Anal. Chem. 1981, 53,
(27) Doue, F.; Guiochon, G. Sep. Sci. 1970, 5, 197-218. (28) Tijssen, R. Sep. Sci. Technol. 1978, 73,681-702. (29) Doue, F.; Merle D’Autiigne, J.; Guiochon, G. Chim. Anal. (Paris) 1971, 53. 363-374. (30) Hofmann, K.; Halasz, I.J . Chromafogr. 1980, 799, 3-22. (31) Ishii, D.; Takeuchi, T. J. Chromatogr. Sci. 1980, 78, 462-472. (32) Martln, M.; Blu, G.; Eon, C.; Guiochon, G. J . Chromatogr. 1974, 172, 399-414. (33) Mohnke, M.; Saffert, VV. I n “Gas Chromatography, 1962”; Van Swaay, M., Ed.; Butterworths: London, 1962; pp 216-221. (34) Desty, D. Adv. Chron’lafogr.(N.Y . ) 1985, 199-228. (35) Novotny, M., Bloomington, IN, unpublished results, 1981. (36) Hirata, Y.; Novotny, MI. J. Chromafogr. 1979, 186, 521-528. (37) Grushka, E. Jerusalem, Israel, unpublished work, 1980. (38) Hershberger, C. W.; Calls, J. 6.; Christian, G. D. Anal. Chem. 1979, 51, 1444.
1325
1325-1335
(39) (40) (41) (42) (43) (44) (45) (46) (47)
Guiochon, G.; Chovin, P. Bull. SOC. Chim. Fr. 1985, 3396-3403. Guiochon, G. Anal. Chem. 1968, 38, 1020-1030. Guiochon, G. Adv. Chromaiogr. (N.Y . ) 1989, 8, 179-270. Nikelly, J. G. Anal. Chem. 1976, 48, 987-989. Halasz, I.; Hartmann, K.; Heine, E. I n “Gas Chromatography 1964”; Goldup, A., Ed.; British Petroleum Institute: London, 1965; pp 38-61. Halasz, I.; Heine, E. Adv. Chromafogr. 1987, 4 , 207-263. Landault, C.; Guiochon, G. I n “Gas Chromatography 1964”; Goldup, A., Ed.; British Petroleum Institute: London, 1965; pp 121-139. Landault, C.; Guiochon, G. Chromafographla 1968, 7 , 119-132. Hirata, Y.; Novotny, M.; Tsuda, T.; Ishli, D. Anal. Chem. 1979, 57, 1807-1809.
RECEIVED for review January 16, 1981. Accepted March 6, 1981.
Measurement of Band Broadening in Size Exclusion Chromatography R. Groh and I. Halfs’z” Ange wandte Physikalischts Chemie, Universltat des Saarlandes, 6600 Saarbrucken, West Germany
The Interstitial band broadening in silica filled columns has been determined experimentally. The pares were filled with water, which is impenetrable to the benzene and polystyrene samples. Total band broadenlng, h, was measured with “dry” CH2C12eluent and the lniterstitlal broadening with “wet” eluent In the same column. The band broadening in the pores was obtained by difference (Ah). The interstitial band broadenlng Is about 10 tlmes smaller than those obtained with accessible pores. The reduced Interstitial band broadening Is at least a factor 2 smaller than roported earller. The diffuslon coefflclents of polystyrenes In CH2Cl2as a function of M, were measured in packed columns by a stopped flow method. It Is shown that restricted dlffuslon is not only a function of M, of the samples but alao of the pore slze distrlbutlon and specific pore volume of the support. Extremely high specific pore volumes, theoretlcidly desirable in SEC, lead In practlce to undesirable high mass transfer terms. Packing structures In dry and slurry packed columns are discussed.
