Environ. Sci. Technol. 2005, 39, 149-154
Analysis of Column Tortuosity for MnCl2 and Bacterial Diffusion Using Magnetic Resonance Imaging MIRA STONE OLSON,† R O S E A N N E M . F O R D , * ,‡ JAMES A. SMITH,† AND ERIK J. FERNANDEZ‡ Program of Interdisciplinary Research in Contaminant Hydrogeology, Department of Civil Engineering, University of Virginia, Charlottesville, Virginia 22904-4742 and Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22904-4741
Subsurface bacteria often have to travel significant distances through tortuous porous media for purposes of groundwater remediation. In modeling such processes, motile bacteria are often represented as suspended colloids, ignoring their individual swimming or diffusive properties. In fact, bacterial migration is much more profoundly affected by the presence of porous media than is that of a chemical contaminant. In this study, we use magnetic resonance imaging (MRI) to perform noninvasive measurements of changes in bacterial concentration distributions across a packed column at a spatial resolution of 330 µm as a function of time. We analyze the diffusive behavior of Pseudomonas putida F1 under static conditions and compare that behavior to the diffusion of a chemical solute and of Escherichia coli NR50. Results indicate that P. putida cells experience a column tortuosity 50 times higher than that predicted from solute diffusion experiments. E. coli cells, which display shorter swimming run lengths in bulk solution than P. putida, seem to be less affected by the constricted pore space. Knudsen diffusion, or reductions in run length because of interactions between the diffusing bacteria and the porous media, may help to explain some of this discrepancy.
Introduction Many motile, soil-inhabiting microorganisms are capable of degrading common organic contaminants (1-3) and may play an important role in subsurface bioremediation. However, for these bacteria to be effective at remediating contaminated aquifers, it may be necessary for them to travel significant distances through tortuous porous media (4, 5). Although the behavior of motile bacteria is notably different than that of abiotic colloids, a lack of adequate information leads many researchers to model bacterial transport as that of suspended colloids, largely ignoring the individual diffusive and swimming properties of the motile cells (6-8). In particular, tortuosity is often assumed to be a property of the porous medium itself, suggesting that motile bacteria would experience the same tortuosity as nonmotile bacteria or * Corresponding author phone: (434)924-6283; fax: (434)982-2658; e-mail:
[email protected]. † Department of Civil Engineering. ‡ Department of Chemical Engineering. 10.1021/es049577x CCC: $30.25 Published on Web 11/25/2004
2005 American Chemical Society
diffusing solutes. This assumption may not be appropriate. In a recent study, Becker et al. (8) suggested that although flagellated and thereby motile bacteria may be more capable of moving into relatively immobile water in response to trapped contaminants, they may also be less likely to travel as far as comparable, nonmotile bacteria. In addition, the behavior of motile bacteria near a surface is not the same as that of nonmotile cells. Motile cells may spend more time swimming or circling around the surface (9, 10) thereby increasing their residence time near soil particles. The diffusive migration of motile bacteria is a result of the swimming properties of individual cells. An analysis of the motile behavior of Pseudomonas putida is presented by Harwood et al. (11). Briefly, cells are lophotrichously flagellated, typically with five to seven flagella inserted at one end of the cell body to form a tuft. This flagellation contrasts with that of Escherichia coli, which are peritrichously flagellated (flagella uniformly distributed over the cell body surface). Motile behavior of free-swimming P. putida cells is characterized by series of unidirectional runs interrupted by abrupt reversals in direction. In contrast to E. coli, changes in direction of the P. putida cells are not accompanied by tumbling, and occasionally cells swim backward for a short distance before resuming movement in the forward direction. Compared with E. coli, P. putida cells swim at higher average linear speeds, change direction less frequently, and therefore exhibit longer swimming run lengths (9, 11). Diffusion in porous media is impeded by tortuous pathways, dead-end pores, interactions with the surface of the porous matrix, and variability in pore length and diameter (12). In addition, Knudsen diffusion (reduced path lengths because of molecule-pore wall collisions) may be important in cases where the mean free path of the