Anal. Chem. 1997, 69, 3293-3298
Measuring Column Void Volumes with NMR T. W. Perkins, T. W. Root, and E. N. Lightfoot*
Department of Chemical Engineering, University of WisconsinsMadison, Madison, Wisconsin 53706
A novel method for measuring resin porosities and column void volumes with fluorine NMR has been developed. In situ measurements of the void volumes accessible to an array of fluorinated probe molecules are used to characterize the pore size distribution of the media. Application of this simple procedure is demonstrated for a commercially packed column and several bulk resins. The porosity distributions obtained by this technique are similar to those obtained by size exclusion chromatography. Unlike chromatographic tracer studies, however, this method does not require packed columns. Characterization of pore size distributions is essential for modeling and designing chromatographic separations and a variety of other commercially important processes, including catalysis, filtration, and oil recovery. Models of these packedbed operations are used to improve the design of processes and to facilitate their extension to other conditions. These models require not only inter- and intraparticle void fractions but also parameters that are dependent on porosities, such as effective diffusivities and rate constants for adsorption, desorption, and mass transfer.1,2 For chromatographic separations, information on pore size distributions can be used to improve the selectivity of a process or to predict the effect of surface modifications on column performance. For example, size-exclusion columns with bimodal pore distributions have been prepared and used to extend the range of molecular weights that can be separated in a single column.3 In addition, columns in which the binding chemistry is varied with pore size have been shown to enhance the resolution of both large and small components in a mixture, increasing the separation efficiency.4 Modification of surface chemistry to change the substrate’s binding characteristics can influence the effective size of the pores. Adsorption of large molecules like proteins or other biopolymers can reduce the effective pore size as well, as shown by the increase in diffusion resistance caused by enzyme immobilization.5 Information on the porosity of the media is needed in order to understand the impact of surface derivatization and adsorption on column performance. Several methods for measuring a pore size distribution (PSD) are available. For chromatographic media, PSD values are typically obtained by tracer experiments, in which the retention times of various probe molecules are used to estimate the (1) Cui, L. C.; Schweich, D.; Villermaux, J. AIChE J. 1990, 36, 86-92. (2) Cui, C. L.; Authelin, J. R.; Schweich, D.; Villermaux, J. Chem. Eng. Sci. 1990, 45, 2611-2617. (3) Northrop, D. M.; Scott, R. P. W.; Martire, D. E. Anal. Chem. 1991, 63, 1350-1354. (4) Smigol, V.; Svec, F.; Frechet, J. M. J. Anal. Chem. 1994, 66, 4308-4315. (5) Wang, Y. J.; Wu, T. C.; Chiang, C. L. AIChE J. 1989, 35, 1551-1554. S0003-2700(97)00271-0 CCC: $14.00
© 1997 American Chemical Society
accessible void volume.6-9 These experiments are simple and convenient, but there are some disadvantages: (1) the resin must be packed into a column and (2) any adsorption of the probe molecule will cause the accessible void volume to be overestimated. Two other common methods are mercury porosimetry10-12 and nitrogen adsorption or desorption with application of the BET isotherm.12 Both of these methods generally work well for rigid materials that maintain their pore structures when dry. Unfortunately, many liquid chromatography resins swell when wetted.6 Measuring the PSD of dry resin, therefore, provides little information on the void volumes of the swollen material actually present in chromatography columns. A promising method for obtaining pore size distribution data for gels and other porous materials utilizes NMR spectroscopy and magnetic resonance imaging.13-19 With most of these techniques, the solvent’s rate of relaxation is measured. Since the rate of relaxation of the fluid in the bulk liquid differs from that at the pore surfaces, models relating the rate of relaxation to the surface to volume ratio of the pores can be evaluated to yield pore size distribution information. These NMR and MRI relaxation studies provide useful information on the structure of porous materials, but interpretation of the results is model dependent and can be difficult. In this paper, a simpler NMR technique is proposed. With this method, the porous resin is equilibrated with a solution containing fluorinated probe molecules of various