Band broadening is of particular interest in size exclusion chromatography (SEC) because separation occurs in a strictly limited range of the elution volume (Ve) V , = V, + KV,,,, = V, -t V , (1) where K is the exclusion chromatographic “partition coefficient” always lying between 0 and 1 (see list of symbols at the end of the paper for the other symbols). Consequently peak capacities (1) in SEC are much smaller than in other forms of chromatography. A further reason for interest in band broadening in SElC is that proper calculation of the molecular weight distributions (MWD) of polymers is only possible, after correction for the different sources of band broadening (2). In this paper an experimental method will be described to measure the interstitial peak broadening in columns packed with porous materials. Band broadening in SEC has been discussed from both a theoretical and an experimental point of view. It is sometimes possible to approach the problem from diametrically opposing points of view and to end up with mathematical treatments which are remarkably similar.
The Theoretical HPLC Approach. The van Deemter equation (3) is the most common expression used to describe the h vs. u curve
B
h = A -I-U
+ CU
(2)
Dimensionless parameters are often used to describe hydrodynamic phenomena. The ones appropriate to chromatography are reduced plate height (bred = h/d,) and reduced linear velocity ( u d= ud,/D). Reduced parameters sometimes permit better comparisons of chromatographic systems with differing d, and D. The simplified form of the van Deemter equation
h = A ’ + C’u (3) is often used in LC, and here in SEC, since the B / u term is only of importance when the flow rate is uninterestingly low. Some experimental h vs. u, curves have a concave ascending branch and therefore cannot be described by the van Deemter equation. In order to account for such h vs. u, curves, several theories have been proposed (4-19). Giddings and Mallik (7) pointed out that the flow velocity can, with advantage, be described in terms of the interstitial velocity (4)
where a is a constant. This can be measured with a totally excluded inert sample (eq 4). Furthermore, they (7) also proposed an obstruction factor (73 describing the ratio of the restricted diffusion inside the particles (D,) and the diffusion in the interstitial volume (Dm),which they assumed to be constant -fa=,
D, = 2Q
(5)
0
The Experimental HPLC Approach. Heitz (20) measured h vs. u curves for olygophenols in columns packed with polystyrene and poly(viny1 acetate) gels (d, = 100-500 pm). All his results were describable by the van Deemter equation. The author states (21) (our translation): “While Giddings (7) assumed, that mass transfer in the stationary phase is rela-
0003-2700/81/0353-1325$01.25/00 1981 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 53, NO. 9, AUGUST 1981
tively unimportant in band broadening, today it can surely be held, that this is the most importapt contribution.” Kelley and Billmeyer (10,12-14) studied band broadening in columns packed with nonporous glass beads, porous silicas, and porous polystyrene gels, all other parameters being held as constant as possible. For the porous packings they concluded that, for silica (d, = 115 pm), mass transfer effects inside the pores predominated, whereas for polystyrene gels (d, = 50 pm) band broadening was mainly caused by interstitial effects. van Kreveld and van den Hoed (18)measured the h vs. u curve of toluene in a column packed with silica (d, = 125 pm). The ascending branch of the curve was linear. In contrast polystyrene samples gave, in the same column, a concave ascending branch. From these experiments, obstruction factors were calculated, which decreased with increasing molecular weight, ranging from y = 0.7 for toluene to ys = 0.1 for polystyrene ( M , = 160000). Knox and McLennan (22,23)measured the h vs. u curves of polystyrene samples on silica (dp = 7 p ) in columns with different lengths. Obstruction factors were calculated, ys = 0.1 for polystyrene of Mw = 20000. They also showed, using a monodisperse oligomer of styrene, that the band broadening due to “kinetic” effects is proportional to L1I2in SEC as it is in retentive chromatography. On the other hand the difference in retention volume of two monodisperse substances is proportional to L. On the basis of these considerations their experimental results were used to calculate polydispersities. Dawkins et al. (24, 25) measured band broadening for spherical silicas and polystyrene gels. They concluded that plate height decreased not only with decreasing d, but also with narrowing particle size distribution. Dawkins and Yeadon (26) describe a method to calculate polydispersities of polymers from plate height measurements at low velocities. T h e Approach of