1781
Anal. Chem. 1984, 56, 1781-1785
backbone away from the adsorbent surface. The separation of oligostyrene stereoisomers in the chromatograms shown in Figures 1 and 2 may be qualitatively explained by the influence of chain tacticity on the planarity of the adsorbed repeat units. Simple geometric models, without the aid of detailed rotational isomeric state calculations, will show that planarity of the adsorbed aromatic rings increases in the order syndiotactic, isotactic, to heterotactic isomers, which parallels the elution order for oligostyrene stereoisomers observed in the previous work (4). Chain tertiary structure also contributes to the planarity of adsorbed units in oligomers longer than pentamer. At large values of N , energy must be expended for both the uncoiling of the oligomer chain and the rotation of repeat units to a conformation that maximizes collectively the planar adsorption of all oligomer units. The rapid decrease in Q, and a, repeat-unit values for r < 5 can be attributed to the rotational barriers of repeat units, The gradual decrease in these values in oligomers longer than heptamer ( r > 5) may reflect the increasing values of chain disorientation entropy. In polymeric chains, the entropy of disorientation is proportional to chain length and generally causes loop and train adsorption conformations, with only a fraction of the repeat units contacting the adsorbent surface ( 8 , 9 ) . There is no abrupt change in the Q, and a, values of oligomers with r between 5 and 12, or other obvious experimental evidence for the adsorption of only a fraction of the repeat units in the oligomers examined. The relatively constant reduction from planar values of Q, and a, in oligomers longer than heptamer (r > 5 ) does provide the possibility for simple adjustments that account for nonplanar adsorption. Typically, subtracting constant values that account for nonplanarity from the readily calculated planar adsorption energies and occupational areas is similar in ap-
proach to adjusting Q, and a, values for localizing solutes (1-3).
CONCLUSIONS Unique polymer characteristics that lead to nonplanar adsorption of oligostyrenes, such as hindered repeat-unit rotation, are observed in oligomers as small as trimer. The functional-group additivity principles used in the solventdisplacement model for small-molecule adsorption chromatography will not accurately explain all aspects of the adsorption behavior of oligomeric solutes, unless these factors that are unique to the sample are indirectly incorporated into the fundamental expressions for retention.
ACKNOWLEDGMENT I thank G. A. Smith for his assistance. Discussions with E. P. Otocka, D. M. Wonnacott, and L. R. Snyder were helpful in the preparation of this manuscript. Registry No. Polystyrene (homopolymer),9003-53-6;silica, 7631-86-9.
LITERATURE CITED (1) Snyder, L. R. Hlgh-Perform. Liq. Chromefogr. 1983, 3, 157. (2) Snyder, L. R. “Principles of Adsorption Chromatography”; Marcel Dekker: New York, 1966. (3) Snyder, L. R. J . Chromatogr. 1980, 784, 363. (4) Mourey, T. H.;Smith, Q. A.; Snyder, L. R. Anal. Chem., preceding (5) (6) (7) (8) (9)
paper In this issue. Snyder, L. R.; Glajch, J. L. J . Chromatogr. 1981. 274, 1. Snyder, L. R. J . Phys. Chem. 1983, 6 7 , 240. Snyder, L. R.; Dale, H. J . Chromatogr. 1984, 13, 344. DIMarzio, E. A,; Rubin, R. J. J . Chem. Phys. 1971, 55, 4318. Lipatov, Yu. S.; Sergeeva, L. M. “Adsorption of Polymers”; Wiley: New York, 1974.
RECEIVED for review April 21,1983. Resubmitted February 6, 1984. Accepted April 19, 1984.
Static Exclusion Method for Determination of Specific Pore Volume Wei Cheng Beckman Instruments, Inc., 1716 Fourth Street, Berkeley, California 94710
A new method named the “statlc excluslon method” (SEM) has been developed to determine specific pore volume of porous materials. I t is based on sire exclusion of some polymer from pore volume in an appropriate solvent. This method has no restriction on particle sire, shape, and rigidity so that It can be applied io mlcroparticies such as catalysts and chromatographlc support materlals. I n addltion to that It provides a simple, rapid and accurate determination.
classical methods for determining specific pore volume of microporous particles are gas absorption and mercury porosimetry. Some other methods were reported such as sizeexclusion chromatography (2,3), centrifugal fiitration (4),and oil titration (5). However, these methods have considerable restriction because of the size and shape of particles and a low accuracy. A simple, rapid and accurate method for determination of specific pore volume, which has no restriction because of particle size and shape, is presented in this paper.
