Modeling the response of passive samplers to varying ambient fluid concentrations of organic contaminants



Passive samplers are an important and versatile adjunct to active sampling methods for air and water. The underlying theory is increasingly mature but is often based on the assumption of constant concentrations with time in the ambient fluid. This theoretical work details the passive sampler response to possible environmental situations in which ambient concentrations vary linearly with time, decrease exponentially with time, constitute a pulsed event, or oscillate regularly with time. The modeled shape of sampler response curve with time is characteristic and can be used to determine how fluid concentrations have changed with time. The sampler elimination rate constant is shown to be crucial in governing sampler response. Thus, by using performance reference compounds and under certain conditions, information from the sampler response can in theory be used to derive specific details of the temporal variation of the contaminant in the ambient fluid. For example, data captured by a passive sampler can be used to characterize a contamination pulse. This extends the versatility of passive samplers. Environ. Toxicol. Chem. 2010;29:591–596. © 2009 SETAC


Passive samplers may be defined as devices that accumulate chemical compounds of interest via passive diffusion from ambient media, which are usually air or water. Advantages such as relative simplicity, inexpensiveness, and lack of energy requirements for sampling have enabled passive samplers to become a very useful complement to more conventional active sampling techniques 1. For organic analytes, passive samplers include biphasic ones such as semipermeable membrane devices 1, 2 and others comprising a single phase such as low density polyethylene 3, 4 and silicone rubber 4. Examples of other passive sampler configurations are solid-phase microextraction fibers 5, polymer-coated glass systems 6, polyurethane foam disks 7, and various single-phase polymeric materials, including those identified above 1, 8, 9. A wide array of compounds may be accumulated, including hydrocarbons and halogenated hydrocarbons, such as polyaromatic hydrocarbons, polychlorinated biphenyls and polybrominated diphenyl ethers 3, 10, and endocrine disruptors 11, together with various pesticides and herbicides 12, 13.

To exploit passive samplers fully it is important to understand their operation. Bartkow et al. 14 noted that, for single-phase samplers or samplers that may be treated as a single phase, accumulation is the result of competing uptake and elimination processes:

equation image(1)

where Cs is the concentration in the sampler, Cf the concentration in the ambient fluid, and ku and ke first-order uptake and elimination rate constants respectively (time−1). It is often considered that Cf is constant with time, meaning that integration of Equation 1 results in

equation image(2)

This approach also assumes that accumulation of the contaminant by the sampler only negligibly depletes the concentration in the ambient fluid. Huckins et al. 1 nominate this as one of the requirements to be met to allow ambient fluid concentrations to be derived by passive samplers.

Early in the deployment of a sampler, when t is small, the elimination process is relatively unimportant; therefore

equation image(3)

Equation 3 describes the linear phase of accumulation, as shown in Figure 1. From Equation 2, the half-life (t1/2), defined as the time required for the sampler to reach half the equilibrium concentration, is given by t1/2 = ln2/ke. The linear phase extends until t = t1/2. As time progresses, the sampler moves into the curvilinear stage and finally approaches equilibrium. At equilibrium, Cs = (ku/ke)Cf. Indeed, some samplers are denoted as integrative (linear) or equilibrium types, depending on the accumulation stage reached during deployment.

Figure 1.

Typical accumulation profile for a passive sampler in the presence of a constant contaminant concentration in the ambient fluid, showing the linear stage; curvilinear stage; and finally, at extended deployment times, the approach to equilibrium.

To relate observed Cs values to Cf, some knowledge of the kinetic characteristics of the sampler is essential. Particularly in the aquatic environment, kinetic data obtained in the laboratory may not be applicable to field situations because of biofouling and site-specific flow regimes. To address this, performance reference compounds (PRCs) may be loaded onto the sampler prior to deployment 15–17. These PRCs are chosen to have physicochemical properties similar to those of the analytes and not occur in the environment of interest. From the exponential loss of a PRC, an elimination rate constant can be determined that is reflective of in situ conditions. This rate constant will be similar to that of the analyte and be related to the uptake rate constant of the analyte through the sampler–fluid partition coefficient (Ksf = ku/ke).