Polymer Chemists. Moore et al. (27) introduced an experimental method for eliminating band broadening caused by polydispersity. The sample is allowed to run only half way along the column, when the direction of flow is reversed and the sample is eluted at the inlet end of the column. Peak broadening caused by polydispersity is thus eliminated, while the kinetic peak broadening is the same as if the sample had passed along the entire column length. Hendrickson (28) used the method of Moore to measure band broadening on polystyrene gels (d, = 40 pm). Excluded substances had the same peak volumes (Le., the same h values), although the ratios of their molecular weight were as much as 100. Consequently it was assumed that interstitial band broadening is independent of the diffusion coefficient of the sample. Partially excluded peaks at constant flow rate showed a maximum in the h vs. M, curve. This must be caused by kinetic (mass transfer) effects, because the effects of polydispersities are eliminated by this technique. The h vs. u curves were, on the other hand, linear. Bly (29, 30) found that the peak volume, when corrected for polydispersity, was constant in the linear part of the calibration curve (log M , vs. VJ. Consequently nl/’ is proportional to V,. Basedow et al. (31) found that the band broadening of nearly monodisperse dextran fractions in water on porous glass at u, = 0.008 mm/s was nearly independent of the molecular weights. The maximum in the h vs. M, curve described by Hendrickson (28) has been confirmed for polystyrene gels by Tung and Runyon (32) and Yau et al. (33) and for dextran gels by Povey and Holm (34). Smit et al. (35) found, however, that h increased monotonicly with increasing M,. The seemingly contradictory findings of Smit (35) and of Tung and Runyon (32), which were both obtained by the
reversed flow technique 0,can be explained by Kubin’s (36) “sphere-model’’ of SEC, which assumes that only a given depth of the porous stationary particle is available to the polymer sample. Both sphere models (36,37)predict a maximum in the h vs. M , curve, as a consequence of two opposing effects. Band broadening is increased as M , increases (because of decreasing of diffusion coefficients) and it is decreased with increasing MW.
Berger (38)has demonstrated, with the aid of radioactively labeled polystyrene samples, that as far as the elution volume is concerned, a polydisperse sample behaves like a mixture of many monodisperse substances. This is also true for h, as long as there are no big differences in diffusion coefficients. From the above it can be seen that the causes of band broadening in SEC are qualitatively well understood. The major ones are, in the interstitial volume, (a) longitudinal diffusion, (b) eddy diffusion, and (c) mass transfer effects and in the pore volume (d) mass transfer effects and (e) polydispersity of the sample. The quantitation of the above contributions to band broadening is extremely difficult. Theory is formulated in terms of independent effects, whereas an experiment always yields the sum of several terms (39). In the main, four experimental approaches have been used to isolate the various contributions to band broadening in SEC: (1)The measurement of h vs. u curves with different combinations of particle size, eluent, and monodisperse sample. This is the classical approach to chromatographic kinetics. ( 2 ) The reversed flow method of Moore (271, which eliminates band broadening caused by the polydispersity of the sample. (3) Experiments with porous and with nonporous packing materials, under as nearly as possible identical chromatographic conditions. Peak broadening in the pores being calculated by difference from measurement in two different columns (10, 12-14). (4) On the basis of theoretical considerations, columns of different length allow the isolation of peak broadening due to polydispersity (22, 23). In this paper, we propose an experimental method which allows the direct measurement of interstitial band broadening in columns packed with porous materials. The same column can also be used to measure the total band broadening. Thereby overcoming the problem of calculating band broadening inside the pores from data obtained with different columns that never pack identically (39). This is done by filling the pores of the stationary phase with a different liquid phase, as in liquid-liquid chromatography (40). Any sample, which cannot penetrate the stationary liquid phase, which fills the pore volume, can have its interstitial band broadening measured. The total peak broadening of polystyrene samples on a porous silica column was measured using “dry” dichloromethane as eluent. If the column is equilibrated with water-saturated dichloromethane (“wet” eluent), the total pore volume is