Anal. Chem. 1999, 71, 1597-1609
Thermal Field-Flow Fractionation Universal Calibration: Extension for Consideration of Variation of Cold Wall Temperature Wenjie Cao,† P. Stephen Williams,‡ Marcus N. Myers,*,§ and J. Calvin Giddings|
Field-Flow Fractionation Research Center, Department of Chemistry, University of Utah, Salt Lake City, Utah 84112
The well-known size exclusion chromatography (SEC) universal calibration allows the characterization of a polymer sample using a set of polymer standards of a different polymer family. This is a polymer-transferable calibration method. A more “universal” calibration method has been proposed for thermal field-flow fractionation (ThFFF) where the calibration constants closely describe the ratio of ordinary (molecular) diffusion coefficient D to thermal diffusion coefficient (or, more correctly, thermophoretic mobility) DT. The universal calibration for ThFFF is therefore instrument or system transferable. Systematic experiments have previously been carried out to verify this calibration method for a wide range of conditions. Results showed that the method holds well provided experimental conditions such as sample size and cold wall temperature Tc are the same for each system. This paper quantifies the effect of cold wall temperature on polymer retention in ThFFF and on the universal calibration constants for the polystyrene-tetrahydrofuran, poly(methyl methacrylate)-tetrahydrofuran, and polyisoprene-tetrahydrofuran systems, respectively. The universal calibration for ThFFF is thereby extended to account for variation of cold wall temperature. The extended calibration will allow for accurate data retrieval from analyses obtained at any cold wall temperature. It will even allow for accurate information retrieval when cold wall temperature is not held constant during sample elution, such as may occur with a change in coolant flow rate or with a programmed decay of ∆T. Field-flow fractionation (FFF) is a family of analytical techniques that exploit the plane Poiseuille velocity profile of a flow of fluid through a thin, parallel-walled channel together with a field applied perpendicularly to this flow (across the space between the parallel channel walls) to separate and characterize dissolved or suspended materials such as polymers, submicrometer particles, and colloids. The concept of FFF was put forward by † Present address: Huntsman Chemical, Huntsman Polymers Corp., P.O. Box 3986, Odessa, TX 79760. ‡ Present address: Department of Chemistry and Geochemistry, Colorado School of Mines, Golden, CO 80401. § Present address: FFFractionation, LLC, 4797 South West Ridge Blvd., Salt Lake City, UT 84118. | Deceased: October 24, 1996.
10.1021/ac981094m CCC: $18.00 Published on Web 03/04/1999
© 1999 American Chemical Society
Giddings in 1966,1 and the various FFF techniques have been developed mainly by Giddings and colleagues thereafter (see, for example, ref 2). These techniques hold some advantage over chromatographic methods in that no packing material or stationary phase is required to detain sample materials relative to the carrier fluid. There is no requirement for sample molecules to partition between phases. Sample particles or molecules need not adsorb at an interface or diffuse in to and out of the pores of a static solid phase. In chromatographic systems, the packing materials or stationary phase may interact adversely with sample materials. For example, particles or molecules may irreversibly adsorb to surfaces, and polymer molecules may encounter strong shear forces that can rupture their chains. Such adverse interactions are much less likely in FFF. In FFF, sample materials are detained relative to the mean fluid velocity by driving them into slower moving streamlines close to one of the parallel channel walls. The majority of the sample molecules or particles migrate through the FFF channel without once encountering or interacting with a surface. They remain in solution or suspension at all times. This is in contrast to chromatographic migration where essentially all molecules or particles must interact with surfaces or interfaces many times during their elution. The FFF family is divided into several different categories, or FFF techniques, depending on the nature of the applied field.2 The technique that makes use of a thermal field, implemented by maintaining a temperature gradient across the channel thickness, is known as thermal FFF (ThFFF). A gravitational or centrifugal field is used in the technique known as sedimentation FFF (SdFFF). A secondary flow of carrier fluid across the channel thickness, requiring the channel walls to be permeable, is used in the technique of flow FFF (FlFFF). An electrical potential difference is maintained across the channel thickness in electrical FFF (ElFFF). Techniques that make use of other types of field are also possible. ThFFF holds many advantages over size exclusion chromatography (SEC) for the separation and characterization of polymers. For example, ThFFF is a better separation technique than SEC for ultrahigh-molecular-weight polymers that are susceptible to shear degradation.3,4 The flow of polymer molecules in solution (1) Giddings, J. C. Sep. Sci. 1966, 1, 123-125. (2) Giddings, J. C. Science 1993, 260, 1456-1465. (3) Giddings, J. C. In Advances in Chromatography; Giddings, J. C., Grushka, E., Cazes, J., Brown, P. R., Eds.; Marcel Dekker: New York, 1982; Vol. 20, Chapter 6, pp 217-258.
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through the packed bed of an SEC column subjects them to extensional shear stress3 and stresses associated with the entangling and locking together of polymer loops.5 Such stresses leading to polymer degradation do not occur in the open FFF channel. On the other hand, low-molecular-weight polymers tend to be better separated by SEC because of difficulties in obtaining sufficient retention for them in the FFF channel. The open channel of ThFFF is also much less susceptible to plugging than a packed SEC column. In fact, ThFFF has been found to be capable of separating particles and colloids6-8 in addition to polymers. It has also been successfully used to measure microgel concentrations in polymer solutions.9 In the case of SEC, these microgels must be removed by filtration prior to injection onto the column and it is therefore not possible to retain the original content of the sample for the analysis. More recently, it has been shown that the polymeric and particulate components of acrylonitrile-butadienestyrene (ABS) plastics may be characterized in a single ThFFF analysis.10 The adjustable field strength (temperature gradient) of ThFFF allows a system to be tuned to the conditions suitable for separation of widely differing samples. Thermal FFF has been used to separate polymers with molecular weights ranging from 1 × 104 to more than 2 × 107.4 Samples may contain components with a wide range of size or molecular weight that require differing field strengths for their efficient separation. For these samples, the temperature gradient may be programmed with time.11-13 Neither SEC nor ThFFF is an absolute polymer characterization technique. Polymer molecular weight cannot be obtained directly from the measured retention volumes without calibrating the system with polymer standards whose molecular weights have been determined previously by some other absolute technique such as light scattering or osmometry. This requirement for calibration may be eliminated by coupling the SEC or ThFFF system to a multiangle light-scattering (MALS) detector14 which yields molar mass information without reference to standards.15 This coupling method has the disadvantages of high cost, however, and calibration remains the most common approach. The purpose of calibration in both SEC and ThFFF is to establish the relationship between molecular weight and retention volume for a set of polymer standards. This relationship is (4) Gao, Y.; Caldwell, K. D.; Myers, M. N.; Giddings, J. C. Macromolecules 1985, 18, 1272-1277. (5) McIntyre, D.; Shih, A. L.; Savoca, J.; Seeger, R.; MacArthur, A. In Size Exclusion Chromatography: Methodology and Characterization of Polymers and Related Materials; Provder, T., Ed.; ACS Symposium Series 245; American Chemical Society: Washington, DC, 1984; Chapter 15, pp 227240. (6) Liu, G.; Giddings, J. C. Chromatographia 1992, 34, 483-492. (7) Shiundu, P. M.; Liu, G.; Giddings, J. C. Anal. Chem. 1995, 67, 2705-2713. (8) Shiundu, P. M.; Giddings, J. C. J. Chromatogr., A 1995, 715, 117-126. (9) Lee, S. In Chromatography of Polymers: Characterization by SEC and FFF; Provder, T., Ed.; ACS Symposium Series 521; American Chemical Society: Washington, DC, 1993; Chapter 6, pp 77-88. (10) Shiundu, P. M.; Remsen, E. E.; Giddings, J. C. J. Appl. Polym. Sci. 1996, 60, 1695-1707. (11) Giddings, J. C.; Smith, L. K.; Myers, M. N. Anal. Chem. 1976, 48, 15871592. (12) Kirkland, J. J., Yau, W. W. Macromolecules 1985, 18, 2305-2311. (13) Giddings, J. C.; Kumar, V.; Williams, P. S.; Myers, M. N. In Polymer Characterization by Interdisciplanary Methods; Craver, C. D., Provder, T., Eds.; ACS Advances in Chemistry 227; American Chemical Society: Washington, DC, 1990; Chapter 1. (14) Wyatt, P. J. Anal. Chim. Acta 1993, 272, 1-40. (15) Wyatt, P. J. LC-GC 1997, 15, 160-168.
