Estimation of Membrane Diffusion Coefficients and Equilibration Times

Diffusion coefficients for low-density polyethylene membranes were measured for He, Ne, H2, O2, and N2 in laboratory experiments and ranged from 1.1 t...
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Environ. Sci. Technol. 2004, 38, 1849-1857

Estimation of Membrane Diffusion Coefficients and Equilibration Times for Low-Density Polyethylene Passive Diffusion Samplers CRAIG E. DIVINE* Department of Geology and Geological Engineering, Colorado School of Mines, Golden, Colorado, 80401 JOHN E. MCCRAY Jackson School of Geosciences, The University of Texas at Austin, Austin, Texas, 78712

Passive diffusion (PD) samplers offer several potential technical and cost-related advantages, particularly for measuring dissolved gases and volatile organic compounds (VOCs) in groundwater at contaminated sites. Sampler equilibration is a diffusion-type process; therefore, equilibration time is dependent on sampler dimensions, membrane thickness, and the temperature-dependent membrane diffusion coefficient (Dm) for the analyte of interest. Diffusion coefficients for low-density polyethylene membranes were measured for He, Ne, H2, O2, and N2 in laboratory experiments and ranged from 1.1 to 1.9 × 10-7 cm2 sec-1 (21 °C). Additionally, Dm values for several commonly occurring VOCs were estimated from empirical experimental data previously presented by others (Vroblesky, D. A.; Campbell, T. R. Adv. Environ. Res. 2001, 5 (1), 1.), and estimated values ranged from 1.7 to 4.4 × 10-7 cm2 sec-1 (21 °C). On the basis of these Dm ranges, PD sampler equilibration time is predicted for various sampler dimensions, including dimensions consistent with simple constructed samplers used in this study and commercially available samplers. Additionally, a numerical model is presented that can be used to evaluate PD sampler concentration “lag time” for conditions in which in situ concentrations are temporally variable. The model adequately predicted lag time for laboratory experiments and is used to show that data obtained from appropriately designed PD samplers represent near-instantaneous measurement of in situ concentrations for most field conditions.

Introduction The use of passive sampling methods to monitor environmental contaminants, particularly trace metals and certain lipophilic compounds (such as organochlorine pesticides, polychlorinated biphenyls [PCBs], and polynuclear aromatic hydrocarbons [PAHs]), has greatly increased over the past decade. For example, much literature exists describing the use of dosimeter-type and semipermeable membrane devices (SPMDs) for evaluating ultralow concentrations and potential bioavailability of contaminants (2-5). These types of samplers are most frequently used for sampling sediment pore water * Corresponding author phone: (720)308-5367; fax: (303)273-3859; e-mail: [email protected]. 10.1021/es034695q CCC: $27.50 Published on Web 02/13/2004

