Determination of Thermal Diffusion Coefficients Using Thermal Field

In this paper, a new approach for determination of thermal diffusion coefficient DT values using thermal field-flow fractionation retention data and M...
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Anal. Chem. 2004, 76, 2382-2386

Determination of Thermal Diffusion Coefficients Using Thermal Field-Flow Fractionation and Mark-Houwink Constants Myhuong Nguyen and Ronald Beckett*

Water Studies Centre, Department of Chemistry, Monash University, Clayton VIC 3800, Australia

In this paper, a new approach for determination of thermal diffusion coefficient DT values using thermal fieldflow fractionation retention data and Mark-Houwink constants is reported. The method utilizes the availability of Mark-Houwink constants from the literature together with thermal field-flow fractionation retention data to calculate DT values for both narrowly and broadly dispersed polymeric samples. The proposed method was tested with thermal field-flow fractionation data from a number of published papers. In general, DT results obtained from the new approach agree well with those reported from the literature. Since Mark-Houwink constants have been extensively tabulated, the new method can be used to generate a broad database of DT values for use in the characterization of polymers and in studies of the thermal diffusion phenomenon. Thermal field-flow fractionation (ThFFF) is one of only a few techniques that can measure the thermal diffusion coefficient DT for polymeric materials dissolved in an organic solvent.1-7 DT values have been successfully used in the characterization of synthetic polymers and copolymers.8 They are also much needed for the study of the thermal diffusion phenomenon in liquids, an interesting phenomenon yet still poorly understood due in part to the lack of available thermal diffusion data. While ThFFF can readily produce DT data for a range of polymer-solvent combinations using narrowly dispersed as well as broadly dispersed polymer standards,2-4,6,7,9 it requires known * To whom correspondence should be addressed. E-mail: ron.beckett@ sci.monash.edu.au. Fax: +61 3 9905 4196. (1) Schimpf, M. E. Thermal field-flow fractionation. In Field-Flow Fractionation Handbook; Schimpf, M. E., Caldwell, K., Giddings, J. C., Eds.; Wiley: New York, 2000. (2) Giddings, J. C.; Caldwell, K. D.; Myers, M. N. Macromolecules 1976, 9 (1), 106-112. (3) Schimpf, M. E.; Giddings, J. C. Polym. Prepr.-Am. Chem. Soc., Div. Fuel Chem. 1986, 27, 158-160. (4) Song, K - C.; Kim, E - K.; Chung, I.-J. Korean J. Chem. Eng. 1986, 3, 171-175. (5) Schimpf, M. E.; Giddings, J. C. Macromolecules. 1987, 20, 1561-1563. (6) Schimpf, M. E.; Giddings, J. C. J. Polym. Sci., Part B: Polym. Phys. 1989, 27, 1317-1332. (7) Nguyen, M.; Beckett, R.; Pille, L.; Solomon, D. Macromolecules 1998, 31, 7003-7009. (8) Schimpf, M. E.; Giddings, J. C. J. Polym. Sci., Part B: Polym. Phys. 1990, 28, 2673-2680. (9) Brimhall, S. L.; Myers, M. N.; Caldwell, K. D.; Giddings, J. C. J. Polym. Sci.: Polym. Phys. Ed. 1985, 23, 2443-2456.

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values of the concentration diffusion coefficient D for the polymersolvent combination under consideration. For narrowly dispersed samples, D values are usually obtained by dynamic laser light scattering (DLLS)10 or size exclusion chromatography (SEC).3,6 Using the basic ThFFF retention DT for narrowly dispersed polymeric samples can be calculated from

DT ) D/λ∆T

(1)

For broadly dispersed samples, Nguyen and Beckett7 proposed an expression for DT in which all digitized points along the ThFFF sample peak are used (see Nomenclature for definitions).

