Anal. Chem. 1994,66, 4054-4062
Thermal Diffusion in Liquid Mixtures and Its Effect on Polymer Retention in Thermal Field-Flow Fractionation Chad A. Rue and Martin E. Schimpf' Department of Chemistry, Boise State University, Boise, Idaho 83725
Polymer retentionin thermal field-flowfractionation(ThFFF) is enhanced by the use of certain carrier-liquid mixtures. The originof enhanced retentionis explored with a focus on thermal diffusion, which is the driving force behind ThFFF. First, we examine thermal diffusion in several binary liquid mixtures; here, the relative tendency of a component to concentrate at the cold wall is correlated to its density and viscous activation energy. Next, we measure polymer retention in several binary carrier liquids. Retention is affected by thermal diffusion of the liquid components if the components have differentsolvating powers for the polymer. When the better solvent partitions to the cold wall, polymer retention is enhanced; when the better solvent partitions to the hot wall, retentionis diminished. These results indicate that a solvent gradient constitutes a significant driving force on the polymer, which may act either in concert or in opposition to thermal diffusion of the polymer, thereby enhancing or diminishing polymer retention, respectively. Thermodynamic arguments confirm the significance of this additional force, and the phenomenon is used to fractionate several polystyrenestandards ranging in molecular weight from 2500 to 160 000.
(1) Giddings, J. C.; Kumar, V.; Williams, P. S.; Myers, M. N. In Polymer
through the Stokes-Einstein equation, an analogous theory for & has yet to emerge. As a result, the dependence of ThFFF retention on polymer properties, such as composition and molecular weight, must be determined empirically. Fortunately, & is (for all practical purposes) independent of molecular weight, at least in homopolymers. Therefore, separation of different molecular weight components is governed primarily by differences in ordinary diffusion. Since D is related to molecular weight through the intrinsicviscosity, only a single calibration point is required to obtain molecular weight information on a polymer, provided the dependence of molecular weight on either diffusion or intrinsic viscosity is known .4 Another advantage of ThFFF is its inherently high resolving power (compared to size exclusion chromatography). However, both the retention and resolving power of polymer components smaller than about 20,000 daltons, drop prec i p i t ~ u s l y .Although ~ this limitation has been overcome by increasing the temperature gradient, extreme measures were required that included pressurizing the channel in order to raise the boiling point of the solvente6 An alternative approach to reducing the molecular weight limitation is to increase the thermal diffusion coefficient through the use of the proper solvent or solvent combination. Unfortunately, this approach is one of trial and error, since thermal diffusion is so poorly understood. A working theory for thermal diffusion, which relates & to physicochemical parameters of the polymer-solvent system, would certainly be useful in this regard; such a theory would also eliminate the need for calibration, provided the appropriate physicochemical parameters were available. Since its discovery7 in 1856, thermal diffusion has been studied in great detail. While thermal diffusion in gases is well understood, progress in liquids and polymer solutions has been slow due to the complexity of the liquid state and problems associated with conventional methods of measurement. In this regard, ThFFF has proven to be an excellent technique for studying thermal diffusion in polymer solutions-precise values of the thermal diffusion coefficient can be obtained with submilligram quantities of polymer. Studies of thermal diffusion in liquids have broad appeal because they lead to a better understanding of the liquid state, while benefiting ThFFF directly by expanding its ability to characterize lower molecular weight polymers, as well as more complex polymers, including copolymers.
Characterization by Interdisciplinary Methods; Craver, C. D., Provder, T., Eds.; ACS Advances in Chemistry Series 227; American Chemical Society, Washington, DC, 1990. (2) Schimpf, M. E. J. Chromatogr. 1990, 517, 405. (3) Schimpf,M. E.; Rue, C.; Mercer,G.; Wheeler, L. M.; Romeo, P. F. J . Coatings Technol. 1993, 65, 51.
(4) Kirkland, J. J.; Rementer, W. Anal. Chem. 1992, 64, 904. (5) Gunderson, J. J.; Giddings, J. C. Anal. Chim. Acta 1986, 189, I. (6)Brimhall, S . L.;Myers, M. N.; Caldwell, K. D.; Giddings, J. C. Sep. Sei. Technol. 1981, 16, 671. (7) Ludwig, C. W e n , Akad. Eer. 1856, 20, 539.
