978
Anal. Chem. 1984, 56,978-985
results in Figure 8b,c clearly illustrate the difficulty in interpreting ratiograms of tailing and overlapping peaks. Clearer data can be obtained if the threshold value is enhanced, but then we lose information. In Figure 8d the elution order is changed, and going to e and f the separation of the two peaks increases again. From ratiogram f it is possible to identify the two solptes.
ACKNOWLEDGMENT We wish to thank Beckman Instruments (Nederland) B.V., Mijdrecht for the loan of the Model 165 multichannel rapid scanning UV-VIS detector. LITERATURE CITED (1) Berridge, J. C. J . Chromatogr. 1982, 244, 1. (2) Watson, M. W.; Carr, P. W. Anal. Chem. 1979, 51, 1835. (3) Glajch, J. L.; Kirkland, J. J.; Squlre, K. M.; Minor, J. M. J . Chromatogr. 1980, 199, 57.
(4) Drouen, A. C. J. H.; Billlet, H. A. H.; Schoenmakers, P. J.; De Gaian, L. Chromatographia 1982, 16, 48. (5) Issaq, H. J.; McKnitt, K. L. J . Liq. Chromatogr. 1982, 5 , 1771. (6) Guiochon, G.; Arpino, P. J. J . Chromatogr. 1983, 271, 13. (7) Ostojlc, Nedjeljko Anal. Chem. 1974, 46, 1653. (8) Yost, R.; Stoveken, J.; MacLean, W. J . Chromatogr. 1977, 134, 73. (9) Krstulovic, A. M.; Brown, P. R.; Douglas, M. R. Anal. Chem. 1976, 48, 1383. (10) Krstulovic, A. M.; Brown, P. R.; Douglas, M. R. Anal. Chem. 1977, 4 9 , 2237. (11) Baker, J. K.; Scheiton, R. E.;Ma, C. J . Chromatogr. 1979, 168, 417. (12) Ll, Kuang-pang; Arrlngton, John Anal. Chem. 1979, 51, 287. (13) Carter. G. T.; Schlesswohl, R. E.; Burke, H.; Yang, R. J . Pharm. Sci. 1982, 71, 317. (14) Mllano, M. J.; Grushka, E. J . Chromatogr. 1976, 125, 315. (15) Webb, P. A.; Ball, D.; Thornton, T. J . Chromatogr. Sci. 1983, 21, 447. (16) Schoenmakers, P. J.; Billlet, H. A. H.; De Galan, L. J . Chromatogr. 1981, 205, 13.
RECEIVED for review October 24,1983. Accepted January 18, 1984.
Theoretical and Practical Limitations on the Optimization of Amperometric Detectors Janean M. Elbjcki, Donald M. Morgan, and Stephen G. Weber*
Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
When steady-state equatlons for hydrodynamic current in amperometric detectors are integrated, they can be used to characterize detectors udng flow-injection experiments. The peak area Is independent of peak shape. This qulck, reliable method gives Information necessary to relate experimental resuits with theory. Once this Is determined, optlmizstion can be considered by examlning the graphical reglon where five inequalities derived from boundary conditions are satlsffed. The waiCJet and channel thin-layer ceUs were studled in detail. It was found that a wall-jet cell is theoretically and experimentally equivalent to a thin-layer cell when small spacers are used. The tendency for current to become less dependent on flow rate with time is traced to the formation of a thin film on the electrode surface.