DT ) A

{

∑h (∆Tλ ) M ∑h

-(1/b)

i

i

w

i

}

b

(2)

This expression for DT is based on the assumption that DT is independent of molecular weight as accepted by the majority of ThFFF workers. The values of constants A and b in this equation were obtained by using an SEC with on-line multiangle laser light scattering (MALLS) detection.7 SEC retention time yields D values and MALLS gives the corresponding M values so that a plot of log D versus log M can be plotted, which should be linear according to the empirical expression11,12

D ) A/Mb

(3)

For studies of particle, and particle size distributions, Mes et al.13 proposed the coupling of ThFFF and on-line multiangle light scattering detection for measuring DT values. In this method, the light scattering detector provides information on the radius of gyration, which can be converted to the hydrodynamic diameter, hence the normal diffusion coefficient D. Although these approaches produce DT data, the use of MALLS is relatively expensive and requires experienced operators. In this (10) Schaefer, D.; Han, C. C. In Dynamic Light Scattering; Pecora, R., Ed.; Plenum Press: New York, 1985. (11) Flory, P. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (12) Cantor, C. R.; Schimmel, P. R. Biophysical Chemistry; W. H. Freeman and Co.: New York, 1980. (13) Mes E. P. C.; Kok W. T.; Tijssen, R. Chromatographia 2001, 53 (11-12), 697-703. 10.1021/ac035311h CCC: $27.50

© 2004 American Chemical Society Published on Web 03/20/2004

paper, we demonstrated how to obtain DT from ThFFF data and Mark-Houwink (M-H) constants, which have been extensively tabulated in the literature. The use of M-H constants in ThFFF calculations was first suggested by Gao and Chen14 in their attempt to develop a universal molecular weight calibration method for ThFFF. They made use of the relationship between M-H constants and D as expressed in the following equation15

D)

(

)

R′T 10πN0 6πηN0 3kMR+1

Table 1. Values of DT for Various Polymer-Solvent Combinations As Reported by Schimpf and Giddings6

1/3

polymer-solvent

DT × 107 (cm2 s K-1)

PS-THF PS-toluene PS-MEK PMMA-THF PMMA-toluene PMMA-MEK PRMS-toluene

1.00 1.03 1.39 1.33 1.63 1.48 1.19

(4)

Substitution of this equation into the basic ThFFF equation (eq 1) and the simple approximation for the retention ratio R applicable for high retention16-18

R ≈ 6λ

(5)

R ) A′/DT(kMR+1)1/3

(6)

gives

where

In this paper, we will show that the use of the calibration line is not necessary. Indeed, DT can be calculated directly from M-H constants and ThFFF retention data for both monodisperse and polydisperse polymeric samples. The approach was tested for a number of polymer-solvent combinations with retention data obtained from various published papers. The results of these calculations are presented and discussed. Theory Development. By comparing eqs 3 and 4, we have

A)

(

)

R′T 10πN0 6πηN0 3k

1/3

(8)

and

A′ )

R'T(10πN0/3)1/3 (πηN0∆T)

b ) (R + 1)/3

(7)

A universal calibration line was constructed as a plot of retention ratio R versus DT(kMR+1)1/3, where DT, R, and k must come from independent sources. As DT is not generally available for most polymers, Gao and Chen suggested the reverse use of their universal calibration line for determination of the DT value for unknown samples. This universal calibration line was constructed using monodisperse standards of some polymer-solvent pair(s) that had known M-H constants and DT values. For an unknown sample (but known M-H constants and molecular weight), the DT value for that particular polymer-solvent pair can be obtained from its ThFFF retention and the calibration line constructed using another polymer-solvent pair that has known DT and M-H constants. Since this approach uses values at peak maximum positions for both the construction of the calibration line and the estimation of DT values, standards as well as samples need to be narrowly dispersed. However, since most synthetic polymer samples are broadly dispersed, it would have limited applications in practice. Furthermore, accuracy may be compromised as the method employs the equation for high retention (eq 5), which is also uncorrected for temperature changes across the channel and sample cloud.

(9)

The A and b values so obtained can then be used to calculate DT for narrowly dispersed samples using eqs 1 and 3 and for broadly dispersed samples using eq 2. The ThFFF retention parameter λ at any elution point can be determined numerically from the retention ratio R using the more accurate form of eq 516-18

R ) 6λ{coth(1/2λ) - 2λ}

(10)

R ) V0/Vr ) t0/tr

(11)

where

In ThFFF, where the parabolic fluid flow profile in the channel is distorted slightly by the temperature gradient, a more complicated expression for R is often employed. 9,19-24 Temperature at the sample center of gravity Tcg (required in eq 8) can be linearly approximated by

Tcg ) Tc + ∆T λ

(12)