filled with water, producing a heavily loaded column (41-44). Since the pore volume is now inaccessible to polystyrene samples, it is possible to measure their interstitial band broadening. EXPERIMENTAL SECTION Supports and Packing Procedures. The silicas used are described in Table I. Supports 1, 8, 9, and 10 were used as delivered. The support for column 2 was dry sieved and freed from fines by repeated sedimentation in water. Columns 3-7 were packed with material prepared from a single batch of Lichrosorb SI 500 of a sieve fraction 63-200 pm. Packings 3 and 4 were obtained by dry sieving. In order to obtain packings 5-1, we ground the original material in an achat mill, support 5 was obtained from the ground material by sieving and 6 and 7 by
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ANALYTICAL CHEMISTRY, VOL. 53, NO. 9, AUGUST 1981
Table I. Properties of Silica Packed Columns support or column no. 1 2 3 4 5 6 7 8 9 10
columns silica
origin
dp,“ pm
Lichrosorb Si 100
e
Lichrosorb Si 500
e
10 150-200 100-125 63-71 25-32
vp,sp,b cm3/g
@J,b
A
L,cm
i.d.,c mm
330
1.0
94
100
0.75
390
27.8 50 102 45 30 23 17.7 27.8 25 25
4 8 4 4 4 4 4 4 4.5 4.5
04’b
m
/g
d,,d
V ~ , c m ~ pm 4.076 25.10 12.302 5.540 4.233 2.995 2.452 3.876 4.030 4.030
9.4 170 110 67 28.5 13.8 6.5 7.4 7.8 5.7
Spherical silica f 7-11 205 1.8 260 Lichrosorb Si 100 e 7 330 1.04 94 Spherisorbi S5W g 5 280 0.54 79 As calculated from the permeability Measured and calculated according to ref 64. Nominal i.d. a Sieve fraction. Kindly donated by K. Unger, Technical University, ( 4 5 ) ,or from our sieve fraction. e E. Merck A.G., Darmstadt. U.K. Darmstadt. g Phase Separation Ltd.,. Queensferry, - , Clwyd, I
countercurrent sedimentation in acetone. Silicas with d, 2 40 gim were dry packed (44, 46). The best results were obtained when the silica was dried overnight at 130 “C and was packed while still warm. Of the various slurry techniques described in the literature (47-51) a modified viscosity method (47) was chosen, with mixtures of cyclohexanol,2-propan01, and carbon tetrachloride as the suspending medium. Colimn packing techniques have been well reviewed (50,51),nevertheless, we found it necessary to establish the optimum conditions for each silica and for each sieve fraction by trial and error. Samples and Eluentri. Polystyrene sample standards (PSt) of narrow molecular weight distribution were obtained from Waters Amociates Inc. (Milford, MA) and from Pressure Chemical Co. (Pittsburgh, PA). 111 this paper we have designated, for example, a PSt with M, = 111000 as PSt 111. Benzene and 1,3-diphenylbutane were also used as samples. The later compound, with a molecular weight of 210, is designated as 0.21. The samples were dissolved in dichloromethane (concentration: 0.1-0.5 % ). The dichloromethane eluent was continuously recycled. To obtain “dry” conditions, we included a moisture control system (52)and a column packed with 5-A molecular sieve in the eluent circuit. The water content of the “dry” dichloromethane so obtained was less than 5 ppm. For the production of watersaturated dichloromethane (“wet” eluent), a temperature-controlled reservoir, where the eluent was stirred in contact with water, was included in the eluent circuit. The water content of the “wet” dichloromethane was approximately 2200 ppm. With a new eluent the whole chromatographic system was always equilibrated for at least 12 h. A heat exchanger was installed between the pump and the injection system. The column itself was enclosed in a thermostated water jacket. The temperature was controlled to 22 f 0.1 “C. The water content of the eluent was determined by Karl Fischer titration (Metrohm, Herisau, Switzerland), before and after each measurement, Apparatus. The equipment was assembled from commercially available and homemade units. Special attention was paid to the connection between injection system, column, and detector to minimize extra column baind broadening, which may yield misleading results in HPLC (!9,53,55,56,62) and in SEC (10,54, 57,58). For the same reason samples were injected on the top of the small volume columns ( 5 H 1 ) through a septum. Samples were introduced onto the larger volume columns by means of a sample loop (either Rhenodyne type 7 120, Kontron Technik, Eching or type U6K, Waters Associates). A Waters M 6OOO syringe pump or an M3 (Orlita, Giessen) membrane pump were used. The UV detector (254 nm, cell volume 8 pL) was homemade (47,62). Columns were made from drilled out (47) stainless steel tubes (nominal i.d. 4 mm or 8 mm) and modified Swagelock fittings. Empty column volumes (V,) were determined by weighing the column empty and filled with water. Presentation of Data. Linear velocities are always quoted in terms of u,, the linear velocity of a totally excluded sample.