sizes. A spectrum of the supernatant solution is compared to that of the equilibrated resin, and, from the change in relative peak areas, the difference in accessible void volumes can be calculated. Like the other NMR techniques, this method is nondestructive, the pore size distribution of the resin can be obtained under the same conditions used during chromatographic separations, and the measurements can be made with packed columns or with raw media. In addition, bias due to adsorption can be reduced, as (6) Jerabek, K. Anal. Chem. 1985, 57, 1598-1602. (7) Jerabek, K. Anal. Chem. 1985, 57, 1595-1597. (8) Guan, H.; Guiochon, G. J. Chromatogr. A 1996, 731, 27-40. (9) Warren, F. V., Jr.; Bidlingmeyer, B. A. Anal. Chem. 1984, 56, 950-957. (10) Mishra, B. K.; Sharma, M. M. AIChE J. 1988, 34, 684-687. (11) Park, C.-Y.; Ihm, S.-K. AIChE J. 1990, 36, 1641-1648. (12) Mikijelj, B.; Varela, J. A.; Whittemore, O. J. Ceram. Bull. 1991, 70, 829831. (13) Davies, S.; Kalam, M. Z.; Packer, K. J.; Zelaya, F. O. J. Appl. Phys. 1990, 67, 3171-3176. (14) Bhattacharja, S.; D’Orazlo, F.; Tarczon, J. C.; Halperlin, W. P.; Gerhardt, R. J. Am. Ceram. Soc. 1989, 72, 2126-30. (15) Chen, S.; Liaw, H.-K.; Watson, A. T. J. Appl. Phys. 1993, 74, 1473-1479. (16) Borgia, G. C.; Brown, R. J. S.; Fantazzini, P. J. Appl. Phys. 1996, 79, 36563664. (17) Gladden, L. F.; Hollwand, M. P.; Alexander, P. AIChE J. 1995, 41, 894906. (18) Merrill, M. R. AIChE J. 1994, 40, 1262-1267. (19) Chen, S.; Qin, F.; Watson, A. T. AIChE J. 1994, 40, 1238-1245.
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discussed below, and, because this technique uses an equilibrium measure, there are no confounding kinetic effects. This spectroscopic technique is discussed in more detail in the following section. Application of this method for the characterization of the pore size distributions of several chromatographic resins is then demonstrated, and the results are compared to those obtained by size-exclusion chromatography. THEORY With 19F NMR, the signal observed following excitation of a fully relaxed sample is proportional to the number of fluorines present. Thus, the spectrum for a sample containing a fluorinated probe and a small fluorinated reference compound will have two peaks with areas that are proportional to the respective 19F concentrations. For a chromatography column equilibrated with this solution, the small reference compound will fill the entire void volume of the column, while the larger probe penetrates only a fraction of the pores. The peak areas now will be proportional to the product of the 19F concentrations and the accessible volumes. From the spectrum of the supernatant solution and that of the equilibrated resin, therefore, the fraction of the resin’s void volume accessible to the fluorinated probe can be calculated:
( ) Vprobe Vref
column
)
( ) /( ) Aprobe Aref
solution
Aprobe Aref
(1)
column
By repeating these simple measurements with probe molecules of different sizes, the relative void volume accessible to each probe can be found. Void volumes can be measured by NMR even under conditions that lead to adsorption of the probe molecules. Signals from bound probes and those in solution are distinguishable by their characteristic rates of relaxation. Bound probes have time constants for spin-lattice relaxation that are approximately an order of magnitude greater than those of probes in solution.20 Thus, with short pulse repetition times, the signal from bound material can become saturated, even while the material in solution is fully relaxed before each pulse. In addition, the spin-spin relaxation time constants, which govern the rate at which the observed signal loses coherence, are much smaller for bound probes.20 Thus, signals from bound probes give short, broad peaks that create a rolling baseline under the sharper peaks produced by the probes in solution, making it possible to identify the contribution to the signal from the molecules in solution. As a result, the NMR technique for measuring void volume distributions is useful even with adsorption of the probe molecule. Significant adsorption may, however, reduce the effective size of the pores and limit the accessible volume available for probes in solution. To express the accessible void volume as a function of the probe size, correlations between probe size and molecular weight are needed. Two classes of probe molecules were used: dextrans and proteins. For dextran chains, the mean radius of gyration was calculated by using the Mark-Houwink equation relating viscosity to molecular weight, (20) Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy; Oxford University Press: New York, 1991.