The specific pore volume is an important parameter to characterize a porous material. Porous materials for catalysis and chromatography, in general, have pore sizes in the range of 20-1000 A (diameter) and particle sizes in the range of 3-1000 fim (diameter or 6 times the ratio of apparent volume to external surface area). For particles larger than 1mm, the pore volume can be determined readily by the simple conventional method ( l ) in , which the apparent volume is measured after coating the particles by a thin layer of wax. The
PRINCIPLE OF THE STATIC EXCLUSION METHOD
0003-2700/84/0356-1781$01.50/0
The present method is based on the complete exclusion of macromolecules added to a suspension of a porous material from a solvent contained in the pores; this exclusion produces a measurable concentration effect which can be used to calculate the pore volume. High molecular weight polymer possesses a large gyration radius, R,, and therefore would be expected to be excluded 0 1984 American Chemical Society
6782
0
ANALYTICAL CHEMISTRY, VOL. 56, NO. 11, SEPTEMBER 1984
by the relatively smaller pores. The relation between gyration radius and molecular weight can be expressed by the formula (6)
R, = kMwa where Mwis weight average molecular weight; k and a have specific values for different polymer-solvent systems. Polystyrene dissolved in “good” solvent such as methylene chloride and tetrahydrofuran (THF) is regarded as random coils and (3)
R, = 0.137Mw0.59 A polystyrene with M, = 2.61 X lo6daltons would have an R, value of 836 A, which is sufficiently large for general exclusion purposes. The chromatographic exclusion value, 4, defined as the minimum pore diameter that allows a polymer with R, unhindered access, was reported to be 2-5 times the R, value (7). However, to assure a complete exclusion in the static partitioning process, a much larger R, is used. Owing to its aromatic ring, which is a common chromophore, polystyrene has a useful second absorption band around 254 nm. Close examination shows that polystyrene in methylene chloride or THF obeys Beer’s law very well in the absorbance range of 0.2-0.7. When a certain volume of a solvent (methylene chloride or THF) is mixed with a certain amount of a porous material, the solvent molecules which are much smaller than the pores are able to permeate through the entire pore volume. U1trasonic cavitation, boiling, and vacuum processes can be employed to achieve a complete solvent permeation. After permeation, a part of the solvent would occupy the pore volume and the rest of the solvent should exist in the interparticle space. Upon addition of some polystyrene solution with the known concentration into the suspension, the concentration of the polystyrene would decrease. The dilution would be attributed to the excess solvent only because the large polystyrene molecules are totally excluded by the pore volume. In the absence of the interaction between the polystyrene and the surface of the porous material, the decrease in the concentration would reflect the volume of the excess solvent accurately. Since the solute is a nonpolar polystyrene macromolecule and the external surface area of a porous material is generally very low, the interaction which is most likely a dispersion force if there is any can be considered as a negligible factor in the static exclusion process. The specific pore volume, V,, can be easily calculated by the formula
(3) where V, is the volume of the added solvent, Vois the volume of the polystyrene solution, Co is the concentration before the dilution, C1 is the concentration after the dilution, and W is the weight of the porous material. Although the concentration of polystyrene can be measured readily on an ordinary spectrophotometer, most porous materials contain some amount of soluble substances which interfere seriously with the absorbance of the polystyrene. To circumvent this problem, the concentration of the polystyrene can be determined by HPLC using a reverse-phase silica-base column which is able to separate the polystyrene from any interfering soluble substance. EXPERIMENTAL SECTION Analytical Procedures. The polystyrene solution was made by dissolving approximately 35 mg of standard polystyrene with