In this paper, for the purpose of demonstration, the proposed (14) Gao, Y.; Chen, X. J. Appl. Polym. Sci. 1992, 45, 887-892. (15) Rudin, A.; Johnston, H. K. Polym. Lett. 1971, 9, 55-55. (16) Giddings, J. C. Pure Appl. Chem. 1979, 51, 1459-1471. (17) Giddings, J. C.; Caldwell, K. D.; Kesner, L. F. In Determination of Molecular Weight; Cooper, A. R., Ed.; John Wiley & Inc.: New York, 1989; Chapter 12. (18) Kesner, L. F.; Giddings, J. C. In Lasers, Molecules and Methods; Hirschfelder, J. O., Wyatt, R. E., Coalson, R. D., Eds.; John Wiley & Sons Inc.: New York, 1989; Chapter 15.

(19) Gunderson, J. J.; Caldwell, K. D.; Giddings, J. C. Sep. Sci. Technol. 1984, 19, 667-683. (20) Schimpf, M. E.; Williams, P. S.; Giddings, J. C. J. Appl. Polym. Sci. 1989, 37, 2059-2076. (21) Van Asten, A. C.; Boelens, H. F. M.; Kok, W. Th.; Poppe, H.; Williams, P. S.; Giddings, J. C. Sep. Sci. Technol. 1994, 29, 513-533. (22) Myers, M. N.; Cao, W.; Chen, C.-I.; Kumar, V.; Giddings, J. C. J. Liq. Chromatogr., Relat. Technol. 1997, 20, 2757-2775.

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Table 2. DT Values Calculated for a Number of Narrowly Dispersed Polymer-Solvent Pairs Using Selected M-H Constantsa 14,27,28 polymer-solvent

k × 103 (cm3 g-1)

R

M

R

λ

Tcg (K)

DT × 107 (cm2 s K-1)b

110 000 470 000 900 000 200 000 670 000 107 000 400 000 107 000 400 000 107 000 330 000 107 000 330 000 107 000 400 000 139 000 380 000 139 000 380 000

0.568 0.241 0.184 0.417 0.197 0.478 0.224 0.478 0.224 0.267 0.156 0.267 0.156 0.433 0.222 0.356 0.22 0.356 0.22

0.127 0.044 0.033 0.083 0.035 0.099 0.040 0.099 0.041 0.049 0.027 0.049 0.027 0.087 0.040 0.069 0.040 0.069 0.040

300.1 296.8 296.3 297.8 296.3 299.0 296.6 298.9 296.6 297.0 296.1 297.0 296.1 298.5 296.6 297.7 296.6 297.7 296.6

0.99 (1.0) 1.22 (22.0) 1.13 (13.0) 1.37 (-1.3) 1.70 (22.5) 1.45 (9.0) 1.67 (25.6) 1.48 (11.1) 1.68 (26.6) 2.40 (47.0) 2.24 (37.4) 2.47 (51.8) 2.33 (43.0) 1.90 (28.2) 1.92 (30.0) 1.44 (20.7) 1.38 (16.2) 1.40 (18.1) 1.35 (13.3)

PS-THF

11

0.725

PS-MEK

39

0.58

PMMA-THF

10.4

0.697

7.5

0.72

7.1

0.73

8.12

0.71

PMMA-MEK

6.8

0.72

PRMS-toluene

7.81

0.73

7.06

0.744

PMMA-toluene

a ThFFF data were from Gao and Chen.14 and Giddings.6

b

Figures in parentheses are percentages and are the deviations from the values reported by Schimpf