2
6
+
8
10
7
14
6
8
20
22
-
JZ[mm’sec
Figure 1. h vs. u, curves for column 1 (SI 100, 10 pm) CH2CIP: cz = 0.454, cP = 0.382.
in “dry”
The shape of the plate height vs. linear velocity curve is the same, whether u or u, is used. However, the constants of the equations describing it, such as the van Deemter equation, will be changed by the transposition. The h values were calculated from the intercept of the tangents to the point of inflection,assuming Gaussian peaks. The constanta of the h vs. u, curves, eq 3, were calculated by the least-squares method.
RESULTS AND DISCUSSION The h vs. u Curves Obtained with “Dry” and with “Wet” Dichloromethane. The change in pore volume experienced on changing from “dry” to “wet” eluent can be expressed in terms of the dimensionless porosities (63-65) €2 = v z / v k (6) Ep = vp,rnax/vk
w,
(8) = + VP,,,,)/Vk The h vs. u, curves for eight polystyrenes, l,&diphenylbutane, and benzene obtained on column 1 (Lichrosorb SI 100, d, = 10 gum) using “dry” CHzClz are shown in Figure 1. The descending branches of this and of most other curves were not determined, because of the very low u, which would have been involved. The h vs. u, curves in Figure 1are linear. The two excluded polymers, PSt 111and PSt 200,together with the two monomers, have the lowest, and very similar, h values, which are almost independent of u,. The partially excluded polystyrenes disclose no obvious relationship between h values, C terms, and M,. The h vs. u, curves of the same samples, on the same column, are, however, quite different, when the eluent is “wet” CHzClz, as shown in Figure 2. The pore volume here is almost totally filled with water and is not accessible for the samples Et
=
€2
+ Ep
(7)
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ANALYTICAL CHEMISTRY, VOL. 53, NO. 9, AUGUST 1981
4.
2s:
15c- h p m l
/-
13
35
A , -FSt
231
2
6
4
8
IC
2
i
Is
@
ic
22
-
2- J$mP sec]
Flgure 2. hvs. u, curves for column 1 in “wet”CH2C12: E, = 0.446, ep = 0.010. The samples are listed In the order of their A’values.
2
5
4
12
C
3
lL
15
3
2C
b z 7 m r sec
Flgure 4. h vs. u, curve for column with “semlwet” CHzC12: water content of the eluent, 0.2%; e, = 0.437, ep = 0.100.
i
I5 n : p :
103
/PSt
\ ‘
I
Y*ox
I 1s2
12
io6
2
1
* M ~ ~ !
Flgure 5. hvs.
CH,CI2:
C,
4
3
6
E:
7
LZ[rnmrsec:
u, curves for column 2 (SI 100, 150-200 pm) In “dry”
= 0.365, ep = 0.443.