Thermal field-flow fractionation (ThFFF) is a separation technique used to characterize the molecular weight and composition of solvated polymer components by their differential migration in a thin channel of flowing liquid.'-3 ThFFF separations are governed by two transport mechanisms acting in opposition: thermal diffusion, which is the movement of mass in response to a temperature gradient, and ordinary (Fickian) diffusion, which is the movement of mass in response to the concentration gradient established by thermal diffusion. The retention of a polymer component in the ThFFF channel is governed by the relative magnitude of these two opposing processes. Thus, retention is proportional to the thermal diffusion coefficient DT of the polymer-solvent system but inversely proportional to the system's ordinary diffusion coefficient D. One advantage of ThFFF is that retention is related precisely to the transport coefficients D and DT. When the two coefficients are known, the retention volume Vr (the volume required to elute a polymer component) can be predicted, or vice versa. Although the magnitude of D is precisely related to physicochemical parameters of the polymer-solvent system
4054
Analytical Chemistty, Vol. 66, No. 22, November 15, 1994
0003-2700/94/0366-4054$04.50/0
0 1994 American Chemical Society
The application of ThFFF to copolymers was largely ignored during the first 20 years of ThFFF development. In the analysis of homopolymers, the independence of thermal diffusion and molecular weight was noted early.8 More recent work established the independence of thermal diffusion and branching configuration in homopolymers;9 however, thermal diffusion is responsible for the compositional dependence of retention.lOJ Only recently have we begun to characterize the retention and thermal diffusion of c ~ p o l y m e r s . ~ J ~ The influence of the carrier liquid (solvent) was also studied in early ThFFF work,8 including a report on mixed solvents in which water was shown to drastically reduce polystyrene retention in dimethyl s ~ l f o x i d e .Further ~~ reports on the use of mixed solvents were absent for over a decade. Then in 1990, Kirkland et al. reported that certain mixtures can be used to raise polymer retention to levels not previously seen.14 An extensive study of polystyrene retention in mixed solvents is contained in the preceding paper by Sisson and G i d d i n g ~ ; ' ~ some mixtures enhance polystyrene retention, while others diminish it. To realize the full potential of solvent mixtures in ThFFF, we must understand the physical basis for enhanced (and diminished) retention, so that the effect of mixing solvents can be predicted a priori. When analyzing polymer retention in mixed solvents, we must consider partitioning of the liquid components under the influence of the thermal gradient in addition to the thermal diffusion of the polymer. Solvent partitioning results in a carrier liquid whose composition varies in the same dimension as the thermophoretic motion of the polymer. It is well-known that a polymer's solubility is sensitive to changes in solvent composition; this sensitivity is used in solvent-precipitation methods for the bulk fractionation of molecular weight components. Therefore, it is reasonable to expect that the transport rates of the polymer in the ThFFF channel will be affected by solvent partitioning, thus altering polymer retention. For example, when one of the liquid components is a significantly better solvent for the polymer, the solvent gradient will cause the polymer to migrate in the same direction as the thermophoretic motion of the better solvent. This solubility-based migration, subsequently referred to as the partitioning effect, constitutes a second force on the polymer, acting in the same dimension as thermal diffusion. If the liquid component that is a better solvent for the polymer is enriched toward the cold wall, the partitioning effect will enhance polymer retention because thermal diffusion of the polymer is also in the direction of the cold wall. The primary goal of this work is to test the validity of the partitioning effect. In doing so, we first characterize the partitioning of solvent mixtures in a ThFFF channel using a temperature gradient of a magnitude similar to that typically employed for the separation of polymers. Next, correlations are made between solvent partitioning and enhanced (or
diminished) polymer retention in the mixtures. The correlations are used to demonstrate the validity of the partitioning effect and to use the effect to increase polymer retention to levels not previously obtained with standard ThFFF methods. We have two objectives in this work: to increase our understanding of thermal diffusion in liquids and polymer solutions and to expand the utility of ThFFF as a polymer characterization technique.
THEORY Thermal Diffusion in Liquids. Although thermal diffusion in liquids has been studied for decades, a unified model has yet to emerge. The phenomenon is more complex than either heat transfer or mass diffusion in that a temperature gradient causes a concentration gradient. Attempts to extend successful kinetic theories of thermal diffusion in gases to the liquid state16-l8 have been unsuccessful due to complications associated with intermolecular interactions. While models based on the thermodynamics of irreversible p r o c e s ~ e s l ~have - ~ ~been successful at predicting the sign of thermal diffusion in a few liquids, they fail to predict even qualitative trends in most liquid mixtures. In fact, different thermodynamic theories predict contradictory results in many cases. Statistical mechanical theories24 have perturbation terms containing complex integrals involving radial distribution functions in the perturbed and unperturbed state, which have not been evaluated for any actual system. As a result, these models lack practical utility and cannot be tested by independent methods. The experimental data also lack consistency, probably due to the wide variability and inherent difficulties associated with conventional methodology. A comprehensive monograph on the dismal state of our understanding of thermal diffusion in liquids is given by Powers.25 Because attempts to present a unified, rigorous theory have been unsuccessful, the approach to defining thermal diffusion in liquids remains phenomenological. This approach states that the flux J of one component in another is governed by the balance of two opposing transport mechanisms. The thermal gradient causes transport of components along coordinate x; as a concentration gradient is formed, mass diffusion acts as a counteracting transport mechanism. The total flux J, resulting from these two effects is derived from Fick's law26 J, = DT E c ( x ) - D dc(x) dx Here & and D are the coefficients of thermal and ordinary diffusion, respectively, dT/dx is the temperature gradient, and c is concentration. The first term on the right side of eq (16) Nernst, W. Z . Physik. Chem. 1889, I, 129. (17) Wirtz, K. Z . Naturforsch. 1948, 3A, 672; Z . Physik. 1948, 124, 348. (18) Prigogine, I.; de Broukere, L.; Amand, R. Physica 1950, 16, 577. (19) Denbigh, K. G.Trans. Faraday Soc. 1952.48, 1. (20) Prager, S.; Eyring, H. J. Chem. Phys. 1953, 21, 1347.
(8) Giddings, J. C.; Caldwell, K. D.; Myers, M. N. Macromolecules 1976,9,106. (9) Schimpf, M. E.; Giddings, J. C. Macromolecules 1987, 20, 1561. (10) Schimpf, M. E.; Giddings, J. C. J. Polym. Sci.: Polym. Phys. Ed. 1989, 27, 1317. (11) Gunderson, J. J.; Giddings, J. C. Macromolecules 1986, 19, 2618. (12) Schimpf, M. E.; Giddings, J. C. J . Polym. Sci.: Polym. Phys. Ed. 1990, 28, 2673. (13) Myers, M. N.; Caldwell, K. D.; Giddings, J. C. Sep. Sci. 1974, 9, 47. (14) Kirkland, J. J.; Boone, L. S.;Yau, W. W. J . Chromatogr. 1990, 517, 377. (15) Sisson, R.; Giddings, J. C. Anal. Chem. 1994, 66, 4043.
(21) Dougherty, E.L., Jr.; Drickamer, H. G. J. Chem. Phys. 1955. 23, 295. (22) Rutherford, W. M.; Drickamer, H. G. J. Chem. Phys. 1954, 22, 1157. (23) de Groot, S.R. The Thermodynamics of Irreversible Processes; Interscience: New York, 1950. (24) Bearman, R. J. J . Chem. Phys. 1959.30, 835. (25) Powers, J. E. In New Chemical EngineeringSeporationr Techniques;Schoen, E. S.,Ed.; Wiley-Interscience: New York, 1962. (26) Grushka, E.; Caldwell, K. D.; Myers, M. N.; Giddings, J. C. In Seporarion and Purification Methods; Perry, E. S., Ed.; Dekker: New York, 1974; Vol.