molecular weight 2.7 X lo6 daltons (Water Associates, Milford, MA) in 100 mL of methylene chloride or THF (UV grade). A 28 (diam) X 61 (high) mm glass vial with a Poly-Seal cone cap was used as the mixing vessel. The dry porous sample was weighed accurately in the tared stopped mixing vial and the weight was controlled approximately to give a total pore volume of 2 mL (estimated by some known information or first analytical run). Then a certain amount of solvent which was just sufficient to wet all porous sample was added gently into the mixing vial by a dropper. The vial with sealing was placed in an ultrasonic cleaner (Bransonic Model 12,50 W) with the water level filled to half of the vial height for about 40 min. The solvent was then allowed to evaporate from the vial under gentle vacuum until the mixture became a thick paste. The vial with sealing was dried with a wiper and weighed accurately. Then about 5 mL of the polystyrene solution was added to the vial and weighed again. The vial with tight sealing was ultrasonicated again for 10 min and was allowed to stand for temperature equilibration and particle settling. In the case of difficult settling due to the fine particles and low density of the sample, filtration of the supernatant was needed and could be carried out on Whatman 50 filtering paper in a 10-mm (diam) stainless steel filter holder. The chromatogramsfor the polystyrene solution before mixing and the clear supernatant were taken on a 150 mm X 4.6 mm i.d. Ultrasphere column (C18,5 wm particle) by using a chromatograph that consisted of a Model 112 pump, Model 160 uv detector, 20-pL injection loop, and Chromatopac-CR1A integrator (all from Beckman, Berkeley, CA, except the integrator which was from Shimadza Seisakusho LTD., Kyoto, Japan). Methylene chloride was used as eluent and the flow rate was 1mL/min. The detection wavelength was 254 nm and operating temperature was ambient. The retention time of polystyrenewas about 0.92 min under these conditions. The injection loop (20 ML)was flushed with a 25-wL syringe, 4 times for each injection. The injection was repeated 5 times for each polystyrene solution. The arithmetic average of the five-peak area obtained on an integrator was used as the representative of the concentration. The specific pore volume was then computed by a modified version of eq 3:
v, =
w,- W,(A,/A, - 1) PW
(4)
where W, and Wo were the weight of the added solvent and polystyrenesolution respectively, A. and A, were the average peak area for the polystyrene solution before and after mixing, respectively, and p was the density of the solvent at the experiment temperature. Since the concentration of the polystyrene was so low (-0.35 g/L), the density difference between the solvent and the polystyrene solution was negligible. Examination of the Linearity between Peak Area and Concentration of Polystyrene Solution. Approximately5 mL of the foregoing polystyrene solution was transferred to a tared mixing vial from a graduated cylinder and the vial was weighed accurately. From the graduated cylinder, 2,4, and 6 mL of the solvent were added, respectively, into a series of the abovementioned mixing vials containing about 5 mL of polystyrene solution. These vials were weighed again. The peak area for these diluted solutions as well as the original polystyrene solution was then determined by foregoing chromatographic procedures. Examination of the Solvent Permeation. About 1.4 g of porous silica with a pore size range of 30-100 A and V, = 1.227 mL/g, which had been dried at 180 OC under vacuum for 6 h, was weighed accurately in a tared and stoppered 10-mL volumetric flask with screw cap. The flask was swirled to wet all silica powders after addition of a sufficient amount of methylene chloride. Then methylene chloridewas added again to the 10-mL scale and the flask with tight capping was weighed accurately. Another volumetric flask filled with methylene chloride only was used as the reference for temperature equilibration. Both flasks were placed in an ultrasonic cleaner with water. After sonication for 10 min, both flasks were taken out of the cleaner and dried by acetone rinse and nitrogen flow. The temperatureequilibration was achieved if there was no further volume change within 10 rnin after the solvent level in the reference flask returned to the 10-mL
ANALYTICAL CHEMISTRY, VOL. 56, NO. 11, SEPTEMBER 1984
1783
o was a f t e r 30 minutes boiling
I
0
l
l
I
I
/
; u
k
2 3 4 4 e l u t i o n time ( m i n u t e )
1
o*02/
m
Figure 1. Chromatogram of the polystyrene solution (methylene chlorlde as solvent)mixed with the silica. Ultrasphere column 150 mm X 4.6 mm i.d., methylene chloride as eluent at flow rate 1 mL/min. accumulative ultrasonic mixing time
(min.)
