Anal. Chem. 2003, 75, 4319-4324
Technical Notes
Partial Analytical Solution of a Model Used for Measuring Oxygen Diffusion Coefficients of Polymer Films by Luminescence Quenching Gudrun Schappacher and Paul Hartmann*
Roche Diagnostics GmbH, Kratkystrasse 2, A-8020 Graz, Austria
Many aspects of optical chemical sensor design would benefit from a better knowledge of the diffusion properties of the analyte in the polymer host. The response times of such sensors to a step change of analyte concentration are of vital interest for many applications of fast-responding sensors. Further, the diffusion properties govern their quenching behavior and their sensitivity. A method for determination of the diffusion constant of oxygen in polymers has been developed and used by several groups in the past. The underlying mathematical model for luminescence quenching by molecules of a gas in a single sensing layer on an impermeable support has not yet been completely derived in an analytical form and still uses tedious numerical methods. We present a partial analytical solution to the problem of modeling the time dependence of luminescence generated by in- or out-diffusion of a gaseous quencher in a polymer film in which a luminophor is immobilized and offer a suitable method to predict sensor response times. Optical chemical sensors have increasingly found promising applications for many analytical problems as a result of their favorable properties.1 One of the recently discussed applications is a method to determine the diffusion constants of gases in polymers by exploiting the quenching of the indicator luminescence in the presence of quenching gas molecules of interest and monitoring and analyzing the luminescence signal over time.2,3 The method was introduced by Yekta et al. in the mid-1990s,2 and its usefulness has been demonstrated by Winnik’s group various times,4-6 and also by Kneas et al. in a recent publication.3 * To whom correspondence should be addressed. E-mail: paul.hartmann@ roche.com. (1) Fiber Optic Chemical Sensors and Biosensors; Wolfbeis, O. S., Ed.; CRC Press: Boca Raton, 1991. (2) Yekta, A.; Masoumi, Z.; Winnik, M. A. Can. J. Chem. 1995, 73, 20212029. (3) Kneas, K. A.; Demas, J. N.; Nguyen, B.; Lockhart, A.; Xu W.; Degraff, B. A. Anal. Chem. 2002, 74, 1111-1118. (4) Jayarajah, C. N.; Yekta, A.; Manners, I.; Winnik, M. A. Macromolecules 2000, 33, 5693-5701. (5) Masoumi, Z.; Stoeva, V.; Yekta, A.; Pang, Z.; Manners I.; Winnik, M. A. Chem. Phys. Lett. 1996, 261, 551-557. 10.1021/ac034098m CCC: $25.00 Published on Web 07/08/2003
© 2003 American Chemical Society
Although in principle expandable to other dynamically quenching gases as well, it has been exclusively used to determine the diffusion constants of oxygen in polymers, since optical oxygen sensors are technically very advanced and since oxygen is an important parameter for analytical and clinical problems. The importance of the choice of the right polymer for optical sensor development has frequently been neglected in the past in favor of research on an appropriate dye. This situation is slowly starting to change; researchers are also focusing on the properties of the polymer matrix, and seeing the properties of the immobilized dye within a greater context. A specific advantage of the discussed “direct” method to determine diffusion constants in such polymers, therefore, is to give valuable information on relevant polymer properties for sensor development. Although it is somehow constrained to polymers that allow for immobilizing luminescent dyes in a homogeneous way, one can investigate directly the diffusion properties of the gas of interest in a sensor system itself. The history of development of methods using luminescence quenching in polymers to determine the diffusion properties of the quencher dates back to the 1960s. An overview of earlier work has recently been given in this journal.3 Frequently, the methods have used specific assumptions as, for example, infinitely small thickness and quenching behavior that can be described by a simple linear Stern-Volmer equation to minimize the mathematical challenge of the associated models.2 This did not lead to satisfactory results, since the real conditions of the employed sensor systems by and large did not meet these crude assumptions. There is a considerable mathematical effort to model the time-evolution of the luminescence in the polymer membrane in more detail, avoiding limiting simplifications, which are, in fact, not met by practically all relevant real world sensor systems.7-9 This leads to severe systematic errors in the resulting diffusion constants, which is also confirmed by the large variety of reported values for one and the same polymer. (6) Lu, X.; Manners, I.; Winnik, M. A. Macromolecules 2001, 34, 1917-1927. (7) Hartmann, P.; Trettnak, W. Anal. Chem. 1996, 68, 2615-2620. (8) Carraway, E. R.; Demas, J. N.; DeGraff, B. A. Anal. Chem. 1991, 63, 332337. (9) Gewehr, P. M.; Delpy, D. T. Med. Biol. Eng. Comput. 1993, 31, 11-21.