Sampling rates (Rs volume time−1) have been variously defined by expressions such as

equation image(4)


equation image(5)

where Vs is the volume of the sampler and Ns is the amount accumulated in the sampler at time t14, 18. The quantity Rst represents the ambient fluid volume cleared of analyte in time t. The analyte is then accumulated by the sampler.

In the environment, fluid concentrations are unlikely to be constant. Rather, they are likely to fluctuate with time. Under such conditions, samplers operating in the linear or integrative mode are considered to afford a time-weighted average ambient fluid concentration 13, 18, 19. Recently, the performance of passive samplers with a contamination pulse in the ambient fluid 13 has been scrutinized. It is uncertain how passive samplers react to contaminant concentrations varying in other ways.

The aim of this work is to show how passive samplers respond to variable ambient fluid profiles and to relate this back to the original ambient behavior of the contaminant. For samplers with added PRCs, it is intended to show how sampler concentrations over time may be used to determine not only the nature but also the specific characteristics of the original ambient behavior of the contaminant with time.


As in previous work 1, it is assumed that sampler accumulation does not measurably affect the fluid concentration. This means that the criterion of negligible depletion of the ambient fluid phase is met 1. For illustrative purposes, consider ambient fluid concentrations varying linearly with time, decreasing exponentially with time, representing a pulse event, and oscillating with time. Derivations of the following equations are provided in the Supplemental Data.

Linear variation with time

If the ambient concentration decreases linearly with time such that Cf = a − bt, where a and b are constants such that a > bt for all t, substitution into Equation 1 and integration affords

equation image(6)

When b is zero, and thus Cf is constant with time, Equation 6 collapses to the familiar Equation 2. Equation 6 is a function in which Cs increases nonlinearly with time, reaching a maximum at

equation image(7)

and is shown in Figure 2. For long deployment times, Cs assumes an inverse linear relationship with t.

equation image(8)
Figure 2.

Response of a passive sampler to a contaminant in the ambient fluid whose concentration decreases linearly with time. Also shown is the concomitant concentration profile in the ambient fluid.

When Cf increases linearly with time, i.e., Cf = a + bt, concentration in the sampler is described by the following expression.

equation image(9)

This function, illustrated in Figure 3, never reaches a maximum, and in fact, for long deployment times, Cs increases linearly with time and in this region is described by Equation 10.

equation image(10)
Figure 3.

Response of a passive sampler to a contaminant in the ambient fluid whose concentration increases linearly with time. Also shown is the concomitant concentration profile in the ambient fluid.

Exponential decrease with time

For situations in which ambient fluid concentrations decrease exponentially with time, if

equation image(11)

where kloss is the first-order loss rate constant (time−1) of the contaminant in the ambient fluid, substitution into Equation 2 and integration results in

equation image(12)

Here, the sampler concentration increases until reaching a maximum at time

equation image(13)

then declines toward zero with time. This is illustrated in Figure 4.

Figure 4.

Response of a passive sampler to a contaminant in the ambient fluid whose concentration decreases exponentially with time. Also shown is the concomitant concentration profile in the ambient fluid.

Contamination event or pulse

Consider a passive sampler exposed to a contamination pulse such that

equation image(14)

where a, b, and c are constants. The (t − c) term introduces a time delay so that t ≠ 0 at the time of maximum concentration of the pulse and b makes z dimensionless. Substitution into Equation 2 and integration means that the sampler responds as

equation image(15)

as shown in Figure 5. Here, erf refers to the (Gaussian) error function, which is defined as

equation image

and describes a sigmoidal shape, with values between −1.00 and 1.00 20. From Figure 5 it can be seen that the sampler response to the symmetrical pulse in fluid concentration is a delayed asymmetrical profile with a long tail at long deployment times. The pulse in ambient fluid concentration reaches a maximum when t=c, but the sampler reaches a maximum concentration at a later time. An approximation of this time is given below in Equation 16.

equation image(16)
Figure 5.

Response of a passive sampler to a pulse event of a contaminant in the ambient fluid. Also shown is the concomitant concentration profile in the ambient fluid.