Relation between methylene chloride permeatlon and ultrasonic mixing time (accumulative time) for 1.400 g of silica with V , = 1.225 mL/g at 21 "C. (The density of methylene chloride is 1.324 g/mL at 21 "C.) Flgure 3.
reasonably chosen to be the mixing period. Ultrasonic mixing offered some particular advantages over the conventional processes. It avoided the protracted evaporation of the solvent in which some fine particles might be entrained by bubbling. It also minimized the change in solution temperature, which might affect the stability of a polystyrene solution. In order to optimize SEM, eq 4 was differentiated to obtain the maximum possible error for the determination of a specific pore volume: r
0
l
1
1
2
1
3
1
4
1
5
i
6
e l u t i o n time ( m i n u t e )
Chromatogram of the polystyrene solutlon (THF as solvent) mixed with polystyrene-DVB resin. Conditions as in Figure 1. Figure 2.
scale. Then the volume in the sample flask was adjusted to the 10-mL scale by adding solvent and weighed again. This procedure was repeated for accumulative mixing times of 10,20,30,40,50and 60 min. At last, the sample flask without cap was placed in a hot water bath to be gently boiled for 30 min to achieve complete penetration. After the sample was cooled down to ambient temperature, the volume in the flask was adjusted to 10 mL by solvent addition and the flask was reweighed. Examination for Reproducibility and Accuracy. Some resin (particle size home-made polystyrene-divinylbenzene (DVB) range 3-10 wm, pore range 20-60 A) was dried at 80 "C under vacuum for 3 h and its specific pore volume was determined repeatly by using the foregoing procedure with THF which was lower than that of the resin. The specific pore volume for some silica samples was analyzed by both the present static exclusion method (SEM) and mercury intrusion method which was performed by Quantachrome Corp. (Syosset, NY).
RESULTS AND DISCUSSION The relation between the dilution factor, i.e., C1/Co, and the peak area ratio, i.e., A1/Ao, showed the excellent linearity for both methylene chloride and THF. The standard deviation was estimated to be about 0.9% , which included the deviation for the weighing and injection volume. On the basis of five runs for each solution, the standard deviation for 20-hL injection volume was estimated to be 0.6%)whereas it was 1.2% for 5-KLinjection volume. The methylene chloride permeation weight vs. ultrasonic mixing time was plotted in Figure 3. Obviously the curve leveled off after 40 min. The weight difference between 40 min and boiling process was 0.0015 g, which was equivalent 1.13 X ml. The relative deviation, Le., AV,/V,, was only 0.0290 and was evidently negligible. Therefore, 40 min was
1
V, = -AW, PW
+
PW
W E- W,(A,/A, - 1)
AW+-
WO PW A
Pw
AAO
Wdo +AAi PWA12
(5)
It was reasonable to assume that the variation of weighing A W , = AWo = A W = 0.002 g; the variation of peak area uo = 0.009&, AAl = 0.009A1, and p = 1.326 g/cm3 (methylene chloride at 20 "C). Then eq 5 can be reduced in the following form:
v, = 0.00151
w + w, + ( W - W,)A,/A, + 0.0135'7% w WA1
(6)
Two useful relationships could be derived as follows (see Appendix): WE =
rv, + k(V* + 1/PSIWP w,+ w, - v,wp
Ao/A1 =
(7)
(8) ws
where k is a dimensionless coefficient describing the ratio of the excess solvent and the apparent volume of a porous sample. k = 0.667 for a random packed bed with spherical particles having the same diameter and k = 1.0-1.5 for the thick slurry of nonuniform particles. pt was the true density for a sample. By use of eq 6, 7, and 5, the maximum variation of the specific pore volume was estimated under various conditions and the results were summarized in Table I. The explicit trend in Table I was that the larger sample pore volume ( WV,) and the lower values of Woand k led to less variation. The variation in percentage was substantially the same for different specific pore volumes if the same sample pore volume, k , and Wowere controlled.
1784
ANALYTICAL CHEMISTRY, VOL. 56, NO. 11, SEPTEMBER 1984
Table I. Calculated Maximum Variation under Various Conditions
mL/g
"PI
w,g
sample pore volume
density," g/cma
k
wo,g
mL/g
A Vp,
AVP/ VP,
1 1 1 1 1 1 1 1 0.5 0.5 0.5
2 2 2 2 2 2 3 1 2 4 6
2 2 2 2 2 2 3 1 1 2 3
2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2 2.2
1.5 1.5 1.5 1.5 1.2 2.0 1.5 1.5 1.5 1.5 1.5
13.26 (10 mL) 9.28 (7 mL) 6.63 (5 mL) 3.98 (3 mL) 6.63 6.63 6.63 6.63 6.63 6.63 6.63
0.131 0.104 0.087 0.069 0.079 0.100 0.071 0.136 0.072 0.049 0.042
13.1 10.4 8.7 6.9 7.9 10.0 7.1 13.6 14.4 9.8 8.4
%
" 2.2 g/cms was the density for the nonporous amorphous silica. Table 11. SEM Analytical Results for the Resin and the Silica
sample resin (THF)" silica (CH2Cl.Jb
1 2 3
2.08 2.02 2.32
4.98 5.13 4.89
3.47 3.74 3,69
2.1020 2.0331 2.031
0.625 0.614 0.605
0.615
1.3
1 2 3
2.11 2.17 1.99
7.45 7.39 8.09
3.99 6.14 6.62
2.1233 1.734 1.8194
1.061 1.002 1.010
1.024
2.6
" THF was used as solvent for the whole process. p = 1.3258 g/cm3.
p = 0.8892 g/cm3.
