J. Phys. Chem. 1995,99, 17294-17296
17294
COMMENTS Comments on Time Lags and Dynamic Delays in DiffusiodReaction Systems Ronald A. Siege1 Departments of Pharmacy and Pharmaceutical Chemistry, University of Celifomia, San Francisco, California 94143-0446 Received: July 19, 1995
Many chemical sensors consist of a series combination of a membrane and a detecting surface, the latter most often being an electrode. The membrane’s purpose is either to select and enrich the analyte concentration or to catalyze the conversion of analyte to another molecular species that can be assayed at the detecting surface. Typically, one of two modes of detection is used.’ In amperometric detection, the detectable species (which often is the analyte itself) is completely eliminated at the detecting surface, and the sensor measures the rate of elimination, which equals the flux of that species to the surface. Detection is usually based on the electrical current generated by electrons donated or received by the electrode as part of the elimination reaction. In potentiometric detection measurement is based on the concentration of the detectable species next to the electrode, with no elimination occurring there. Typically, the voltage at the detecting electrode relative to a standard reference electrode is measured. Although the terminology for detection derives from the use of electrodes, alternative situations exist which can be treated similarly. The well-known case of diffusion of a chemical across a membrane into a perfect sink is analogous to the amperometric conditiom2 The recent advent of surface detection methods such as attenuated total reflectance Fourier transform infrared (ATR-FTIR) spectroscopy permits the repertoire of potentiometric detection to be ~ i d e n e d .Although ~ techniques such as ATR-FTIR do not involve measuring a voltage, it is the concentration of the detectable species itself, and not the flux of that species, that is determined. We generalize the term “potentiometric” to include any assay in which concentration at the sensor surface is measured and reserve the term “amperometric” for situations in which flux is the measured quantity. The presence of the membrane prevents instantaneous detection of changes in analyte concentration in the media being assayed, due to delays arising from transport and reaction processes in the membrane. (Even in the absence of the membrane, some delay due to boundary layers and interfacial processes occurring at the detecting surface can be expected.) A useful summary parameter for such delays is the rime lag, tL,2 which usually is defined in terms of an amperometric experiment, in which the time-integrated flux at the detecting surface (or the integrated electrical current) is plotted uersus time following introduction at time zero of analyte to the external medium, with external analyte concentration held constant for a suitable period of time. The asymptote of the integrated flux curve (or integrated electrical current) is a straight line intersecting the time axis at tL. The time lag is invariant with respect to external analyte concentration provided all transport and reaction processes in the membrane are linear, i.e. concentration-independent. 0022-365419512099- 17294$09.0010
Recently, Baker and Gough4 introduced a new parameter called the dynamic delay, QD, which they define by considering exposure of an amperometric sensor to a linear ramp, starting at time zero, of analyte concentration. A plot of the resulting instantaneous flux (or current) at the detecting surface yields an asymptote that intersects the time axis at QD. These authors then consider a uniform membrane containing a catalyst which partially removes the analyte in a linear (first order) manner. Measurement is based on the flux of the remaining analyte at the detecting surface. Provided both diffusion and reaction processes are linear, Baker and Gough demonstrate by solving the relevant partial differential equations that t L = QD for this particular system. The purpose of this note is twofold. First, we wish to point out that the result just quoted should apply for any membrane system supporting linear transport and reaction of analyte and detectable reaction products. This may be important since many sensor systems involve multilaminar membranes, with the various layers fulfilling distinct structural and chemical functions. In the second part of this note we consider the time lag (or equivalently the dynamic delay) under potentiometric conditions. It will be shown that the time lag for a given membrane is generally not be the same for potentiometric detection as for amperometric detection. However, it will also be demonstrated that the potentiometric time lag is equal to another quantity that is defined under amperometric conditions. On the Equality oft^ and SD. We consider the membrane system as a linear, time invariant black box, with input being the time (t) profile of extemal analyte concentration, C(t), and output being the flux of analyte at the detecting surface, J ( t ) . Formally we may write J(*>= UC(*)I
(1)
where L is a linear, time invariant operator. The amperometric lag time t L is derived from the curve
when J, is in response to a step input, C,(t) = C*H(t), where C* is a constant concentration and H(t) is the Heaviside function. Equation 2 can be written in an operator form,
where Z represents the operation of integration. From the theory of linear, time invariant system^,^ it is known that the response of such a system to the time integral of an input is just the time integral of the response to that input. Now a ramp concentration profile, Cr(t) = RtH(t), is just proportional to the time integral of a concentration step. Combining these observations, we find that
where “s” and “r” subscripts correspond to step and ramp inputs, respectively. In other words, the instantaneous flux in response 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 47, 1995 17295
Comments to a ramp function input is proportional to the integrated flux in reponse to a step function input. Since both t~ and 6~ are independent of the level of input, it follows that t L = 6 ~ Notice . that this line of argument also holds when the detectable species is not the analyte itself, provided that the system is linear with respect to both the analyte and all species leading to detection. Baker and Gough point out that measuring the time lag or dynamic delay after challenging the system with different levels of analyte can provide a test for system linearity. This observation is generalized by the present analysis to linear membrane systems of otherwise arbitrary structure.