 2004 American Chemical Society

and surface water (i.e., rivers, estuaries, harbors) and have occasionally been used for groundwater characterization in support of contaminant toxicology studies. Typically, these types of devices consist of a semipermeable membrane (such as dialysis-type cellulose or a synthetic polymer) surrounding a thin sequestering (or “trapping”) phase, such as hexane, triolein (a prominent fish lipid), or activated carbon. Contaminants diffuse through the membrane and are retained in the trapping phase until analysis. After the SPMD is removed, the contaminants must typically be extracted from the trapping phase before analysis. Because the contaminant uptake pattern for SPMDs is similar to some aquatic organisms (e.g., 6, 7), SPMDs are increasingly employed in environmental toxicology studies to assess the bioavailability of contaminants. Passive diffusion (PD) samplers are similar to dosimetertype and SPMD sampling methods in that they generally consist of a sealed container with a semipermeable membrane material that is deployed over a period of time at a discrete location to measure resident analyte concentrations. The analytes of interest diffuse from the surrounding media across the sampler membrane (according to the local concentration gradient) until the analyte concentration in the sampler is equilibrated with the concentration outside the sampler. After equilibration, the PD sampler is retrieved, and the concentration measured in the sampler is assumed to directly relate to in situ analyte concentrations at the deployment location. However, unlike dosimeter-type devices and SPMDs, PD samplers do not contain a trapping phase and are instead filled with simply water or air. Because the trapping phase is efficient at retaining contaminants, SPMDs may permit detection of ultralow contaminant concentrations. However, water- or gas-filled PD samplers are less expensive to construct and easier to analyze because no extraction step is required. Furthermore, unlike SPMD measurements (which are not necessarily based on equilibration), the correlation between the contaminant concentration measured in the PD sampler and the surrounding environment is straightforward: for conditions without phase partitioning (i.e., water-filled PD samplers deployed in groundwater), the concentrations are equivalent, and for conditions with phase partitioning (i.e., gas-filled PD samplers deployed in groundwater), the concentrations are directly related by the Henry’s law constant of the contaminant. Passive diffusion-type sampling methods have been used in a variety of applications for many years, including dissolved gas measurement in seawater and groundwater (e.g., 8-12). Generally, PD samplers offer several potential advantages for monitoring many common groundwater contaminants, as compared to conventional sampling techniques (e.g., 1315). For dissolved constituents with large Henry’s Law constants, such as dissolved gases and most volatile organic compounds (VOCs), analyte loss due to volatilization can be greatly minimized. For large contaminated sites with ongoing monitoring programs, significant cost savings may be realized due to reduced sampling-related labor. Furthermore, waste generation and disposal costs may be reduced because well purging is not required prior to sampling. Passive diffusion samplers can also provide additional characterization data compared to conventional purge-and-sample techniques. For example, multiple PD samplers can be deployed in wells with large screen intervals to further characterize contaminant stratification at a relatively small scale (4, 16). This technique can be used in tandem with conventional purgeand-sample techniques to identify zones with high chemical flux at heterogeneous sites (e.g., 16). VOL. 38, NO. 6, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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Because of the potential costs savings and technical advantages noted above, use of PD samplers for groundwater monitoring at contaminated sites is greatly increasing, and water-filled PD samplers utilizing low-density polyethylene (LDPE) membranes are currently available commercially from two companies, Columbia Analytical, Inc. (Kelso, WA) and EON Products, Inc. (Snellville, GA). Laboratory and field compatibility tests indicate that LDPE membranes are suitable for most fuel and solvent VOCs; however, it has been demonstrated that a LDPE membrane is generally not suitable for measuring inorganic ions, metals, pesticides, explosives, semivolatile organics, alcohols, organic acids, and ethers (including methyl-tert-butyl ether [MTBE]) with PD samplers (for results of LDPE compatibility tests for specific constituents, see refs 1, 14, 15, 17). However, it should be noted that dialysis-type samplers utilizing other membrane material (i.e., cellulose acetate) have been used to monitor pH, major ions, nutrients, and metals for several years, particularly in the fields of limnology and oceanography (e.g., 4, 18). To date, LDPE membranes have not been specifically evaluated for measurement of dissolved gases commonly used as groundwater tracers (He, Ne, etc.) or those of environmental interest (O2, H2, N2, etc.). Depending on the application, PD samplers may be gasor water-filled. For gas-filled PD samplers designed to measure soil gas concentrations, equilibrium analyte concentrations measured in the PD samplers are assumed to directly represent in situ soil gas concentrations. Likewise, analyte concentrations measured in water-filled PD samplers deployed in groundwater (or surface water) are assumed to be equivalent to dissolved-phase concentrations at the measurement location. Gas-filled PD samplers can also be used to measure gases and VOCs dissolved in water (e.g., 1, 9). In this case, the temperature-dependent Henry’s Law constant (H) relates the measured gas-phase PD sampler concentration (Cg) to the in situ water-phase concentration (Cw) for each constituent by the following linear gas-water partitioning relationship.