Therefore, there is a lag time of

equation image

Oscillating fluid concentration with time

In situations in which ambient fluid concentrations vary regularly, consider Cf described by

equation image(17)

where a, b, and c are constants such that c ≥ a, meaning that Cf never becomes negative. The amplitude of the oscillation is a, and the period is bt. The constant c represents a phase shift to account for Cf when t = 0. From substitution and integration of Equation 2, the following relationship between Cs and time is derived.

equation image(18)

This expression, illustrated in Figure 6, describes a curve of the same shape as the familiar accumulation function represented by Equation 2 and Figure 1 but with a wave feature superimposed.

Figure 6.

Response of a passive sampler to a contaminant in the ambient fluid whose concentration varies regularly with time. Also shown is the concomitant concentration profile in the ambient fluid.


Linear variation with time

When contaminant concentrations in the ambient fluid are variable, a time-weighted average concentration (equation image) has been defined for passive samplers operating in the linear or integrative mode 21, 22 as

equation image(19)

As shown in Equation 19, equation image can also be related to the amount accumulated in the sampler, sampling rate, and deployment time. Defined in this manner, equation image = a − (1/2)bt for situations in which Cf decreases linearly with time. Booij et al. 18 found an expression for the sampler response equivalent to Equation 6 and also derived this equation for equation image. It is apparent that equation image, as defined, is not a good measure of Cf = a − bt, and the difference grows greater with time. Gourlay-Francé et al. 23 have also noted a discrepancy between equation image and Cf. The reason for it is that Equation 19 is premised on Cs being a linear function of time.

Even at relatively short deployment times, when the loss process is minimal,

equation image

Defining an integrative time period for the sampler of t < t1/2 may be problematic unless b is very small and thus Cf is effectively constant with time. This is because t1/2 is defined as the time required for Cs to reach half its equilibrium concentration. In this situation, with a linearly decreasing ambient fluid concentration, before the sampler concentration reaches a maximum (and dCs/dt = 0), the function is essentially nonlinear. After the maximum is reached, the sampler never attains equilibrium and is always trying to respond to an ambient concentration that is continually changing with time. Furthermore, expressions such as Equation 5 in which sampling rate is related to Cf are inappropriate, because Cf varies with time, yet sampler characteristics and Rs should not.

Whereas an individual passive sampler deployed and sampled after a particular time period affords equation image, which may be only an approximation of Cf, a series of samplers deployed together and exposed for different times would produce a curve such as that in Figure 2. So would a sampler such as a polyethylene strip from which sections were detached at various times. This could be used to deduce how Cf actually changes with time. Because the slope of the linear portion of Figure 2 at long deployment time is (ku/ke)b = Ksfb, knowledge of the sampler–fluid partition coefficient (Ksf) of the compound allows b to be calculated. Values of Ksf are typically a function of the octanol–fluid partition coefficient that may be measured directly or estimated 4, 17, 18. When the concentration in the sampler is at a maximum, dCs/dt = 0, so, from Equation 1, ku(a − bt) = keCs or Ksf(abt) = Cs. Substituting the appropriate values of Cs and time from the maximum of the sampler response curve (Equation 6) allows derivation of a and therefore the original relationship between Cf and time, i.e., Cf = a − bt. This approach, however, is not applicable to the situation in which there is a linear increase in contaminant concentration in the ambient fluid with time, because the sampler response function Equation 9 never reaches a maximum.

Exponential decrease with time

When the contaminant concentration in the ambient fluid decreases exponentially with time, calculation of the time-weighted average, equation image, is of limited relevance. The sampler response is illustrated in Figure 4, and the time when it reaches a maximum (Equation 13) is a function of ke and kloss only. Employment of a PRC allows ke to be derived, so from Equation 13kloss may be obtained. The maximum concentration attained by the sampler is

equation image(20)

so knowledge of Ksf for the compound of interest allows the parameter a to be derived. Thus, from the sampler response and use of a PRC, the original concentration variation in the ambient fluid is deduced.