diffusing agent is comparable to the characteristic pore diameter (13). Comparing the effective pore diffusion of a diffusing solute to its bulk aqueous diffusion is complicated and often inconsistently defined but is generally analyzed in terms of porosity, tortuosity, and constrictivity. Van Brakel and Heertjes (14) designate porosity and tortuosity as properties of the pore space in an absolute sense, but constrictivity as a function of the type of transport. However, when comparing bacterial diffusion to solute diffusion, even the definition of the available pore space may not be entirely consistent and must therefore not be assumed constant. Because of the impracticality of experimentally separating the effects of constrictivity and tortuosity, researchers often introduce an “effective” tortuosity factor, which includes the effects of both tortuosity and constrictivity (12). For this reason and because of the added complexity introduced when studying the diffusion of self-propelled bacteria, we will employ an effective tortuosity factor for the duration of this paper, incorporating both characteristics of the available pore space and of the transport mechanism of the diffusing agent. The purpose of this study is to examine the differences between the diffusive migration of motile P. putida F1 cells through a packed column and the transport of a diffusive solute through the same column under no-flow conditions. Calculations of the column tortuosity on the basis of bacterial experiments suggest that motile cells experience a much greater tortuosity than that which would be predicted on the basis of studies of a diffusive solute. Results were also compared with similar experimental studies of E. coli to evaluate how differences in cell swimming properties may affect the predicted column tortuosity. We believe that cellular interactions with the porous matrix, including Knudsen diffusion, may help to explain some of the differences in VOL. 39, NO. 1, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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transport behavior. We also believe that interactions with the porous media may interrupt the typically long swimming run lengths exhibited by P. putida, thereby impeding its diffusive migration.
Experimental Section Diffusing Solute. Manganese chloride (MnCl2) was chosen as the diffusive solute for this study because manganese is a contrast agent, dramatically reducing water relaxation times thereby detected and quantified using MRI. All experiments were run at a concentration of 0.038 mM MnCl2. Bacteria. Pseudomonas putida F1, obtained from Dr. Caroline Harwood at the University of Iowa, is a native soilinhabiting bacterial strain and was selected for these experiments because of the availability of a monoclonal antibody specific to P. putida (15) and because its swimming properties have been extensively studied (11, 16-19). Cells were grown as previously detailed (20) and inspected for motility at 400× using a Zeiss Std 16 microscope. Bacteria were suspended in a 10% (vol/vol) dilution of random motility buffer (21), a phosphate buffer that does not support growth, for all experiments. Immunomagnetic Labeling. A purified monoclonal antibody specific to P. putida, developed and provided by Dr. Maribel Ramos-Gonzalez at the Estacion Experimental del Zaidin CSIC (15), was diluted to a strength of 1:500, bound with 50-60-nm magnetite particles (Immunicon Corp., Huntington Valley, PA), and attached to suspended P. putida F1 cells harvested at an OD590 of 1.0, as previously described (20, 21). Cells were incubated for 2 h with the conjugated antibody and then filtered to separate labeled bacteria from excess antibody. Labeled cells were resuspended in 10% (vol/ vol) random motility buffer (21). Consistent with Sherwood et al. (21), we observed no difference in the swimming behavior of labeled cells versus unlabeled cells under a microscope in terms of swimming speed and fraction of total population of cells swimming. Column Assembly. Column assembly and packing is described in detail elsewhere (21). Briefly, glass-coated polystyrene beads (SoloHill Engineering, Inc., Ann Arbor, MI) with a size distribution of 250-300 µm were wet-packed into a specially designed plastic column with an inside diameter of 1.5 cm and two adjustable flow adapters (BioRad, Hercules, CA). The column was designed to permit impinging flow via four equidistant holes drilled along the midpoint of the column. Nylon frits across the inside of the holes served to retain the porous media. The column was packed to a final bed length of 8 cm by filling it with buffer and adding the prewetted beads. Following assembly, the column was pretreated with unlabeled