Cg ) CwH

(1)

For constituents with high Henry’s Law constants, gasfilled PD samplers are particularly advantageous because they permit collection of significantly greater analyte mass than water-filled PD samplers and, therefore, facilitate lower analytical detection limits. Despite the many potential advantages associated with LDPE PD samplers, several important fundamental and practical considerations remain that have not been previously examined and reported in the peer-reviewed literature, particularly with respect to their use at contaminated sites (e.g., 19-21). For example, one important difference between measurements made with PD samplers and other conventional techniques is that PD samplers do not measure “realtime” in situ concentrations. Because equilibration is a diffusion-type process (and therefore not instantaneous), the PD sampler must be deployed long enough for equilibration to occur. Furthermore, the concentration measured by a PD sampler represents a time-integrated measurement of recent in situ concentrations. Although several empirical studies of equilibration times have been performed with LDPE PD samplers for selected VOCs (e.g., 1, 20), LDPE membrane diffusion coefficients (Dm) for these constituents have not been previously estimated, and physically based models have not been applied to predict equilibration times for LDPE PD samplers. Furthermore, the significance of lag time between temporally variant in situ concentrations and PD sampler concentrations has not been previously evaluated. In this paper, we use an analytical diffusion model to estimate LDPE Dm values for several gases from laboratory 1850

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equilibration experiments. Additionally, Dm values are estimated from experimental data previously presented by Vroblesky and Campbell (1) for several commonly occurring VOCs. The analytical model is also used to predict equilibration times for various sampler dimensions and membrane thicknesses. Finally, we present a numerical model that can be used to predict sampler concentrations and evaluate the equilibration lag time when in situ resident concentrations are temporally variant. Laboratory experiments with varying in situ concentrations are used to evaluate the performance of this numerical model. Additionally, the numerical model is used to assess the potential significance of lag time for commercially available LDPE PD samplers in a hypothetical field scenario. Sampler Equilibration Time: Mathematical Models. The following solution to the governing differential equation for describing the helium and neon concentrations within a gasfilled PD sampler (utilizing a silicone membrane) deployed in groundwater is presented by Sanford et al. (9)

[ (

CS(t) ) CrH 1 - e

)]

-DmAt ∀Lm

(2)

where Cs(t) is the concentration within the sampler at time t after PD sampler deployment [M L-3]; Cr is the resident concentration at the measurement location [M L-3]; Dm is the effective constituent diffusion coefficient for the membrane material [L2 T-1]; H is the solute dimensionless Henry’s Law constant [ ]; A is the surface area of the PD sampler membrane [L2]; t is the deployment time [T]; ∀ is the internal volume of the PD sample [L3]; and Lm is the membrane thickness [L]. This solution assumes independent solute transport behavior following Fickian-type solute diffusion across a thin membrane, a constant Cr concentration at the measurement location, and a well-mixed reservoir in the PD sampler. As presented, eq 2 incorporates phase partitioning and is applicable for gas-filled PD samplers deployed in water; however, it can be modified for conditions under which no phase partitioning occurs (for a water-filled PD sampler deployed in groundwater or a gas-filled PD sampler deployed in gas-phase systems) by treating the Henry’s Law constant (the gas-water partition coefficient) as unity. As clearly shown in eq 2, equilibration time is specifically dependent on the constituent Dm value and the PD sampler dimensions (A, ∀, Lm). Therefore, PD samplers can be constructed to minimize equilibration times by using a thin membrane and increasing the A/∀ ratio. When the sampler dimensions are known, eq 1 can be used to estimate LDPE Dm values in equilibration time experiments. For example, Sanford et al. (9) used eq 2 to estimate Dm values for He and Ne for silicone. In some cases, eq 1 is inappropriate, because Cr may vary temporally due to natural diurnal/seasonal fluctuations, changing pumping rates, remediation system operation, and nonsteady plume conditions. The following is an approximate numerical solution for Cs when Cr varies temporally,

Ci+1 ) Cis + ∆Cis s

(3a)

where the change in PD sampler concentration over the timestep interval (∆Cis) is approximated from the governing differential equation by

∆Cis )