method was tested using simplified forms of ThFFF retention expressions (eqs 1 and 10) as well as the more accurate forms of these equations, which account for temperature and carrier viscosity variations across the channel.7,22,24 It should be noted that eqs 1, 2, and 4 were derived for both lipophilic and hydrophilic polymers; therefore, the proposed approach should be applicable for both polymers. However, due to their weak retention, there is only limited ThFFF data reported for hydrophilic polymers in the literature.25 As a result, only ThFFF data for lipophilic polymers were used for method validation in this paper. Although thermal/hyperlayer FFF was reported for fast separation of ultrahigh molecular weight polymer samples in the past,26 its use is scarce and its theory is not fully developed yet, due to the complex nature of the hydrodynamic lift forces. Accordingly, calculation of DT values for thermal/hyperlayer ThFFF will not be discussed in this paper. RESULTS AND DISCUSSION ThFFF retention data from several published papers7,14,22,24 together with M-H constants listed from the literature14,27,28 were used to calculate DT for a number of polymer-solvent combinations using the approach described above in the theory development section. DT calculated using the proposed methods were then compared with those reported by Schimpf and Giddings (Table 1).6 The percentage differences between these values were included in all tables of DT (the numbers in parentheses). As M-H (23) Martin, M.; Van Batten, C.; Hoyos, M. Anal. Chem. 1997, 69, 1339-1346. (24) Cao, W.; Williams, P. S.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1999, 71, 1597-1609. (25) Kirkland, J. J.; Yau, W. W. J. Chromatogr. 1986, 353, 95-107. (26) Giddings, J. C.; Li, S. T.; Williams, P. S.; Schimpf, M. E. Makromol. Chem., Rapid Commun. 1988, 9 (12), 817-823. (27) Brandrup, J.; Immergut, E. H. Polymer Handbook, 3rd ed.; John Wiley & Sons: New York, 1989. (28) Yau, W. W.; Kirkland, J. J.; Bly, D. D. Modern Size Exclusion Chromatography; Wiley: New York, 1979.

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Table 3. DT Values Calculated for PS-Toluene Using Six M-H Pairs Listed in the Polymer Handbooka 27 k ×103 (cm3 g-1)

R

DT ×107 (cm2 s K-1)b

(a) M ) 110 000, λ ) 0.106, Tcg ) 299.2 K 17 0.69 1.17 (13.4) 7.5 0.75 1.22 (18.1) 13.4 0.71 1.17 (13.6) 7.54 0.78 1.07 (3.8) 8.45 0.748 1.18 (14.4) 10.5 0.73 1.17 (14.1) (b) M ) 670 000, λ ) 0.031, Tcg ) 296.3 K 17 0.69 1.04 (0.8) 7.5 0.75 1.04 (1.2) 13.4 0.71 1.03 (-0.3) 7.54 0.783 0.90 (-12.9) 8.45 0.748 1.01 (-1.8) 10.5 0.73 1.02 (-1.1) a ThFFF retention data was from Gao and Chen.14 While M-H constant pairs are different, resulted DT values are quite similar. b Figures in parentheses are percentages and are the deviations from the values reported by Schimpf et al.6

constants and D and DT coefficients are dependent on temperature, the sample temperature Tcg must be close to the temperature at which the M-H constants were determined. The sample molecular weight must also be within the range of molecular weights that the constants were measured. While Myers et al.,22 Cao et al.,24 and Nguyen et al.7 used retention parameters that had been corrected for the variation of thermal conductivity and solvent viscosity across the channel, Gao and Chen14 used the simplified equation (eq 5) for the calculation of their retention parameter. To improve the comparison of DT using the proposed method, the retention parameters in Gao and Chen were recalculated using eq 10. Sample temperatures were also recalculated in accordance with the new retention parameters using eq 12.

Table 4. DT Values Calculated for Narrowly Dispersed PS-THF Systems, at ∆T ) 50 °Ca M 28 500 170 600 575 000 900 000 1 290 000 28 500 170 600 575 000 900 000 1 290 000

Tc (K)

λ

Tcg (K)

DT × 107 (cm2 s K-1)b

(a) k ) 11 × 10-3 cm3 g-1, R ) 0.725 0.237 290 301 0.92 (-8.0%) 0.086 294 298 0.90 (-10.0%) 0.043 297 299 0.90 (-10.0%) 0.033 297 298 0.90 (-10.0%) 0.026 298 299 0.93 (-7.0%) (b) k ) 14 × 10-3 cm3 g-1, R ) 0.7 0.237 290 301 0.086 294 298 0.043 297 299 0.033 297 298 0.026 298 299

0.93 (-7.0%) 0.92 (-8.0%) 0.92 (-8.0%) 0.93 (-7.0%) 0.97 (-3.0%)

a ThFFF retention data was from Myers et al.22 of different MW. Figures in parentheses are percentages and are the deviations from the values reported by Schimpf et al.6

b

Table 5. DT Values Calculated for Narrowly Dispersed PS-THF Systemsa M 46 000 92 300 217 000 440 000 827 000 1 310 000 46 000 92 300 217 000 440 000 827 000 1 310 000