1598 Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
subsequently to be applied to the interpretation of retention data for unknown samples. For ThFFF, calibration with a set of polymer standards of the same type (chemical as well as structural nature) as the sample is required, and of course, the standards and the sample have to be eluted by the same solvent and, prior to this work, under the same temperature conditions. This was also the case for SEC at first. However in 1967, Grubisic et al.16 made a significant breakthrough for SEC calibration. They found it is possible to apply the calibration constants obtained for a set of polymer standards of some arbitrary type to polymer samples of any other type having the same chain structure. It is required that the dependence of intrinsic viscosity on molecular weight (as described by the Mark-Houwink constants) be known for the two polymer types in the solvent being used as carrier and that the unknown sample be run at the same temperature as the calibration. This polymer-transferable SEC calibration method is referred to as “universal calibration”. The advantage of this universal calibration is that well-characterized, narrowly distributed polymer standards may be used for all analyses. It takes care of the situation where polymer standards are not available for the polymer of interest, although the relevant Mark-Houwink constants must still be known. The requirement that the temperature for calibration and sample runs be the same is not a great drawback for SEC. (It is of course necessary to use Mark-Houwink constants consistent with the common temperature.) Thermal FFF however may exploit changes in hot and cold wall temperatures to control sample retention and resolution. The current design of ThFFF system utilizes a flow of coolant (often cold water from the main water supply) to remove heat from the cold wall and maintain the temperature gradient between it and the electrically heated hot wall. The temperature of the cold wall Tc is therefore dependent on the temperature and the flow rate of the coolant and the temperature of the hot wall Th. (For the very thin channels used in FFF, the flow rate of the carrier solvent will have negligible influence.) Any fluctuation in water pressure or temperature will therefore influence Tc. If the temperature gradient is programmed with time without any compensation in circulating flow rate, then Tc will decrease as Th decreases. A full calibration of the polymersolvent system, as carried out in this work, may be used to account for any temperature conditions extant during a sample elution. It will simply be required that both Tc and Th are recorded at short time intervals along with the channel flow rate. It must be emphasized that the SEC universal calibration method is not column transferable. The pore characteristics of SEC packing materials are complex and may even change with time during use. Columns must therefore be recalibrated periodically and also must be recalibrated if any of the experimental parameters, such as solvent type or temperature, is changed. SEC universal calibration also requires the information concerning the polymer samples to be well defined. If the structure and conformation are unknown, the method cannot be applied without accepting the possibility of error in the results.15 Universal calibration for ThFFF is quite different from that for SEC. The concept of a universal calibration for ThFFF was (16) Grubisic, Z.; Rempp, P.; Benoit, H. J. Polym. Sci., B: Polym. Lett. 1967, 5, 753-759.
proposed by Gao and Chen,17 but the potential breadth of its application (with regard to system and operating condition transferability) was addressed later by Giddings.18 It depends on the fact that the open FFF channel bounds a simple and predictable fluid flow pattern, and elution in the normal mode of FFF is therefore governed by fundamental physical processes. The calibration constants determined for a given polymer-solvent combination describe the molecular weight and temperature dependence of certain physicochemical constants. It requires standards of the same type as the unknown polymer, but once determined, these calibration constants should be instrument and channel transferable. The calibration is therefore truly universal although applicable only to a specific polymer-solvent combination. RETENTION THEORY Influence of Temperature. In the normal mode of FFF, sample molecules are driven across the channel thickness toward the so-called accumulation wall by their interaction with the applied field. There is a consequent buildup of concentration near this wall which results in a back diffusion, opposing the influence of the field. In the ideal or standard model, the field-driven velocity is constant across the channel thickness for each component of the sample (although this velocity may differ for the different components), and the coefficient for back diffusion is also independent of position across the thickness (i.e., diffusion coefficient D is assumed to be independent of concentration over the range found within the channel, and there is no other positiondependent local condition that has an influence on D). The net result is a concentration profile that decays exponentially from a maximum value at the accumulation wall:
c(x) x/w ) exp c0 λ
(
)
(1)
in which c(x) is the concentration at distance x from the accumulation wall, c0 is the concentration corresponding to x ) 0, w is the channel thickness, and λ is the so-called retention parameter, given by
λ ) D/|u|w
(2)
where D is the constant ordinary diffusion coefficient, and |u| is the constant field-driven velocity toward the accumulation wall. In the standard model, the fluid velocity profile between the parallel walls is parabolic, and the retention equation is then given by2
R)
V0 1 - 2λ ) 6λ coth Vr 2λ
(
)
(3)
where R is the retention ratio, V0 is the void (channel) volume, and Vr is the retention volume, or the volume of fluid required to carry the sample component through the channel from inlet to outlet. (17) Gao, Y.; Chen, X. J. Appl. Polym. Sci. 1992, 45, 887-892. (18) Giddings, J. C. Anal. Chem. 1994, 66, 2783-2787.
