Ind. Eng. Chem. Res. 2003, 42, 2853-2860
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Aqueous Ion Transport Studies in Stainless Steel Membranes Richard L. Ames,*,†,‡ Elizabeth A. Bluhm,† Annette L. Bunge,‡ Kent D. Abney,† J. Douglas Way,†,‡ and Stephen B. Schreiber† Los Alamos National Laboratory, NMT-2/CINC, Mail Stop E511, Los Alamos, New Mexico 87545, and Department of Chemical Engineering, Colorado School of Mines, 1500 Illinois Street, Golden, Colorado 80401
Aqueous ion transport through unmodified, acid/base resistant, stainless steel membrane material was investigated to determine the feasibility of using the material as a basis for an actinide separation unit operation. The Mott Metal Corporation membrane material used for testing was a sintered 316L stainless steel membrane having a particle size cutoff rating of 0.5 µm and an average pore size of 2.2 ( 0.5 µm. Radiotracer transport experimentation was conducted with 45Ca, 137Cs, 241Am, 152Eu, and 239Pu cations at varying pH and at dilute cation concentrations. Infinite-dilution diffusion coefficients for the cations were measured and had the same order of magnitude (10-6 cm2 s-1). The infinite-dilution diffusivities compared well with values found in the cited literature. These results confirmed bulk diffusion as the controlling mechanism for cation transport through the water-saturated stainless steel membrane pores and support the hypothesis that the membrane, in the unmodified form, does not add any contribution to the removal of selected cations under aqueous conditions. In addition, the infinitedilution diffusion coefficient for plutonium in solution as Pu(OH)3+ was determined for the first time to be 3.3 ( 1.3 × 10-6 cm2 s-1. Introduction Chemically robust selective membranes for the separation of radioactive and hazardous metal cations from wastewater effluents are an underdeveloped alternative to classical ion exchange and solvent extraction methods. Suitable membrane materials containing wellordered, homogeneous pore structures have recently become available. For example, track-etched polycarbonate and Anopore γ-alumina membranes can be chemically modified to cleanly separate hydrophobic and hydrophilic molecules in aqueous solution.1-3 As an alternative support material, commercially available Mott stainless steel membranes contain large pores formed by compressing and sintering metal powder under high temperatures and hydrogen atmosphere.1 The nonuniformity of pore diameters and path lengths in the Mott membranes creates many difficulties in physical characterization and theoretical modeling of transport.4 However, these membranes are chemically robust and can be coated with a thin layer of gold or silicon oxide and then chemically modified with molecular recognition sites to enhance the specificity of the chemical separations.5 Transport of uncharged solutes through mesoporous γ-alumina has been studied, but electrostatic interactions between cationic solutes and charged membrane surfaces are expected to be competitive with diffusion in determining transport rates. We recently reported on monovalent and divalent cation transport across Anopore membranes, also describing the interdependence between cation diffusion coefficients and alumina surface properties.6,7 Although our ultimate goal is to prepare membrane structures with very selective molecular or ionic actinide recognition proper* To whom correspondence should be addressed. Tel.: (505) 667-3637. Fax: (505) 665-1780. E-mail:
[email protected]. † Los Alamos National Laboratory. ‡ Colorado School of Mines.
ties, the charged surface of stainless steel itself could influence the transport of cations through the Mott membranes. To date, little work has focused on bare stainless steel membranes and their inherent transport properties in aqueous solutions. In this paper, we continue to further probe the relationship between surface chemistry and cation transport by using a new membrane support made from 316L stainless steel. Stated differently, the purpose of this investigation was to determine whether the unmodified porous stainless steel membrane surface chemistry contributes to the selective removal of actinide and nonactinide ionic constituents from nitric acid waste streams. Fickian diffusion diffusivities or diffusion coefficients (D) have been experimentally measured for an enormous variety of materials, not only dissolved solutes in liquids (D values on the order of 10-5 cm2 s-1), but also gas-phase solutes (10-2 cm2 s-1) and solid solutes in other solids (10-9 cm2 s-1).8-11 In a nonagitated, dilute, pure aqueous environment, the diffusion coefficient for transport of a solute (in our case, a dissolved ion) is defined as the infinite-dilution diffusion coefficient (D0). When a porous membrane barrier is placed in the path of the ion, diffusion occurs through the water-saturated pores of the material.12 A simple model for predicting an effective diffusion coefficient (Deff) from D0 when adding an impermeable object in the path of the diffusing solute was first predicted by Maxwell.13 In this model, it is assumed that the ratio of D0 to Deff is a function of barrier geometry; more specifically, he concluded that D0/Deff for a barrier consisting of a series of spheres occupying the volume fraction φ is given by by eq 1
φ 1+ D0 2 ) Deff 1 - φ
(1)
Models that transcend the Maxwell model have attempted to account for a number of other barrier
10.1021/ie020787x CCC: $25.00 © 2003 American Chemical Society Published on Web 05/01/2003
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Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003
Figure 1. Transport aqueous test apparatus.
