J. Phys. Chem. 1994, 98, 13432-13433
13432
COMMENTS Comments on the Relationship between Time Lag and Dynamic Delay in Diffusion-Reaction Systems
4’ = k6,2/D,,
= CIC,,,
i? = R 6,’
(1
+ ~)/D,C,,,
C, is taken as the maximum concentration obtained by the ramp input. The solution for this set of equations can readily be found by using the separation of variables technique5 for this transient problem with a time varying input at the boundary$
Dale A. Baker and David A. Gough* Department of Bioengineering, University of Califomia, San Diego, La Jolla, Califomia 92093-0412
C(zd =
Received: June 20, 1994
sinh[4(1 - 211 sinh RT - 2 8
w
n=l
Leypoldt and Gough’ gave an expression for the penetrant time lag in a diffusion-reaction system as follows:
{ 1 - exp[-((nn)’
nn
[(nn)’
+ 4’1’
X
+ 4*) z]} sin(nnz) (6)
For determining the ramp dynamic delay of the system, we are interested in the flux at z = 1 , appropriately normalized by the steady state flux at the maximum concentration, in the quasistatic limit as t 0. This is
-
where D, is the penetrant diffusion coefficient in the membrane, 6, is the membrane thickness, 6 is the equilibrium coefficient for the local free and bound analyte within the membrane, and q5 = kdm21D,, in which k is a first-order reaction rate constant. This and related time lag expression^^.^ have proven useful for analyzing diffusion and reaction of analytes in biosensor membranes that contain immobilized catalysts and for other membrane systems. Recently, Baker and Gough4suggested that the dynamic delay of such biosensors, or temporal displacement of the signal in response to a concentration ramp challenge, is equivalent to the time lag given in eq 1 . This suggestion of equivalency, based on a physical argument, leads to the use of the time lag to describe a dynamic delay and dynamic error characteristic of all continuously operated biosensors. The purpose of this Comment is to show that the dynamic delay, 6 ~is, indeed equivalent to the time lag, L. The system under consideration is a thin membrane containing homogeneously distributed catalyst, separating two compartments. The diffusing penetrant reacts within the membrane according to a first-order kinetic expression with a rate constant k. Reversible immobilization or partitioning of the penetrant within the membrane may also occur. Initially, the membrane is devoid of penetrant, and at time t = 0 one face of the membrane is exposed to a linear concentration ramp, Rt, where R is ramp rate. The other face is maintained at zero concentration. A conservation equation for this system in nondimensional form with the appropriate initial and boundary conditions is as follows:
noting that the cos(nzz) = (- 1)” and, as previously given from complex variable theory,’,’ that
Substitution of eq 8 into eq 7 yields the following expression for the ratio of the fluxes: j,
om, = R{z - ZD}
This shows that the original ramp input, it,has been simply shifted in time by a dimensionless delay ZD, given by
Substituting eq 5 in the above expression to solve for the dimensional time t equal to the dynamic delay, d ~ yields ,
The right-hand side of this expression is identical to the expression given in eq 1 for the penetrant time lag in a diffusion-reaction system. This means 6, = L; that is, the dynamic delay of a diffusion-reaction biosensor system responding to a ramp input is equivalent to the time lag. The expression for the dynamic delay in eq 1 1 is also identical to eq 1 when the reaction rate is zero and binding of the analyte within the membrane is negligible. In this case, t~ l/3 as q5 0, and we have simply
-
-
6 , = L = 6,’16D, C(0,z) = Rz
(4)
where the following nondimensional parameters and variables are used: 0022-365419412098-13432$04.5010
(9)
(12)
In this limit of no reaction and negligible binding of analyte, eq 9 for the transient flux response, quasi-static limit can be rewritten as
0 1994 American Chemical Society
J. Phys. Chem., Vol. 98, No. 50, 1994 13433
Comments 1.o
0.8 h
Io:
0.6
0.4
’
/ )onse
Draper.8 The ramp rate independence of the response is particularly important for experimentally testing such diffusionreaction systems with a ramp input. Practically, this means any convenient ramp rate, R, can be used to determine the ramp dynamic delay or the equivalent time lag. Also, challenging the system with ramps of multiple rates provides a method to verify the linearity, which is assumed in the above derivation: if a nonunique value for the dynamic delay is obtained, the system can be considered nonlinear and a different model will be needed. The equivalence shown here between the dynamic delay and time lag provides a basis for description of dynamic membrane processes.
Figure 1.
References and Notes
This result, along with the input ramp, is presented graphically in Figure 1 as a function of the dimensionless time, t. By plotting eq 13 in this way, it is readily seen that the dynamic delay is independent of the ramp rate, R. This is in accordance with the rate independence previously described for simple lumped parameter systems responding to ramp inputs by
(1) Leypoldt, J. K.; Gough, D. A. J . Phys. Chem. 1980, 84, 1058. (2) Gough, D. A.; Leypoldt, J. K. AIChE J . 1980, 26, 1013. (3) Siegel, R. A. J . Membr. Sci. 1986,26, 251. (4)Baker, D. A,; Gough, D. A. Anal. Chem., in press. ( 5 ) Ozisik, M. N. Heat Conduction; John-Wiley and Sons: New York, 1980. (6) Ojalvo, I. U. Int. J . Heat Mass Transfer 1962, 5, 1105. (7) Carrier, G. F.; Krook, M.; Pearson, C. E. Functions of a Complex Variable; McGraw-Hill: New York, 1966. (8) Draper, C. S. Znstrument Engineering; McGraw-Hill: New York, 1952; Vol 3.