2.
Analytical Chemistty, Vol. 66,No. 22, November 15, 1994
4055
1 defines the transport of a component toward one wall due to the thermal gradient. Thesecond termdefines the transport away from the wall due to ordinary diffusion. Under steadystate conditions, these two transport processes are exactly balanced, J x = 0, and rearrangement of eq 1 yields 1 dc(x) --DrdT --c ( x ) dx
D dx
a differential equation whose solution is (3) where co is the concentration at the accumulation wall. Thus, a component's highest concentration is at the accumulation wall, decreasing exponentially away from the wall. The ratio &/D is referred to as the Soret coefficient. In binary mixtures, a similar equation is defined for each component, and the two equations are combined to yield the following extent of separation in an open cell2'
However, by carefully controlling the experimental variables, we were able to make the qualitative comparisons required to examine both the validity of the partitioning effect and its influence on ThFFF retention. ThFFFRetention. Indilute polymer solutions, the opposing forces of thermal and ordinary diffusion result in a polymer concentration that decreases exponentially away from the cold wall. The center of mass of the profile is defined by its distance 1from thecold wall. For mathematical convenience, parameter 1 is expressed in the dimensionless form X = I /w, which is related to the transport coefficients by
A=
7
= w2/Da2
D D,AT
(6)
Retention parameter X is related to the thermophoretic force on the polymer by29 Fth= k T/ Xw
(7)
Using the velocity profile of the carrier liquid in the FFF channel, the retention volume V,can be related to X as follows: R=
p/V,= 6X(coth(l/2X) - 2X)
(8)
Here, V'is the void volume of the channel, Vr is the volume of carrier liquid required to flush a polymer component through the channel, and R is termed the retention ratio. In the limit of high retention, the term in parentheses on the right side of eq 9 approaches unity, and eqs 7 and 8 can be combined to yield V, = 6V'AT(D,/D)
(9)
Thus, under constant experimental conditions, a polymer's retention volume varies directly with its Soret coefficient. Thermodynamics of Polymer Solutions. The classical thermodynamic theory of polymer solutions, developed independently by F l ~ r y ~ O and - ~H~ ~ g g i n s , defines ~ ~ - ~ the ~ reduced molar Gibbs free energy of mixing as AGJRT = X , In 4,
+ X 2 In 42 + gX142
(10)
where Xi and 4i are the mole fraction and volume fraction of component i, respectively, and subscripts 1 and 2 refer to the solvent and polymer, respectively;parameter gis the polymersolvent interaction coefficient. Partial differentiation with respect to the number of moles of solvent and polymer yields the reduced partial molar Gibbs free energy of mixing of solvent and polymer, respectively
(5)
where w is the thickness of the cell (or ThFFF channel). Assuming a diffusion coefficient of cm2/s, eq 5 predicts a relaxation time on the order of 10 s for the ThFFF channel used in this work. Unfortunately, we measured a continuous increase in solvent partitioning with residence time in the channel, even for mixtures that were in the channel for 6 min. As we discuss in the results section, we could not keep liquids in the channel longer than 6 min without losing satisfactory precision in our measure of solvent partitioning. Therefore, we did not attempt to obtain Soret coefficients in this work. (27) Tyrell, H. J. V. In DiffsionandHeot F1ow inLiquids; Butterworths: London,
1961. (28) Von Halle, E. AEC Research and Development Report 1959, K-1420.
4050
c=-
Fth
(4)
Here CH and Cc are the concentrations of one of the components at the hot and cold walls, respectively, and AT is the temperature difference between the hot and cold walls. The Soret coefficient of a liquid mixture is obtained from the equilibrium compositions of the binary liquid mixture at the hot and cold walls of the thermal diffusion cell, using eq 4. Values of the Soret coefficient for liquid mixtures have been measured in the range of 0.001-0.1 K-I. For a mixture with an initial concentration ratio of 10:90 and a Soret coefficient of 0.01, the more dilute component will be enriched to 13:87 at the hot wall and diluted to 7:93 at the cold wall when the temperature difference is 50 K. Relaxation Time. In order to obtain precise values for the Soret coefficient, it is important to allow sufficient time for the liquid to partition to its steady state concentration profile under the influence of the temperature gradient. In this work, solvent partitioning was measured after exposing mixtures to a temperature gradient while they flowed through a ThFFF channel. The amount of time that a mixture is exposed to the temperature gradient is dictated by its flow rate in the channel. According to the calculations of Von Halle,28 the approach to steady state in static thermal diffusion cells is exponential with a relaxation time 7 approximated by
D wD,(dT/dx)
Analytical Chemistry, Vol. 66,No. 22, November 15, 1994
Here, A' is the change in chemical potential associated with dissolution and N = V2/Vl, where V is molar volume. (29) Giddings, J. C.Science 1993, 260, 1456. (30) Flory, P. J. J. Chem. Phys. 1942, 10, 51. (31) Flory, P.J. J. Chem. Phys. 1944, 12, 425. (32) Flory, P. J. PrinciplesofPolymer Chemistry;Cornell University Press: Ithaca, NY, 1953. (33) Huggins, M. L. J. Phys. Chem. 1942, 46, 151. (34) Huggins, M.L. J , Am. Chem. Soc. 1942, 64, 1712. (35) Huggins, M. L. Physical Chemistry of High Polymers; Wiley: New York, 1958.