λ

Tcg

DT × 107 (cm2 s K-1)b

(a) k ) 11 × 10-3 cm3 g-1, R ) 0.725 0.204 301.1 1.02 (2.0%) 0.140 298.6 0.99 (-1.1%) 0.079 296.1 1.06 (6.0%) 0.050 295 1.10 (10.0%) 0.033 294.3 1.16 (16.0%) 0.024 294 1.19 (19.0%) (b) k ) 14 × 10-3 cm3 g-1, R ) 0.7 0.204 301.1 0.140 298.6 0.079 296.1 0.050 295 0.033 294.3 0.025 294

1.03 (3.0%) 1.00 (0.0%) 1.09 (9.0%) 1.13 (13.0%) 1.20 (20.0%) 1.24 (24.0%)

a ThFFF retention data was from Nguyen et al.7 of different MW. Figures in parentheses are percentages and are the deviations from the values reported by Schimpf and Giddings.6

Table 6. DT Values Calculated for Narrowly Dispersed PMMA-THF and PS-THF Systems with ThFFF Data from Cao et al.a 24 k × 103 (cm3 g-1)

11 14.0

R

DT × 107 (cm2 s K-1)a,c

(a) PS-THF System, M ) 556000, Tc ) 295.5 K, Tcg ) 299 K 0.725 1.18 (18.0) 0.70 1.21 (21.0)

DT × 107 (cm2 s K-1)b,c

1.02 (2.0) 1.05 (5.0)

11 14.0

(b) PS-THF System, M ) 1 000 000, Tc ) 295.5 K, Tcg ) 298 K m3 g-1) 0.725 1.18 (18.0) 1.02 (2.0) 0.70 1.22 (22.0) 1.05 (5.0)

10.4 7.5

(c) PMMA-THF System, M ) 570 000, Tc ) 296 K, Tcg ) 299 K 0.697 1.59 (19.5) 1.38 (3.8) 0.72 1.61 (21.0) 1.39 (4.5)

a D were obtained from retention parameters that were calculated T from reported retention ratio R using eq 10. b DT* values were obtained using corrected retention parameter as reported. cFigures in parentheses are percentages and are the deviations from the values reported by Schimpf and Giddings.6

Table 7. DT Values for Polydisperse PS-THF Systema Mw

DT × 107 (cm2 s K-1)b

(a) k ) 11 × 10-3 cm3 g-1, R ) 0.725 100 000 1.10 (10.0) 250 000 1.09 (9.0) 498 000 1.15 (15.0) 1 000 000 1.34 (34.0) (b) k ) 14 × 10-3 cm3 g-1, R ) 0.7 100 000 1.12 (12.0) 250 000 1.12 (12.0) 498 000 1.19 (19.0) 1 000 000 1.40 (40.0) a ThFFF retention data was from Nguyen and Beckett.7 b Figures in parentheses are percentages and are the deviations from the values reported by Schimpf Giddings.6

b

Retention parameters and sample temperature from Myers et al. and Cao et al. were used as reported. For the data in Nguyen et al., retention parameters were used as reported, but the sample temperature, which they assumed to be the same as the channel cold wall temperature (Tc), was adjusted by the approximation of eq 12. The results shown in Tables 2-6 are for narrowly dispersed samples. For broadly dispersed samples, retention data from Nguyen et al. were used and the results are listed in Table 7. For most polymer-solvent combinations (PS-THF, PStoluene, PS-MEK, PRMS-toluene), the DT values obtained using the new method agree well with those reported in the literature with the difference between the calculated DT and literature values being less than 20% in most cases. However, the DT values obtained for PMMA, especially the PMMA-toluene pair, tend to have higher deviation from the literature values than other polymers. The deviation may be due to either the M-H constants or the ThFFF retention data. As two different pairs of M-H constants were used, but very similar DT values resulted, it would be more likely the differences are due to the ThFFF data. In Table