-DmA∆ti i (Cs - HCir) ∀Lm

(3b)

where ∆ti is the magnitude of the time-step (note that superscripts i and i + 1 refer to parameter values at specific time-step intervals). As before, this solution can be used for

FIGURE 1. Comparison of the numerical model (eqs 2a and 2b) to the analytical solution (eq 1). For this comparison, Cr remained constant and was used to normalize the predicted PD concentration. Model parameter values were A ) 350 cm2, ∀ ) 220 mL, Lm ) 0.01 cm (4 mil), ∆t ) 0.042 day, and Dm ) 0.5 × 10-7 cm2 sec-1, 5 × 10-7 cm2 sec-1. conditions without phase partitioning by treating the Henry’s Law constant as unity. This numerical solution compares well with the analytical solution (eq 2) for steady Cr conditions (Figure 1). This figure also illustrates the relative significance of the magnitude of the solute Dm value by modeling equilibration times for typical commercial PD sampler dimensions where Dm varies by an order of magnitude (0.5 × 10-7 cm2 sec-1 and 5 × 10-7 cm2 sec-1). The mathematical models presented above assume that solute diffusion through the LDPE membrane is the ratelimiting step for sampler equilibration. For conditions in which diffusion is the dominant solute transport mechanism through the aquifer or sediment, a concentration gradient may develop outside the PD sampler, reducing the concentration gradient (and therefore, solute flux) across the membrane. For these cases, equilibration time is controlled both by diffusion through the membrane and diffusion through water to the membrane surface. Model results presented by Harrington et al. (22) and Webster at al. (23) show that equilibration times can be significantly longer and are strongly dependent on sampler dimensions for these general conditions. Of course, if the in situ concentrations at the membrane interface (i.e., the effective Cr) are known or can be predicted, use of eqs 3a and 3b for predicting PD sampler concentrations is still appropriate. Additionally, the models presented in this paper assume a constant Dm value; however, under some field conditions, a biofilm may develop on the membrane surface, reducing the effective Dm over time. In most cases, however, sampler equilibration will occur long before the development of a significant biofilm layer. Therefore, the potential development of a biofilm is not anticipated to have any practical effect on equilibration time and sampler performance. On the basis of these and other potential controls on equilibration, a minimum 2-week deployment time has been suggested as a rule-of-thumb for commercially available LDPE PD samplers (1, 14). However, it is important to note that this generic guideline is not quantitatively based on any specific field or modeling results. Rather, it was drawn from general results of studies using LDPE PD and dialysis-type samplers (with different/unknown sampler dimensions, membrane thicknesses, and Dm values) and represents a conservative recommendation of minimum deployment time. In fact, empirical evidence from several laboratory and

FIGURE 2. Schematic of constructed PD sampler. (a) A rubber band is wrapped around the neck of a glass sample vial to serve as seal gasket. The vial may be gas- or water-filled (water-filled samplers may be constructed while submerged in deionized water to prevent entrapment of air bubbles). The vial may also be filled with inert sand or glass beads to reduce the internal volume (∀), thereby decreasing equilibration time. (b) A sheet of LDPE membrane is placed over the mouth of the sampler. (c) and (d) A nylon cable tie is placed over the LDPE membrane and is tightened around the vial neck and seal gasket. Excess membrane and cable tie material are trimmed. Once the PD sampler is retrieved, a cap with a thick silicone septum is screwed over the top of the vial to prevent analyte loss before analysis. field studies has verified that equilibration occurs relatively quickly (several days) for LDPE PD samplers deployed in advection-dominated systems (1, 13-17), and this is likely to be the most common application of these types of samplers for monitoring groundwater at contaminated sites. For these cases, the presented models (eqs 2, 3a, and 3b) are appropriate, and they provide a physically based method for predicting equilibration times for PD samplers. VOL. 38, NO. 6, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 3. Results of equilibration time experiments and estimated Dm values for He (a), Ne (b), H2 (c), O2 (d), and N2 (e). Vertical error bars represent one relative standard deviation for Cr measurements.