characteristics that can affect the transport of an ion. These characteristics include membrane pore size, pore tortuosity, pore density (porosity), impurities in the membrane pores (primarily air or other gases), and general surface chemistry.14,15 More specifically, Cussler presented a model that can be used to predict the ratio of D0 to Deff assuming that the manner in which the membrane affects transport of the solute is due to the membrane tortuosity (τ) and the membrane volume (or void fraction, )12
D0 τ ) Deff
(2)
In eq 2, the tortuosity is defined as the actual pore length divided by the thickness of the membrane material.14 Using this relationship, if the membrane material does not influence the diffusion of solute ions, aside from physical tortuosity and porosity effects, the value of D0 determined from an experimentally determined Deff value should be identical to that found for the solute in the literature. For this study, empirical tests conducted in a small-scale batch transport test apparatus allowed the determination of effective rates of diffusion for specific cations at very low solute concentrations. Analysis of the impact of the stainless steel barrier material on aqueous cation diffusion was conducted by comparing experimentally determined D0 values to ionic D0 values found in the literature. Judging from physical characterization data for the Mott stainless steel membrane (large pore size), our premise was that bulk diffusion would dominate and the unmodified membrane would not have selectivity for aqueous cation separations.4 The results presented herein provide verification as to the validity of this hypothesis, as well as a baseline for future work in membrane surface modifications and their subsequent characterization via metal ion transport studies. Experimental Section Effective diffusion coefficients calculated from changes in feed and receptor cell cation concentration as a function of time were determined from radiotracer cation activities in both cells of the transport apparatus depicted in Figure 1. The stainless steel membrane material being examined was located in the bridge between the two cells. The stainless steel membrane material chosen for these studies was manufactured by Mott Metal Corporation (Farmington, CT) and is described by the manufacturer as a sintered 316L stainless steel membrane with a pore diameter between 2 and 3 µm. The membrane thickness was measured to be between 0.123 and 0.124 cm. In prior characteriza-
tion studies, we determined the average pore size (2.2 ( 0.5 µm), the average porosity (24.9 ( 5.2%), and the membrane tortuosity (1.3 ( 0.07).4,16 Experimentation was conducted to determine the transport characteristics of main group (Cs+ and Ca2+), lanthanide (Eu3+), and actinide (Am3+ and Pu4+) metal cations through the membrane material. A. Membrane Cleaning Process. Prior to testing, the membrane material was cleaned using a three-step process. The procedure consisted of solvent and oil removal in an ozone generator for 20 min, purication of the membrane surface with 25% HCl for 5 min, followed by a deionized (DI) water wash in an ultrasonic cleaner for 24 h. B. Chemicals and Radiotracers. Metal cations used for transport experiments were chosen according to charge (monovalent to tetravalent), the desire to test cations found in the LANL aqueous processing facility environment (actinides), and results from previously published alumina membrane transport studies.6,7 Transport tests were conducted with Cs+, Ca2+, Eu3+, Am3+, and Pu4+ as radiotracers in solutions with varying pH. The solution pH values ranged from 5 to 9 for cesium and calcium, from 2 to 3.5 for americium, and from 1 to 3 for plutonium. Radiotracer feedstock was stored in acidic solution, and the acidity of the stock feeds was taken into account during preparation of the experimental solutions. Europium transport experiments were conducted only at pH 2, as a limited amount of the radiotracer was available. Radiotracer concentrations were very low and varied between 10-7 and 10-12 M. Commercially available, reagent-grade chemicals were used in all experiments. C. Transport Apparatus and Procedure. Transport experiments were conducted in the transport apparatus shown in Figure 1. The apparatus used consisted of Teflon feed and receptor cells of equal volume joined by a bridge containing the vertically mounted stainless steel membrane disk. The bridge halves (also Teflon) were threaded into each cell wall, and the membrane was sealed between the bridge sections with Viton O-ring gaskets and held in place with a circular clamp. A Teflon friction fit cover was used to minimize evaporation of solution during experimentation. It should be noted that, if significant density differences exist between feed and receptor cells, a horizontally mounted diaphragm cell system is required for aqueous transport experiments similar to those described in this article. However, low cation concentrations and additional constraints (such as complete mixing and the elimination of bulk flow between the cells) gave rise to the assumption that density differences were very small and allowed the use of a vertically mounted diaphragm apparatus to generate accurate diffusion data.17 Prior to each transport experiment, a cleaned membrane was placed in the apparatus. Prior to addition of the radiotracer stock, both the feed and receptor sides of the transport experimental apparatus were filled with 100 mL of deionized water, and the pH was adjusted with dilute NaOH or HNO3. Finally, the calculated volume (typically 0.05-0.5 mL) of radiotracer stock was added to the feed cell, and the feed cell pH was again adjusted to the desired value. The pH’s of both cells were to be identical after all pH adjustments were completed. Aliquots were periodically removed from both the feed