Parameter N is not necessarily equal to the degree of polymerization, but of the same order of magnitude. The parameters x and {are the Flory-Huggins parameters for the solvent and polymer, respectively. In practice, x is the parameter that is obtained experimentally because the chemical potential of the solvent can be measured through the solvent’s activity. Phenomenologically, parameters x and { are related to g by
solvent systems.37 Thedifference in AH1 , between polystyrene dissolved in EtB and CyH was 320 cal/mol of monomeric units (at 303 K). For 90 000 molecular weight polystyrene, this difference amounts to 460 RT. Based on this information and the calculations outlined above, even a slight solvent gradient in the ThFFF channel can be expected to establish a potential gradient that is significant enough to cause polymer migration in the direction where the better solvent is enriched.
A typical concentration of polymer solution injected in the ThFFF channel is 2 mg/mL. However, the injected concentration cinj is increased by thermal diffusion to c cinj/X. For the levels of retention used in this work, the resulting volume fraction of solvent in the compressed polymer zone is on the order of 41 0.98. For 90 000 molecular weight polystyrene, the degree of polymerization is 865. Substituting these values into eq 17 yields A(Ap2/RT)s01 = 830 A{. While empirical values of { do not exist, X-values can be found in the l i t e r a t ~ r e .For ~ ~ dilute solutions of polystyrene, x varies from 0.4 in a good solvent, such as ethylbenzene (EtB), to 0.5 in the €)-solvent cyclohexane (CyH). Assuming similar values for {, this translates to a difference in chemical potential of 83 R T for polystyrene dissolved in EtB compared to CyH. This is a conservative estimate because A t is greater than Ax when a good solvent is compared with a poor solvent. Large differences in the energy of mixing between a good and poor solvent have been confirmed with calorimetry measurements. For example, Bianchi and co-workers measured the heats of solution (AHs,~)in several dilute polymer-
EXPERIMENTAL SECTION The ThFFF instrument has been previously described;1° it is similar to the FFFractionation Model T-100 (Salt Lake City, UT), except that temperature control is achieved using a proportional counter built in-house. The controller is based on an operational amplifier functioning as an astable multivibrator, which governs the on and off times of the hot-wall heaters. To enhance stability, a thermistor is used to drive a feedback loop that alters the reference point of the multivibrator and hence the on and off times. The temperature stability using this controller is better than fO. 1 K. All solvents were HPLC grade. Two channels were used in this work. Both channels were formed with polyester (Mylar) spacers with a tip-to-tip length of 46 cm and a breadth of 2.0 cm. For measuring polymer retention, we used a 102-pm-thick channel with a resulting void volume of 0.90 mL; the injected polymer solutions had a concentration of 1 mg/mL. A variable-wavelength UV detector was used for sample detection, except in aromatic solvents, where we used a refractive index detector. In measuring the partitioning of solvent mixtures, we used a 254-pm-thick channel (void volume 2.25 mL) with a single inlet and two outlets, one at each (hot and cold) wall. Binary solvent mixtures were exposed to a temperature gradient (AT 47 K) by flowing them through the channel. The composition of solvent layers at the hot and cold walls were sampled at the channel outlets. This was achieved by adjusting the back pressure at each outlet with needle valves, so that the fraction of the total flow eluting from each outlet (subsequently referred to as the split ratio) was precisely controlled. Flow rates and split ratios were continuously monitored with a stop watch and burette for each effluent. The sampling port for each wall consisted of a t-union with a rubber septum, placed just upstream from the needle valves. The effluent was sampled by penetrating the septum with a microliter syringe. The liquid collected in the syringe was injected directly into a gas chromatograph (GC) for analysis. This arrangement precluded selective evaporation of one component prior to analysis. All experimental data reported in this work represent an average of at least four replicated experiments. The polystyrene standards used in this work came from either Supelco, Co. (Bellefonte, PA) or Pressure Chemical Co. (Pittsburgh, PA). In performing ThFFF on polymers, we used the stop-flow technique. In this technique, the flow of carrier liquid through the channel is halted for 1 min after sample injection to allow the sample to relax into its steadystate distribution at the accumulation In computing the retention ratio R from eq 8, Vr was calculated by subtracting the volume of the inlet and outlet tubing from the
(36)Brandup, J.; Immergut, E. H. PolymerHandbook, 3rded.;Wiley: New York,
(37) Bianchi, U.; Cuniberti, C.; Pedemonte, E. Rossi, C. J . Polym. Sci. 1969, 7 ,
r = g + 42(ag/a42)
(13)
x = g - (1 - 42)(ag/a42)
(14)
Equations 13 and 14 can be combined to obtain
In the lim 42
-
r- x = w e 4 2
(15)
0, the derivative of eq 14 yields
dxld4, = dg/d42
(lim 42
-
(16)
0)
Empirical evidence indicates that dx/d& is positive in a poor solvent and negative in a good solvent. Consequently, { is larger than x in a poor solvent and smaller than x in a good solvent. Since x decreases with the goodness of the solvent, the difference in {-values (A{) between a good and poor solvent is larger than the corresponding difference in X-values (Ax); Le., A{> Ax, The importance of this observation will become apparent below. Since we are interested in the localized motion of the polymer, we will focus on eq 12 to examine the motion of polymer in response to a solvent gradient. More specifically, we are interested in comparing a polymer molecule’s chemical potential in a good solvent vs a poor solvent. The difference in the reduced partial molar free energy of mixing between two solvents [A(Ap2/RT)lso1 is contained in the third term on the right side of eq 12 [NAP2/RT)lsol = W 2 A l
(17)
-
-
1989.
855.
Analytical Chemistry, Vol. 66, No. 22, November 15, 1994
4057
54
I
I
I
I
8
0.0
0.4
0.8
I .2
1.6
Flow Rate ( d i m i n )
Figure 1. Partitioningof DCE to the cold wall as a function of flow rate in a mixture of 49.7 vol % DCE in CyH. The split ratio was held constant at 4.2 vol %. Measured partitioning continues to increase as the flow rate is decreased below 0.4 mL/min.
total volume of carrier liquid required to flush the sample from injector to detector. The void volume P was equated to the retention volume of a polymer standard in the absence of a field.