Analytical Chemistry, Vol. 75, No. 16, August 15, 2003 4319
Figure 1. Schematic representation of the preferred sensor arrangement.
However, the mathematical model behind this direct method has meanwhile improved considerably. Mills10 took a big step by taking into account the nonlinear dependence of the luminescence intensity on the quencher concentration. As a result, the signal response times (usually given as t90) of in-diffusion and outdiffusion are different from each other and different from the time required to reach, on average, 90% of the equilibrium quencher concentration in the polymer host. Yekta et al.2 have taken up this approach and developed it further to analyze luminescence response curves in setups for determination of diffusion constants. Recently, Kneas et al.3 have expanded the model to account for more realistic Stern-Volmer quenching models. They replaced the linear Stern-Volmer constraint in the model by using a nonlinear solubility relation, the Langmuir isotherm, which is better suited to describe the quenching characteristics of the sensor systems under consideration. In addition, they have provided algorithms for also including absorption effects in membranes of higher optical thickness, in which prefilter effects can no longer be neglected. All of these methods have been developed analytically up to a certain point that a further analytical derivation of the model seemed to be too complicated, and at this point, they instead switched to numerical solutions.2,3 This worked reasonably well for a number of model sensor systems, and the elaborated numerical algorithms have been used to investigate a number of real systems in the past. However, an analytical description of the model required for analysis of the data is called for, since it offers more versatility, computational speed, and the possibility to use common data-fitting algorithms. In this work, we present a partial analytical solution to the problem of modeling the time dependence of luminescence generated by in- or out-diffusion of a gaseous quencher in an optically dilute polymer film in which a luminophor is immobilized. This equation can be advantageously exploited to determine diffusion constants of gases in polymers. We have tested the validity of the found partial analytical solutions with the help of an all-numeric simulation of the timedependence with the help of a commercial software package and demonstrate the fitting abilities by using simulated noisy data. THE ANALYTICAL MODEL Consider a system having a single sensing layer that is permeable to the gas of interest, the layer being supported at one side by a solid support or, more generally, by an impermeable medium (Figure 1). Opposite to the support, the layer is exposed (10) Mills, A.; Chang, Q. Analyst 1992, 117, 1461-1466.
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to the gas of interest. This also represents a simple, yet frequently encountered, case for optical sensors, for example. For determination of diffusion constants or response, times the sensing layer is initially kept at a constant gas concentration C0. A step change of the concentration C1 at its surface is the starting trigger for recording the time evolution of the luminescent signal with the help of proper equipment (e.g., a fluorometer). We assume diffusion-controlled gas transport within the sensing layer, that is, the applicability of Fick’s first and second laws of diffusion, and that Henry’s law (eq 1) is valid. The equilibrium concentration within the sensing layer is
C ) RP
(1)
where R is the solubility, and P is the partial pressure of the gas adjacent to the sensing layer. If C0 is the initial concentration of the gas inside the film, the spatial and temporal shape of the concentration C in the film is derived from Fick’s law of diffusion13 (a problem similar to thin film electrochemistry),
[
C(x, t) ) C0 + (C1 - C0) 1 -
(
4
∞
∑
(-1)n
π n)0 2n + 1
exp -(2n + 1)2
) (
Dπ2t 4l2
)]
(2n + 1)πx
cos
2l
(2)
where D is the diffusion constant, l is the layer thickness, and t is the time after the step change. Further, we assume that a luminescent dye is uniformly distributed within the sensing layer and that all dye molecules are exposed to the same intensity of irradiation (dilute sample, negligible absorption). Following a step-change of the gas concentration at the sensor surface, the intensity i(x, t) of a luminescent sensor layer is not uniform throughout the layer thickness until equilibrium is reached. The signal St measured in the detector over time is given by the integral2
St )
∫ i(x, t) dx
1 l
l
0
(3)
In the system under consideration, the dye’s luminescence is dynamically quenched in a way that is usually accounted for by the Stern-Volmer equations.11 They allow us to calculate i(x, t) dependent on the concentration, C, of the quenching gas. Solution for the One-Component Stern-Volmer Equation. In few simple cases, a one-component Stern-Volmer equation applies to the sensor system of interest.