Contamination event or pulse

From Equation 16, the sampler reaches a maximum concentration at a time

equation image

later than the contamination peak in the ambient fluid. It is a function of the elimination rate constant only. As a corollary, if ke changes, then this time lag of the sampler changes. Minimal time lag would be observed with relatively hydrophobic compounds possessing a small elimination rate constant. Overall, however, with ke from a PRC, this time period is easily obtained. The peak concentration in the ambient fluid is a, whereas the maximum concentration attained in the sampler is

equation image(21)

To determine the value of a from the sampler response, from Ksf, ku can be derived, and therefore rearrangement of Equation 21 affords a. The time of the contamination peak in the ambient fluid may be found from Equation 16 and knowledge of ke. Through the use of a series of passive samplers deployed at the same time but exposed for different time periods, or subsampling over time from a sampler such as a polyethylene strip, when the contamination pulse occurred in the ambient fluid and its magnitude can be found. It is evident from Figure 5 that the sampler begins accumulating contaminant at the same time as the ambient pulse begins. There is no lag time for commencement of accumulation in this treatment, but one is likely to be observed in actual deployments 1, 24. This is a result of consideration of the sampler as a relatively simple single compartment or phase. Lag times are observed operationally but in terms of modeling are generally evidenced when the sample is treated as a multiphase environment with multiple diffusion barriers.

Oscillating fluid concentration with time

The response of a sampler to a contaminant whose concentration oscillates in the ambient fluid is illustrated in Figure 6. Focusing on the kesin(bt) − bcos(bt) term of Equation 18, which imparts the wave form into the expression, the period of oscillation in the sampler concentration is the same as in the ambient fluid, i.e., bt. Information about concentration variation of Cf can thus be gained immediately.

The oscillation is phase shifted in the sampler, however. Working in radians, Cf first reaches a maximum when bt = π/2 or t = π/2b. In the sampler response, it is when

equation image(22)

Knowledge of ke from a PRC allows calculation of b and therefore the period of oscillation in the fluid and the magnitude of the phase shift. From Equation 17, contaminant concentration in the fluid varies between a + c and −a + c. The amplitude of the sampler response is best estimated at long deployment times as the sampler concentration reaches the plateau of Figure 6. Here, Cs will oscillate around a mean value of Ksfc, allowing derivation of the variable c. The maximum (or minimum) of this oscillation is

equation image

which means that the parameter a can be derived, so the original oscillating function in the fluid (Equation 17) can be fully characterized.


An individual passive sampler deployed for a specific time can give only a time-weighted average concentration that is an approximation of the true fluid concentration at any time if the fluid concentration is variable with time. The shape of a passive sampler response curve to varying ambient concentrations of a contaminant is diagnostic and can be used to infer the nature or form of the ambient variation. This has been theoretically demonstrated for linear and exponential ambient concentration decreases with time as well as pulse events and situations in which ambient fluid concentrations vary regularly with time. Through the use of a PRC and multiple passive samplers deployed over a range of times, or a PRC and a single passive sampler subsampled at different times, the specific relationship of Cf with time can be determined. Details of a pulse event can be captured by the sampler or samplers, for example, highlighting the importance and versatility of passive samplers. If passive samplers can concentrate analyte, then their response with time can be used to infer ambient fluid concentration changes that might be difficult to measure directly as a result of analytical sensitivity or reproducibility problems.

The importance of the elimination rate constant as a governor of sampler response is striking. As a caveat, however, the use of a PRC affords ke, but inherent in this approach is the assumption that ke does not change with deployment time. In practice, this means that for air-based passive samplers for example that are air-side limited, wind speed (and temperature) would have to be constant over the range of deployment times. Similarly, for water-based passive samplers that are water-side limited, flow velocity and the extent of biofouling would have to remain relatively constant. If these conditions are reasonably well met, passive samplers can be very useful tools for investigating trace contaminant behavior in ambient fluids in which concentration is a function of time. Passive samplers may be described as biomimetic under certain conditions. To the extent that organisms may be treated as a single compartment, this work provides insight into their response to varying contaminant concentrations in ambient fluids.


Supplemental Data. Derivation of equations. (41.99 KB PDF)