P. putida F1 cells to minimize sorption of labeled bacteria and flushed with plain buffer. Magnetic Resonance Imaging Protocol. MRI protocols followed those of Olson et al. (20) and Sherwood et al. (21). Experiments were conducted in a 1.75 T, 12-cm bore horizontal magnet (Nalorac Cryogenics Corporation, Martinez, CA) and spectrometer (TecMag, Inc., Houston, TX) with gradient coils (Magnex Scientific, U.K.) of up to 20 G/cm. The imaging protocol was a T2-weighted, one-dimensional, x-slice spin-echo sequence, programmed using MacNMR version 4.5.9 software (Tecmag, Inc.), with 30 TE times and a spatial resolution of 330 µm. T2 values were converted to concentrations, c, according to the following correlation (21):
Retardation Factor Experiments. Tracer experiments for both MnCl2 and P. putida F1 were conducted to determine the retardation factors characterizing manganese and bacterial transport through the packed column. Experiments were run separately with 3H-H2O as the conservative tracer in both cases. Columns were packed and pretreated with unlabeled bacteria as described above. MnCl2 tracer experiments were run by flowing an influent solution of 0.038 mM MnCl2 at 0.075 µCi/mL 3H-H2O at a flow rate of 1 mL/min for 50 min. Effluent samples were collected every minute. Five tenths milliliters of the effluent sample was used for T2 analysis using MRI and the remaining 0.5 mL was analyzed for 3H-H2O concentration using a Packard liquid scintillation counter. Tracer experiments for the labeled bacteria were conducted by introducing an influent solution of labeled bacteria at 0.075 µCi/mL 3H-H2O at a flow rate of 1 mL/min (1.3 cm/min) for 11 min and then immediately switching to a solution of plain buffer for 40 min at the same flow rate. A pulse experiment was more practical for the bacteria because of limitations in labeling large volumes of bacteria at one time. Effluent samples were collected every minute and analyzed for total bacterial, labeled bacterial, and 3H-H2O concentrations. Total bacterial concentrations were quantified by optical density at 590 nm using a Beckman DU-7 spectrophotometer. Labeled bacterial and 3H-H2O concentrations were analyzed in the same manner as in the MnCl2 tracer experiments. Breakthrough curves of normalized effluent concentration profiles were plotted versus time and retardation factors were calculated by comparing the time required for the reactive compound (MnCl2 or labeled P. putida F1) to reach c/co ) 0.5 to the time required for the conservative tracer (3H-H2O) to reach c/co ) 0.5 (23). Mathematically,
R)
vconservative vreactive
(2)
where vconservative and vreactive represent the velocities of the conservative and reactive tracers, respectively. Diffusion Experiments. MnCl2 diffusion experiments were conducted to quantify the tortuous diffusion of manganese through a uniformly packed column. Following pretreatment of the column with unlabeled bacteria, a 0.038 mM solution of MnCl2 was introduced onto one-half of the column at a flow rate of 2 mL/min as plain buffer was pumped onto the other half at the same flow rate. Both streams exited the column at its midpoint, creating an initial step change in MnCl2 concentration at the midpoint of the column. Once the step change was established, flow was halted and experiments proceeded under no-flow conditions for 15-16 h. Spatial manganese concentration profiles were collected using MRI. Solute transport is represented as follows:
∂c Deff ∂2c R ) ∂t ∂x2
(3)
(1)
where c is the concentration of manganese (mM), R is the retardation factor for manganese (determined from breakthrough curves), is porosity, and Deff is the effective diffusion coefficient of manganese (cm2/s). Observed data were matched to the solution of eq 3 by adjusting the lumped effective diffusion term, Deff/.
where R is the relaxivity constant and T2,o is the T2 value for pure water. Verification of the linear relation between concentration and inverse T2 is provided by Olson (22).
Random Motility Experiments. Bacterial random motility experiments were performed to examine the diffusion-like migration of P. putida F1 cells and were conducted in the same manner as the MnCl2 diffusion experiments, with
c)
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(
1 1 1 R T2 T2,o
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TABLE 1. Transport Parametersh D0 or µ0 (cm2/s) 1.2 × 10-5
a
P. putida 1.3 × 10-5 F1
b
MnCl2
E. coli NR50
FIGURE 1. Breakthrough curves of normalized effluent concentrations of 3H-H2O (b) and Mn2+ (2) versus time. Influent solution was introduced at 0 min and continuously flowed at 1 mL/min throughout the sampling period. Dotted lines show the breakthrough time of c/co ) 0.5 for both 3H-H2O and Mn2+.