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[η] ) ΚΜa
(2)
and a relationship between viscosity and mean radius of gyration,21
[η] ) Φso R3η
( ) 〈s2〉 3/2 o M
(3)
where parameters K ) 0.0978 mL/g, a ) 0.50, and Φso ) 3.67 × 1024 mol-1. An a of 0.5 corresponds to conditions where the chain length has its unperturbed value,22 giving Rη a value of 1. Using a probe diameter that is twice the mean radius of gyration gives
d ) 0.60M0.5
(4)
where d is the diameter in angstroms. For proteins, a similar expression relating molecular weight and size is available:23
d ) 1.42M1/3
(5)
Thus, globular proteins generally have a more compact structure than dextran chains of the same molecular weight. The relative accessible void volumes can be expressed in terms of the column’s interstitial porosity, b, the probe-dependent intraparticle porosity, p,i, and the maximum intraparticle porosity, p,max:
Vprobe b + (1 - b)p,i ) Vref b + (1 - b)p,max The maximum intraparticle porosity can be obtained from NMR by comparing the small probe areas in the supernatant and resin spectra. Once the maximum intraparticle porosity is known, the interstitial porosity can be found by applying this NMR technique to a probe molecule that is large enough to be excluded from all intraparticle pores. Both of these parameters can also be found using size-exclusion chromatography (SEC). With values for these parameters, the relative accessible void volumes for different species can be used to determine the variation in the intraparticle porosity with probe size. The intraparticle porosity distribution can, in turn, be used to describe the pore size distribution of the resin. Interpretation of SEC or NMR data in terms of pore structure is complicated by the need to account for steric exclusion effects that reduce the volume accessible to the probe molecule. One method for addressing these exclusion effects is to model the structure of the pore network.6,7,24-26 Data for accessible volume as a function of probe size can then be converted to pore size distributions, as demonstrated by Scott and Knox.27,28 (21) Young, R. J.; Lovell, P. A. Introduction to Polymers, 2nd ed.; Chapman & Hall: New York 1991; pp 151-166. (22) Brandrup, J.; Immergut, E. H. Polymer Handbook, 3rd ed.; John Wiley & Sons: New York 1989. (23) Creighton, T. E. Proteins: Structures and Molecular Properties, 2nd ed.; W. H. Freeman and Co.: New York, 1984. (24) Ackers, G. K. Adv. Protein Chem. 1970, 24, 343-446. (25) Casassa, E. F. J. Phys. Chem. 1971, 75, 3929-3939. (26) Giddings, J. C.; Kucera, E.; Russell, C. P.; Meyers, M. N. J. Phys. Chem. 1968, 72, 4397-4408. (27) Knox, J. H.; Scott, H. P. J. Chromatogr. 1984, 316, 311-332. (28) Knox, J. H.; Ritchie, H. J. J. Chromatogr. 1987, 387, 65-84.
A second approach that eliminates the need to impose a specific pore model has been developed by Halasz and Martin.29 This empirical method assumes that each probe molecule has an exclusion limit that marks the boundary between completely inaccessible and completely accessible pores. Due to steric effects that hinder access of the probe into pores, the exclusion limit is 2-3 times the pore diameter. The pore size distribution can be found by differentiating the porosity versus probe size data and shifting the resulting curve by a factor of 2-3 to make the mean pore size match the value determined by other techniques. Although the mean pore size can be well described, neglecting the pore size dependence of probe exclusion results in reported pore size distributions that are too broad.27 In this paper, we employ another empirical approach that is also independent of specific models of pore geometry. This fitting function developed by Ackers30 is used for column calibrations and comparing the performance of different columns.31 The void volume is assumed to consist of a normal distribution of penetrable volume elements:30,32
Φ(d) )
(
)
-(d - B)2 A exp 2σ2 σx2π
(7)
where Φ(d) is the void volume accessible only to probes of diameter d. Integrating this volume distribution function for pores accommodating probes larger than di gives the accessible porosity,
p,i )
( )
∫ Φ(x) dx ) A2 erfc ∞
di
di - B σx2
(8)
and the maximum intraparticle porosity is given by
p,max )
∫ Φ(x) dx ) A ∞
0
(9)
With data for p,max and p,i versus di, the values of A, B, and σ are readily found. Hence, from simple NMR measurements of the relative accessible void volume as a function of probe size, the variation in intraparticle porosity with probe size can be found. This information can then be used with a simple description of the pore size distribution function to characterize the porosity of the chromatography resin. This technique is demonstrated for several chromatography resins in the following sections. EXPERIMENTAL SECTION Materials. Several chromatographic resins from Pharmacia Biotech (Uppsala, Sweden) were used in this study, including Butyl Sepharose 4 Fast Flow, and Sephadex resins G-100, G-75, G-50, and G-25. Sodium fluoride (Sigma, St. Louis, MO) was used as the reference compound, and this small molecule was assumed to penetrate all of the pores. The larger probes included 19F-labeled tryptophan, 19F-labeled cytosine, fluorinated bacitracin, and the following fluorinated proteins: conalbumin, ovalbumin, and hen egg white lysozyme (Sigma). These fluorinated com(29) Halasz, I.; Martin, K. Angew. Chem., Int. Ed. Engl. 1978, 17, 901-908. (30) Ackers, G. K. J. Biol. Chem. 1967, 242, 3237-3238. (31) Nozaki, Y.; Schechter, N. M.; Reynolds, J. A.; Tanford, C. Biochemistry 1976, 15, 3884-3890. (32) Harlan, J. E.; Picot, D.; Loll, P. J.; Garavito, R. M. Anal. Biochem. 1995, 224, 557-563.