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2855 Table 1. Cation Effective Diffusion Coefficient Summary
pH
average effective diffusion coefficient (10-7 cm2-s-1)
+1
5-9 5 7 9
8.7 ( 2.4 7.3 9.0 9.1
19
+2
5-9 5 7 9
6.0 ( 1.9 5.4 5.2 7.1
11
+3
2 3.5
6.0 ( 1.6 2.0 ( 1.6
4 3
+4
1-3 1 2 3
4.4 ( 1.2 4.8 4.3 4.0
12
2
4.9 ( 3.6
3
hydrated ionic radius (10-12 m)
ionic charge
cesium
25020,21
calcium
60020,21
americium
45122
plutonium
-
cation
europium
45822
+3
total no. of runs
|
V
|
dAr ∂A ) SDeff dt ∂x x)L
|
|
Af - Ar ∂A ∂A ) ) ∂x x)0 ∂x x)L L
(3) (4)
where V is the cell volume, S is the cross-sectional area of the membrane in contact with the feed and receptor solutions, Deff the cation effective diffusion coefficient, and L is the membrane thickness. Assuming further
(5)
Substituting eq 5 into eqs 3 and 4 and subtracting eq 4 from eq 3 results in eq 6
2SDeff d(Af - Ar) )(Af - Ar) dt VL
(6)
Solving eq 6 for the initial conditions (Af ) Af0 and Ar ) 0 at t ) 0) provides a relationship between both the feed and receptor cell cation activities and the cation effective diffusion coefficient, as shown by eq 7
(
)
Af0 VL ln ) Defft 2S Af - Ar
and receptor cells for analysis over the duration of each experiment, with the sample volume removed for each run depending on the activity of the radiotracer in solution. Solution activities were measured by counting the sample on either a NaI gamma counter (Packard NaI Gamma Counter model 5000 for Cs) or a liquid scintillation counter (Wallac Liquid Scintillation Counter model 1414 for Ca, Am, Pu, and Eu radiotracers). Solution pH was measured at the beginning and at the end of each experiment, and both the feed and receptor cells of the apparatus were continuously mixed with a magnetic stirrer at a constant rate of 300 rpm. Individual experimental testing periods ranged from 2 to 3 weeks. Standard procedures were developed for assembly and cleaning of the transport apparatus during and after each experiment, for membrane cleaning, and for transport test operation and sampling to ensure repeatability of cation transport experiments. Multiple transport experiments were conducted with each of the five cations (Table 1 lists the total number of runs conducted for each cation). D. Effective Diffusion Coefficient (Deff)/Activity (A) Relationships. Transport characteristic comparisons were performed by measuring the change in the radiotracer cation activity in both the receptor and feed cells as a function of time and then calculating an effective diffusion coefficient as described by Cussler.12 Assuming that cation transport within the membrane is by Fickian diffusion alone and that local equilibrium applies at the membrane interfaces with the feed and receptor, differential mass balances for the amount of cation in the feed (Af) and the receptor (Ar) are related to that present in the membrane (A) as described by eqs 3 and 4
dAf ∂A ) -SDeff V dt ∂x x)0
that the concentrations in the membrane are at quasisteady state leads to the generation of eq 5
(7)
As indicated by eq 7, data plotted with respect to time will have a slope that is equal to Deff and a zero intercept. However, an analysis of initial transport data identified small differences in the initial volumes of the receptor and feed cells (on the order of a few volume percent). These small cell volume differences likely produced bulk flow for a period of time at the beginning of each transport experiment, thereby skewing the slope and causing the intercept to have a value other than zero. The initial volume differences were caused by volume measurement error, the addition of acid or base to adjust pH, and the addition of stock radiotracer solution. From a mass balance, the volume differences between the feed and receptor vary in time as
-
dVf dVr ) )Q dt dt
(8)
in which the volumetric flow rate Q is related to the volume difference by
Q)
K (V - Vr) Scell f
(9)
The constant K, representing the permeability of the membrane to flow, was determined experimentally to be 0.422 cm2 h-1 as described in the Appendix.4 Equations 8 and 9 can be combined to give
-
d(Vf - Vr) 2K (V - Vr) ) dt Scell f
(10)
Equation 10 can then be integrated and rearranged to obtain
t)
( )
Scell ∆V0 ln 2K ∆V
(11)
which describes the time required for the initial volume difference ∆V0 to dissipate to ∆V. For a given ∆V, the ratio of the bulk and diffusion fluxes is given by
Jbulk LK∆V ) Jdiff ScellSDeff
(12)
With given experimental procedures, it seems likely that
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in all of our dilute experimental conditions, the published cation infinite-dilution diffusion coefficients were used as the basis of comparison. Results and Discussion
Figure 2. Determination of effective diffusion coefficient (Deff) for Cs at pH 7. The indicated regression line has a slope of 9.9 × 10-7 cm2/s and an intercept of -0.050 cm2.