There have been attempts of various sorts to optimize the amperometric detector commonly used in liquid chromatography. Many physical factors must be taken into account to accomplish this. Some rather general notions have allowed the semiquantitative optimization of detectors, such as the fact that current density in an electrochemical detector decreases at points along the electrode farther from the entrance of the solution to the cell and that the noise in the measurement is proportional to the area of the electrode (1). With this latter assumption it can be shown, for example, that there is no benefit to having a channel wider than an electrode and that there is no best shape for a rectangular electrode (2)in channel-type thin-layer cells. Recently other workers have attempted to deal with the much more difficult question of the relative merits of various detector geometries (3-7). Efforts in this direction have not been uniformly acceptable because of several difficulties; e.g., the best equation for current due to convective diffusion has not been employed, comparisons between theory and experiment have been made 0003-2700/84/0356-0978$0 1.50/0
Table I. Equations Describing Various Cell Geometries electrode type tubular planar planar
flowa
eq
I/
1 2, 3 4, 5
// 1
equations
ref
= 1.6I n FC( D A jr)”’” 8, 9 = 0.68nFCDZ13~‘116(A/b)112~11z 10 = 1.47nFC(DA/b)Z13~113 = 0.903nFCD2/3v- 1 1 6 A 3 1 4 ~ l 1 2 = 0. gggnFCDZl 3 v - 51 12, - I I ZA 31 8 u7 .1 4
9 11 12
a // means flow parallel to the electrode surface, 1 means means flow perpendicular to the electrode surface.
in which the theory is for steady-state current and the experiments measure peak current, and a particular theoretical equation for current may be extrapolated into regions in which it is analytically useless or the equation no longer correctly describes the system. In this paper we will theoretically determine which equation is most probably applicable to a particular geometry. We will indicate the theoretical and practical limitations on the application of this equation to a given cell, and we will show how one may experimentally determine whether a particular equation for steady-state current i s in fact being followed by one’s detector by making measurements on peaks.
THEORY Equations describing current at electrodes in flowing s t r e w s have been derived for a variety of geometries. There are five which describe amperometric detectors in use. These are listed in Table I. Equation 1 (in Table I) is for tubular electrodes with fully developed laminar flow passing parallel to the surface of the electrode. Equation 2 is for a flat plate 0 1984 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 56, NO. 6, MAY 1984
979
1 0 .{ 0 -
9.0
diameter of input conduit, cm electrode area, cm2 channel height or thickness, cm intercept concentration, mmol L-' bulk concentration, mmol L-' interfacial concentration, mmol L-' surface concentration, mmol L-' diffusion coefficient, cm' s - ' diffusion coefficient of analyte in polymer, cm2 s-' Faraday's constant, C mol-' current density, P A cm-' total diffusion limited current, pA flux, mol cm-' s - ' numerical constant distribution coefficient electrode length, cm total mass injected, @mol number of electrons transferred per mole a conglomeration of Barameters, cm3 s-' slope of log i vs. log U radius of tubular electrode, cm dimensionless electrode length radius of a circular electrode, cm average volume flow rate, cm3 s-' velocity, cm SS' dead volume, cm3 channel width, cm electrode width, cm distance in the direction of flow, cm entry length, cm KD,T;lDS, skin friction coefficient, g cm-' diffusion layer thickness, cm Prandtl boundary layer thickness, cm viscosity, g cm-' s-' kinematic viscosity, cm2 s-' density, g cm-3 electrode in the presence of developing laminar flow parallel to the plate. Equation 3 is for a flat plate in an enclosed rectangular channel, with fdly developed laminar flow parallel to the surface. Equation 4 is for an electrode against which is flowing a stream which is larger in diameter than the electrode, and finally eq 5 is for an electrode against which is flowing a stream much smaller in diameter than the electrode. Equations 2, 4, and 5 describe cases in which the Prandtl boundary layer's development is critical, and for these equations to apply there must be no physical impediment to the growth of the Prandtl layer. Since the objective of this work is to determine the factors that affect optimization efforts, let us consider which theoretical expressions apply to typical detectors. For tubular electrodes with fully developed laminar flow the case is clear, eq 1 applies. For the flat electrodes in channel cells, eq 2 or 3 may apply, depending on the state of development of the flow. This case has been dealt with in detail (13) so that only the conclusion will be stated. An initially uniform flow must travel a distance x , inside a rectangular channel before becoming fully developed.