Deff/E or µ0eff/E (cm2/s)
2.6 ( 0.5 × 10-6 c
v (cm/s)
6.8 ( 0.2 × 10-6 na 7.1 ( 0.2 × 10-6 7.6 ( 0.5 × 10-6 1.2 ( 0.3 × 10-7 d 44 × 10-4 1.7 ( 0.4 × 10-7 d 1.5 ( 0.6 × 10-7 3.0 ( 0.7 × 10-7 e 24 ( 5 × 10-4 g
R 3.5 ( 0.6 f
1
1
a Ref 24. b Ref 17. c Unpublished data. d Originally published in ref 20. e Original data from ref 21. f Ref 11. g Ref 9. h D0 and µ0 represent the bulk diffusion and bacterial motility coefficients, respectively. Deff/ and µ0eff/ represent the lumped effective diffusion and motility coefficients, averaged for each experimental trial, and are presented with one standard deviation. v represents bacterial swimming speed, and R is the measured retardation factor.
labeled cells replacing the MnCl2 solution. Bacterial transport is represented as follows:
∂b µ0,eff ∂2b ) ∂t ∂x2
R
(4)
where b is the concentration of bacteria (cells/mL), R is the retardation factor for the labeled bacteria, is porosity, and µ0,eff is the effective motility coefficient of bacteria (cm2/s). Observed data were matched to the solution of eq 4 by adjusting the lumped effective random motility term, µ0,eff/. Mathematical Description of Column Tortuosity. The effective diffusion coefficient can be related to a bulk aqueous diffusion coefficient according to the following correlation (21):
Deff D0 ) τ
(5)
where D0 is the bulk aqueous diffusion coefficient and τ is the tortuosity of the porous medium. For the diffusion-like migration of bacteria, the effective motility coefficient, µ0,eff, and the bulk motility coefficient, µ0, can be substituted for Deff and D0, respectively.
Results and Discussion Retardation Factor Experiments. Duplicate MnCl2 tracer experiments were run to examine the retardation of manganese in the packed column relative to a conservative tracer. Effluent concentrations of both manganese and 3H-H2O were plotted versus time (Figure 1). On average, manganese eluted 3.5 times more slowly than the conservative 3H-H2O and eq 2 was employed to calculate a retardation factor for manganese (Table 1). We have assumed that manganese sorption is an equilibrium process, with no kinetic limitations. Although this distinction is not definitive from the shape of the manganese breakthrough curve in Figure 1, we believe that the initial delay in measurable manganese concentration indicates that equilibrium sorption is a reasonable assumption. Results from the bacterial tracer experiment are presented in Figure 2. Compared with breakthrough of the conservative 3H-H O, there seems to be no apparent retardation either 2 of total bacteria or of labeled bacteria. Both total bacterial concentration and labeled bacterial concentration were measured to examine the possibility of exchange between presorbed unlabeled cells and newly introduced labeled cells on the surface of the beads. Results indicate that there was no apparent exchange between labeled and unlabeled cells and no sorption of labeled cells onto the pretreated beads.
FIGURE 2. Breakthrough curves of normalized effluent concentrations of 3H-H2O (/), total bacteria (b), and labeled bacteria (4) versus time. Influent solution was pumped at 1 mL/min for 11 min and then replaced with plain buffer, which continued at 1 mL/min throughout the sampling period. Dotted lines show the expected breakthrough time of c/co ) 0.5 for 3H-H2O and P. putida F1 (total and labeled). Diffusion Experiments. Triplicate MnCl2 diffusion experiments were conducted to examine the diffusion of manganese through the packed column. Concentration profiles were collected every 18 min for 15-16 h and each temporal profile was fitted individually to the numerical solution of eq 3 for an independent prediction of the lumped effective diffusion coefficient, Deff/. For each experiment, results were averaged and are presented in Table 1 along with ( one standard deviation of the averaged fitted diffusion coefficient, Deff/. Concentration data from the first profile, collected at 18 min, was used as the initial input for modeled simulations to account for any deviation of the stagnation point from the midpoint of the column. Results from one experiment are depicted in Figure 3. For clarity, only three temporal concentration profiles are included. Random Motility Experiments. Triplicate random motility experiments demonstrated the migration of P. putida F1. Experiments were conducted at cell concentrations on the order of 108-109 cells/mL. Concentration profiles were collected every 35 min and fitted separately to numerical solutions of eq 4 for individual predictions of the random motility coefficient for each temporal profile, as described for MnCl2 experiments. Representative concentration profiles with averaged simulated results ( one standard deviation of the fitted lumped random motility coefficient, µ0,eff/, for one experiment are shown in Figure 4. Averages of fitted values for the random motility coefficients from all experimental trials are listed in Table 1. Although the system is made more VOL. 39, NO. 1, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 3. Dimensionless concentration profiles for MnCl2 in a packed column at 18 min (9), 454 min (2), and 928 min (×) after onset of a diffusion experiment. Data at 18 min are used as initial input for model simulations (solid line). Dark curves are averaged best-fit solutions to model eq 3 with Deff/E ) 6.8 ( 0.2 × 10-6 cm2/s at 454 min (dashed line) and 928 min (dotted line). Lighter curves (dashed and dotted) represent the standard deviation of Deff/E.