RESULTS AND DISCUSSION Thermal Diffusion in Liquids. One of the first mixtures reported to enhance polystyrene retention combines DCE with CyH.lS We studied this mixture in detail because the effect is relatively large and polystyrene is soluble in all proportions of the mixture. Several different compositions were exposed to a temperature gradient by flowing them through a ThFFF channel, and the partitioning of components was measured, as described above. Equation 5 predicts that the partitioned solvents should reach their steady-state distribution within a short distance from the inlet of the channel, so we can expect the outlet compositions to be invariant with changes in flow rate provided the split ratio is held constant. To test this, we measured the outlet composition of a DCE-CyH mixture (49.7 vol 5% DCE) at several different flow rates, while maintaining a split ratio of 4.0 f 0.1%. As Figure 1 illustrates, partitioning of DCE to the cold wall apparently continues to increase as the flow rate is decreased below 0.4 mL/min. At this flow rate, the liquid is in the channel for 6 min. It was not possible to reduce the flow rate further and continue to maintain a stable split ratio. (In fact, it was difficult to maintain stable split ratios below 0.6 mL/min.) Several sources for the unexpected dependence of partitioning on flow rate should be considered. For example, it is possible that turbulence at the outlet causes partial remixing of the separated components. However, the Reynold’s number for fluid flowing in the channel at 0.6 mL/min in the region of the outlet, where the channel tapers to a width of 1 mm (the diameter of the outlet tube), is less than 20. Thus turbulent flow is not expected, even near the outlet. Another possibility is that steady-state partitioning is simply not reached after 6 4058
Analytical Chemistry, Vol. 66,No. 22, November 15, 1994
min in the channel, even though eq 5 predicts a relaxation time (in static cells) on the order of 10 s. In order to test the latter possibility, we measured the retention ratio of a polystyrene standard (MW 90 000) at several flow rates in a 50 vol % mixture of DCE and CyH. The retention ratio increased by only 3% as the flow rate was increased from 0.1 to 0.8 mL/min, indicating that steady-state partitioning of the liquid components is complete within a short distance from the inlet. A third possible cause for the dependence of solvent partitioning on flow rate is the existence of a turbulent boundary layer next to the channel walls. The turbulence of the boundary layer increases with the velocity of the carrier liquid and contributes to the ill-defined lift forces that reduce retention in steric-mode FFFe3* In order to calculate the Soret coefficient of a liquid mixture, accurate values of the solvent composition at each wall must be known. Because of the flow rate dependence, however, accurate wall compositions could only be obtained by varying both the split ratio and flow rate, so that multivariate regression methods could be used to obtain the solvent composition as both variables approach zero. The time required to produce such data would preclude the examination of several different solvent mixtures. Since our primary reason for studying thermal diffusion in liquids is to determine whether enhanced polymer retention in such mixtures can be explained by solvent partitioning, we opted to settle for qualitative information rather than spending a great deal of effort to obtain Soret coefficients for each mixture. Certainly it would be useful to have precise values of the Soret coefficients, but our primary objectives can be met by simply identifying which component of a mixture concentrates at the cold wall. As it turns out, the qualitative information also revealed two correlations which are discussed later in this section. Although it was difficult to keep the split ratio constant from one solvent mixture to the next, it remained steady for any given mixture as long as the flow rate through the channel was at least 0.6 mL/min. Therefore, all subsequent partitioning experiments were carried out with this flow rate. Although we did not attempt to use the same split ratio for all measurements, we did keep the ratio small (below 5%) in most cases, so that only a thin layer next to the appropriate wall was collected for analysis. However, in characterizing the thermal diffusion of DCE-CyH mixtures, we varied the split ratio over a range of values in order to obtain more detailed information. Plots of the outlet composition as a function of the split ratio were extrapolated to a split ratio of zero to obtain accuratevalues of the solvent composition at each wall. A pair of such plots, one for each (hot and cold) wall, are illustrated in Figure 2. Note that as the split ratio is increased so that liquid further from the wall is sampled, the measured concentration of the enriched component decreases exponentially, as expected from eq 4. By fitting the outlet composition to an exponential function of the split ratio, the extrapolated intercept values had relative uncertainties of less than 1% in all cases (r2 > 0.99). Note also that solvent partitioning is greater at the cold wall (compared to the hot wall), where the thermal energy is lower and therefore less diffusional remixing occurs. (38) Williams, P. S.;Koch, T.;Giddings, J. C. Chem. Eng. Commun. 1992, 111, 121.
81.5
,
I
Table 1. Summary
solvent paip
0
10
0
10
20 Split Ratio (5%)
30
20
30
40
'
io
Split Ratio (70)
Flgure 2. Measured concentration of DCE eluting from the outlets as a function of the split ratio in mixtures of 78.9 voi % DCE in CyH. The dashed lines represent 95 % confldence intervals: top, depletion (hot) wail; bottom, accumulation (cold) wail.