i(x, t) )
i0 1 + KsvC(x, t)
(4)
(11) Stern, O.; Volmer, M. Phys. Z. 1919, 20, 183-188. (12) Hartmann, P.; Leiner, M. J. P.; Schappacher G.; Meisterhofer, E. Abstracts of the Europt(r)ode V conference, Lyon, April 2000, p 157. (13) Crank, J. The Mathematics of Diffusion, Clarendon Press: Oxford, 1955.
i0 is the intensity of luminescence in the absence of the quencher, and Ksv is the Stern-Volmer quenching constant. Thus,
∫ 1+K
1 St ) l
l
0
1
0
0
St )
i0 svC(x,
t)
dx
( ( (
∫ i/ 1+K l
)
l
Thus, the analytical solution for the average signal for sufficiently large times is
[
C0 + (C1 - C0) 1 -
sv
2
exp -(2n + 1)
) (
Dπ t
2
4l2
∞
4
(-1)n
∑ 2n + 1 π n)0
2l
) ]))
St )
dx (5)
There is no way to obtain an exact analytical solution of this integral, and so prior investigators3 preferred to use an approximation for small times and used to solve the resulting integral numerically. This had the consequence of rather tedious algorithms, which have been applied to extract the solution of the problem by comparing iteratively the model with the data. The following approach can largely help to simplify the task. Since the exponential term in the sum of eq 2 quickly vanishes, for sufficiently large times, the following approximation of the concentration is valid.
[
(
( )
(x ) x A + Bt 2
2
A - Bt
/ A2 - Bt2
for A2 > Bt2 (8a)
and
(2n + 1)πx
cos
4 arctan π
( )])
4 arctan hyp π
(x ) 2
Bt - A
2
/xBt2 - A2
for A2 < Bt2 (8b)
These equations can be used to fit time-dependent emission data of sensing layers showing a linear Stern-Volmer characteristic following a step-change of a gas from C0 to C1 at its surface. The fitted parameter is D/l2. To obtain the diffusion constant, the film thickness l has to be determined by separate measurements. The Stern-Volmer constant Ksv may be fitted independently as well, but since it is accessible also by steady-state quenching experiments, this can be advantageously exploited for the accuracy of D/l2 in the fit. There is a discontinuity for the case A2 ) Bt2 at time
t)
4 Dπ2t πx C(x, t) ≈ C0 + (C1 - C0) 1 - exp - 2 cos π 2l 4l
A + Bt
(
)
4Ksv|C1 - C0| 4l2 ln 2 π(1 + KsvC1) πD
(9)
(6)
Thus, for sufficiently large times, St can be written as
St ) 1 l
∫
l
0
[
(
i0
1 + Ksv C0 + (C1 - C0) 1 -
( )
( )])
4 Dπ2t πx exp - 2 cos π 2l 4l
dx )
∫
1 l
l
0
i0 A - Bt cos
( ) πx 2l
dx
(7a)
with
A ) 1 + KsvC1
(7b)
( )
4 π2Dt Bt ) Ksv(C1 - C0) exp - 2 π 4l
We can easily solve the integral eq 7a by substituting
π x ) 2 arctan y 2l
and normalizing i0 ) 1.
(7c)
where the eqs 8a and 8b are not defined. Some real-life sensor systems can be described by a linear Stern-Volmer equation with sufficient accuracy at least in a limited range of gas concentrations7 so that the use of eqs 8a and 8b is justified for these cases. Numerical Verification. We have verified these expressions numerically by simulating the time course of luminescence of such a system with the help of MATLAB software package.12 This was done by transferring Fick’s differential equations into a finite difference equation and solving it stepwise in adjacent small elements of the sensing layer. The diffusion constant, film thickness, and Stern-Volmer parameters have to be fed as input parameters in this case. Figure 2 shows the excellent agreement of the simulation with the calculated data from eqs 8a and 8b and also gives an impression of the time constraint of this solution. It shows the calculated time dependence of the integrated signal St and the numerically obtained emission intensity (in a.u.) of a stepchange of oxygen pressure from P0 ) 0 hPa to P1 ) 200 hPa (Figure 2a, in-diffusion case) and from P0 ) 200 hPa to P1 ) 0 hPa (Figure 2b, out-diffusion case), respectively. In practice, it is more convenient to work with partial pressures instead of concentrations and include the (unknown) gas solubility in a pressure-related Stern-Volmer quenching constant, which can be determined experimentally: KsvC ) KsvRP ) Ksv,pP. Also shown is the calculated total concentration change (averaged over the film thickness and expressed as partial pressure change, PO2av) in the polymer (see eq 10). In Figure 2a, the discontinuity of eqs 8a and 8b is at very small times and, therefore, not shown in the graph. In Figure 2b, the discontinuity is at t ∼ 1.33 s. Analytical Chemistry, Vol. 75, No. 16, August 15, 2003
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Figure 2. Dependence of average partial pressure PO2av ([) and integrated luminescence Intensity (0) on time t for a step change of oxygen pressure from (a) P0 ) 0 hPa to P1 ) 200 hPa and (b) from P0 ) 200 hPa to P1 ) 0 hPa, respectively. D ) 1.9 × 10-7cm2/s, l ) 5 µm, Ksv,p ) 0.05 hPa-1, intensity and signal arbitrarily normalized to 200 (data are typical for an oxygen sensor based on a porphyrin dye dissolved in a polystyrene matrix).7 Solid lines: calculated signal St according to eq 8a (Figure 2a,b) and eq 8b (Figure 2b only), respectively.
