;Co))- H)’ = SSQZ
(10)
;=I
was minimized with respect to Co. Here the Ei values are the measured data points and the g(f(ui;Co))are the predicted measured data points assuming Co is the true unknown standard concentration. The variable H is the shift parameter. First, an initial guess for Co is made. Then since H appears in a “linear” way in eq 10, it can be determined immediately with the standard linear least-squares method. The determination of the shift parameter effectively shifts the predicted data set to approximately the same potential values as the measured data set. Next, a one-dimensional nonlinear parameter estimator uses this value of H and the initial guess of Co to compute the SSQz value of eq 10. The parameter estimator then uses this information to predict another estimate of the minimizing Co. This process eventually converges to a Co value inside the range of possible Co values specified by the calibration run or it assigns Co the value of one of the concentration extremes of the calibration file. The two methods result in a total of 18 estimates of the unknown concentrations to be determined for each calibration analysis pair, If calibrations are interleaved with determinations, there are then two calibration-analysis pairs for every unknown sample or 36 estimates of the ion concentration in the unknown.
Since the electrode outputs in the non-Nernstian region are erratic, any set of 36 estimates may (and usually will) cover a range of 1to 2 orders of magnitude in concentration estimates. How then does the scientist select the “correct” estimates? At the start of this project, it was thought that the magnitude and form of the titration curves would contain the information required to assess the quality of the electrochemical cell output. Therefore, the problem was one of selecting the relevant parameters to be evaluated and the evaluation rules. Obviously a number of parameters were evaluated. However, many of those tested were found to bear no simple relationship to the quality of the concentration estimates, e.g., the sum of squares of the residuals and the constancy of the estimated concentration. The parameter found to contain much good information was the difference profile obtained by subtracting the calibration curve from the unknown curve. For the No-Shift method there is only one difference pattern. However, for the Shift method there is a difference pattern every time eq 10 is evaluated. The difference pattern used for the analysis is the one obtained when the SSQ value is minimum. In practice, the difference profile of the two curves was found to be the most useful parameter for assessing the quality of the concentration estimate. Figure 6 shows four of the many types of difference profiles that were obtained when performing routine analyses. The difference profile in each figure has a characteristic shape. Each is jagged and nonmonotonic but has some systematic parameters associated with it. Several parameters associated with the difference patterns were evaluated with respect to their utility in selecting quality estimates of the unknown ion concentration. The two most useful parameters were the concavity and slope (sign and magnitude) of the difference pattern. These parameters were evaluated as follows. A computer program calculated a least-squares fit of the arcs of 11 different circles of decreasing radii to each difference pattern. For each fixed radius a nonlinear least-squares fitting procedure is carried out to determine where the center of a circle with this radius would be located in order to achieve the best least-squares fit to the data. This method is very much like laying a circle template on a data set and then moving the template around until the best fit to a section of data is achieved. The circle providing the best fit to the pattern was chosen. As for example, in Figure 7 the circle in Figure 7B was chosen since it was a better fit than either circle 7A or circle 7C. Next the tangent to the selected circle was calculated at the midpoint of the difference pattern. The slope of the tangent, sign and magnitude, provided two other useful parameters. In addition, it was found that the ratio of No-Shift to Shift estimates is also a good parameter, Le., when the ratio is very nearly equal to 1 both estimates are good, when it is >1the Shift estimate is best, and when it is