1.01
*Methylenechloride was used as the solvent for the whole process.
Vpmwas the specific pore volume determined by mercury intrusion analysis.
There was considerable difference in variation between sample pore volume of 1and 2 mL while only small difference between 2 and 3 mL. Considering the low consumption of an analytical sample, 2 mL of sample pore volume was appropriate to SEM. Although a low value of W, would decrease the variation to a considerable degree, it would result in a difficulty in particle settling as well as an enhancement of the interaction effect (if there was any). As a practical compromise, 5 mL of solvent was chosen for the value of W,. Similarly, the practical k value could be controlled in the range of 1.2-1.5, which correspond to thick slurry. When these analytical parameters were controlled, the maximum variation of V , was less than lo%, which was better than that of all conventional methods. The analytical results for the polystyrene-DVB resin and the silica determined by SEM and mercury intrusion analysis were summarized in Table 11. Evidently, SEM data were highly reproducible and had a standard deviation less than 3% although the resin probably swelled in THF to some degree. The specific pore volume determined by SEM was in good agreement with that from mercury porosimetry, which indicated the accuracy of SEM. As indicated before, the chromatogramsfor the polystyrene solutions mixed with the resin and silica showed two peaks (shown in Figures 1and 2). The first peak with the shorter retention time was due to the polystyrene solute and the second peak was due to some unknown soluble impurities contained in the samples. (THF showed a peak also if methylene chloride was used as eluent.) It was observed that the second peak was extremely intense for the resin sample and the separation between the two peaks was still attained. In general, the soluble impurities contained in a porous material were some organic substances with low molecular weight and low polarity instead of macromolecules comparable to the huge polystyrene molecules. The low polarity small molecules would have a long retention time on the RP C-18 column whereas the huge polystyrene molecules could retain on the external surface of the column material only because of size exclusion effect. As a result of the combination of
reverse-phase and size-exclusionmechanism, the resolution between the polystyrene and soluble impurities was better on the C-18 column than by ordinary size exclusion chromatography. It was plausible to mix the polystyrene probe solution and a dry porous sample directly without using a solvent. An increase in the concentration of the polystyrene would be expected if the polystyrene molecules would still be excluded while the solvent molecules could permeate through the pore volume. However, the concentration of the polystyrene after mixing consistently showed a slight decrease, which contradicted the expectation and would lead to an unrealistic negative V,. Since a macro polystyrene molecule in the solvent was a random coil instead of a hard sphere, the polystyrene molecules may change shape to partially squeeze into some pores during the direct mixing process, which would cause a large increase in the entropy. Hence, a certain amount of the polystyrene molecules was immobilized on the porous sample and possibly obstructed some pores, which became inaccessible for the solvent molecules. As a result, the concentration of the polystyrene in the bulk solvent decreased to a degree instead of increasing. The major technical advantage of SEM is that the interstitial volume of the determined particles does not interfere with the pore volume because the interstitial space is so large that the polystyrene macromolecules can travel entirely. In contrast, there is uncertainty in distinguishingthe interstitial volume from the pore volume for both mercury porosimetry (MP) and the gas adsorption method (GAM), which poses a considerable variation in specific pore volume. This uncertainty is mainly dependent of the particle size, the size distribution, and isotropism of the particle shape. The fast and easy analytical repetition as well as large sample weight for SEM (about 20-fold of that for MP and GAM)offers another advantage to minimize random error and enhance the determination accuracy. Although SEM is somewhat similar to size exclusion chromatography (SEC),SEM has also a number of advantages
ANALYTICAL CHEMISTRY, VOL. 56, NO. 11, SEPTEMBER 1984
over SEC in the determination of V,. (1) AE mentioned before, SEM does not have any restriction on particle size and shape, which is a very important requirement for SEC. Even for uniform particle size with ideal sphericity, there is inevitably theoretical error depending on the used solute, the particle size, and packing condition in SEC (8). (2) The factors affecting the determination are minimized in SEM. There are only three such factors, i.e., weighing, volume injection, and absorbancedetection. In contrast, many factors in SEC such as flow rate, back pressure, temperature, solute, assymmetry of peak, extracolumn volume, and interaction between the solute and the chromatographic material affect the determination (9, IO). (3) SEM simplifies the required equipment and reduces analytical time. An independent determination takes about 2 h and several determinations can be performed at the same time which is much faster than all other methods. The present SEM, which uses polystyrene as the probe solute and methylene chloride as well as THF as the solvent, can be employed to determine specific pore volume for many porous materials. Moreover, the present method would be extended to some porous materials which methylene chloride and THF do not apply to. This alternative SEM may use an aqueous solution of some water-soluble polymers such as poly(ethy1ene oxide), dextran, and so on. In addition, SEM could be applied to determination of pore distribution for porous materials by using a series of probe polymers of different sizes. A similar mechanism was mentioned elsewhere ( I I ) , but the determination of pore structure is beyond the scope of the present paper.