2, the ThFFF data for PMMA were not corrected for temperature variation across the channel. As shown later (Table 6), the temperature correction significantly reduces the error for the PMMA-THF pair. Correction parameters for PMMA-toluene are not available in the literature; thus, it is not possible to verify whether the temperature variation has stronger effects on toluene than on THF. Table 3 illustrates that although there are differences in the M-H constants for a particular polymer-solvent combination reported in the literature, very similar values of DT for the PStoluene system were obtained when using each pair of the constants. The effect of molecular weight on DT is probably not significant (Tables 2, 4, and 6), although a small increase in DT as molecular weight increases was observed (Tables 5 and 7) for the PS-THF combination using data reported by Nguyen et al. This trend is consistent with that reported in the original paper,7 which provided a detailed discussion of this observation. According to Nguyen and Beckett,7 the trend is due to the greater value of b (mass (29) Ko, G. H.; Richards, R.; Schimpf, M. E. Sep. Sci. Technol. 1996, 31(8), 10351044.

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selectivity) obtained by ThFFF than the commonly accepted value of ∼0.55 for polystyrene-THF combination. Ko et al.29 stated that the greater value of ThFFF mass selectivity b is due to the fact that D is effected by temperature more strongly than DT. In ThFFF, the Tcg of higher MW samples is smaller than that of lower MW samples under the same field condition; consequently, the mass selectivity b of ThFFF is not only due to molecular weight differences but is also enhanced by the temperature differences in sample temperature Tcg. When the temperature effect on DT/D is removed, the b value is reduced; hence, the trend of increase in DT value for higher MW samples will be diminished. Comparison of the DT values for the broadly dispersed samples (Table 7) to those for the narrowly dispersed samples (Table 5) shows good agreement between these two data sets. These values were obtained using the same ThFFF channel. Thus, the new method has proved to be reliable for polydisperse samples as well. Calculation errors can generally be reduced by using more accurate retention expressions that are corrected for the influence of temperature on the parameters involved. The data in Table 6 illustrate the improvement in DT obtained by including temperature correction in the calculation. Excellent agreement was achieved between literature DT values and those obtained with the M-H method using more accurate retention calculations. It appears that the larger DT values obtained in Table 2 were due to the use of simplified ThFFF retention calculations. In summary, the results validate the proposed method for determination of DT using M-H constants and ThFFF retention data. When temperature-corrected ThFFF retention parameters were employed, excellent agreement was achieved between literature DT values and those calculated using M-H constants. Even when using simplified retention equations, reasonable DT values were obtained for most polymer-solvent pairs; however, in this case, the DT values were usually slightly overestimated. CONCLUSIONS The proposed method for calculating DT of polymer-solvent systems developed in this paper is applicable for both narrowly and broadly dispersed samples. The method does not require any other information except the ThFFF fractograms and the availability of the MH constants values, which have been extensively tabulated. When temperature-corrected retention parameters were used, excellent agreement between calculated and literature DT values was achieved. For retention parameters obtained using the simplified retention equations, as commonly practiced in previous research, DT values tend to be overestimated.

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PS PMMA PRMS THF MEK M-H FFF ThFFF SEC MALLS DLLS values

GLOSSARY polystyrene poly(methyl methacrylate) poly(R-methyl)styrene tetrahydrofuran methyl ethyl ketone Mark-Houwink constants field-flow fractionation thermal field-flow fractionation size exclusion chromatography multiangle laser light scattering dynamic laser light scattering if not indicated, were at 298 K

Symbols DT D k R R′ T Tc Tcg ∆T η ηTHF ηtoluene ηMEK M Mw A, b λ R Vr V0 tr t0 hi

thermal diffusion coefficient (cm3 s K-1) concentration diffusion coefficient (cm2 s) Mark-Houwink constant (cm3 g-1) Mark-Houwink constant gas constant (8.314 J K-1 mol-1) (sample) absolute temperature (K) temperature at the cold wall of ThFFF channel (K) temperature at sample center of gravity (K) temperature field (K) solvent viscosity (mN s m-2) 0.46 0.556 0.42 (at T ) 293 K) molecular weight weight average molecular weight constant in eq D ) A/Mb ThFFF retention parameter ThFFF retention ratio ThFFF retention volume (mL) ThFFF void (channel) volume (mL) ThFFF retention time (s) ThFFF void time (s) detector signals at digitized point ith (arbitrary unit)

Received for review November 6, 2003. Accepted February 16, 2004. AC035311H