(See Table I1 for a list of variables.) For electrodes close to the channel entrance and shorter than x,, eq 2 applies; for electrodes that begin downstream from the entrance at least x,, eq 3 applies. For a typical channel thin-layer cell (cTLC) x , = 10 pm; thus generally eq 3 applies. For flat, round electrodes with perpendicular flow, eq 4 or 5 applies. An initial criterion is the ratio of electrode diameter to stream diameter. For a small carbon fiber electrode and a typical chromatographic stream diameter, eq 4 would apply. For large electrodes and small streams eq 5 applies. As a
8.0 7.0 6.0
5.0 4.0 3.0
2.0 1.o
0.0 0.0
0.5
1.0
l o g (Flow ratel
1.5
2.0
(ul/sec)
Flgure 1. Percent wall-jet character for the rTLC using a 0.005 cm spacer, 0.300 cm diameter glassy carbon electrode, 0.050 cm diameter jet nozzle, and flow rates ranglng from 0.2 mL min-' to 3.0 mL mln-'. Under these condltlons the behavior fo the cell is that of a
thin-layer cell. practical matter, however, the dimensions of the flow cell may not allow unimpeded growth of the Prandtl layer (14), and then even eq 5 will not describe the electrode. The consideration of an electrode with apparent wall-jet geometry but with impeded development of flow follows. From the work of Matsuda (I5),Yamada and Matsuda (I2), and Levich (IO)one can determine quantitatively when the cell begins to restrict Prandtl layer development. We seek conditions under which 6o < b for some significant fraction of the electrode. The diffusion layer concept states that
i = nFDCO/G 60
=
(7)
(;>'/"6
Thus one has for a wall-jet electrode
where x is the radial distance out from the center of the electrode. At a distance out from the center of R/2'I2 the fluid will have traversed 50% of the electrode's area. At this point if a0 < b, at least half the electrode will be describable by eq 5. We arbitrarily choose a cell with an electrode that is at least half described by eq 5 as a wall-jet cell. Then to obtain at least one-half wall-jet character, one needs
b > 6.108(v/ D ) 3 / 4 ~ 1 / 2 ( R / & ) 5 / 4
(10)
or log b
> 0.2871 + 3/4
log ( v / U )
+ 1/2
log a
+
5 / 8 log A (11) Using some typical numbers (v = 0.01 cm2 s-', 0 = 0.02 cm3 s-', a = 0.03 cm, A = 0.05 cm2), one has log b > -1.513; b 2 0.03 cm. This corresponds to about 0.012 in., a very thick spacer. For typical spacers, b = 0.0125 or b = 0.0051 cm, a fairly small portion of the electrode follows wall-jet behavior. Figure 1 shows the percent wall-jet character as a function of flow rate for a "wall-jet" cell of typical dimensions. Even for high flow rates this cell is only 10% wall-jet-like. Yamada and Matsuda (12)provide the conceptual framework for developing an equation for current when ?>> io b by showing a relationship between ~ ( x ) the , skin friction coefficient, and current. To determine ~ ( x we ) must know the velocity d x ) = PCL(dU/&')y=o
(12)
980
ANALYTICAL CHEMISTRY. VOL. 56. NO. 6. MAY 1984
profile. The cell exhibits developed flow emanating radially from a point at the center of the electrode. The velocity in the radial direction x is just the flow rate in the radial direction divided by the cross sectional area exposed to flow.