initial differences in volume could have been significant. For example, assuming an average effective diffusion coefficient of 5 × 10-7 cm2 s-1, the cation flux from bulk flow would be less than 10% of the flux from diffusion if ∆V were less than approximately 0.2 mL. From eq 11, the time in hours required for bulk flow to dissipate to less than about 10% of the diffusive flux (i.e., for ∆V ) 0.2) depends on ∆V0 in milliliters as follows
t ) 37.3 + 53.4 log ∆V0
(13)
Thus, we estimate that bulk flow will be negligible after 37 h if ∆V0 ) 1 mL and after 75 h if ∆V0 ) 5 mL. Consequently, the least-squares data regressions used to determine Deff did not include data taken at times less than 48 h. Figure 2 illustrates the method used to determine Deff for Cs+ at an initial pH of 7.0. For this example, Deff ) 9.9 × 10-7 cm2 s-1, which is one of 19 Cs experiments included in the data summarized in Table 1. E. Comparing Experimental and Referenced Cation Diffusion Coefficients. The diffusion of an electrolyte is dependent on both the cation and the anion in the aqueous solution. More specifically, because of electrical neutrality, the rate of diffusion of an ion pair is a function of ion valance, concentration, and the diffusivities of both ions. To determine the appropriate D0 value, Cussler presented eq 14 (derived from the Nernst-Planck flux relationships), which estimates the diffusion coefficient from known diffusivities for ions with charges of 1 or larger12,18
D0 )
D01D02(z12c1 + z22c2) D01z12c1 + D02z22c2
(14)
where zi is the ion valance and ci is the ion concentration in solution (M). For the aqueous conditions used in our experiments, eq 14 was used to determine the reference D0 value with which all experimentally determined D0 values were compared. For example, with acidic experiments, nitric acid was used to adjust the pH from the initial pH (5.8) of the DI water. For an experiment using Ca2+ at the pH of 5, the concentrations of calcium and nitrate were determined to be 2.50 × 10-11 and 8.46 × 10-6 M, respectively. Using eq 14, the infinite-dilution diffusion coefficient was determined to be 7.9 × 10-5 cm2 s-1, which is identical to the coefficient of the cation alone. Because the less common cation coefficient dominated
A. Effective Diffusion Coefficients for Radiotracer Cations. Effective diffusion coefficients were determined using the data analysis methods previously described. The required concentration of the cation under investigation varied depending on the counting efficiency and sensitivity of the radioactivity being monitored. Table 2 lists the initial feed cell radiotracer concentration and activity for each set of transport experiments.19 Table 1 provides a summary of the radiotracer cation overall effective diffusion coefficient results, the average effective diffusion coefficient at each pH for a given cation, the hydrated ionic radius, and the ionic charge. As can be seen from the data, where the pH has been varied, a small difference in the average effective diffusion coefficient is observed. However, these data are not statistically different, so the overall effective diffusion coefficient used for calculations is the average over the entire pH range for each cation (first row listed for each cation). Statistical error is reported for a 95% confidence interval, as multiple transport experiments were conducted at each initial pH.23,24 With the exception of Eu, effective diffusion coefficient statistical variance ranged from 27% for Am to 32% for Ca. The europium effective diffusion coefficient error was higher predominantly because only three experiments were conducted. Experimental errors in the determination of the stainless steel filter structural parameters were included in the transport data error analysis. B. Cation Speciation during Transport. Cation transport experiments were conducted at varying initial pH. Transport experiments completed with 137Cs and 45Ca as the radiotracers were conducted at near neutral conditions (at initial pH’s of 5, 7, and 9). Alkali metal hydroxides are strong bases; thus, all cesium was considered to be monovalent (Cs+) under these experimental conditions.25 Given the low calcium concentration and the system pH, it was assumed that all calcium was in solution as the divalent metal ion.26 Transport experiments completed with 152Eu as the radiotracer were conducted only at pH 2 because of the limited amount of the material. Given the very small europium concentration and the equilibrium constant for the reaction of the trivalent cation with hydroxide, it was assumed that all europium was present in solution in the trivalent state.26 The transport experiments completed with 241Am as the radiotracer were conducted under acidic conditions (at initial pH’s of 2, 3.5, and 5). The effective diffusion coefficient for pH 3.5 was approximately half that at pH 2, and no cation transport was evident at pH 5. The explanation for this phenomenon is that, as the hydroxide ion concentration increases (increase in pH), a larger percentage of the Am precipitates as insoluble hydroxide, limiting or eliminating americium transport across the stainless steel membrane. At pH 2, all americium was assumed to be in solution as the trivalent metal ion (Am3+) in that the concentration of Am(OH)+ was determined to be far below the total Am concentration, given the trivalent cation and hydroxide reaction equilibrium constant.26