FIGURE 4. Dimensionless concentration profiles for motile P. putida F1 in a packed column at 35 min (9), 456 min (2), and 913 min (×) after onset of a random motility experiment. Data at 35 min are used as initial input for model simulations (solid line). Dark curves are averaged best-fit solutions to model eq 4 with µ0,eff/E ) 1.2 ( 0.3 × 10-7 cm2/s at 456 min (dashed line) and 913 min (dotted line). Lighter curves (dashed and dotted) represent the standard deviation of µ0,eff/E.
TABLE 2. Calculated Parameters for Column Tortuosity (τ), Adjusted Bacterial Motility with Knudsen Diffusion (µpore), and Adjusted Column Tortuosity with Inclusion of Knudsen Diffusion (τ*) MnCl2 τ µpore (cm2/s) τ* (with Knudsen diffusion)
1.7 ( 0.3 na na
P. putida F1 E. coli NR50 87 ( 16 6.2 × 10-6 42 ( 8
8.7 ( 0.7 1.8 × 10-6 6.1 ( 0.5
complex by the need to saturate bacterial attachment sites on the beads prior to the onset of experiments, we feel confident that this condition is being adequately met because of the shape of the concentration profiles. As time progresses, we see increased bacterial migration into the initially cellfree region of the column. If the labeled cells were attaching to the beads, we would expect to see increased bacterial accumulations at unsaturated sites as opposed to distinct migration into the cell-free region. We therefore further validate our assertion that there is no measurable sorption of labeled cells onto the pretreated beads. Calculation of Column Tortuosity. Values for the tortuosity of the packed column were calculated for both MnCl2 and P. putida F1 by rearranging eq 5 and are presented in Table 2. Reported tortuosity values and their associated errors represent averaged calculations including all three experimental predictions of Deff/ or µ0,eff/. Values used for the bulk diffusion coefficient of MnCl2 and the bulk motility coefficient of P. putida F1 are reported in Table 1. A tortuosity prediction 152
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of 1.7 for MnCl2 diffusion is within the expected range of values for a packed bed of uniform particle diameter (24) and differs from previous predictions by Sherwood et al. (21) only because of the inclusion of the retardation factor for manganese. Results indicate that bacterial cells do not experience the same column tortuosity as a chemical solute diffusing through the same porous medium. In fact, we see more than a 50-fold increase in the tortuosity experienced by the P. putida F1 cells compared with that experienced by the manganese. This difference itself is not unanticipated; however, its magnitude is greater than expected. Although it is possible that aerotaxis to oxygen gradients could refocus the bacteria, resulting in a reduced apparent tortuosity, we have not included the effect of aerotaxis in our analysis because cells were in a resting state with no carbon source and were unlikely to be consuming much oxygen. To verify that the smaller size of the diffusing manganese was not the reason for its lower tortuosity, we examined data from a random motility experiment using nonmotile E. coli NR50 collected by Sherwood et al. (21) in the same experimental system as that used in this study. Assuming that a nonmotile colloid having the same size as a typical bacterium is expected to have a diffusion coefficient of 1.5 × 10-10 cm2/s (25), we calculated the lumped effective diffusion coefficient of nonmotile cells assuming a column tortuosity of 1.7 using eq 5. This predicted lumped effective diffusion coefficient of 8.8 × 10-11 cm2/s matches closely with experimental motility profiles collected over 736 min (21), which essentially demonstrate no measurable migration of the nonmotile cells (data not shown). This indicates that nonmotile cells experience a column tortuosity consistent with that measured for manganese, thereby validating the column tortuosities measured using this complex system. One factor that we believe contributes to the high tortuosity for P. putida cells is their long run length. P. putida cells may be 3 orders of magnitude larger than a typical solute (26) and may travel 88 µm before changing direction (11), much further than the mean free path of a diffusing solute. Therefore, the bulk behavior of the bacterial cells is affected more noticeably in a porous medium than the bulk behavior of a chemical solute in the same medium. For further evaluation, we compared the diffusive migration of P. putida F1 to earlier reports of E. coli NR50 migration in the same experimental system as that used in this study (21). Bulk swimming