60
U(X)
- y/b)
= --Cv/b)(l 2ux b
(13)
To find T one differentiates with respect t o y and seta y = 0 r(x) =
6p U
2uxb2
The total current is given by (11)
This is exactly the result one obtains for a channel cell in which the rectangular electrode is exactly as wide as the channel. For thin spacers then, the so-called wall jet behaves e x a d y like an ideal thin-layer cell. In a practical sense neither exactly meets the requirements of satisfying the conditions entirely. The thin-layer cell may have a velocity distribution across the width of the cell (16),and the wall jet has a finite entry region; first it is disklike (eq 4) over an area ra2/4, then it is wall-jet-like (eq 5) until 6, = b, and then finally the flow is fully developed, and it becomes thin-layer-like (eq 3). It is appropriate to point out at this time that the tubular and channel cells are effectively identical except for a multiplicative constant: For a tube ( r = tube radius)
For a channel
Thus, for practical cells, one equation applies to all. EXPERIMENTAL SECTION Apparatus. In experiment A the electrochemicaldetector was a channel thin-layer cell (cTLC) of standard design, employing a glassy carbon working electrode, and a spacer thickness of 0.005 cm d e f the ~ 0.20 cm wide channel The reference and auxiliary electrodes were in a Kel-F holder (Bioanalytical Systems) downstream from the electrode. The detector was used to oxidize 2,4-toluenediamine (TDA). The potential (750 mV vs. Ag/AgCI, 3.0 M NaCI) was on the plateau of the TDA oxidation hydrodynamic voltammogram. The solvent was 0.1 M phosphate buffer pH 5.51methanol 75/25 (v/v). For experiment B, the apparatus consisted of a fluid pumping system, a low-pressure six-port Teflon rotary valve, and an electrochemical detector cell. For the first set of experiments a cTLC, TG5A (BAS, Purdue Research Park, West Lafayette, IN) of standard design employing a glassy carbon working electrode was d. The reference was housed in a Kel-F holder which could be conveniently screwed into the cell exit port. A stainless steel auxiliary electrode located downstream completed the circuit. All components of the wall-jet cell were the same as above except for the electrochemical detector cell (see Figure 2). In thisrase the working electrode was a 0.3 cm diameter glassy cnrbon rod centered in a 1 in. diameter Teflon plug with a silver wire glued to the hack of the rod for electrical contact. A stainless steel auailiary electrode and a 3.0 M NaCl filled Ag/AgCI reference electrode were also used. Solution was introduced through a 0.35 cm diameter inlet conduit centered in another Teflon plug lwted helow the working electrode. The inlet conduit was positioned so that solution hit the center of the glassy carbon and exited
Flgure 2. The walkjet cell desbn incorporates the following components: (A) stahless steel auxHiary electrode: (8) Ag/A@l(3.0 M NaCI) referenceelecwcde; (C) K e K reference electmde holder; (D) Rvegort KeCF piece: (E) aluminum ring; (F) cylindrical Teflon plug: (G) soluikm exit pori; (H) glassy carbon electrode; (I) polypropylene sleeve; (J) TeAar spacet; (K) cylindrical Teflon plyl: (L) Temn tubing *om pump; (MITeflon tubing to waste container.
radially by way of four Teflon tubes spaced SOo apart around the working electrode. These four lines were attached to a five-port' Kel-F union block. The fifth port located at the top of the block housed the saturated Ag/AgCI reference electrode. The two Teflon cylinders, one containing the entry port, and the other containing the working electrode and the four exit porta, were held apart by a washer-shaped spacer. These three pieces were held in place hy a hollow polypropylene 1 in. diameter cvlindrical sleeve outside of them. Procedure. Experiment A, was conducted in two parts. In the first case, the cell was used to detect TDA as it was eluted from a reversed-phase column (5 pm Spherisorh ODS, 25 cm long 5 mm i.d., packed by HPLC Technology). The TDA was injeeted with a Valco Imp injedor as a 1GpL slug containing 15 ng of TDA. The solution was pumped with a Waters M-45 solvent-delivery pump. In the second experiment, the column was replaced with a cylindrical cell containing a magnetic stirring bar. Solvent flowed into the bottom of the cylinder and left the top of the cylinder