pounds were also used as size-exclusion chromatography probes, as were dextrans of well-defined molecular weights (Polysciences, Inc., Warrington, PA). Bacitracin and the proteins were fluorinated by the method of Fanger and Harbury.33 Protein was dissolved in 0.5 M Na2HPO4 (pH ) 8.5) at a concentration of 15-20 mg/mL. Excess S-ethyl thiotrifluoroacetate (Lancaster, Windham, NH) was added, and the pH of the solution was kept at 8.0 ( 0.2 with the addition of 1.0 M sodium hydroxide. The solution was passed through a 0.45 µm filter, dialyzed against 0.02 M sodium phosphate buffer (pH ) 8), and then dialyzed against 18 MΩ deionized water three times before being lyophilized. Incorporation values ranged from 0.14 to 6.3 mmol of 19F per gram of labeled protein. NMR Instrumentation and Procedures. The NMR spectroscopy was done on a Bruker Avance DMX 400 wide-bore spectrometer located at the Nuclear Magnetic Resonance Facility at Madison. For the Butyl Sepharose column measurements, an imaging probe equipped with a 25 mm 19F coil was used for both the solution calibrations and the column void fraction measurements. For the Sephadex resin studies, the solution and resin samples were run in 5 mm tubes, and so a more sensitive 5 mm 1H/19F combination probe was used instead. The NMR samples were prepared by dissolving fluorinated protein in 0.1 M sodium phosphate buffer (pH ) 6.8) containing 0.015 M NaF. For lysozyme, conalbumin and ovalbumin concentrations of 15 mg/mL were used. For bacitracin, 19F-labeled tryptophan, and 19F-labeled cytosine, similar fluorine levels could be achieved at concentrations of only 5 mg/mL. For each resin/supernatant pair, a small amount of resin was placed in a 5 mm NMR tube. The amount of resin needed to produce 0.4 mL when swollen was roughly 21 mg for G-100, 27 mg for G-75, 37 mg for G-50, and 81 mg for G-25. Between 1.5 and 1.8 mL of labeled protein/sodium fluoride solution was passed through a 0.45 µm filter and then added to the resin. The tube was placed in a warm water bath for several hours to allow the resin to swell, and periodic gentle agitation was used to facilitate equilibration. After the resin was allowed to settle, the supernatant was withdrawn, leaving 0.4 mL of equilibrated resin. The supernatant was then filtered, and 0.4 mL was placed in another 5 mm NMR tube. For the preswollen Butyl Sepharose resin, the amount of slurry needed to make 0.4 mL of settled resin was placed in a beaker. The resin was allowed to settle, and then the supernatant was withdrawn. To replace the original buffer, 2 mL of phosphate buffer was added to the resin. The resin was allowed to settle, and then the supernatant was removed. This step was repeated twice, and then 2 mL of the fluorinated probe solution was added to the resin. After equilibration, 1 mL of supernatant was removed. The resin and remaining supernatant were then placed in a 5 mm NMR tube. After the resin settled, the supernatant was removed, and 0.4 mL was placed in a second NMR tube. For the commercially packed Butyl Sepharose column, equilibration was achieved by pumping 10 mL of protein solution through the 1 mL column at a flow rate of 1 mL/min. The FIDs were obtained with short, high-power pulses (tp e 10 µs, tip angle e 40°) and adequate recycle delay for full relaxation. A signal-to-noise ratio of at least 8 was obtained by using 1000 or more scans. The TFA and NaF peaks are separated by 44 ppm, so sweep widths of 65 ppm were used to prevent (33) Fanger M. W.; Harbury, H. A. Biochemistry 1965, 4, 2541-2545.