ThFFF deviates from the standard model in several ways. Most importantly, the fluid velocity profile deviates from the symmetric parabolic profile of the standard model. This is because the fluid viscosity varies with the temperature across the channel thickness. In this case, the profile may be approximated by a third-order polynomial equation of the form19
[
( )]
x 2 x x + 2ν ν(x) ) 6〈ν〉 (1 + ν) - (1 + 3ν) w w w
()
3
(4)
where ν may be obtained by equating the shear rate at the accumulation wall (6〈ν〉(1 + ν)/w) with that predicted from a rigorous treatment of the velocity profile.20-23 The approach is justified by the fact that the retained polymers are concentrated in the region of the accumulation wall during their elution. The details of calculation of ν are given in ref 24. If the concentration profile in ThFFF conformed to eq 1 with λ independent of x/w, the retention equation would be given by19
R ) 6λ{ν + (1 - 6λν)[coth(1/2λ) - 2λ]}
(5)
This is not the case however. The ordinary diffusion coefficient D is strongly temperature dependent, as described by the StokesEinstein equation:
D ) kT/6πηRh
(6)
where k is the Boltzmann constant, T is the absolute temperature, η is the fluid viscosity, which decreases with increase of T, and Rh is the hydrodynamic radius of the polymer molecule in solution, which tends to increase somewhat with temperature.25 In a temperature gradient, the induced velocity |u| is given by
|u| ) DT (dT/dx)
(7)
where DT is the thermal diffusion coefficient, or more correctly the thermophoretic mobility, and dT/dx is the local temperature gradient. The direction of induced motion is generally toward the colder wall. Local temperature gradient is inversely dependent on the local thermal conductivity of the fluid, which varies with the local temperature. Therefore, dT/dx varies across the channel thickness.20 This is a relatively small effect however. For example, the thermal conductivity for ethylbenzene changes less than 10% between 293 and 343 K.21 Studies of the temperature dependence of thermal diffusion (see, for example, ref 21) have shown that DT increases quite strongly with T, although not as strongly as (19) Martin, M.; Giddings, J. C. J. Phys. Chem. 1981, 85, 727-733. (20) Gunderson, J. J.; Caldwell, K. D.; Giddings, J. C. Sep. Sci. Technol. 1984, 19, 667-683. (21) Brimhall, S. L.; Myers, M. N.; Caldwell, K. D.; Giddings, J. C. J. Polym. Sci.: Polym. Phys. Ed. 1985, 23, 2443-2456. (22) 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. (23) Belgaied, J. E.; Hoyos, M.; Martin, M. J. Chromatogr., A 1994, 678, 8596. (24) Schimpf, M. E.; Williams, P. S.; Giddings, J. C. J. Appl. Polym. Sci. 1989, 37, 2059-2076. (25) Novotny, V. J., J. Chem. Phys. 1983, 78, 183-189.
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D. Theories for the thermal diffusion effect differ greatly in their conceptual basis and often predict contradictory results.26 However, those theories of Emery and Drickamer,27 Ham,28 and Khazanovich29 all predict DT to be proportional to some diffusion coefficient divided by the square of absolute temperature. The different theories require this diffusion coefficient to refer to the polymer in solution, to the self-diffusion of the solvent, or to a polymer segment, respectively. Combining eqs 2 and 7, we see that λ is given by
will approach Tc. However, it is not immediately obvious how in general λapp may be assigned to a particular xeq and, hence, Teq. It has been commonly assumed that λapp corresponds to the local value for λ at the center of gravity of the zone across the channel thickness or, to a good approximation, at a distance of λappw from the accumulation wall.21,26,33-38 However, for a model where λ is assumed to vary linearly with x, Martin et al.32 have obtained the following approximate empirical equation relating xeq/w to λapp and ν.
λ ) D/DT(dT/dx)w
xeq/w ) 2λapp - 2.1365(1 + 2ν)λapp2 - 6.1678(2 - ν)λapp3
(8)
(11) and is therefore related to the ratio of D/DT. (Note that a rigorous derivation of the expression for λ would account for thermal expansion of the solution across the channel thickness (see ref 30). Following common practice, we have ignored this relatively small effect.) The various diffusion coefficients referred to in the different theories for DT may be expected to have similar temperature dependencies, and it follows that D/DT would in each case be expected to vary approximately with the square of absolute temperature. Such a variation has in fact been confirmed by Whitmore31 for polystyrene in both the poor solvent cyclohexane and the good solvent toluene. Therefore we expect
D/DT ) (D/DT)298(T/298.15)m′
(9)
where (D/DT)298 is the ratio of normal to thermal diffusion coefficients at 298.15 K and m′ is expected to have a value of ∼2. Considering the relatively small dependence of dT/dx on local temperature, λ may be expected to vary across the channel thickness approximately in parallel with D/DT. According to the theories for thermal diffusion mentioned above, λ may therefore be expected to vary with the local temperature raised to some power of ∼2. It follows that, for any given value of R, the solution of eq 5 yields some weighted mean value for λ. This apparent value λapp must be equal to the local value for λ at some point within the thickness of the zone. Martin et al.32 defined this point as the equivalent point and its distance from the accumulation wall as xeq. It follows that
λapp )
( ) D DT
1 (dT/dx) eq eqw
(10)
where (D/DT)eq is the ratio of D to DT at the local temperature Teq of the equivalent position xeq and (dT/dx)eq is the local temperature gradient at xeq. It is apparent that as retention increases and λapp decreases then xeq will approach zero and Teq (26) Schimpf, M. E.; Giddings, J. C. J. Polym. Sci., Part B: Polym. Phys. 1989, 27, 1317-1332. (27) Emery, A. H., Jr.; Drickamer, H. G. J. Chem. Phys. 1955, 23, 2252-2257. (28) Ham, J. S. J. Appl. Phys. 1960, 31, 1853-1858. (29) Khazanovich, T. N. J. Polym. Sci., Part C: Polym. Symp. 1967, 16, 24632468. (30) Hovingh, M. E.; Thompson, G. H.; Giddings, J. C. Anal. Chem. 1970, 42, 195-203. (31) Whitmore, F. C. J. Appl. Phys. 1960, 31, 1858-1864. (32) Martin, M.; Van Batten, C.; Hoyos, M. Anal. Chem. 1997, 69, 1339-1346.
1600 Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
Interestingly, in the limit of high retention, it may be seen that xeq/w converges to 2λapp rather than to λapp. This improved approach has recently been applied to experimental data.39,40 It was shown40 that while the classical approach based on assigning xeq/w to λapp resulted in an apparent molecular weight dependence of DT for a certain polymer-solvent system, the improved approach based on the empirical equation of Martin et al.32 resulted in DT that was independent of molecular weight. This improvement in the method of data reduction can apparently have significant consequences. Influence of Molecular Weight. The ordinary diffusion coefficient for a polymer molecule in solution at some given temperature can be related to its molecular weight by the equation,41
D ) AM-b
(12)
where A and b are constants reflecting the polymer chain stiffness in the solvent and the interaction between the polymer chain and the solvent. Some studies have found DT for many polymer-solvent systems to be independent of molecular weight (see, for example, refs 26, 33, and 42), as predicted by Brochard and de Gennes.43 Other studies claim to have found a weak molecular weight dependence in some systems (see, for example, refs 44 and 45). (33) Giddings, J. C.; Caldwell, K. D.; Myers, M. N. Macromolecules 1976, 9, 106-112. (34) Song, K.-C.; Kim, E.-K.; Chung, I.-J. Korean J. Chem. Eng. 1986, 3, 171175. (35) van Asten, A. C.; Venema, E.; Kok, W. Th.; Poppe, H. J. Chromatogr. 1993, 644, 83-94. (36) van Asten, A. C.; Kok, W. Th.; Tijssen, R.; Poppe, H. J. Chromatogr., A 1994, 676, 361-373. (37) van Asten, A. C.; Kok, W. Th.; Tijssen, R.; Poppe, H. J. Polym. Sci., Part B: Polym. Phys. 1996, 34, 283-295. (38) van Asten, A. C.; Kok, W. Th.; Tijssen, R.; Poppe, H. J. Polym. Sci., Part B: Polym. Phys. 1996, 34, 297-308. (39) Van Batten, C.; Hoyos, M.; Martin, M. Chromatographia 1997, 45, 121126. (40) Myers, M. N.; Cao, W.; Chien, C. I.; Kumar, V.; Giddings, J. C. J. Liq. Chromatogr. Relat. Technol. 1997, 20, 2757-2776. (41) Flory, P. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (42) Schimpf, M. E.; Giddings, J. C. Macromolecules 1987, 20, 1561-1563. (43) Brochard, F.; de Gennes, P.-G. C. R. Seances Acad. Sci., Ser. 2 1981, 293, 1025-1027. (44) Meyerhoff, G.; Rauch, B. Makromol. Chem. 1969, 127, 214-223. (45) Gaeta, F. S.; Perna, G.; Scala, G. J. Polym. Sci.: Polym. Phys. Ed. 1975, 13, 203-222.