where it passed to the deteetar. Solutions (2 pL) containing 15 ng of TDA were injected through a silicone rubber Beptum directly into this cylinder. For the first experiment, reasonably symmetrical peaks were obtained while in the second experiment the peak shape was an abruptly increasing leading edge followed by an exponentially decaying tail. The data were acquired with a microcomputer and peak areas were determined by Simpson's rule integration. The flow rat^ were changed in a random fashion to avoid confounding time and flow rate. Before experiment B was started, the glassy carbon electrode was polished with a 0.5-pm diamond compound, wiped clean with methylene chloride, and potentiostated overnight at a slow flow rate. The detector was used to oxidize samples of hexacyanoruthenium(lI), R U ( C N ) ~in~a, 0.1 M aqueous solution of sodium perchlorate which were injected into the flow stream. The peaks were then integrated. Constant potentials were maintained by the use of an LC4B amperometric detector (BAS. Purdue Research Park. Went Lafayette, IN) at +1.2 V for experiment 1, and + L O V for experiment 2 vs. Ag/AgCI 3.0 M NaCI. A lower potential, +0.90 V, was used in all the other cell design experiments to eliminate some of the noise problems. All potentials fell on the plateau region of a potential vs. current curve for the oxidation of Ru(CN)d-. RESULTS AND DISCUSSION Experiment A. The theoretical slope, q. of a log i h.log 0plot for a cTLC is 1/3 (13). However, consider the effect of assuming q = 112. Figure 3 demonstrates that reasonably straight lines of peak current vs. VI2are obtained for both chromatographic and exponential peak shapes. Notice, however, that there is a peak shape dependence of the current of a factor of as much as 2 which is not predicted hy any steady-state current equation and that there is in both cases an entirely unreasonable intercept (i at 0 = 0). The actual dependencies of i on 0 may be inferred from a log-log plot such as Figure 4 (17). For the chromatographic peaks, the power of 0 is about 0.05, and for the exponential peak it is about 0.20. In fact, because id # idr .Uti these plots are meaningless. By analcgy to flow-sensitive gas chromatography detectors (18) one can consider the electrochemical detector as a mass sensitive detector, and then one can compare peak
ANALYTICAL CHEMISTRY, VOL. 56, NO. 6, MAY 1984
a81
_ I _
Table 111. Evidence of Electrode Fouling for cTLC spacer expt flow rate range, thickness, no. mL/min cm 0.0125 0.0125 0.0125 0.0125
0.2-2.00 0.2-2.00 0.1-3.00 0.1-3.00
1 2 3 4
experimentala
theoretical
bo
4
bo
4
2.25 i: 2.88 x 10.' 2.26 * 1.35 X lo-' 2.26 * 8.33 X lo-' 2.23 i: 1.35 X lo-'
-6.46 X lo-' i: 2.53 X lo-' -6.85 X lo-' i: 1.08 X lo-' -6.96 x 10-l i. 7.11 x lo-' -7.05 X lo-' * 1.09 X
2.05 2.05 2.05 2.05
-21s -213 -213 -213
a b o is the intercept and q is the slope of the linear least squares line from plots of log area vs. log flow rate. The errors are 95% confidence limits. -~ -
3'2
10.
--
C.O
4.0
5.0
6.0
7.0
2.0 0.6
8.0
1.15
j
1.10
L-,
i
000
*
0.6
0
* * "
0.8
" '
'
1.0
'
"
"
"
1.2 ? . 4
" '
1.6
1.8
Log(Average Flowrate)
Flgure 4. The same data as in Figure 3 In log,,, coordinates: chromatographic peaks, soli dnmonds; exponentlal peaks, open diamonds. area to integrated steady-state equations. Consider the general case in which one has an equation for steady state current to an electrode in the following form:
i, = knFC!P1-qOq
(19)
where k is a numerical constant n, F and C have their usual meanings, P is a composite of diffusion coefficient, kinematic viscosity, and cell dimensions, and 0 is the average volume flow rate. One would use C in millimolar (pmol ~ m - ~P) will ; have units cm3 s-l, as will 0, for i in pA. Rearranging we have
is, = knFP1-qUq-1(Ci7)
,
0.8
,
I
"
, , 1.0
,
"
.
.
'
,
1.2
"
"
7
i
,
1.4
1.6 1.8
Log(Average Flohrate)
(ullsec)
Figure 3. The peak heights (nA) due to 15-ng injections of TDA as a function of the square root of 0 (pL si-'): chromatographic peaks, solld circles; exponential peaks, open circles.
1.20
"
L
:.0 2.0 3 . 0
s q u a r e root(F!owrate)
v
r
(20)
C o is the mass flow rate dmldt. One can then integrate i,, d t to obtain relating the number of coulombs passed as the quantity m of solute is electrolyzed. This can be used for any peak shape.