Experimental Methods Water-dissolved gases such as He and Ne offer specific advantages as groundwater tracers at contaminated sites (e.g., 24, 25), and dissolved H2, O2, and N2 are often measured as indicators of biological activity in environmental studies. Dissolved gases in ground and surface water are difficult to measure by conventional sampling methods. For example, their typically large Henry’s Law constants and relatively low aqueous solubilities often lead to poor analytical precision and relatively high detection limits (e.g., 26). Gas-filled PD samplers offer a potential method improvement, because volatilization loss during sample collection and preparation is eliminated and greater analyte mass is collected (9). Therefore, a series of laboratory experiments were conducted to determine LDPE Dm values for He, Ne, H2, O2, and N2. The results of these experiments can then be used to estimate equilibration times for LDPE PD samplers. For the He, Ne, and H2 experiments, simple gas-filled PD samplers were constructed using inexpensive materials. The mouth of a 5-mL glass vial (A ) 0.72 cm2, ∀ ) 5.3 mL) was covered with a 0.0018-cm [0.7 mil]-thick LDPE membrane (Film Guard, Tyco Plastics, Minneapolis, MN) and sealed with a rubber-band gasket and nylon cable tie. Thin membrane-suitable LDPE is widely available as drop sheeting 1852

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in the painting supply section of most hardware stores, and is therefore easy and inexpensive to obtain. The sampler design and construction steps are shown in Figure 2. Initially, the gas-phase composition within the samplers was equivalent to atmospheric composition, and therefore, the initial concentrations of these three trace gases were assumed to be negligible. A number of samplers were concurrently deployed in water-filled vessels with dissolved gas concentrations maintained near the aqueous solubility for the respective gas. Gas concentrations were maintained by bubbling (via an airstone) a constant flow of gas with a specific partial pressure of the analyte gas through the continuously stirred water column. This method for maintaining dissolved gas concentrations is similar to those described by others (9, 25). The PD samplers were then removed at various intervals after deployment to characterize equilibration times. Immediately after removal, a cap with a thick silicone septum was screwed over the LDPE membrane to preserve analyte concentrations within the sampler. Sample aliquots were removed through the septum via a syringe and injected into a gas chromatograph (GC) for analysis. Gas-phase PD concentrations were converted to equivalent dissolved-phase values by application of Henry’s Law and were then normalized by the average Cr value for

FIGURE 4. Results of equilibration time experiments (from data presented by Vroblesky and Campbell (1), shown in plot (a) and estimated Dm values for EDB (b), benzene (c), TCE (d), toluene (e), and PCE (f). Vertical error bars represent one relative standard deviation for Cr measurements. comparison and interpretation. The LDPE Dm values were estimated by fitting eq 1 to the experimental data. Because O2 and N2 have large background concentrations, relatively low concentrations of these constituents were maintained in the vessels rather than near-solubility concentrations. The low concentrations were achieved by bubbling a constant flow of helium through the vessel to reduce the equivalent partial pressures of O2 and N2. The equilibration time was measured by the rate at which the PD sampler concentrations dropped from background values (water in equilibrium with the atmospheric) to the lower concentrations maintained in the charging vessel. The measured PD sampler concentrations were normalized for comparison and interpretation by

Cs ) 1 -

Cm - C r Cbg - Cr

(3)

where Cm is the measured concentration in the PD sampler and Cbg is the initial background concentration. Helium, Ne, H2, O2, and N2 were analyzed with a GC (Schimazdu 8A) equipped with a molecular sieve column (60/80 mesh [5 Å], Ar carrier gas) and a thermal conductivity detector (TCD). Gas-phase PD concentrations were measured