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2857 Table 2. Radiotracer Activity Specifications cation
radiotracer cation concentration (M)
feed cell activity (counts m-1 mL-1)
radioactivity monitored
maximum caustic or acid concentration
2.5 × 10-11 1.1 × 10-7 9.3 × 10-12 2.0 × 10-7 5.3 × 10-7
4.4 × 105 2.8 × 106 9.8 × 103 1.1 × 103 1.8 × 104
β γ β R R
1 × 10-5 M NaOH 1 × 10-5 M NaOH 0.01 M HNO3 0.01 M HNO3 0.1 M HNO3
45Ca 137Cs 152Eu 241Am 239Pu
Table 3. Comparison of Experimental D0 Values with D0 Values from Referenced Literature experimental D0 (10-6 cm2/s)
cation Cs+ Ca2+ Eu3+ Am3+ Pu(OH)3+
average D0 from the experimental literature from 2P -7 2 Deff (10 cm /s) (cm2/s 10-6) porositya 8.7 ( 2.4 6.0 ( 1.9 4.9 ( 3.6 6.0 ( 1.6 4.4 ( 1.2
10.028 7.929 5.1830 6.2525 N/A
4.6 ( 1.6 3.1 ( 1.2 2.5 ( 1.9 3.1 ( 1.1 2.2 ( 0.8
from DI water porosityb 7.0 ( 2.613 4.5 ( 2.0 3.7 ( 2.9 4.5 ( 1.8 3.3 ( 1.3
a 2-Propanol-determined porosity of 24.9%. b Deionized-waterdetermined porosity of 17% (∼8% pore volume air).
Transport experiments completed with 239Pu as the radiotracer were also conducted under acidic conditions (at initial pH’s of 1, 2, and 3). Experimentation above pH 3 was avoided, as hydrolysis begins to precipitate Pu from solution at pH 3 and higher. However, when researching stability constants for the hydrolysis of plutonium, it was discovered that, even in these highly acidic conditions, a single hydroxide is associated with Pu4+ to give Pu(OH)3+.25,26 In addition, at the Pu concentrations (∼10-8 M) for our system, additional hydrolysis does not occur below pH 3, and precipitation of the colloidal plutonium hydroxide does not occur below pH 7.5.27 Given this information, it was then assumed that all plutonium in solution was as Pu(OH)3+. C. Discussion. Transport experiments were conducted to determine whether the stainless steel membrane hinders the diffusion of the selected cations under dilute aqueous conditions. Ideally, if the membrane does not affect transport, the diffusion rate through the solution in the membrane pore will be identical to that in solution without the membrane present. To determine the effect of the membrane on cation transport, the experimental infinite-dilution diffusion coefficient was determined and compared to the infinite-dilution diffusion coefficient found in the literature. The method used to calculate the infinite-dilution diffusion coefficient from transport data in this study involves using the experimental diffusion coefficient (Deff) results for each cation, the actual membrane porosity (determined using 2-propanol as the solvent), and the tortuosity. Equation 2 shows the relationship between Deff, D0, and the membrane characteristics. In Table 3, D0 estimates (fourth column) are presented and include the propagation of error associated with the tortuosity, porosity, and experimental effective diffusion coefficient. Table 3 also includes bulk infinite-dilution diffusion coefficients found in the literature for all cations with the exception of Pu(OH)3+, which has not been previously published. Results indicate that the infinite-dilution diffusion coefficients from the referenced literature are consistently higher than the range calculated for each cation experimental infinite-dilution diffusion coefficient (determined from Deff). Because this result was not consistent with our hypothesis, an investigation was initiated into identifying the cause. An evaluation of
membrane preparation procedures revealed that we had failed to remove all of the air from the test membranes, essentially reducing the pore volume available for transport and, in turn, reducing the porosity of the material. The analysis is presented below. Error Caused by Air in Stainless Steel Membrane Pores. Prior to initiating a transport experiment, each assembled apparatus was leak tested, and the membrane was submerged in water (DI) for 1 day. After 1 day, the radiotracer stock was added to water, and the transport experiment was initiated. Other than the leak test during the presoak step, no precautions were taken to guarantee that all solution or gases in the membrane were displaced with water, and it was possible that some air might have remained in the membrane, essentially reducing the pore volume available for