Response Times. For a simulation of the sensing properties of a potential system, the response time is of interest. The total (integrated) concentration change for a single layer with time has been given in an analytical form by13
Mt M∞
)1-
∞
(
)
Dπ2t exp -(2n + 1)2 π2 n)0 (2n + 1)2 4l2 8
∑
1
(10)
Mt denotes the total amount (either given as concentration, or as partial pressure) of diffusing substance that enters the layer during time t, and M∞ is the corresponding amount during a infinite time. From eq 10, the time required for a 90% total concentration (or pressure) change can be given as
t90,p ) ln
( )
l2 8 4l2 ≈ 0.85 2 2 D 0.1π π D
(11)
For a 5-µm thin layer of, for example, polystyrene penetrated by oxygen (D ) 1.9 × 10-7 cm2/s) on a solid support, being 4322
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initially at P0 ) 0, the response time to an immediate exposure to air (PO2 ∼ 200 hPa) leads to t90,p ) 1.1 s (Figure 2), regardless if the gas diffuses into or out of the sensing layer. The response times of luminescence, on the other hand, are different from the concentration behavior and depend also on the direction of gas transport.10 For in-diffusion of the quenching gas, it is considerably shorter (in our example, t90,s ) 0.28 s), and for out-diffusion, it is considerably longer (t90,s ) 2.3 s) than t90,p. Unfortunately, no analytical expression for t90,s can be derived from eqs 8a or 8b. However, with the help of the capabilities of modern state-of-theart office software packages (e.g., the “goal seek” function of MS Excel), numerical values of the corresponding response times can be obtained easily. Since the response time of the in-diffusion case (Figure 2a) is shifted to shorter times, it happens that the majority of the signal change falls into a region of small times at which the chosen approximation of eq 6 suffers from inaccuracy. On the other hand, in the out-diffusion case (Figure 2b), the response time is shifted to longer times, making the approximate solution very accurate for most of the observed signal change. The approximation chosen in eq 6 does not produce any errors in eq 8a, but the considered range for eq 8b requires some examination before a fit is done and possibly truncation of the data set at early times. In conclusion, the out-diffusion case is far better suited for further analysis. For the experimentalist it does not matter at all whether the in- or the out-diffusion case is considered; it is just a matter of changing the gas lines of the experimental setup. Example Fits to Noisy Data. To demonstrate the ability of the proposed model to fit real data, we have generated a theoretical out-diffusion signal curve with the help of the MATLAB routine and added normally distributed noise of different levels (0.1-5% of the maximum signal). The parameters of eq 8a were fitted to the set of data by assuming that the values for Ksv,p, P1, and P0 (used for Figure 2b) are known. The signal intensity was normalized to reach the steady-state signal S∞ ) 1 (in practice, S∞ after equilibrium can be easily obtained by steady-state quenching experiments). A commercial software package was used (Origin 3.5). Fits were done on the complete data range between t ) 0 and ∼3 s (data intervals ∆t ) 0.01 s). It is quite interesting to note that the software was able to recognize the constraint of the valid range and the discontinuity of eq 8a automatically. Therefore, no initial estimate of the range was necessary. Table 1a gives an overview of the results obtained for D/l2 for data with increasing noise levels, while