by direct injection of PD sampler gas, and water-phase concentrations (for Cr determination) were measured by headspace analysis of vessel water samples (e.g., 25). To ensure quality control, laboratory-grade chemicals, gases, sample vials, and other materials were used for the experiments. During chemical analyses (both gas and VOC) multipoint calibration curves were constructed, and spike samples with known analyte concentrations were periodically analyzed to evaluate analytical precision. As noted earlier, Vroblesky and Campbell (1) present the results for equilibration tests with LDPE PD samplers for several VOCs, including ethene dibromide (EDB), benzene, toluene, trichloroethene (TCE), and tetrachloroethene (PCE). However, LDPE Dm values, which can permit estimation of equilibration time for specific sampler dimensions, were not estimated from these data. Therefore, the data from these experiments were reproduced by digitizing concentration values presented in Figure 4 of Vroblesky and Campbell (1) and were used to estimate Dm for these compounds. In these experiments, the PD samplers were constructed of waterfilled, flexible 1.5-in.-diameter LDPE tubes (Lm ) 0.01 cm [4 mil]). This type of PD sampler is also referred to as a PD “bag” sampler and is similar to commercially available LDPE PD samplers. With PD bag samplers, the sample fluid is VOL. 38, NO. 6, 2004 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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TABLE 1. Summary of Estimated LDPE Dm Values LDPE Dm Values Estimated in This Study Dma (21 °C) H (21 °C)b VOC Dma (21 °C)

gas

105

benzene

2.2 × 10 -7

Ne H2

1.2 × 10 -7/ 1.6 × 10 -7c 1.1 × 10 -7 1.3 × 10 -7

89 52

toluene TCE

O2 N2

1.9 × 10 -7 1.4 × 10 -7

31 60

PCE EDB

4.4 × 10 -7 2.5 × 10 -7/ 1.9 × 10 -7c 2.4 × 10 -7 1.7 × 10 -7

He

alkane

Previously Estimated LDPE Dm Valuesd Dm 25 °C Dm 50 °C

hexane heptane nonane dodecane pentadecane

1.3 × 10 -7 8.8 × 10 -8 5.3 × 10 -8 2.4 × 10 -8 8.0 × 10 -9

3.3 × 10 -7 2.6 × 10 -7 1.4 × 10 -7 9.9 × 10 -8 5.3 × 10 -8

Dm 70 °C 5.8 × 10 -7 3.4 × 10 -7 1.8 × 10 -7 1.2 × 10 -7

a All D 2 -1 b Henry’s Law constant (H) m values reported in cm sec . values for gases are dimensionless. Calculated from temperature relationships given by CRC (27). c The first LDPE Dm value presented is estimated from analysis of constant-Cr experiments by eq 1. The second value is estimated from changing-Cr experiments by eqs 2a and 2b. d From Aminabhavi and Naik (28).

transferred from the bag to glass vials immediately after retrieval and sealed to prevent analyte loss. Although the sampler volumes varied between 50 and 70 mL, the sampler diameters were the same for each experiment. Consequently, the A/∀ ratio remained constant (A/∀ ) 1.05 cm-1), and this effective parameter value was used for Dm estimation with eq 2. Measured PD concentrations were normalized to average Cr values for comparison and interpretation. In addition to the experiments described above in which Cr remained constant and Dm was estimated with eq 2, two equilibration experiments were conducted in which Cr varied and the data were modeled with the numerical solution presented earlier (eqs 3a and 3b). These experiments were primarily designed to test the performance of the numerical solution. One experiment evaluated dissolved He equilibration with gas-filled PD samplers in which sampler dimensions and analytical methods were similar to the dissolved-gas experiments described above. The other experiment evaluated TCE equilibration with water-filled PD samplers constructed of glass vials with LDPE membranes. For this experiment, vials with two different volumes were used: sampler type 1; ∀ ) 43 mL, A ) 3.2 cm2, Lm ) 0.0018 cm [0.7 mil]; and sampler type 2; ∀ ) 24 mL, A ) 3.2 cm2, Lm ) 0.0018 cm [0.7 mil]. Dissolved TCE concentrations were measured with a GC (Schimadzu 17A) equipped with a capillary column (Phenomenex ZB-624, He carrier gas) and flame ionization detector (FID). For these water-filled PD samplers, an aliquot of liquid was directly injected into the GC, where it was then volatized at high temperature in the injection port. For both experiments, PD sampler concentrations were normalized by the initial Cr values (He experiment, 9.8 mL L-1; TCE experiment, 480 µg L-1).