Flgure 3. Calibration curves for column 1 with “dry” and “moist”
CH2C12: (solid line) “dry” CHPC12(st
A
I3
L rm- sf.,
1‘
*
I
Z-
Figure 9. Low velocity h vs. u, curves for column 8 (SI 100, 7 pm) in “dry” CH,CI,: E, = 0.463, E,, = 0.400; (solid symbols) “static” method, (open symbols) “dynamic” method.
01
30
3
1
M
J
2 ; .
F
Figure 10. h,, vs. u,, (quasi) for column 2 (SI 100; 150-200 km) in “wet” CH,CI,.
Longitudinal diffusion only takes place during the residence of the sample in the interstitial volume, consequently
B,
- D,-
Ei
€2
51
Combining eq 9 and 11 OPSt 5’
I
OPSt 4
21
where ~i is the total porosity available to a given sample and is a function of M,. If B , c , / Eis~ plotted against M, in log-log format, from the slope of the straight line it can be calculated, that
D,
N
MW -0.6
X 10-4)Mw-0.6
h = const urn
(15) Knox (16, 76) used a similar empirical form, when describing interstitial band broadening in terms of reduced parameters. = const u,ed1/3
(16)
In Figures 10-12 bred is plotted against u,dJM,o.6 in log-log terms, that is, a quasi-reduced presentation. Only data obtained with “wet” CHzClzare presented, and consequently here u, = u. If D, is assumed to be about (5 x 10-4)Mw4.6cm2/s, then Ured
= constant u,d,&P6
GI
1
‘3
00
L z dp M,363
Flgure 11. h,, vs. urd (quasi) for column 4 (SI 500; 63-71 pm) in “Wet” CH2C12.
(14)
Band Broadening in the Interstitial Volume. All the more sophisticated attempts (over and above the A and B terms in eq 2) to describe band broadening in the interstitial volume, resemble the coupling term of Giddings (6). However, Giddings himself (39) has written: “The coupling expression is simply an approximation to a very complex interaction between diffusion and convection. The process has been formulated rigorously, but the boundary conditions for real granular beds are so complex, that meaningful solutions have not yet been obtained.” We shall try to describe our experimental results in terms of empirical equations with only a few constants. It is not likely that each of these constants will be ascribable to a single physical effect. Synder (15) described total band broadening in adsorption chromatography by the empirical equation
bred
301
(13)
This is in good agreement with the figure calculated by Knox (23) if we used the Wilke-Chang eq (71). From‘Figure 12 it can also be calculated, for polystyrene samples in CHzClz eluent, that
B , E , / E=~ (6
*
1
(17)
+
A
PSt 2,2 ^SI 103
c PSI 51 0
X’
3:
PSt
‘C 2 p
:‘I
‘OC “eo VY’”
Flgure 12. h, vs. u, (quasi)for column 6 (SI 500; 13.8 pm) in “wet” CH2CIz.
Thus to obtain u,d from the abscissas of our plot, it must be multiplied by a constant factor, whose value is about 2000. The plots in Figures 10-12 show a considerable amount of scatter, not all of which can be attributed to experimental error. They also diverge from linearity. Therefore we concur with other authors (77, 78), who stated that the concept of reduced parameters alone is not sufficient to describe band broadening in the interstitial volume of a packed column. The corresponding equation to eq 16 of Knox for our quasi-reduced velocity is
bred
= A*(~,d$4,~.~)”
(18)
The value of A* is a good measure of the efficiency of a column (79), because it defines plate height when u , d N 6 = 1. The
ANALYTICAL CHEMISTRY, VOL. 53, NO. 9, AUGUST 1981
Table 111. The Constants of Equation 18 together with for Various Columns
eZ
/
24i
col-
1331
umn no. packing 1 SI100 2 3 4 5 6 a
SI 500
d,, pm
m
10 150-200
A* 5.6 2.8
0:LO
0.14
0.454 (s) 0.365 (d)
100-125 63-71 25-32 13.8
6.7 4.9 5.5 3.8
0.09 0.11 0.15 0.10
0.393 (d) 0.352 ( d ) 0.466 (s) 0.501 (s)
EZa
Key: d = dry packing; s = slurry packing.