transport (reducing the porosity). To check this possibility, gravimetric tests were conducted to determine the porosity of membranes soaked only in water. The modified porosity was determined to be 17.0 ( 4.8%, from which the conclusion was made that there was approximately 8% air (percentage of total volume including the membrane and void volume) present in the DI-water-soaked stainless steel membranes (the porosity measured using 2-propanol minus that measured using water).4 Judging from eq 2, the difference in porosity contributed significant error in the calculation of D0 and biased the coefficient below the value that would have been determined otherwise. To test this hypothesis, a set of three Cs transport experiments (pH 7) was conducted with membranes that were saturated with DI water by a procedure that was developed to ensure that a high percentage of the air trapped in the membrane was removed and replaced with DI water. The procedure involved submerging the membrane in 2-propanol for 3 days and then installing the membrane in a transport cell, where 200 mL of DI water was then gravity fed through the membrane to remove the watermiscible solution. Results from the Cs transport tests using the water-saturated membranes were that the average Deff increased as compared to results from tests using membranes soaked only in water. In addition, the average calculated D0 value (8.7 ( 2.4 × 10-6 cm2 s-1) was identical to the D0 value from the literature (10 × 10-6 cm2 s-1).28 The conclusion drawn from this analysis was that the D0 values calculated from the transport experiments performed with membranes soaked only in DI water were based on an inappropriate porosity (porosity determined for membranes soaked in 2-propanol where the air had been removed). For the majority of the transport experiments (where membranes were soaked only in water), the porosity value of 17.0% would be more appropriate. Infinite-dilution diffusion coefficients recalculated using eq 2 and 17% for (see righthand column of Table 3) were all within the range of the literature values, with the exception of that for Ca (low by ∼11%). This information strongly supports the hypothesis that diffusion is the rate-controlling trans-
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port mechanism of solutes for these stainless steel membranes. Elimination of Resistances from Electrostatic Wall Effects and Apparatus Bridge. In addition to diffusion effects caused by air entrapment in the stainless steel membranes, we considered the effects of membrane surface charge or poorly mixed solution located in the bridge of the transport apparatus. The analysis of these possible problems is presented below. Mulder and Schmid claim that, if the pore diameter is greater than several hundred times the diameter of the diffusing molecule, it can be assumed that wall effects do not have a significant influence on the diffusion of the ionic species in solution.31,32 In the case of the stainless steel membranes, the ratio of the membrane pore diameter to the diameter of the largest molecule under investigation (calcium, which has a hydrated molecular radius of 6.0 × 10-4 µm) was estimated to be 1700. In addition, in solution, as the ions flow along the charged surface of a membrane, the distance of influence of a charged surface on an aqueous ion can be estimated by the Debye length, which, in turn, can be estimated from the total ion concentration and ionic valence using33
0rRT
λD )
F
∑i zi ci
2
(15)
2
The Debye length for the conditions of the transport experiments outlined in this document was estimated to be on the order of nanometers. Given the experimentally determined mean pore diameter for the stainless steel membrane (2.2 µm), the pore diameter is approximately 2 orders of magnitude larger than the Debye length. Given both the large ratio of the pore diameter to the hydrated cation diameter and the estimated Debye length information, it was assumed that wall effects due to electrochemical conditions of the stainless steel membrane transport experiments were not significant. A second concern was the result of an inspection of the experimental transport apparatus, which revealed that, in the horizontal, cylindrical sections of the bridge on either side of the membrane, the solution might be possibly poorly mixed, thus contributing to the total diffusive resistance. A series of transport experiments was completed with dilute 14C-labeled sucrose to estimate how the sucrose effective diffusion coefficient changed when the membrane thickness or resistance was doubled, holding all other variables constant. From these data, the relative resistance in the cylindrical bridge can be estimated.12,30 If the poorly mixed regions in the bridge on either side of the membrane contribute an additional mass-transfer resistance, the total resistance Rn ) Deff/L calculated from eq 8 would include the effective resistance of n membranes (i.e., Dm,eff/nLm), as well as a resistance through the bridge represented by a mass-transfer coefficient (kB) as follows