Results and Discussion Estimation of LDPE Dm Values and Sampler Equilibration Time. The results of the dissolved gas and VOC constant-Cr equilibration experiments are shown in Figures 3 and 4, and the estimated LDPE Dm values are presented in Table 1. Generally, LDPE Dm values ranged from 1.1 to 1.9 × 10-7 cm2 sec-1 for the gases and from 1.7 to 4.4 × 10-7 cm2 sec-1 for the VOCs. The values are generally consistent with Dm values of alkanes for LDPE geomembranes (ref 28, provided in Table 1 of this study) and He and Ne for silicon (4 × 10-7 cm2 sec-1 for both He and Ne, ref 9). They are higher than values 1854

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measured by Xiao et al. (29) for high-density polyethylene (HDPE) for several VOCs, including benzene and TCE (2.04.6 × 10-8 cm2 sec-1). In aqueous solutions, larger molecules are generally expected to be associated with lower diffusion coefficient values, and this general trend is observed when comparing the gas or VOC Dm values to the previously measured alkane Dm values (Table 1). However, there is not a strong size-Dm correlation among the measured gases and VOCs, and the gas values are lower than the values for the higher-molecular-weight VOCs. The reason for this behavior is unknown; however, it is important to recognize that LDPE is not compatible with certain species (ions, MTBE, etc.), and therefore, molecular weight is not the only control on Dm. Furthermore, experimental error and uncertainty may mask a weak size-Dm relationship, if present. The uncertainty associated with these Dm estimates is difficult to quantify, as it results from a variety of errors and uncertainties, including measurement error, actual membrane thickness, and minor temporal variations in actual Cr values (especially for the dissolved gases). On the basis of measurement precision and the model fit to the data, the uncertainty with these measurements is estimated to be approximately (30%. For most practical applications, this Dm uncertainty likely results in acceptable equilibration time uncertainty, which can be evaluated with eq 1 for a particular PD sampler construction. The Dm values were measured at ∼21 °C and are expected to be slightly lower at typical groundwater temperatures. Aminabhavi and Naik (28) measured Dm values of LDPE geomembranes for several alkanes at 25, 50, and 70 °C. On the basis of an empirical logarithmic relationship between temperature and Dm for these experiments, the extrapolated LDPE Dm values are only anticipated to be ∼17% lower at 15 °C than at 21 °C. Assuming a similar relationship for the constituents measured in this study, actual equilibration times are expected to be only 15-20% longer for field conditions with slightly cooler temperatures, due to slightly lower in situ Dm values (shallow groundwater temperatures in the conterminous United States are commonly ∼14 to 18 °C). This is supported by empirical equilibration data for TCE reported by Hare et al. (20) in which PD bag samplers reached 95% equilibrium within ∼3 days at 10 °C. As discussed earlier, equilibration time is a function of the temperature-dependent Dm, sampler dimensions (A and ∀), and Lm for advection-dominated systems. By inspection of eq 1, it is clear that these parameters have equivalent influence: doubling Dm is equivalent to doubling A, halving ∀, or halving Lm. Therefore, equilibration times (defined when Cs g 0.95Cr) for several realistic sampler dimensions, including various constructed (Figure 2) and commercially available LDPE PD samplers, are provided in Table 2. On the basis of the results of this study, a reasonable range of LDPE Dm values for the dissolved gases and VOCs evaluated is ∼(1-4) × 10-7 cm2 sec-1. Using this Dm range, eq 2 predicts that only 2-3 days are necessary for commercial LDPE PD bag samplers to equilibrate, and this is consistent with previously reported equilibration times for these samplers in empirical laboratory and field studies (1, 14, 16). If a relatively thin membrane (