Analytical Chemistry Chemical Cytometry Quantitates Superoxide

Figure 4. The volume fraction of absorbed analyte, φ, approaches
its asymptotic value exponentially upon exposure to an analyte, in
this case p-xylene vapors, at an activity of 0.054.
Figure 5. The measured characteristic response time, τmeas, isthe
y intercept of the line obtained by plotting A ) ∫0 t ∆φ(s)/φ∞ ds as a
function of B ) ∆φ(t) ) φ∞ - φ(t).
is a mathematically complex problem, but for practical purposes
the kinetic data can be fit by the simple exponential expression
φ(t) ) φ ∞ (1 - e -t/τmeas ) (5)
Here t is time and τmeas is the measured response time of the
chemiresistor to the analyte in question. Fitting to this form
will prove useful in correcting the observed kinetic data for
the time it takes for the analyte concentration to reach the
steady state in the flow cell. Despite this, we can actually obtain
the response time in a model-independent fashion. To do so,
we simply plot the integral A ) ∫0 t ∆φ(s)/φ∞ ds against B )
∆φ(t)/φ∞, where ∆φ(t) ≡ φ∞ - φ(t). We can operationally define
τmeas as the y intercept of this curve in the limit as ∆φ(t) f 0,
but if eq 5 is a reasonable fit, the result will be a straight line
whose y intercept is easily obtained. The data in Figures 5 are
indeed linear, so eq 5 is actually quite a good description of the
raw kinetic data.
RESULTS
In the following we give the theoretical expression for the true
sorption kinetics, which is shown to be independent of the analyte
activity. We then develop an expression for the measured sorption
kinetics, which is a convolution of the true sorption kinetics and
the kinetics of filling the flow cell. We then show how the flow
cell fill kinetics can be determined from the measured sorption
kinetics and how the fill time can be used to extract the true
6972 Analytical Chemistry, Vol. 82, No. 16, August 15, 2010
sorption time from the measured time. Both the measured and
true sorption kinetics are shown to be independent of the activity,
but strongly dependent on the analyte saturation vapor pressure.
Predicted Sorption Time. From Raoult’s law, the partial
pressure of a chemical species, P, in a gas is equal to its volume
fraction, φ vap, times the total pressure, or φvap ≡ V/Vtot ) P/Ptot.
Recall that the analyte activity is a ) P/P*; therefore, a ) φvap/
φvap*, where φvap* is the volume fraction of analyte vapor at
saturation. From the linearized form of the Flory-Huggins
equation (eq 3), φ ) a/e 1+� , the partition coefficient, K, is then
K ≡ φ 1
)
φvap φvap *e 1+�
At high inlet stream fluxes the response time of an FSCR is
diffusion limited, and from Fick’s second law of diffusion for the
case of a semi-infinite slab, the sensor’s response time can be
expressed as
τd ∝ K d2 1
)
Dt Dtφvap *e 1+�d2
(6)
(7a)
where d is the thickness of the composite and Dt is the diffusion
coefficient of the analyte into the silicone. Note that even for
analytes of identical chemical affinity there is discrimination
based on sorption kinetics, due to variations in their diffusivity
and saturation volume fraction. From the ideal gas law, the
saturation vapor pressure is given by P* )Fφvap*RT/Mw, so
the sorption time can also be written as
τ d ∝ RT
P*
F
M w D t e 1+�d2
(7b)
The saturation vapor pressure varies over a wide range, so the
sorption kinetics should be a very useful method of discrimination.
Measured Sorption Time. The goal is to determine the true
sorption time, including any dependence this time might have on
the analyte activity. Unfortunately, the measured sorption time is
a convolution of two factors: the time it takes for the analyte
activity in the flow cell to reach its steady-state value (flow cell
fill time) and the true mass sorption time. In a constant-flow-rate
apparatus, and for any particular analyte, neither of these factors
should be dependent on the analyte activity, so the measured
sorption time should be independent of the analyte activity. The
data in Figure 6 show that this is indeed the case and also show
significant differences in the sorption kinetics for different analytes,
as expected. At low concentrations (where the slope of the
response curve is low) and for long response times, sensor drift
becomes an issue and response times are difficult to reliably
measure; such is the case for low concentrations of undecane.
This issue, however, can be addressed by using a sensor with
high sensitivity at low analyte concentrations. 23
The data in Figure 6 are an average for five sensors, each
tested simultaneously in the same flow cell. Because of sensorto-sensor
variations in polymer thickness, each chemiresistor had
a somewhat different response time. For example, the five sensors
exposed to mesitylene at an activity of 0.058 have an average
response time, τmeas,of48± 13 s. To account for these thickness