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Table 1. Summary of Interstitial and Maximum Intraparticle Porosities for Columns Packed with G-100, G-75, G-50, and G-25 and for a Commercially Packed Butyl Sepharose Column porosity
resin
bed volume (mL)
interstitial, settled, b
interstitial, packed, b
intraparticle, p,max
Sephadex G-100 Sephadex G-75 Sephadex G-50 Sephadex G-25 Butyl Sepharose
8.3 7.7 8.6 13.7 1.0
0.36 0.36 0.42 0.32 0.40
0.32 0.24 0.30 0.24 0.36
0.82 0.70 0.59 0.47 0.82
clipping the peaks. For samples containing 19F-labeled cytosine peaks, the sweep width was increased to 122 ppm. The transmitter frequency was centered, and the pulse length was adjusted to ensure uniform excitation over the frequency range of interest. At least 16K data points were acquired, giving a digital resolution of 0.03 ppm. Chromatography Procedures. The chromatography equipment consisted of a Waters 600E multisolvent delivery system (Millipore Corp., Milford, MA), with a Waters 700 satellite WISP for making automatic injections and a Waters 410 differential refractometer for monitoring the column effluent. Six columns were used: a commercially packed Butyl Sepharose column with a bed volume of 1 mL and five columns packed with Sephadex resins with bed volumes ranging from 7 to 14 mL. For the Butyl Sepharose column, 20 µL samples were injected, and the mobile phase flow rate was set to 0.5 mL/min. For the Sephadex columns, 50 µL samples were injected, and the mobile phase flow rate was increased to 1.0 mL/min. The mobile phase for all of the runs was 0.1 M sodium phosphate buffer (pH ) 6.8). Retention times, based on the center of mass of the peak, were measured for sodium fluoride and each of the dextrans and the fluorinated proteins. From the retention times of these molecules, the accessible void volume of the column as a function of probe size was calculated. For each resin, the void volume of the largest dextran (M ) 2 × 109) was used to estimate the interstitial porosity of the packed bed, and the intraparticle porosity was calculated from the void volume associated with NaF. RESULTS AND DISCUSSION The void volumes accessible to probes of various sizes were investigated for a series of resins. Each resin was equilibrated with a solution containing sodium fluoride and a fluorinated probe molecule. NMR spectra for the supernatant and equilibrated resin were obtained, and the void volume accessible to the probe (relative to the accessible volume of NaF) was calculated using eq 1. The void volumes of these resins were then characterized with size-exclusion chromatography. Retention times were measured for a series of well-characterized dextrans, as well as sodium fluoride and the fluorinated probe molecules. These data were used to calculate the volume accessible to each probe as well as the corresponding value of the intraparticle porosity. The interstitial and maximum intraparticle porosities are summarized in Table 1. Comparison of the NMR and SEC results is shown in Figure 1 for Sephadex G-100, G-75, G-50, and G-25. For these Sephadex 3296
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resins, good agreement between the two techniques was observed, indicating that this NMR technique provides a reasonable method for obtaining porosity distribution information for chromatographic resins. Two interesting features are evident. First, the SEC porosities for the dextrans are higher than those for proteins with similar radii of gyration, presumably because the dextran chains are more asymmetric. This sensitivity to the shape of the probe molecule has been previously noted: asymmetric proteins were found to have longer retention times than expected based on Stokes radii, possibly because of end insertion into gel pores.31 The difference in porosities observed for globular proteins and dextrans illustrates the difficulty in assigning appropriate characteristic lengths for use in column calibrations.26,34 This difficulty underscores the need for a simple method for measuring porosities so that the key parameters needed for a modeling column performance can be readily obtained. Comparing the SEC and NMR data for fluorinated proteins reveals a second interesting feature. The NMR data show a steep drop in intraparticle porosity with probe size, while the SEC data show a more gradual decline. In addition, the mean pore sizes from the NMR technique are slightly higher than those with SEC. These trends may reflect actual changes in the pore structure with mobile phase flow rate. Resin compression, and the corresponding reduction in void volume, increases with flow rate. Thus, the void volumes measured under static conditions with the NMR technique would be higher than those measured by SEC. Additional research is required to examine the significance of these trends. The Sephadex resins, which differ in the degree of crosslinking, show systematic variations in their porosities. To examine these variations, the NMR data for the Sephadex resins were fit using the empirical function of Ackers, and the distribution of accessible volume elements with probe size was found, as shown in Figure 2. An increase in cross-linking results in a decrease in the total porosity and a reduction in the size of the probe able to penetrate half of the available void volume. Earlier SEC data for these resins show the same trends.30 For the Butyl Sepharose system, the uncertainties associated with NMR experiments were much higher than those with the Sephadex resins. Part of the increase in uncertainty is due to the reduced sensitivity of the larger NMR coil used in these experiments. The signal-to-noise ratio was also lower because adsorption of the fluorinated probes lowered the fluid-phase concentrations. This reduced the signal coming from probes in solution, while increasing the contribution to the baseline roll from adsorbed material. Signal from adsorbed material, and the low solution phase concentrations, gave artificially high intraparticle porosities for several probes. These erroneous results show that, when adsorption is occurring, it may be desirable to modify the pulse sequence to improve suppression of signal from the bound material, to increase the number of scans to improve the signalto-noise ratio, or to use time domain analysis to improve quantitation of the liquid-phase peaks. The NMR and SEC results for Butyl Sepharose are shown in Figure 3. The agreement between the bulk resin and prepacked column results indicates that the NMR procedure for obtaining pore size distribution information works with unpacked resins as (34) Le Maire, M.; Aggerbeck, L. P.; Monteilhet, C.; Andersen, J. P.; Moller, J. V. Anal. Biochem. 1986, 154, 525-535.