Table 1b gives the same results for the case in which the quenching constant Ksv,p was fitted,as well (this option is another advantage over numerical fitting algorithms). From Table 1, it is apparent that for data showing the considered noise levels, fits of eq 8a can be done in a straightforward way and give accurate results for D/l2, more so if Ksv,p is already accurately known from steady-state quenching experiments. Convergence of the fits was satisfactory; no side-minimums have been observed. Because of an inherent limitation of the software package used (it was not able to calculate arcus-hyperbolicus functions at all), we cannot demonstrate the fitting of eq 8b at this point. However,
Table 1. Results for D/l2 Obtained from Fits of Eq 8a to a Set of Simulated Out-Diffusion Data with Added Normally Distributed Noise noise level,
D/l2, s-1
%a
Ksv,p,
0.1 0.2 0.5 1 2 5
Part ab 0.7677 ( 0.0001 0.7674 ( 0.0004 0.7670 ( 0.0007 0.766 ( 0.002 0.763 ( 0.004 0.758 ( 0.007
0.1 0.2 0.5 1 2 5
Part bc 0.7692 ( 0.0008 0.768 ( 0.002 0.766 ( 0.004 0.763 ( 0.008 0.75 ( 0.02 0.73 ( 0.04
hPa-1
(1) A12 > B1,t2 and A22 > B2,t2
4 St ) vi arctan π
(x
A1 + B1,t 2
A1 - B1,t
4 (1 - vi) arctan π
0.0503 ( 0.0002 0.0501 ( 0.0005 0.0498 ( 0,0009 0.049 ( 0.002 0.047 ( 0.004 0.045 ( 0.007
(x
4 St ) vi arctan hyp π
we do not expect any serious problems for the fits with more stateof-the-art software packages, except that a sufficient part of the data at early times of the signal change has to be truncated. Solution Using the Two-Component Stern-Volmer Equation. In practice, the so-called “linear” one-component Stern Volmer model is frequently not met by optical gas sensors. The most successful “nonlinear” models commonly used to describe the luminescence quenching characteristics are the dual sorption (Langmuir isotherm) model14 and the mathematically equivalent two-component model15 (which we choose for further analysis).
[
1 + Ksv1C(x, t)
+
1 - vi 1 + Ksv2C(x, t)
]
(12)
Ksv1, and Ksv2 are Stern-Volmer constants, and vi is the fractional coefficient. In this case, the integrated signal St becomes simply
St ) viSt1 + (1 - vi)St2
(13)
where
i0 l
∫ 1 + K 1 C(x, t) dx, l
0
A2 - B2,t
j ) 1, 2
)x
A2 + B2,t 2
A2 - B2,t
B1,t2 - A12
(x
/ A22 - B2,t2 (15b)
2
)x
A1 + B1,t
4 (1 - vi) arctan hyp π
/ A22 - B2,t2 (15a)
/ B1,t2 - A12 +
B1,t - A1
(x
2
)x
2
(3) A12 < B1,t2 and A22 < B2,t2
/ B1,t2 - A12 +
A2 + B2,t 2
)x
2
B2,t - A2
/ B2,t2 - A22 (15c)
where
Aj ) 1 + KsvjC1
j ) 1, 2
( )
4 π2Dt Bj,t ) Ksvj(C1 - C0) exp - 2 π 4l
(15d) j ) 1, 2 (15e)
Again, i0 ) 1. The fitting behavior of these sum expressions is not expected to add unmanageable complexity to the problem, since preferentially D/l2 is still the only adjustable parameter, once the SternVolmer parameters vi, Ksv1, and Ksv2 are determined in a separate steady-state quenching experiment. If, however, the latter parameters are determined from the same transient diffusion experiment, as well, the situation will become more complex. A more difficult problem is also encountered if the absorption within the sensing layer is no longer negligibly small (also known as prefilter effect). Kneas et al.3 have shown that this is the case for optical densities higher than OD ) 0.1. Under these conditions, the integral eq 3 for the overall signal becomes
St ) Stj )
2
2
(x
)x
A2 + B2,t
A1 + B1,t
4 (1 - vi) arctan π
4 St ) vi arctan hyp π
i(x, t) ) i0
(x
/ A12 - B1,t2 +
(2) A12 < B1,t2 and A22 > B2,t2
a 1σ in % of max. signal. b K -1 sv,p ) 0.05 hPa ; P0 ) 200 hPa; P1 ) 0 hPa. c Ksv,p fitted as well. Theoretical target result: D/l2 ) 0.77 s-1.