For a given temperature, we may therefore assume the following relationship between DT and molecular weight,
DT ) BMβ
(13) λappθc∆T ) φcM-n
where B and β are constants dependent on the polymer-solvent combination and β tends to be small or possibly zero. Combining eqs 12 and 13 we see that
D/DT ) (A/B)M-(b+β) ) φ′M-n′
(14)
where φ′ ) A/B and n′ ) b + β. From the previous discussion, we can see that φ′ must be dependent on temperature (see eq 9). Any temperature dependence of n′ must be determined from experiment. UNIVERSAL CALIBRATION Molecular Weight Dependence. It was stated earlier that the purpose of calibration in both ThFFF and SEC is to establish the relationship between polymer molecular weight and retention volume. It was stressed earlier that for both techniques it has been a requirement that the unknown sample be eluted under the same temperature conditions as the standards. Under these conditions, the calibration for ThFFF is based upon a consideration of eqs 10 and 14. Combining the equations we see that
1 λapp ) φ′eqM-n′ (dT/dx)eqw
(15)
where φ′eq is the value of φ′ at temperature Teq. (If n′ is found to be temperature dependent, then its value in eq 15 must be consistent with temperature Teq.) Now even if Tc and the temperature drop across the channel thickness ∆T are held absolutely constant, xeq and Teq will be functions of M simply because λapp is a function of M. It has been pointed out46 that dT/ dx will vary little in the region close to the cold wall where the sample components are concentrated. We may therefore substitute (dT/dx)c (i.e., the value of dT/dx at the cold wall) for (dT/dx)eq without incurring significant error. The temperature gradient at the cold wall is given by20
( ) ( dT dx
range with change in M, and in a smooth and monotonic manner. If eq 17 holds, we can therefore expect to obtain the good empirical relationship
)
1 dκ ∆T ∆T ∆T ) 1+ ) θc κ dT 2 w w c c
λappθc∆T ) φ′eqM
(17)
We mentioned above that Teq will vary with M. It follows that φ′eq would have a different value for each of the standards. A calibration plot is not therefore directly obtainable using an equation of the form of eq 17. However, the Teq will vary over a relatively small (46) Sisson, R. M.; Giddings, J. C. Anal. Chem. 1994, 66, 4043-4053.
where φc and n are empirical constants corresponding to cold wall temperature Tc. The empirical constant n will differ only slightly from the physicochemical constant n′, with n tending to be a little larger than n′. This is because as M increases D/DT decreases according to eq 14, and there is an additional decrement to D/DT due to the lower Teq associated with the higher and more strongly retained M (see eq 9). The net result is an enhanced mass selectivity. This effect was recently discussed by Ko et al.47 The θc factor of eq 18 is often ignored, the effective result being that it is incorporated into the empirical constant φc. Since θc is a function of ∆T, the resulting value for φc would then be expected to have a dependence on ∆T. For typical experimental conditions, θc fortunately does not vary greatly with ∆T, however. The influence of cold wall temperature on retention, and the importance of using the same Tc for calibration and sample runs, were the subjects of a recent study.40 The θc factor was ignored in the calibration procedure considered in this work, and yet the temperature difference ∆T (ranging from 50 to 80 K) was shown to have little influence on the empirical calibration constants obtained for some fixed Tc. This does tend to justify the practice, although the significance of θc depends on the relative change in κ across the channel thickness. Wider ranges of ∆T may show the need to include θc in the calibration in order to obtain consistency in θc. Some solvents may also exhibit significantly greater changes in thermal conductivity with temperature than others, and for these the inclusion of θc would be more critical. The modified form of the calibration function, as recommended by Giddings,18 follows from eq 18 (where here we retain the θc factor) and is given by
λappθc∆T ) φk(M/10k)-n
(19)
φk ) φc(10)-kn
(20)
where
(16)
where κc is the thermal conductivity at temperature Tc, dκ/dT is the rate of change of κ with T, and θc represents the quantity in parentheses. From eqs 15 and 16 we obtain the following relationship -n′
(18)
and φk has an implied association with a certain cold wall temperature Tc. The value of k is chosen to avoid long extrapolation to obtain the empirical constant φk. The universal calibration constants φk and n are obtained from the intercept and slope of a plot of log(λappθc∆T) versus log(M/10k), as given by
log(λappθc∆T) ) log(φk) - n log(M/10k)
(21)
There are no references to channel dimensions in the calibration equation above. It was mentioned earlier that the empirical constant n tends to differ only slightly from the physicochemical constant n′. This would also be true of constants φk and φ′k, where (47) Ko, G.-H.; Richards, R.; Schimpf, M. E. Sep. Sci. Technol. 1996, 31, 10351044.