CCl4-DCE CCl4-BEN CCl4-DOD DCE-CyH DCE-BEN CyH-THF CyH-EtOH CyH-BEN CyH-EtB CyH-MEK CyH -D0D EtOH-THF EtOH-EtAc EtOH-BEN EtOH-EtB EtOH-MEK THF-BEN THF-EtB THF-DOD Et AC-BEN EtAC-EtB BEN-EtB EtB-MEK MEK-DOD DOX-ACN
ol Partltlonlng In Binary Solvent MMurer
component enriched at cold wall CCl4 Cc4
cc& DCE DCE neither CYH CYH CYH CYH CYH EtOH EtOH EtOH EtOH EtOH THF THF THF neither neither neither EtB MEK DOX
inlet compb8
split ratid
xi
(%)
47.0% CCl4 50.9% ccl4 5 1.7% cc4 57.2%DCE 57.1% DCE 43.7% CyH 42.1 % CyH 49.5% CyH 43.5% CyH 52.4% CyH 5 1.9% CyH 53.8% EtOH 52.5% EtOH 53.2% EtOH 61-6%EtOH 54.75 EtOH 51.5%THF 52.9%THF 51.9%THF 49.9% EtAc 50.0% EtAc 50.6% BEN 51.6% EtB 52.9% MEK 69.8% DOX
2.5 3.8 1.9 2.2 3.7 1.0 6.8 2.3 10.0 11.O 3.4 5.6 1.0 3.3 1.9 11.5 3.3 11.4 2.6 6.4 6.4 2.4 2.1 12.6 2.8
cold wall compbs
enrichment factor
xo
49.2% cc4 56.7% CC14 5 9 . 1 % ~ 60.6% DCE 60.0% DCE 43.7% DCE 42.81CyH 50.4%CyH 43.9% CyH 53.8% CyH 55.0% CyH 54.7% EtOH 53.7% EtOH 53.6% EtOH 64.6% EtOH 55.6% EtOH 52.5%THF 53.5% THF 55.O%THF 50.0% EtAc 50.0% EtAc 50.7% BEN 52.4% EtB 53.6% MEK 70.1%DOX
$(%I
4.7 11.4 ~14.3 ~ 5.9 5.3 0 1.4 1.8 1.4 2.7 6.0 1.7 2.3 0.7 4.9 1.6 1.9 1.1 6.0 0 0 0 1.6 1.3 0.4
a Key: DCE, 1,2-dichloroethane;BEN, benzene; DOD, n-dodecane; CyH, cyclohexane; THF, tetrahydrofuran; EtOH, ethanol; EtB, ethylbenzene; MEK, 2-butanone;EtAc, eth 1acetate; DOX,p-dioxane; ACN, acetonitrile. b Standard error is 0.2k c Compositions are in volume percent.
Table 2. Order of Thermal DMwlon In Solvent Mlxtures
carbon tetrachloride 1,2-dichloroethane tetrahydrofuran and cyclohexane benzene, ethylbenzene, and ethyl acetate 2-butanone n-dodecane
0
20
80
Val-% W E
Flgure 3. Gradient in DCE at the cold wall as a function of the DCE content in mixtures of DCE and CyH. Maximum partitioning occurs in mixtures containing 55-60 vol % DCE.
reported15 to yield the greatest enhancement in polystyrene retention. (Note that while ref 15 reports maximum retention at 75 vol % DCE, the greatest deviation from linearity in plots of R vs composition occurs at 6 0 ~ 0 1 %DCE). This correlation between the DCE gradient and retention enhancement lends further support to the partitioning theory. In addition to DCE-CyH, we examined the partitioning of several other solvent mixtures. The enrichment $c of a component at the cold wall is given by $c
We examined DCE-CyH mixtures ranging in composition from 18 to 87 vol 5% DCE. In every mixture, DCE concentrates at the cold wall, while CyH concentrates at the hot wall. Since DCE is a much better solvent for polystyrene, its enrichment at the cold wall is consistent with the partitioning theory, that is the theory that the partitioning effect is responsible for observed enhancement in polystyrene retention. Figure 3 plots the gradient in DCE concentration at the cold wall, obtained from plots similar to those in Figure 2, as a function of the DCE content in DCE-CyH mixtures. The gradient is largest in mixtures containing 55-60 vol % DCE, which is within experimental uncertainty of the composition (60 vol % DCE)
x,- xi
= -x 100 Xi
where Xi and X o are the inlet and outlet concentrations of the enriched component, respectively. Table 1 summarizes the partitioning data. With the exception of ethanol (to be discussed separately below), partitioning of the solvents can be placed in the order displayed in Table 2. Solvents higher in Table 2 concentrate at the cold wall when combined with any solvent lower in the table. For example, C C 4 always concentrates at the cold wall when mixed with all other liquids in Table 2; DCE concentrates at the cold wall when combined with other solvents in the table besides CCl4, and so forth. With few exceptions, each solvent was tested with all those Analytical Chemistry, Vole66,No. 22, November 15, 1994
4059
in adjacent layers. However, DCE could not be combined with tetrahydrofuran (THF) because the resulting mixture attacked the chrome plating on the channel walls. (We found, as a general rule, that any combination of an oxygenated solvent with a chlorinated solvent should be avoided.) Two other mixtures containing adjacent components were also not examined; these were mixtures of T H F and ethyl acetate (EtAc) and EtAc mixed with 2-butanone (MEK). We also looked at the partitioning of several components that are not on adjacent levels in Table 2. In each case, the results were consistent with the order. For example, if we compare the behavior of solvents combined with benzene, we find that CCl4 showed the greatest partitioning toward the cold wall, followed by DCE, and then CyH and THF. Care must be taken in such comparisons because the split ratios are not always matched. For example, compare the behavior of two separatecombinations containing benzene (BEN), namely, CCl4-BEN and CyH-BEN. With a split ratio of 3.8%, the measured enrichment of CC14 at the cold wall was greater than CyH, where a lower split ratio (2.3%) was used. In this case we say that CC14 partitions more strongly than CyH when combined with BEN. If the split ratio used in CyHBEN had been higher than that for CC14-BEN, a comparison of these two mixtures would be more ambiguous because measured partitioning diminishes as the split ratio is increased. Of course, some ambiguity remains because of the flow rate dependence; therefore, any comparison assumes a similar dependence on flow rate in all the mixtures. A comparison of the behavior of mixtures containing CyH also agrees with the order displayed in Table 2. Thus, CyH does not partition at all with T H F (they are on the same level), and the magnitude of its enrichment at the cold wall increases as we go down Table 2 from BEN and EtB, to MEK, and finally to n-dodecane (DOD). Several other cross-checks can be made as well; in each case, the integrity of the solvent order displayed in Table 2 is maintained. While most of the solvents fall into a consistent order, ethanol (EtOH) is an exception. For example, EtOH is enriched at the cold wall when combined with T H F but it concentrates at the hot wall when combined with CyH. Yet T H F and CyH do not partition as a mixture. Therefore, a position cannot be found in the order for EtOH. There is another inconsistency with EtOH-although it concentrates at the cold wall when mixed with THF, it undergoes less partitioning than T H F when mixed with benzene. The reason for the anomalous behavior of EtOH is unclear, although its unique ability (among the liquids we examined) to form hydrogen-bonded networks may be a factor. It is possible that mixtures containing EtOH have a significantly greater relaxation time to steady state, in which case our partitioning data cannot be used to establish even qualitative comparison of its thermophoretic behavior with other solvents. It is worth noting that EtOH and other hydrogen-bonding solvents yield anomalous behavior in the thermal diffusion of polymer solutions as wel1.6339 We looked at the partitioning of a mixture containing acetonitrile (ACN) and dioxane (DOX) because it was reported that polystyrene retention is neither enhanced nor (39) Kirkland, J. J.; Yau, W . W. J . Chromafogr. 1986, 353, 95.