Figure 5. Peak area (pC) vs. 0 (pL si-'), log,, coordinates: chromatographlc peaks, crosses; exponential peaks, squares. Thus the most informative display of peak data is the log (area) vs. log 0 plot shown in Figure 5. The least-squares slope of the best line drawn through the points is -0.72 f 0.01 ( n = 56) as compared to the theoretical value of -0.67. The overlap of the two sets of data indicates the independence of the area on the peak shape as predicted. The disagreement between theory and experiment is most probably due to some degree of electrode fouling (see below). Experiment B. Initial experimenr s indicated that the sensitivity was decreasing by a few percent an hour. Observation of the electrode surface indicated that electrode fouling was occurring. The results for four consecutive determinations of the slope of the log Q vs. log 0 plot are shown in Table 111. Note that the absolute values of the slopes increase steadily. This is expected if the electrode is covered by a layer through which solute can diffuse, but which does not allow bulk flow to occur. The following treatment of the fouling problem uses the simplifying approximation that the concentration is a linear function of distance inside the diffusion layer (2). For the case in which a solute must diffuse from the bulk to a nonelectroactive surface and then diffuse through the nonelectroactive material to the electroactive surface, there will be two diffusion layer thicknesses, one in solution and the other the thickness of the surface layer. Since the solute's activity coefficient will not generally be the same in bulk solution and in the surface layer, there will be a change in analyte concentration from the right (+, solution) to the left (-, surface layer) side of the interface where Ci is the concentration at the interface, - indicates just to the surface layer side, and + indicates just to the solution side. The concentration of analyte a t the electrode surface C, is related to the applied potential. Note that if one calculates the analyte's concentration at the electrode surface based on the Nernst equation where the E" used is in the bulk electrolyte, then C, will be K times the calculated concentration.
062
ANALYTICAL CHEMISTRY, VOL. 56, NO. 6, MAY 1984
Table IV. Summary of Experimental and Theoretical Results flow rate spacer range, thickness, mL/min cm
expt cell type
cTLC
no.
5
0.1-2.35 0.1-3.00 0.1-2.35 0.1-3.00 0.1-3.00 0.7-2.35 0.7-3.50
6
rTLC
7 8'
9 wall jet a
10 lla
0.0051 0.0051 0.0051 0.0051 0.0051 0.3175 0.3175
experimental b bo
4
2.09 t 2.27 t 1.86 i 1.64 i
1.41 X lo-' 1.03 x lo-' 1.48 x l o - ' 1.93 X 10.' 1.48 * 3.12 x lo-' 1.35 t 2.77 X l o - ' 1.52 t 3.22 X lo-'
Electrode not cleaned before experiment.
theoretical
-6.98 X 10-1t -6.91 x 10.' t -7.27 X lo-' i -8.21 X l o - ' t -7.49 x l o - ' i -2.09 X 10.' t -3.39 X 10.' t
1.27 X 10.' 8.71 x 1.32 x 10.' 1.67 X lo-' 2.56 x lo-' 2.04 X lo-' 2.24 X lo-'
bo
q
symbol
2.31 2.31 2.31 2.31 2.31 1.14 1.14
-2/3 -2/3 -2/3 -2/3 -213 -1/4 -1/4
oc
b , is the intercept and u is the slope of a plot of log Q vs. log
c.
cc cd
od OS CS
The conservation of mass requires that the flux of the analyte be equal just to the right and just to the left of the interface
J+ = J-
(23)
D(CO- Cj,+)/S, = Dp(Ci,-- C , ) / &
(24)
a1 and a2 are the diffusion layer thicknesses in the surface layer and in the flowing solution, respectively, divided by the cell thickness, thus they are dimensionless. From eq 22 and rearranging one obtains
Ci,+ = (DpC,/Bl + D C o / & ) / ( D # / &
+ D/&)
-
0 m
0
(25) log ( f l o w r a t e )
The current is given by (2)
(ul/secl
Figure 6. Comparison of data for the three types of cell. See Table
IV for symbol identification, slopes, intercepts, and experimental condltlons.
agreement with eq 3 of Table I. When z layer limits mass transport. Recall that
where r is a dimensionless electrode length
r = xDW,/Ub
(27)
x being the distance traversed along the electrode. rL is r at the end of the electrode
rL = L D W J U b
(28)
Recalling that for sufficiently thick spacers, small electrodes, and rapid flow (2) J2
=
(29)
log ( 1