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the latter is a physicochemical constant corresponding to φ′c(10)-kn′ (where φ′c is the value of φ′ at temperature Tc), provided the value for k does avoid long extrapolation. In fact, the differences between n and n′, and between φk and φ′k would tend to be duplicated from system to system. It follows that φk and n may be regarded as universal calibration constants, applicable to any instrument and channel, provided the same cold wall temperature is used. Temperature Dependence. The development of the calibration procedure described above was based upon a consideration of eqs 10 and 14. Consider now eqs 9 and 10 for some fixed M. Substituting eq 9 into eq 10 results in
1 λapp ) (D/DT)298(Teq/298.15)m′ (dT/dx)eqw
(22)
Following the same arguments as used earlier concerning the small variation of dT/dx close to the cold wall, and making the appropriate substitution, we obtain
λappθc∆T ) (D/DT)298(Teq/298.15)m′
(23)
The equivalent temperature Teq is related to Tc via the approximate equation
Teq ) Tc + (dT/dx)cxeq
(24)
and at high retention (following the model of Martin et al.32) this reduces to
Teq ) Tc + 2λappθc∆T
(25)
in which θc is a function of both Tc and ∆T. For the purposes of calibration, it is not useful to relate retention to Teq which, under constant conditions, varies with the molecular weight. (This point will be discussed in detail later.) Equation 25 shows that Teq is strongly correlated with Tc, however. From a consideration of eqs 18, 19, and 23, assuming orthogonal molecular weight and temperature dependence, we may expect an empirical equation of the form
(
λappθc∆T ) φ298
)
(
Tc m -n Tc M ) φk,298 298.15 298.15
)( ) m
M 10k
-n
(26) to be a good descriptor of retention. The power m would be expected to differ slightly from the m′ of eq 23, and the empirical constants φ298 and φk,298 are referenced to a standard temperature of 298.15 K, rather than some arbitrary cold wall temperature. The extended universal calibration equation for ThFFF that accounts for variation of Tc is seen to require just the three empirical constants φk,298, m, and n. Experiments were carried out to determine the applicability of this empirical calibration approach to different polymer-solvent systems. These are described below. EXPERIMENTAL SECTION Newly built ThFFF channels (in-house codes 20 and 21) were used for the experiments reported here. These channels exhibited 1602 Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
Table 1. Molecular Weights of Polymer Standards and Their Nominal Polydispersities
PMMA
PS
PI
Mp (×10-3)
Mw (×10-3)
Mn (×10-3)
Mw/Mn
37 100 280 570 34.3 99.4 273 556 1000 114 210 293 1050
37 97 270 570 33.5 96 257 546 944 108 208 293 963
35 91 255 550 32.6 92.4 245 536 880 104 205 287 860
1.04 1.04 1.05 1.03 1.03 1.03 1.05 1.02 1.07 1.04 1.02 1.02 1.12
a more uniform temperature along their length than earlier models. With these instruments, both ∆T and Tc were controllable to within (1 K. For all experiments reported here, ∆T was maintained at 50 K. The channels had dimensions of length L ) 33.5 cm (tip-tip), breadth b ) 2.0 cm for the polystyrene (PS) experiments (channel 20) and 1.6 cm for the poly(methyl methacrylate) (PMMA) and polyisoprene (PI) experiments (channel 21), and nominal thickness of the Teflon-coated polyimide spacer w ) 127 µm. An SSI II (FFFractionation, LLC, Salt Lake City, UT) pump was used to deliver carrier (THF) at the flow rate of 0.1 mL/min for all experiments reported. A Varex evaporative light-scattering detector, model ELSD IIA (Varex Corp., Burtonsville, MD), was used to detect void and polymer standard peaks. The detector signal was recorded by an Omniscribe chart recorder (Houston Instrument, Austin, TX) and an IBM-compatible PC. In-house data collection and processing software was used. PS, PMMA, and PI polymer standards were obtained from Polymer Standards Service-USA (Silver Spring, MD). Their nominal parameters are listed in Table 1. HPLC-grade tetrahydrofuran (THF) was obtained from Fisher Scientific (Fair Lawn, NJ) and used as solvent for the polymeric samples and as carrier. Coefficients for the polynomial describing the temperature dependence of THF viscosity and the values for κ at 293 K and dκ/ dT are listed in ref 22. Sample stock solutions (of concentration 1 mg/mL) were made up at least 24 h before the experiment to ensure good solution. A small amount (∼0.001 mg/mL) of the antioxidant tetrakis[methylene(3,5-di-tert-butyl-4-hydroxyhydrocinnamate)]methane (commercially supplied as Irganox 1010 by Ciba-Geigy Corp., Hawthorne, NY) was added to the stock solutions. This not only inhibits THF peroxide formation and polymer degradation but also gives a good signal for void peak detection. The position of the void peak indicates the retention volume of a nonretained material, which is required for data processing. The stock solutions were diluted to between 0.05 and 0.20 mg/ mL before injection. A Rheodyne (Cotati, CA) model 7125 syringe loading sample injector was used for sample injection. This had a sample loop volume of 20 µL so that the absolute sample mass injected was between 1 and 4 µg. Sample mass corresponded to 2 µg for each of the PMMA standards, 1 µg for the PS standards, and 4 µg for the PI standards; the differing required sample mass being a result of their different detectabilities.
regression are nevertheless listed in Table 5. Quadratic fits to the data, of the form
log(λappθc∆T) ) log φ6 - n log(M/106) × {1 + n2 log(M/106)} (27)
Figure 1. Fractograms of PMMA polymer standards in THF at different cold wall temperatures as labeled. Sample size, 2 µg for each standard.
The total dead volume for this ThFFF system was ∼26.4 µL, and this was subtracted from overall retention volumes. The least-squares surface fitting of retention data to the calibration functions was accomplished using TableCurve 3D, version 2, Jandel Scientific Software (AISN Software Inc., Mapleon, OR). RESULTS Figure 1 shows a set of ThFFF fractograms of a mixture of PMMA polymer standards in the solvent THF. The temperature drop across the channel thickness (∆T) was held at 50 °C for all four fractograms, but cold wall temperature Tc was set at 52, 42, 32, and 23 °C, as labeled. It is apparent that polymer retention time increases as cold wall temperature is reduced. This is consistent with a decrease in λ and hence in D/DT with a lowering of temperature (see eqs 8 and 9). The same trend was observed for the PS-THF and PI-THF systems. The measured values of R and calculated λapp (obtained by solution of eq 5) for the PMMA, PS, and PI standards at different cold wall temperatures, in all cases with ∆T held at 50 °C, are listed in Tables 2-4, respectively. The molecular weight dependence of the retention data for the three systems was first examined. Retention data for each polymer-solvent system at each Tc were plotted in the form of the conventional calibration plot for ThFFF of log(λappθc∆T) versus log(M/106) (see eq 21). The data for the three systems are shown in Figures 2-4, in which the points represent the measured ThFFF data. In the case of PMMA-THF and PS-THF (Figures 2 and 3), the straight lines are not the best-fit lines for the individual cold wall temperatures, nor are the quadratic curves shown in Figure 4 for the PI-THF system. They are consistent with twodimensional least-squares surface fitting of the full data sets, described below. The φ6 and n constants for best-fit linear calibration plots for each system at each Tc are listed in Table 5 however. For PMMA-THF and PS-THF, the data fit the linear relationship very well at all Tc (see Figures 2 and 3). Furthermore, the slopes of the individual calibration plots as listed in Table 5 are independent of Tc. (The slightly greater variation in n shown for PS-THF is certainly within experimental error.) The PI-THF data do not conform to the linear relationship at any of the Tc considered (see Figure 4). The φ6 and n constants for linear