Time Lags and Dynamic Delays for Potentiometric Measurements. The time lag for a potentiometric system can be defined in terms of the integrated concentration of detectable species at the detecting surface, in response to a step external analyte concentration, Cs(t). Similarly, the dynamic delay will be defined by analogy to the amperometric case, and the logic of the previous section leads to the conclusion that tL = QD for potentiometric detection. Generally, the potentiometric time lag will not equal the amperometric time lag for the same membrane. Amperometric measurements involve elimination of the detectable species (analyte or reaction product) at the detection surface, while no such elimination occurs for potentiometric measurement. Therefore, the boundary conditions for the transport and reaction equations differ for the two cases: and this gives rise to the difference in the respective time lags, as illustrated in Figure 1. In a previous contribution? this author showed how twoport electrical network theory8 can be adapted to calculate amperometric time lags. A similar but condensed analysis will be made now for the potentiometric case. We confine our analysis to the case where the detectable species is the analyte and not some secondary species. The two-port approach considers an experiment in which the membrane is placed between two compartments, 1 and 2, and exposed to respective analyte concentration-time profiles C I( t ) and C2(t). As a result, there are respective fluxes Jl(t) and J2(t) of analyte into the membrane. Taking Laplace tranforms of these profiles we obtain the functions c~(s),c2(s),11(s), and &(s). Fluxes and concentrations in Laplace space can be related through an admittance matrix (Yu), such that
EXtWlMll
Medium
lp
In the amperometric case, with flux occuning from side 1 to side 2, we assume compartment 2 to be a perfect sink, Le. = 0 and therefore 1 2 = Y21c:1.In ref 7 it is shown that the time lag corresponding to amperometric case is given by
tL=--
(i%)
= - -( "
y21
s=o
(6)
y 2 I ) s=o
In the absence of external forces or active transport, Y12 = Y21?s8 and it is straightforward to show that the time lag for flux in the reverse direction (2 1) is the same as in the forward direction (1 2). Two amperometric "time lead" parameters, corresponding to fluxes in the forward and reverse directions, are also calculated using similar equation^:^
-.
-
For asymmetric membranes, Y I Iand Y22 are not necessarily
Time
Figure 1. (a, top) Buildup of concentration gradient in membrane with increasing time, t, for amperometric detection. (b, middle) Buildup of concentration gradient in membrane with increasing time, t, for potentiometric detection. (c, bottom) (-) Time-integrated flux of analyte to detection surface for amperometric case. (- - -) Time-integrated analyte concentration at detection surface for potentiometric case. Asymptotes of amperometric and potentiometric curves. The timeintercepts of these asymptotes are t L and t p , respectively. (-0-)
equal. Thus, while the amperometric time lag is independent of direction of flux, the time lead parameters for the two directions may differ. The time leads can be combined with the time lag to calculate directionally dependent mean first passage times for analyte molecules traversing the membrane,
For the potentiometric case, with analyte originating in compartment 1, we assume a vanishingly small comp-artment 2, such that flux into that compartment is zero, Le., J2 = 0. This corresponds to an impemeable detection surface. We then find from eq 5 that = - ( Y Z I / Y ~ ~ ) By C ~ analogy . to the development of eq 6, we derive the potentiometric time lag in the forward direction,
17296 J. Phys. Chem., Vol. 99, No. 47, 1995
5)
tp+ = - d(ln
y22
Comments (9)
s=o
Combining eqs 6-9 leads to the simple result that -
tp+
= t-