Rn )
1 nL + k B Dm
(16)
As a result, kB and Dm,eff/L can be calculated from the resistances measured for single- (R1) and double- (R2)
membrane systems as follows
1 ) 2R1 - R2 kB
(17)
L ) R2 - R1 Dm,eff
(18)
The resistances for the apparatus bridge (1/kB) and the membrane (L/Dm,eff) were determined to be approximately 7.4 × 103 and 3.5 × 105 s cm-1, respectively, from R1 (3.55 × 105 s cm-1) and R2 (7.03 × 105 s cm-1). This result verifies that 1/kB was relatively insignificant (by a factor of 50) compared to L/Dm,eff. With effective diffusion coefficient statistical variability greater than 27%, we can state that the resistance through any unstirred layers in the bridge did not significantly contribute to the total transport resistance of the system. Summary and Conclusions A series of transport experiments was completed with five cation solutes, Cs+, Ca2+, Eu3+, Am3+, and Pu [as Pu(OH)3+], for the purpose of testing the hypothesis that diffusion is the rate-controlling transport mechanism through the Mott stainless steel membrane material. Using the transport data generated and basic membrane characterization information, effective infinitedilution diffusion coefficients (D0) were generated for each cation. These data were then compared with referenced D0 values in water (no membrane) for the purpose of determining the impact of the membrane on cation transport across the stainless steel material. The results indicated that there were no statistically significant differences between the experimentally determined D0 values and the D0 values in water (found in the literature) for Cs+, Eu3+, and Am3+. The D0 value from referenced literature was slightly higher than the range calculated for the D0 for Ca2+ only. In summary, results from the comparison of infinitedilution effective diffusion coefficients determined experimentally and those found in the literature verify that molecular diffusion through stainless steel membrane pore structures is dominant and support the hypothesis that the unmodified membrane does not contribute to the removal of selected cations under aqueous conditions. In addition, the infinite-dilution diffusion coefficient for Pu [as Pu(OH)3+], which was not available in the literature for comparison to our results, was experimentally determined to be (3.3 ( 1.3) × 10-6 cm2 s-1. Safety Considerations All experimentation, radiotracer storage, and waste disposal was conducted in accordance with the Department of Energy and Los Alamos National Laboratory instructions and regulations. These instructions require that operations occur in appropriately ventilated hoods or gloveboxes with strict compliance with mandated personal protective equipment and equipment protection policies. Material accountability, containment, and waste disposal was determined on the basis of the material form (liquid, organic, corrosive, etc.) and quantities and type of radioactive material. Acknowledgment Funding was provided by the Department of Energy (DOE) and the Plutonium Stabilization and Scrap Recovery Program at Los Alamos National Laboratory.
Ind. Eng. Chem. Res., Vol. 42, No. 12, 2003 2859
The Los Alamos National Laboratory is operated by the University of California under Contract W-7405-ZNG36. Dr. J. Douglas Way was also supported by the DOE Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division under Grant DE-FG03-93ER14363.
Vf0 - Vf ) )
K Scell
∫0∆tVf dt
KVf0 Scell
[
( (
∫0∆texp - SKtcell
) Vf0 1 - exp -
Nomenclature A ) activity in solution in the membrane (cpm) Ar ) activity (cpm) in the receptor cell at time t (s) Af ) activity (cpm) in the feed cell at time t (s) Af0 ) activity (cpm) of the feed at t ) 0 Ci ) ion concentration (mol m-3) ci ) ion concentration (mol L-1) D ) general diffusion coefficient of a solute in a given solvent (cm2 s-1) D0 ) diffusion coefficient with no membrane (cm2 s-1) Deff ) measured diffusion coefficient with membrane (cm2 s-1) Dm ) effective diffusion coefficient of membrane alone (cm2 s-1) F ) Faraday constant (96 500 C mol-1 electron-1) hcell ) height of feed or receptor cell (cm) J ) cation- or C14-labeled flux (A cm-2 s-1) Jbulk ) flux of solution by bulk flow (mL cm-2 h-1) Jdiff ) flux of solution by Fickian diffusion (mL cm-2 h-1) K ) bulk flow constant (cm2 h-1) kB ) bridge mass-transfer coefficient (cm s-1) L ) membrane thickness (cm) M ) molar concentration (mol L-1) R ) universal gas constant (8.3143 J mol-1 K-1) Rn ) total resistance measured experimentally with n membranes (s cm-1) S ) cross-sectional area of membrane in contact with the cell solutions (cm2) Scell ) cell cross-sectional area (cm2) T ) temperature (K) t ) time (s or h) Vi ) volume of solution in the feed (i ) f) or receptor (i ) r) cell (cm3) Vf0 ) volume of solution in the feed cell at t ) 0 (cm3) x ) distance across the membrane (cm) zi ) ion valence ∆t ) time interval for a volume of solution to flow out of the feed cell (h) ∆V ) difference in solution volume between feed and receptor cells (mL) ∆V0 ) difference in solution volume between feed and receptor cells at t ) 0 (mL) ) porosity (void volume/total volume - unitless) 0 ) permittivity of vacuum (8.85 × 10-12 C V-1 m-1) φ ) sphere void fraction λD ) Debye length (m) τ ) tortuosity