were significantly better. The resultant best-fit values for φ6, n, and n2 are also listed in Table 5. For this system, the slopes of these curves at M ) 106 are remarkably high (between 0.74 and 0.75 for Tc of 289, 298, and 309 K). The slopes fall to between 0.25 and 0.32 at M ) 105. It is not clear why this should be so. The much lower slope of 0.593 observed for M ) 106 at Tc ) 322 K may also be significant. The solvent THF is not a good solvent for PI and the θ temperature may fall within the range of Tc used in these experiments. The influence of solvent on retention in ThFFF is the subject of another study, and we shall not speculate further here. The same data were then plotted in the form of log(λappθc∆T) versus log(Tc/298.15) in order to examine the cold wall temperature dependence of retention for each polymer standard. These plots are shown in Figures 5-7. The points represent the measured data, and again, the straight lines are not best fits for each of the individual polymer standards. They conform to the least-squares surface fits for each data set. It is apparent that the data plotted in this form do conform very well to a linear dependence (see eq 26). The relationship may be written as
log(λappθc∆T) ) log φ298 + m log(Tc/298.15)
(28)
where φ298 depends on polymer molecular weight and, of course, the solvent. The best-fit values for φ298 and m are listed for each individual polymer standard in Table 6. For PMMA-THF and PSTHF, the values of m are apparently independent of polymer molecular weight. The lowest molecular weight PS standard yields a slightly lower value for m than the others, but greater measurement error in λ is associated with less retained materials. This apparent deviation in m for this polymer standard may be attributable to such error. It is also apparent that the PS-THF system is much more sensitive to changes in Tc than PMMATHF. In the case of PI-THF, m appears to increase with molecular weight, with the result that the larger polymers are more strongly influenced by changes in Tc than the smaller polymers. This may be another aspect of THF being a poor solvent for PI. For PMMA-THF and PS-THF, the orthogonal dependence of retention on molecular weight and Tc allows a least-squares surface fitting of the data according to the logarithmic form of eq 26. For PMMA-THF, we obtain φ6,298 ) 1.08((0.01), m ) 1.49((0.10), and n ) 0.625((0.003), where the values in parentheses represent one standard deviation. For PS-THF, we obtain φ6,298 ) 1.35((0.02), m ) 3.04((0.17), and n ) 0.587((0.005). The standard deviation of the fit for PMMA was 0.005 95, which corresponds to only 1.4% in λ, or 2.2% in M. For PS, the standard deviation of the fit was 0.0111, corresponding to 2.6% in λ, or 4.4% in M. The straight lines drawn in Figures 2, 3, 5, and 6 are consistent with these two-dimensional surface fits. The agreement of the measured data with the calibration equations of the form Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
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Table 2. Values of R and λapp for PMMA-THF System at Different Cold Wall Temperatures, with ∆T Held Constant at 50 K Tc (K)
ν
Mp (×10-3)
R
λapp
xeq/w
Teq (K)
296
-0.148
305
-0.139
315
-0.129
325
-0.120
37 100 280 570 37 100 280 570 37 100 280 570 37 100 280 570
0.644 ( 0.002 0.421 ( 0.002 0.234 ( 0.001 0.158 ( 0.001 0.664 ( 0.005 0.440 ( 0.003 0.248 ( 0.002 0.166 ( 0.001 0.680 ( 0.002 0.459 ( 0.002 0.257 ( 0.001 0.173 ( 0.001 0.705 ( 0.003 0.480 ( 0.003 0.273 ( 0.003 0.185 ( 0.002
0.176 0.0964 0.0492 0.0323 0.185 0.101 0.0519 0.0337 0.191 0.106 0.0536 0.0349 0.204 0.111 0.0567 0.0372
0.233 0.167 0.0932 0.0626 0.234 0.173 0.0978 0.0651 0.233 0.179 0.101 0.0673 0.229 0.184 0.106 0.0715
307.2 304.0 300.4 299.0 316.2 313.2 309.6 308.1 326.1 323.5 319.8 318.2 336.0 333.8 330.0 328.4
Table 3. Values of R and λapp for PS-THF System at Different Cold Wall Temperatures, with ∆T Held Constant at 50 K
a
Tc (K)
ν
Mp (×10-3)
R
λapp
xeq/w
Teq (K)
295.5
-0.148
304
-0.140
315.5
-0.129
323
-0.122
34.3 99.4 273 556 1000 34.3 99.4 273 556 1000 34.3 99.4 273 556 1000 34.3 99.4 273 556 1000
0.701 ( 0.003 0.446 ( 0.002 0.274 ( 0.003 0.186 ( 0.001 0.135 ( 0.001 0.733 ( 0.003 0.506 ( 0.002 0.312 ( 0.001 0.212 ( 0.001 0.154 ( 0.001 0.759 ( 0.004 0.536 ( 0.005 0.338 ( 0.003 0.226 ( 0.009 0.168 ( 0.001 0.779 ( 0.002 0.571 ( 0.005 0.366 ( 0.004 0.254 ( 0.003 0.184 ( 0.002
0.207 0.104 0.0585 0.0384 0.0274 0.226 0.121 0.0672 0.0438 0.0311 0.243 0.130 0.0729 0.0466 0.0339 0.258 0.142 0.0796 0.0525 0.0370
0.232 0.177 0.109 0.0738 0.0534 0.221 0.196 0.123 0.0835 0.0603 0.204a 0.204 0.132 0.0884 0.0655 0.184a 0.214 0.142 0.0987 0.0711
306.6 303.9 300.7 299.0 298.0 314.6 313.4 309.9 308.0 306.9 325.2 325.3 321.8 319.7 318.6 331.8 333.2 329.8 327.7 326.4
Breakdown of eq 11 at high λapp.
of eq 26 is extremely good. It should be remembered that these calibrations were carried out by assuming the nominal molecular weights for the standards. Any random error in the nominal values would result in some increment to the standard error of the fit. In the case of PI-THF, a least-squares fit of the data to an equation of the form
log(λappθc∆T) ) log φ6,298 + m log(Tc/298.15) n log(M/106){1 + n2 log(M/106)} (29)
was carried out. This allowed for quadratic dependence of log(λappθc∆T) on log(M/106) and linear dependence on log(Tc/ 298.15), but forced an orthogonality on these dependencies. This simplifies the resulting fit and subsequent data reduction, while accounting for the major dependencies of retention. The result was φ6,298 ) 2.32((0.05), m ) 3.68((0.25), n ) 0.706((0.039), and n2 ) 0.299((0.061). The standard deviation of the fit was 1604 Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
0.0177, corresponding to 4.1% in λ, or 5.9% in M (at 106) and 14% in M (at 105). The greater error in M at lower molecular weights is a consequence of the reduced selectivity in this region for the PI-THF system. The quadratic curves in Figure 4 and the straight lines drawn in Figure 7 are in accord with this surface fit. Even for this system, a reasonably good calibration was obtained. The fit could have been improved by including cross terms in log(M/ 106) and log(Tc/298.15) (i.e., terms in the product of these independent variables). To fully realize the benefits of universal calibration in ThFFF, the parameters φk,298, m, n, and perhaps n2, must be system transferable. The channel flow rate V˙ , cold wall temperature Tc, and the temperature drop ∆T must be monitored throughout sample elution. The sample mass must also be known (the effect of sample mass on calibration parameters is the subject of another study48). The effect of random error in nominal values for (48) Cao, W.-J.; Myers, M. N.; Williams, P. S.; Giddings, J. C. Int. J. Polym. Anal. Charact. 1998, 4, 407-433.
Table 4. Values of R and λapp for PI-THF System at Different Cold Wall Temperatures, with ∆T Held Constant at 50 K Tc (K)
ν
Mp (×10-3)
R
λapp
xeq/w
Teq (K)
289
-0.155
298
-0.146
309
-0.135
322
-0.123
114 210 293 1050 114 210 293 1050 114 210 293 1050 114 210 293 1050
0.544 ( 0.002 0.461 ( 0.002 0.400 ( 0.001 0.196 ( 0.001 0.578 ( 0.003 0.497 ( 0.007 0.436 ( 0.004 0.216 ( 0.001 0.626 ( 0.002 0.554 ( 0.009 0.486 ( 0.004 0.254 ( 0.002 0.675 ( 0.001 0.600 ( 0.003 0.549 ( 0.006 0.327 ( 0.003
0.136 0.109 0.0901 0.0409 0.148 0.119 0.100 0.0450 0.166 0.137 0.115 0.0531 0.188 0.153 0.134 0.0697
0.211 0.183 0.159 0.0784 0.220 0.194 0.172 0.0857 0.229 0.211 0.189 0.0998 0.232 0.221 0.208 0.127
299.1 297.8 296.6 292.7 308.5 307.3 306.2 302.1 319.9 319.1 318.0 313.7 333.1 332.6 331.9 328.0
Figure 2. Universal calibration plots for PMMA in THF at different cold wall temperatures. See text for explanation of the straight-line fits.