Figure 1. Comparison of intraparticle porosities from SEC and NMR as a function of probe size for Sephadex resins: G-25, G-50, G-75, and G-100. The lines shown are cumulative pore distributions given by eq 8, with parameters obtained by least-squares fitting to the NMR data.
Figure 2. Accessible void volume distributions for a series of Sephadex resins, based on the cumulative pore distributions shown in Figure 1.
Figure 3. Comparison of intraparticle porosities from SEC and NMR as a function of probe size for Butyl Sepharose. The line is the cumulative pore distribution given by eq 8, with parameters obtained by least-squares fitting to the NMR data.
well as packed columns. Comparing the NMR and SEC results illustrates the difficulty in measuring void volumes in adsorbing systems. As seen for the Sephadex resins, the drop in intraparticle porosity with probe size is steeper with the NMR technique than with SEC. But unlike the Sephadex results, the NMR method
gives a lower apparent mean pore size. This difference can be readily explained by considering the effect of adsorption on the two measurements. With SEC, adsorption increases the retention time and inflates the apparent void volume. With the NMR technique, adsorption during the equilibration process leads to a Analytical Chemistry, Vol. 69, No. 16, August 15, 1997
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small reduction in the volume accessible to the liquid-phase probes, thereby lowering the apparent void volume slightly. Thus, adsorption skews both measurements, and the difference between them shows the significance of the adsorption. The true porosity distribution will lie between the SEC and NMR results. The NMR technique provides a convenient means of characterizing the properties of resins. Like SEC, this technique is simple, quick, and inexpensive. Unlike SEC, the NMR technique does not require a packed column and can be done with a very small amount of resin. The primary disadvantage of the NMR technique is that it requires probe molecules that can be observed by NMR. By creating a mixture of probes that have slightly different resonance frequencies, however, the NMR technique can be used to measure the accessible void volume for an entire series of probes in a single step. This relatively simple extension of the basic NMR technique would make the routine characterization of porous materials easier and more convenient. CONCLUSIONS The use of a simple NMR technique for studying pore size distributions in porous media has been demonstrated. This procedure was used for a series of resins as well as a prepacked chromatography column. These examples show that the same technique can be used to study inter- and intraparticle voids in
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packed beds or to characterize the pore size distributions of bulk media. The advantage of this approach is that it provides a direct measure of the void volume distribution under the conditions of greatest interest. ACKNOWLEDGMENT We thank K. Biodrowski, H.-Y. Liu, and J. Torres for their fluorination work. Funding for this study was provided by NSF Grants CTS 9204436 and CTS 9522528. This work made use of the National Magnetic Resonance Facility at Madison, which is supported by NIH Grant RR02301 from the Biomedical Research Technology Program, National Center for Research Resources. Equipment in the facility was purchased with funds from the University of Wisconsin, the NSF Biological Instrumentation Program (DMB-8415048), NSF Academic Research Instrumentation Program (BIR-9214394), NIH Biomedical Research Technology Program (RR02301), NIH Shared Instrumentation Program (RR02781 and RR08438), and the U.S. Department of Agriculture. Received for review March 11, 1997. Accepted May 27, 1997.X AC9702711 X
Abstract published in Advance ACS Abstracts, July 1, 1997.