4060
Analytlcal Chemistry, Vol. 66, No. 22, November 15, 1994
cyclohexane
0
i IC 0.6
0.8
1.0
1.2
1.4
1.6
Solvent Density (g/mL)
Flgure 4. Correlationbetweendensity (293 K) and the relativetendency of solvents to partition to the cold wall in binary mixtures.
diminished in this mixture40 (i.e., retention is a linear function of solvent composition). Consistent with the partitioning theory, the mixture did not significantly partition (see Table 1). Additional studies on ACN and DOX were not attempted because of their limited miscibility with other solvents already examined. We are currently attempting to correlate thermal diffusion in liquids to physicochemical parameters. Such correlations (or their absence) will be used to test the myriad of theories on thermal diffusion in liquids. While a detailed summary of these efforts is beyond the scope of this paper, a cursory look reveals that density is a significant factor. The correlation is easily illustrated by plotting the partitioning order displayed in Table 2 as a function of solvent density (at 293 K); such a plot is illustrated in Figure 4. With the exception of CyH, the relative tendency of a solvent to partition to the cold wall follows the order of increasing solvent density. An analysis of the variance signifies the role of density at the 99% confidence level. Variance analysis also indicates that solvent partitioning is affected (99% confidence) by the solvent's activation energy for viscous flow, which is the coefficient in the exponential dependence of viscosity on temperature. Both correlations are consistent with previous studies of thermal diffusion in polymer solution^,^ where a significant fit was found in regressing polymer & values to a function of the polymer density and theviscous activation energy of the solvent. Since partitioning of the solvents is correlated to density, we must consider the existence of convective motions in the fluid. Convection is a macroscopic displacement caused by density gradients and is a process separate from thermal diffusion. With homogeneous liquids, convection is negligible in thin ThFFF channels, even though a small density gradient is induced (across the thin dimension of the channel) by the temperature gradient. However, in mixed solvents, much larger density gradients are possible due to solvent partitioning. Under the appropriate conditions, convective motion may ensue, leading to instabilities in the concentration profiles (40) Sisson, R.; Giddings, J. C. Presented at 2nd Infernational Symposium on Field-Flow Fracrionafion,Salt Lake City, UT, 1991.
Table 3. Summary of Polystyrene Retentlon In Mlxed Solvents' %BENb %DCE Rc.d %EtB %DOD Re 0 0.489 (0.002) 0.737 (0.002) 100 0 100
26.3 53.4 78.5 100
73.7 46.6 21.5 0
0.723 (0.008) 0.680 (0.005) 0.633 (0.003) 0.593 (0.004)
%CC14 %DOD Re 100 0 0.749 (0.005)
88.3 77.2 67.3 57.0 48.4
11.7 22.8 32.7 43.0 51.6
0.636 (0.002) 0.541 (0.002) 0.486 (0.002) 0.445 (0.002) 0.428 (0.002)
86.3 72.5 59.6 47.4
13.7 27.5 40.4 52.6
%THF %DOD 100 79.0 64.5 49.2 44.1
0 21.0 35.5 50.8 55.9
075
I
0.55
%
0.452 (0.002) 0.423 (0.002) 0.419 (0.004) 0.441 (0.002)
Re 0.567 (0.002) 0.463 (0.002) 0.411 (0.002) 0.384 (0.002) 0.377 (0.002)
a AT47 K:flow rate0.2mL/min. Comoositionsareinvolumeuercent. Numbers in parentheses ark one standard error. Molecula; weight 47 500. e Molecular weight 100 000.
established by thermal diffusion. Normally, the ThFFF channel is oriented with the cold wall down. In this case, when the denser liquid thermally diffuses to the cold wall, convective forces will reinforce partitioning due to thermal diffusion. However, if the denser liquid were to thermally diffuse up to the hot wall, convection would lead to remixing of the liquid components. In fact, we cannot rule out the possibility that convective instabilities completely counteract any partitioning due to thermal diffusion in the majority of cases where the denser liquid thermally diffuses to the hot wall. Therefore, the observed correlation between partitioning and density may be an artifact of convective motion in the channel. The issue of convective motion may be resolved by measuring solvent partitioning in a ThFFF channel whose orientation has been reversed, so that the cold wall lies above the hot wall; such studies are currently underway. Polymer Retention. On the basis of the partitioning data summarized in Table 1, we chose several mixtures in which to examine polymer retention. Table 3 summarizes the retention data in these mixtures. We chose mixtures of BEN and DCE because the better solvent (BEN) partitions to the hot wall, and therefore, diminished retention of polystyrene is expected. As illustrated in Figure 5, retention is clearly diminished in BEN-DCE mixtures, with the greatest deviations from linearity occurring in mixtures containing small amounts of the better solvent. Our next goal was to find a solvent combination that displays a large amount of enhanced retention. A strong candidate was CC14-DOD because this mixture showed the greatest amount of partitioning of all the mixtures we examined. However, EtB-DOD also partitioned strongly, and EtB yields the highest retention of any homogeneous carrier liquid. Furthermore, EtB is a better solvent for polystyrene, making the transfer of partitioning to retention enhancement more efficient. Finally, T H F is nearly as good a solvent for polystyrene in terms of both retention and solvating power, and its partitioning with DOD exceeds that of EtB. We examined retention over a range of compositions in each of these solvent pairs. Figure 6 contains plots of retention vs composition for each pair. As expected, the curvature is greatest in mixtures of EtB and DOD, where the greatest difference is solubility occurs. However, retention is greatest in a mixture of 44 vol % T H F in DOD. Although the data indicates that retention will continue to increase with the
0
20
40
60
80
100
Vol-% Benzene Figure 5. Retention ratio vs composition in mixtures of DCE and BEN, illustrating the deviation from linearity toward diminished retention. The polymer is 90 000 molecular weight polystyrene.