molecular weights of standards has been mentioned. However, there may be systematic error in these molecular weights. Evidence for this was revealed in a recent study. Calibration curves for a group of PS standards from Polymer Laboratories and a second group of standards from Polymer Standards Services were obtained by alternately injecting samples containing 1 µg of each standard into one of our laboratory ThFFF channels (in-house code 20) at Tc ) 298 K, ∆T ) 50 K, and V˙ ) 0.1 mL/min. The value of n was the same for the two polymer sources, but φ6 was 13% higher for the Polymer Laboratories group. As noted in the Experimental Section, the data in this paper were obtained using only Polymer Standards Services standards. It is apparent that calibration may be performed only with respect to the nominal molecular weights of a given set of standards. DISCUSSION We shall now present a justification of our approach to calibration in terms of the empirical constants φk,298, m, and n (and, if necessary, n2), rather than in terms of the constants φ′k,298, m′,
Figure 3. Universal calibration plots for PS in THF at different cold wall temperatures. See text for explanation of the straight-line fits.
and n′ (and, again if necessary, n′2). First, let us consider the simplest application of calibration where a set of standards is run under constant temperature conditions, and the unknown sample is subsequently eluted under the same conditions. As was mentioned earlier, it is not possible in this case to obtain a calibration in terms of φ′eq and n′ (see eq 17) because Teq varies with M, and φ′eq would have a different value for each molecular weight. For this situation, the calibration must be in terms of φc and n or in terms of φk and n (see eqs 18, 19, and 21). It is only when a set of standards is run at several different cold wall temperatures that values of φ′k,298, m′, and n′ (and, if necessary, n′2) may be obtained. In fact, when these retention data are available, there is no great difficulty in obtaining values for these constants, although one does have to make use of the approximate equation of Martin et al.32 (eq 11, above) to obtain xeq/w, and hence Teq, for each standard eluted under each temperature condition (i.e., each combination of Tc and ∆T). These calculations were carried out for our sets of retention data, and the results for xeq/w and Teq are listed in Tables 2-4. Note that, in the case of PS-THF, there are a couple of instances where eq Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
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Figure 4. Universal calibration plots of PI in THF at different cold wall temperatures. See text for explanation of quadratic fits.
Figure 5. Cold wall temperature dependence of λappθc∆T for PMMA in THF. Sample size, 2 µg. See text for explanation of the straightline fits.
Table 5. ThFFF Universal Calibration Constants O6, n, and n2 at Different Cold Wall Temperatures for PMMA-THF, PS-THF, and PI-THF Systems polymer-solvent
Tc (K)
φ6
n
n2
PMMA-THF
296 305 315 325 295.5 304 313.5 323 289 298 309 322 289 298 309 322
1.07 1.12 1.16 1.23 1.28 1.47 1.58 1.76 2.06 2.28 2.68 3.45 2.01 2.21 2.60 3.39
0.624 0.626 0.626 0.626 0.596 0.588 0.588 0.576 0.555 0.550 0.529 0.455 0.738 0.750 0.741 0.593
0.286 0.308 0.329 0.266
PS-THF
PI-THFa
PI-THFb
a
Linear fit. b Quadratic fit to eq 27.
11 is seen to have broken down for relatively high λapp. This occurs for λapp greater than ∼0.2 (depending on the value of ν), where xeq as predicted by eq 11 passes through a maximum. No unnecessary approximations were made in calculating Teq for each xeq, and the variation in κ across channel thickness was taken into account, as explained in ref 20. Least-squares surface fittings of the PMMA-THF and PS-THF data sets to equations of the form
log(λappθc∆T) ) log φ′6,298 + m′ log(Teq/298.15) n′ log(M/106) (30)
were carried out. For the PMMA-THF system, this resulted in φ′6,298 ) 1.07((0.01), m′ ) 1.46((0.09), and n′ ) 0.612((0.003), with a standard deviation of the fit of 0.00 57 (corresponding to 1.3% in λ, or 2.1% in M). For PS-THF, the following values were obtained: φ′6,298 ) 1.32((0.01), m′ ) 3.05((0.14), and n′ ) 0.5651606 Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
Figure 6. Cold wall temperature dependence of λappθc∆T for PS in THF. Sample size, 1 µg. See text for explanation of the straight-line fits.
((0.004), with a standard deviation of the fit of 0.009 52 (corresponding to 2.2% in λ, or 4.0% in M). We see that in each case, as expected, n′ is slightly lower than the value for n reported in the previous section. What is important is the fact that the goodness of fit for the two approaches is not significantly different. The prediction of λapp either from Tc and M (using calibration constants φk,298, m, and n) or from Teq and M (using calibration constants φ′k,298, m′, and n′) would be equally good. It might then be argued that, since φ′k,298, m′, and n′ are physicochemical constants, the small amount of additional work required to obtain them is justified. For a given polymer-solvent system, φ′k,298 is a physicochemical constant equal to D/DT for M ) 10k and T ) 298.15 K, and m′ and n′ describe its dependence on temperature and molecular weight, respectively. This is not the whole story, however. We shall consider in a moment the relative difficulty of
obtain the M eluted at the discrete tr.49 For each of the small set of discrete M, this entails prediction of R over the intervals in time at which Tc, ∆T, and V˙ are recorded. The retention time is then obtained by solving the integral equation
V0 )
Figure 7. Cold wall temperature dependence of λappθc∆T for PI in THF. Sample size, 4 µg. See text for explanation of the straight-line fits. Table 6. Constants O298 and m for Different Polymer Standards in THF for Best Fit of Retention Data to Eq 28 polymer-solvent
M (×10-3)
φ298
m
PMMA-THF
37 100 280 570 34.3 99.4 273 556 1000 114 210 293 322
8.41 4.61 2.36 1.54 10.1 5.16 2.89 1.89 1.35 7.03 5.69 4.76 2.18
1.52 1.51 1.47 1.47 2.35 3.28 3.25 3.14 3.19 2.98 3.17 3.66 4.91
PS-THF
PI-THF
∫ RV˙ dt tr
0
(32)
For our chosen approach to calibration, this may be accomplished relatively simply. Values for λapp are obtainable directly from the relevant calibration equation (either eq 26 or 29). With a calculation of ν, R may then be obtained using eq 5. An equivalent approach is not possible for the alternative calibration method. We cannot obtain values for λapp from the relevant calibration equation (either eq 30 or 31) because Teq is not known. A different and, as we shall see, more complicated approach is required. This is described below. As it is written, eq 30 is a special case of the general relationship between λ and T for a given polymer-solvent system at given M, ∆T, and w. (This of course cannot be said of eq 26.) Taking the antilog, the general form of eq 30 may be written
λθc∆T ) φ′k,298(T/298.15)m′(M/10k)-n′
(33)
which shows directly how λ varies with T across the channel thickness. It follows that once φ′k,298, m′, and n′ are known, then λc (the value of λ at the cold wall temperature) may be predicted. We may now consider an approximate equation for R, derived by Martin et al.32 for their model of linear variation of λ with x/w:
(1 + ν) (1 + 3ν) R ) 6λc - 12λc2 + (1 - 2δ) (1 - 2δ)(1 - 3δ) ν (34) 72λc3 (1 - 2δ)(1 - 3δ)(1 - 4δ) where δ is defined by the equation
predicting R and retention time tr for given experimental conditions using the two approaches to calibration. In the case of PI-THF, a least-squares surface fit of the data to an equation of the form
log(λappθc∆T) ) log φ′6,298 + m′ log(Teq/298.15) n′ log(M/106){1 + n′2 log(M/106)} (31) was carried out. It was found that φ′6,298 ) 2.20 ((0.04), m′ ) 3.59 ((0.22), n′ ) 0.661 ((0.036), and n′2 ) 0.298 ((0.060), with a standard deviation of the fit of 0.005 57 (corresponding to 3.8% in λ, or 5.8% in M (at 106) and 15% in M (at 105)). Again, we see that the fit is equally good for the two approaches to calibration. Now let us consider in more detail the procedure for data reduction in relation to each of the two approaches to calibration. Data reduction involves the association of molecular weight M with a series of closely spaced discrete retention times tr. This may be accomplished by predicting retention times for a relatively small set of discrete M and then carrying out an interpolation to
λ ) λc + δ(x/w)
(35)
Note that we prefer to use the symbol λc to represent λ at temperature Tc rather than λ0 corresponding to x/w ) 0 (as used by Martin et al.30), as λc better reflects the possibility that Tc varies during sample elution. We know that λ does not, in general, vary linearly with x/w across the full thickness of the channel. However, during elution the sample is of course concentrated close to the accumulation wall, and we may approximate the variation of λ in this region by a linear function in x/w. It is simply a matter of equating the rate of change of λ with x/w at the accumulation wall, as predicted for our more general model, with δ. From eq 33 we see that
λ ) λc(T/Tc)m′
(36)
A consideration of eq 36 and eq 16 allows the derivation of the (49) Williams, P. S.; Giddings, M. C.; Giddings, J. C., in preparation.