PSI 103 10’
s+-
- ’A-
-
t-LSH6
4.
2
4
6
8
IC
2
JZ[mrnsec]
Figure 14. Ah vs. u, curves for column 3 (SI500, 100-125 pm).
Table IV. AA’ and AC’ (Equation 3) for Column 2 (SI 100,150-200fim) AA’, pm AC’, ms benzene PSt 0 . 6 PSt 2.2 PSt 4 PSt 10.3 2cc/ 217 pm
25-32 Cim
Flgure 13. A’(from eq 5) vs. log Mwplots in “wet” CHzCtzfor columns 1, 2, and 5 and for a colunin filled with broken glass. The figure to the left of each plot is the value of its intercept.
packing structure mainly determines A*. The values of the constants for this equation are given in Table 111. As can be seen the exponent m lies between 0.09 and 0.15 and is therefore more than a factor of 2 less than the value 0.33 proposed by Knox (16), which is most likely a consequence of differing packing technique. The Quality of Column Packing. As can be seen from Table I11 E , is lower, consequently the packing density is higher, for those columns which were dry packed than is the case for those which were slurry packed. For dry packed column E, approaches the value of 0.36, calculated by Le Fevre (BO) for randomly packed spheres. The relatively high E, = 0.393 in column 3 is probably a consequence of is small aspect ratio (6), which is less than 40 (14, 37, 81-86). From the systematic v 0.45. The difference in the packing structure is not understood.
-
ACKNOWLEDGMENT Our special thanks are due to F. Hampson for extensive discussions during the preparation of the manuscript. GLOSSARY A eddy diffusion constant A‘ eddy diffusion constant A* constant A’ for “dry” eluent minus A‘for “wet” eluent AA’ B longitudinal diffusion constant B, longitudinal diffusion constant b constant constant mass transfer constant mass transfer constant C’ C, mass transfer constant AC‘ C’ for “dry” eluent minus C’ for “wet” eluent Acgdj A c ’ adjusted for K containing term D diffusion coefficient Dm D in interstitial volume Ds D in the pores dc column i.d. mean particle diameter d, height equivalent to the theoretical plate h hred h l d , reduced plate height Ah h for “dry” eluent minus h for “wet” eluent K exclusion chromatographic partition coefficient k constant L column length number averaged molecular weight Mn weight averaged molecular weight M W m constant n number of theoretical plates specific surface area 08,
2
ANALYTICAL CHEMISTRY, VOL. 53, NO. 9, AUGUST 1981
P t2
U U2 ured ve
vk
vo
tpvy" a
4
P
€2
2 Ci
EP et €2
K
urn QV
eJM4
M,/.M,, polydispersity elution time of a totally excluded inert peak time averaged linear velocity of the eluent linear velociity of a totally excluded inert sample u d p / D reduced linear velocity elution volume empty column volume elution volume of a totally penetrating inert sample pore volume pore volume available to smallest inert sample interstitial volume constant
WD)
D,/D, obstiruction factor d,/d aspect ratio v e / S k total porosity available to a polymer of a given molecular size Vp/ vk pore porosity e, + eP total porosity as measured by the smallest inert samde V,/ vk interititial porosity ep/% = VpIlJz standard deviation of the molecular weight distribution standard deviation in volume units EV caused by MWD diameter of a pore
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RECEIVED for review October 28, 1980. Accepted April 27, 1981. This paper is part of the Ph.D. Thesis of R. Groh, Universitat des Saarlandes, Saarbrucken, 1979. The authors are indebted to the Deutsche Forschungsgemeinschaft, Bad Godesberg, for financial support.