ACKNOWLEDGMENT I thank Nelson Cooke, Ben Archer, Richard Hatch, and Richard Meagher for their review of this paper. Barbara Gervase’s help in some experimental work and Greg Swalley’s help in literature searching are gratefully acknowledged.
APPENDIX After complete permeation, the added solvent includes internal portion, i.e., pore volume portion, and external portion, i.e., excess solvent portion. It can be expressed as follows: = vpwp (All
w,
+ we
where We is the excess solvent portion. A coefficient k is defined as the ratio of the excess solvent volume to the apparent sample volume, i.e., the sum of the pore volume and the solid volume of a sample. Hence one obtains:
We/P = kvn
1785
(A2)
where Va is the apparent volume of a sample. On the basis of fundamental relation, one also has
Vn = W/Pn
(A3)
where pa is the apparent density of a porous sample and pt is the true density of the sample, Le., the density for nonporous sample. By use of eq 2 and 3, eq 2 becomes
We = kWp(Vp + 1/PJ
(A5)
Upon insertion of eq 5 into eq 1, the desirable formula can be obtained:
w, = rv, + wp+ l/Pt)lWP
(A61
Evidently, the peak area ratio should be equal to the concentration ratio:
AO/A1 = co/c1
(A7)
The relation between Co and C1after dilution should be
co = Cl(W0 + W,)/Wo
(A8)
When noted that We = W,- VpWp (from eq l ) , the foregoing formula is then obtained
Ao/A1 =
w, + w, - vpwp WO
(A91
Registry NO. Polyh&omopolymer), 9003-53-6;(styrene) (divinylbenzene) (copolymer), 9003-70-7; silica, 7631-86-9.
LITERATURE CITED Scheidegger, A. E. “The Physics of Flow Through Porous Medla”; Unlverslty of Toronto Press: Toronto, 1 9 8 0 Part I. Nlkolov, R.; Werner, W.; Haiasz, I. J . Chromatogr. Sei. 1980, 78, 207-215. Werner, W.; Halasz, I . J . Chromtogr. Scl. 1980, 78, 277-282. Freeman, D. H.; Schram, S. B. A d . Chem. 1981, 53, 1235-1238. Thomas, J. M.; Thomas, W. J. ”Introductlon to The Princlples of Heterogenous Catelysls”; Academic Press: New York, 1987. Yau, W. W.; Glnnard, C. R.; Kirkland, J. J. J . Chromatogr. 1978, 749, 485-487. Crlspln. T.; Haiasz, I . J . Chromatogr. 1982, 239, 351-382. Schou, 0.;Larsen, P. HRC CC, J . High Resolut. Chromtogr. Chromatogr. Commun. 1982, 4 , 515-518. Yau, W. W.; Kirkland, J. J.; Bly, D. D. “Modern Size-Exclusion Liquid Chromatography”; Wiley: New York, 1979; Chapter 7. Aubert, J. H.; Tlrreil, M. J . Ll9uM ChfOm8tOg7f. 1983, 6 , (Suppl 2), 219-235. Kuga, S. J . Chromatogr. 1981, 206, 449-461.
RECEIVED for review January 20,1984. Accepted April 6,1984.