properties for E. coli NR50 are reported in Table 1. Original concentration profiles for E. coli random motility through the packed column were obtained from Sherwood et al. (21). We then analyzed the concentration profiles and fitted for a lumped effective random motility coefficient, µ0,eff/ , in a manner identical to that employed for all other analyses in this study (Figure 5). Results are included in Table 1 and the calculated tortuosity is reported in Table 2. Although we did not measure a retardation factor for E. coli, we assumed it to be identical to that measured for P. putida since both cell samples were labeled in the same manner and the columns were similarly pretreated with unlabeled cells. In contrast to the average run length of 88 µm reported for P. putida cells, E. coli cells typically experience run lengths of approximately 17-24 µm (9) before changing direction. Not surprisingly, the predicted tortuosity describing E. coli diffusive migration through the packed column is closer to that of manganese than is the predicted tortuosity for P. putida cells. However, neither of the bacterial tortuosity predictions match that experienced by a chemical solute, so we infer that there are other interactions slowing the migration of the bacteria, presumably affecting P. putida more profoundly than E. coli. The observed effective diffusion of P. putida F1 and E. coli NR50 is relatively consistent (Figures 4 and 5, Table 1), with E. coli exhibiting an effective motility coefficient of
TABLE 3. Representative Pore Size Distributiona
FIGURE 5. Dimensionless concentration profiles for motile E. coli NR50 in a packed column at 45 min (9), 367 min (2), and 734 min (×) after onset of a random motility experiment. Data at 45 min are used as initial input for model simulations (solid line). Dark curves are averaged best-fit solutions to model eq 4 with µ0,eff/E ) 3.0 ( 0.7 × 10-7 cm2/s at 367 min (dashed line) and 734 min (dotted line). Lighter curves (dashed and dotted) represent the standard deviation of µ0,eff/E. approximately twice that of P. putida. However, both the bulk motility coefficients (Table 1) and therefore the tortuosity predictions (Table 2) differ by an order of magnitude. Thus, the relative deviation of the effective motility from the bulk motility is much more dramatic between the two strains. It is also important to realize that despite the fact that the bulk motility of these motile strains is dramatically reduced in porous media, motility is still an advantage to bacteria migrating through porous media because their effective motility through porous media remains orders of magnitude greater than that of nonmotile bacterial strains. Incorporation of Knudsen Diffusion. One factor that we believe plays a role in bacterial diffusion as opposed to solute diffusion is Knudsen diffusion, or reductions in run length because of interactions between the cells and the packed bead surfaces. Knudsen diffusion is negligible for contaminant diffusion because the mean free path of a diffusing chemical is much smaller than a characteristic pore diameter so molecule-pore wall collisions are infrequent; however, bacterial run lengths are comparable to typical pore diameters and may therefore be cut short resulting in a reduced diffusivity. Previous researchers have defined Knudsen diffusion for use with bacterial migration and have adapted correlations to describe bacterial motility as follows (17, 27, 28):
1 1 1 ) + µpore µ0 µK
(6)
where µpore represents the adjusted random motility coefficient, µ0 represents the bulk motility coefficient, and µK represents the Knudsen diffusion term. Knudsen diffusion is defined as
µK )
vdp 3
(7)
where v represents cell swimming speed and dp represents pore diameter. Adjusted pore motility coefficients for both P. putida and E. coli were calculated from eq 6 using the bulk motility coefficients listed in Table 1 and are included in Table 2. The Knudsen diffusion was determined using a representative pore size distribution for 250-300-µm sand (Table 3) (29). For comparison, an average pore diameter of 93 µm was predicted for 250-300-µm spherical beads and was found to yield similar results (calculations not shown). Calculations accounting for Knudsen diffusion effects on the column
a
fraction of pores
pore diameter (µm)
0.026 0.021 0.036 0.053 0.069 0.080 0.090 0.079 0.081 0.084 0.078 0.067 0.024 0.018 0.013 0.007 0.172
248.5-739.5 159.0-248.5 120.2-159.0 107.6-120.2 101.6-107.6 97.9-101.6 94.8-97.9 92.4-94.8 89.6-92.4 87.2-89.6 84.7-87.2 79.5-84.7 54.0-79.5 41.4-54.0 31.7-41.4 29.8-31.7