vi
)x
2
i0 l
-dcdx
∫ 1 +10K l
0
svC(x,
t)
dx
(16)
(14)
svj
which can be approximated quite analogously to eq 5. Depending on the values of Aj and Bj,t (j ) 1, 2) we have to distinguish four cases for St. However, if we assume that Ksv1 > Ksv2, only three of them are relevant:
d is the molecular absorption coefficient of the indicator dye, and cd is its concentration in the film. For this case, we have not found an analytical solution, so the numerical approach chosen in ref 3 is the only choice, so far, for optically thick layers. However, the only motivation for using films (14) Li, X.-M., Wong, K.-Y. Anal. Chim. Acta 1992, 262, 27-32. (15) Lehrer, S. S. Biochemistry 1971, 10, 672-680.
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of higher optical density for diffusion constant measurements is the limited sensitivity of the detectors used in the experiments. This should be solvable by modern techniques or oversampling, for example, to avoid high optical densities. DISCUSSION A general drawback of this method is the high dependence of the diffusion constants on the accuracy of the layer thickness determination. Although Stern-Volmer parameters entering eqs 8a and 8b or eqs 15a-c can easily be determined in separate steady-state experiments with sufficient precision, the error of the thickness affects the accuracy of diffusion constants because of its quadratic influence. Especially for sensing layers in the low micrometer region, effort should be made to use more sophisticated means for film thickness determination. However, with modern commercially available equipment (e.g., white light interferometry), it is possible to measure polymer film thickness of some micrometers with an accuracy of ∼1-5%. Another shortcoming of the method is the lack of information obtained for the permeability and solubility of the gas in the polymer.16 In principle, once the diffusion constant is known, the permeability and solubility can be derived from the Stern-Volmer quenching constant. This has been described and used for oxygen sensor systems in the past. However, the quenching constant also depends on the unquenched decay time, the quenching probability of the encounter pair, and the van der Waals radii of the dye and the quencher.5 For an accurate determination of permeability and solubility, this is not very convenient, especially since the quenching probability for a system under test is not known a priori.17 The problem is further complicated because the properties of real world sensor systems frequently follow a two-component SternVolmer model and nonexponential excited-state decay.8,18 Overall, this complex situation still prevents Stern-Volmer quenching constants of sensors from being predicted on the basis of diffusion parameters and decay time information alone.3 (16) Rharbi, Y.; Yekta, A.; Winnik, M. A. Anal. Chem. 1999, 71, 5045-5053. (17) Timpson, C. J.; Carter, C. C.; Olmsted, J., III J. Phys. Chem. 1989, 93, 41164120. (18) Hartmann, P.; Leiner, M. J. P.; Lippitsch, M. E. Anal. Chem. 1995, 67, 88-93. (19) Vieth, W. R. Diffusion in and through polymers; Hanser: New York, 1996.
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For additional determination of solubility and permeability, a related method described by Winnik et al.16 appears to be more suitable. This method is analogous to the well-known and frequently used permeation and time-lag methods, which are described in detail elsewhere.19 They use an oxygen sensor to replace the conventional and rather inaccurate pressure sensors, but maintain the concept of time-lag methods using a polymer membrane separating two chambers with selectable gas fillings. This modified time-lag method using optical sensors in addition to pure polymer layers offers more versatility (in terms of polymers) and, under the precondition that the sensor response to oxygen is very fast as compared to the time scale of the diffusion experiment, a significantly simpler mathematical description in expense of a more complicated experimental setup. However, this method becomes more and more inaccurate for sensors with a high diffusion constant (or high permeability).16 Furthermore, it does not provide the possibility to probe the diffusion properties in the polymer directly within the sensing layer, which is of certain value in the development process of optical sensor systems and represents the strengths of the described direct method.3 CONCLUSIONS Both of the two employed methods for measuring diffusion coefficients in polymer films based on luminescence quenching, the direct method as described herein and the time-lag method, have their strengths and weaknesses, and in practice, the method of choice will be highly dependent on the problem to solve. The presented partial analytical model is expected to help to improve the mathematical handling of the direct method by eliminating remaining numerical complexity, which facilitates its broader use. ACKNOWLEDGMENT We thank M.J.P. Leiner and E. Meisterhofer (both of Graz) for fruitful discussions and for providing the MATLAB code for the numerical verification.
Received for review January 31, 2003. Accepted May 22, 2003. AC034098M