A corresponding relation, t p - = t+, is derived for potentiometric detection in the reverse direction, Le., with membrane flipped over. Four interesting facts are gleaned from these results. First, potentiometric and amperometric time lags are not equal. Second, while the amperometric time lag is symmetric with respect to membrane orientation, the potentiometric time lag is not. Third, the potentiometric time lags are directly related to parameters that can be measured or calculated under amperometric conditions. Fourth, the forward potentiometric time lag equals the reverse mean first passage time, and vice versa. A list of general formulas for calculating parameters, including time lags and mean first passage times, for membranes in series given the amperometric case is given in ref 7. These formulas are derived using transmission matrices,8which in turn are related to admittance matrices. Equation 10 and its reverse counterpart then allow potentiometric time lags to be calculated as a bonus.I0 It is interesting to compare t~ and rp for the reactive membrane considered by Baker and Gough and by other^.^,^ Using eqs 21.3 and 21.4 of ref 4, we find that”
where 4 = h(dD)’/*is the Thiele modulus, defined in terms of h, the membrane thickness, D,the analyte diffusion constant, and K , the first-order rate constant for removal of analyte. The relation in eq 1 1 is plotted in Figure 2. Of note are the’limits tp/t~ 3 as 4 0 and t p / t L 1 as C#J -. The first limit corresponds to the case where there is no analyte rgmoval; in this case it is well-known that t~ = h2/6D and tp = t = h2/2D. In the second limiting case all analyte is removed before reaching the detecting surface, hence, both the sink and noflux boundary conditions hold there, explaining the equality of t L and t p . Finally, it is worth pointing out that with increasing Thiele modulus both t L and t p decrease, Le., response becomes f a ~ t e r . 4This ~ ~ ~can ~ be explained by noting that, for the analyte to be detected, it must reach the detecting surface without having been removed by the membrane. As the membrane becomes more reactive (increasing 4), only the “fastest” molecules are
-
-
-
-
Thlele Modulua 0 Figure 2. Ratio of potentiometric time lag, rp, to amperometric time lag, rL, as a function of the Thiele modulus, 4, for a uniform membrane
system which catalyzes degradation of the analyte.
detected, explaining this trend. Of course, this benefit occurs at the cost of reduced sensor sensitivity due to the removal of analyte.
References and Notes (1) Gough, D. A.; Leypoldt, J. K. In Applied Biochemistry and Biotechnology; Wingard, L. B., Katchalski-Katzir, E., Goldstein, L., Eds.; Academic Press: New York, 1981; Vol. 3, pp 275-306. (2) Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon Press: Oxford, 1975. Frisch, H. L. J. Chem. Phys. 1962, 36, 510. Chen, J.-S.; Fox, J. L. J. Chem. Phys. 1988, 89, 2278. Chen, J.-S.; Rosenberger, F. Chem. Eng. Commun. 1991, 99, 77. (3) Fieldson, G. T.; Barbari, T. A. Polymer 1994, 34, 1146. Farinas, K. C.; Doh, L.; Venkatraman, S.; Potts, R. 0. Macromolecules 1994, 27, 5220. (4) Baker, D. A.; Gough, D. A. J. Phys. Chem. 1994, 98, 13432. (5) Liu, C. L.; Liu, J. W. S. Linear Systems Analysis; McGraw-Hill: New York, 1975. (6) For both the amperometric and potentiometric conditions, constant inhomogeneous Dirichlet conditions hold at the membraneholution interface. At the detection surface, homogeneous Dirichlet and homogeneous Neumann conditions are indicated for the amperometric and potentiometric cases, respectively. (7) Siegel, R. A. J. Phys. Chem. 1991, 95, 2556, and corrections in: Siegel, R. A. J. Phys. Chem. 1992, 96, 3182. (8) Desoer, C. A.; Kuh, E. S. Basic Circuir Theory; McGraw-Hill: New York, 1969. Bunow, B.; Aris, R. Math. Biosci. 1975, 26, 157. (9) Leypoldt, J. K.; Gough, D. A. J. Phys. Chem. 1980,84, 1058. Siegel, R. A. J. Membr. Sci. 1986, 26, 251. Chen, J.-S.; Rosenberger, F. J. Phys. Chem. 1991, 95, 10164. (10) One corollary of this result is that extemal mass transfer effects on time lag are more simply described for the potentiometric case than for the amperometric case. See eqs 18.4 and 18.2, respectively, in ref. 7 . (11) Since the membrane is uniform, we may suppress the direction subscript in tp.
JP95234 1I