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approximate expression for δ:
δ≈
dλ (dx/w )) c
m′λcθc∆T Tc
(37)
Once a value is determined for ν, it is then possible to predict R using eqs 33 (with T ) Tc), 34, and 37. We see that it is possible to make use of either calibration to extract molecular weight information from experimental fractograms, but it is evident that our chosen approach allows for a simpler and more direct method of data reduction. Furthermore, this is achieved without a compromise in accuracy. Our chosen approach also has the advantage of consistency with the method that is forced upon the situation where calibration and sample runs are obtained under identical constant-temperature conditions, as explained above. The only advantage of the approach described above is that the constants φ′k,298, m′, and n′ are possibly more closely related to the underlying physicochemical constants and their variation with M and T than are φk,298, m, and n. CONCLUSIONS This investigation has shown that, in the case of PMMA and PS in the solvent THF, the relative change in retention (as measured by λapp) with cold wall temperature is apparently independent of polymer molecular weight. This is shown by the fact that, for a given polymer-solvent system, measured values for m are independent of M. The universal calibration for ThFFF could therefore be extended to account for changes in Tc with measurement of this single additional parameter m. For PI in THF, the dependence of this relative change with Tc is not a strong function of polymer molecular weight, and the assumption of constant m does not result in a great deterioration of accuracy in data obtained using such a calibration. The influence of cold wall temperature on retention is significant. For example, a change in Tc from 25 to 45 °C with the PS-THF system would result in almost a 20% reduction in retention volumes. This illustrates the importance of continuously monitoring Tc as well as ∆T during sample elution. With the use of the extended universal calibration it is no longer necessary that Tc be held constant and identical to the temperature at which some specific calibration run was carried out. This will greatly simplify experimental practice and eliminate errors resulting from unavoidable differences in Tc between calibration and unknown sample runs. The special problems associated with variation of Tc during field-programmed ThFFF were discussed earlier. In the past it has been necessary for calibration and sample runs to be carried out under exactly the same conditions where it is attempted to reproduce temperature variations. With the use of the extended universal calibration, this will no longer be necessary. The resulting accuracy of molecular weight information should be greatly improved. The variation of Tc during programmed runs is not likely to be reproducible between laboratories, since it will be dependent on coolant temperature and flow rate among other things. This will not be a problem when the extended universal calibration is used to transform fractograms to molecular weight distributions. It will be possible to exploit the full flexibility of ThFFF without compromising accuracy of retrieved data. 1608 Analytical Chemistry, Vol. 71, No. 8, April 15, 1999
ACKNOWLEDGMENT This work was supported by Grant CHE-9322742 from the National Science Foundation. LIST OF SYMBOLS A
common diffusion constant defined by eq 12
b
channel breadth
b
common diffusion exponential constant defined by eq 12
B
empirical constant defined by eq 13
c(x)
concentration at distance x from accumulation wall
c0
concentration at the accumulation wall
D
the common (concentration relaxation) diffusion coefficient
DT
the thermal diffusion coefficient
k
Boltzmann constant
k
constant introduced in eq 19
L
channel length
M
polymer molecular weight
Mp
peak average molecular weight of polymer
Mn
number-average molecular weight of polymer
Mw
weight-average molecular weight of polymer
m
empirical calibration constant defined by eq 26
m′
physicochemical constant defined by eq 9
n
empirical calibration constant defined by eq 18
n2
empirical calibration constant defined by eq 27
n′
physicochemical constant defined by eq 14
R
relative zone velocity, or retention ratio
Rh
the hydrodynamic radius
T
absolute temperature
Tc
temperature of the cold wall
Teq
equivalent temperature
Th
temperature of the hot wall
|u|
field-induced velocity toward accumulation wall
v(x)
fluid velocity as function of x
〈v〉
mean fluid velocity
V˙
flow rate in milliliters per minute
V0
void (channel) volume
Vr
retention volume
w
channel thickness
x
distance from the accumulation wall
xeq
distance of equivalent point from accumulation wall
β
exponent for thermal diffusion dependence on M as defined by eq 13
δ
linear rate of change of λ with x/w
∆T
temperature difference of cold and hot wall of ThFFF channel
η
fluid viscosity
θ
theta temperature of polymer solution
θc
temperature gradient correction at cold wall (see eq 16)
κ
fluid thermal conductivity
κc
fluid thermal conductivity at temperature Tc
λ
retention parameter defined by eq 2
λapp
apparent value for λ obtained by solution of eq 5
φ′
physicochemical constant defined by eq 14
φ′eq
constant defined by eq 17
φk,298
ThFFF universal calibration constant referred to Tc of 298.15 K
λc
value for λ at temperature Tc
ν
velocity profile distortion parameter
φ′c
value for φ′ at temperature Tc
φc
empirical calibration constant corresponding to cold wall temperature Tc, defined by eq 18
φ′k
constant equal to φ′c(10)-kn′
φk
ThFFF universal calibration constant defined by eq 20 with implied correspondence to some Tc
φ298
empirical calibration constant corresponding to temperature of 298.15 K, defined by eq 26
Received for review October 3, 1998. Accepted January 25, 1999. AC981094M
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