Appendix The bulk flow constant K was determined by measuring the time interval ∆t required for a known volume of water to drain from a single cell connected to the membrane remaining in the bridge section of the apparatus. According to eqs 8 and 9 written without cell 2, Vf decreases from its initial value Vf0 according to
(
Vf ) Vf0 exp -
)
K∆t Scell
(A1)
The cumulative volume of fluid that drains from the cell (Vf0 - Vf) during the time interval ∆t is
) )]
K∆t Scell
dt (A2)
which can be rearranged to give K in terms of the experimental observations as follows
K)
( )
Scell Vf0 ln ∆t Vf
(A3)
Finally, for Vf0 ) 150 mL and at t ) 51 h and Vf ) 50 mL in our cell (Scell ) 19.6 cm2), we calculated the bulk flow constant (K) to be 0.422 cm2 h-1. Literature Cited (1) Data Sheet, Mott Metal Corporation, Farmington, CT. Unpublished results. (2) Furneaux, R. C.; Rigby, W. R.; Davidson, A. P. The Formation of Controlled-Porosity Membranes from Anodically Oxidized Aluminum. Nature 1989, 337, 147. (3) Li, F.; Zhang, L.; Metzger, R. M. On the Growth of Highly Ordered Pores in Anodized Aluminum Oxide. Chem. Mater. 1998, 10, 2470. (4) Ames, R. L.; Bluhm, E. A.; Bunge, A. L.; Abney, K. D.; Way, J. D.; Schreiber, S. B. Physical Characterization of 0.5 Micron Cutoff Sintered Stainless Steel Membranes. J. Membr. Sci. 2003, 213/1-2, 13. (5) Hulteen, J. C.; Jirage, K. B.; Martin, C. R. Introducing Chemical Transport Selectivity into Gold Nanotubule Membranes. J. Am. Chem. Soc. 1998, 120, 6603. (6) Bluhm, E. A.; Bauer, E.; Chamberlin, R. M.; Abney, K. D.; Young, J. S.; Jarvinen, G. D. Surface Effects on Cation Transport across Porous Alumina Membranes. Langmuir 1999, 15, 8668. (7) Bluhm, E. A.; Schroeder, N. C.; Bauer, E.; Fife, J. N.; Chamberlin, R. M.; Abney, K. D.; Young, J. S.; Jarvinen, G. D. Surface Effects on Metal Ion Transport Across Porous Alumina Membranes. 2. Trivalent Cations: Am, Tb, Eu, and Fe. Langmuir 2000, 16, 7056. (8) Fick, A. E. U ¨ ber Diffusion. Poggendorff’s Ann. Phys. Chem. 1855, 94, 59. (9) Cussler, E. L. Multicomponent Diffusion; Elsevier Scientific: New York, 1976. (10) Reid, R. C.; Sherwood, T. K.; Prausnitz, J. M. Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977. (11) Shelby, J. E. American Society for Metals, Handbook of Gas Diffusion in Metals and Melts; American Society of Metals International: Materials Park, OH, 1996. (12) Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems, 2nd ed.; Cambridge University Press: Cambridge, U.K., 1997. (13) Maxwell, J. C. Treatise on Electricity and Magnetism; Clarendon Press: London, 1873; Vol. I. (14) Falla, W. R.; Mulski, M.; Cussler, E. L. Estimating Diffusion through Flake-Filled Membranes. J. Membr. Sci. 1996, 119, 129. (15) Aris, R.; A Problem of Hindered Diffusion. Arch. Ration. Mech. Anal. 1986, 95(2), 83. (16) Ames, R. L. Stainless Steel Membrane Characterization and Aqueous Transport Studies. M.S. Thesis, Colorado School of Mines, Golden, CO, 2001. (17) Toor, H. L. Convection and Transport in an Inclined Diaphragm Cell. Ind. Eng. Chem. Fundam. 1967, 6 (No. 3), 454. (18) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications; John Wiley and Sons: New York, 1980. (19) Parrington, P. R.; Knox, H. D.; Breneman, S. L.; Baum, E. M.; Feiner, F. Nuclides and Isotopes, 15th ed.; General Electric Company and KAPL, Inc.: Schenectady, NY, 1996. (20) Emsley, J. The Elements, 2nd ed.; Clarendon Press: New York, 1991.
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(29) Robinson, R. G.; Stokes, R. H. Electrolyte Solutions; Butterworth: London, 1960. (30) Latrous, H.; Ammar, M.; M′Halla, J. Determination of SelfDiffusion Coefficients of 152Eu3+ Ion in Aqueous Solutions. Radiochem. Radioanal. Lett. 1982, 55/1, 33. (31) Mulder, M. Basic Principles of Membrane Technology, 2nd ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1996. (32) Schmid, G. Electrochemistry of Capillary Systems with Narrow Pores, I. Overview. J. Membr. Sci. 1998, 50, 151. (33) Schaep, J.; Vandecasteele, C.; Peeters, B.; Luyten, J.; Dotrotremont, C.; Roels, D. Characteristics and Retention Properties of a Mesoporous γ-Al2O3 Membrane for Nanofiltration. J. Membr. Sci. 1999, 163, 299.
Received for review October 4, 2002 Revised manuscript received March 14, 2003 Accepted March 19, 2003 IE020787X