fraction of DOD in such mixtures, further increases in the DOD content were limited by the solubility of the polymer. Sisson and Giddings also measured higher retention volumes for polystyrene in mixtures of T H F and DOD than in any other solvent or solvent c o m b i n a t i ~ n .However, ~~ they reported maximum retention in mixtures containing 30 vol % DOD, with a decrease in V, as the amount of DOD was increased beyond 30 ~ 0 1 % .The single apparent difference in our work is the use of a higher field. Thus, the discrepancy may result from an increased role in solvent partitioning with field strength. However, more work is needed to sort out such secondary effects and apparent inconsistencies. Figure 7 illustrates the separation of five polystyrene standards, ranging in molecular weight from 2500 to 179 000, in a mixture containing 45 vol 7% T H F in DOD. While molecular weights below 2500 have been retained previously,14 the ThFFF channel had to be pressurized in order to increase the temperature gradient without boiling the carrier liquid. This fractogram represents the lowest molecular weight polymer ever resolved from the void peak without channel pressurization.
CONCLUSIONS Thermal field-flow fractionation has proven to be a useful tool in our studies of thermal diffusion. Data on the thermal diffusion of 12 liquids in 25 binary combinations reveal a correlation between thermal diffusion and both density and viscous activation energy of the solvent components. Both correlations are consistent with studies of thermal diffusion in polymer solutions. However, the correlation with density may be an artifact induced by convective forces within the channel, which lead to instabilities in the motion of the liquid when the denser components thermally diffuse to the upper (hot) wall. Future work using different channel orientations, designed to clarify the role of convection, are underway. The correlation between thermal diffusion and the viscous activation energy of the solvent is consistent with kinetic Analytical Chemistry, Vol. 66, No. 22, November 15, 1994
4081
0.701 0.80 I
I
t
R
I
0.60
.I’
*,” 0 40
0.50
I
0.48
1
60
50
40
70
80
90
100
Vol-% CCl,
0
I
2
3
4
5
6
7
vr (m
0.42
Figure 7. Separation of polystyrene components using a mixture of 45 voi % THF In DOD as the carrier liquid. From left to right, the first peak Is the void peak: successive peaks contain polystyrenestandards of the following molecular weights, in thousands (K): 2.5K, 20K, 47K, 97K, and 179K.
i
0.40
60
50
40
70
80
90
100
Vol-% Ethylbenzene
1
0.60
0.40
t
0.35 40
/’*
,,,.” I
50
,
,
,
I
60
,
,
,
I
70
,
I
, I I
80
I
,
I
90
%
I
I
1
I
100
Vol- % Tetrahydrofuran
Flgure 6. Retention ratio vs solvent composition in solvent mixtures that enhance retention. The polymer is 90 000 molecular weight polystyrene: top, CCI,-DOD: mlddie, EtB-DOD; bottom, THF-DOD.
approaches to thermal diffusion, which are based on the reduction in diffusive motion associated with molecules moving into a cooler environment. In the theory of rate processes, the temperature dependence of diffusion is conceived as an activation energy required for a solvent molecule to jump into an adjacent hole in the liquid lattice.41 As indicated by the work of Eyring et al.,42this activation energy is correlated to the solvent’s viscous activation energy. The enhanced retention of polymers in certain solvent mixtures can be explained by a synergistic effect between the thermal diffusion of both polymer and solvent. In this effect, the solvent partitions under the influence of the temperature gradient and the polymer partitions in the resulting solvent gradient. This solvent-induced partitioning acts as a driving (41) Hilby, G.; Wirtz, K. Z. Phys. 1943, 44, 369. (42) Glasstone; Laidler; Eyring. H. Theory ofRare Processes; McGraw-Hill, Inc.: New York, 1941.
4062
Analytical Chemistry, Vol. 66, No. 22, November 15, 1994
force on the polymer that is separate from the thermophoretic force on the polymer. The two forces can act either in concert or in opposition, resulting in retention that is either enhanced or diminished, respectively. When the liquid component that is a better solvent for the polymer is enriched at the cold wall, solvent partitioning forces the polymer in the same direction as thermal diffusion of the polymer, and polymer retention is enhanced. When the better solvent component is enriched at the hot wall, the solvent-induced motion acts in opposition to the polymer’s thermophoretic motion and retention is diminished. An additional factor in the retention of polymers in mixed solvents, which has not been addressed here, is the influence of the ordinary diffusion coefficient D. Thus, retention will be enhanced or diminished if polymer D-values do not change with solvent composition in a linear fashion. The effect of ordinary diffusion on retention in solvent mixtures is currently being studied by Sisson and Giddings. However, in the work described above, enhanced retention is only observed when the better solvent partitions in the same direction (to the cold wall) as the polymer, strongly supporting the view that the enhancement arises from the partitioning effect.
ACKNOWLEDGMENT We thank Professor J. Calvin Giddings at the University of Utah for his valuable input. Thanks go to Richard Sisson for sharing his retention data in mixed solvents. This work was funded by the Idaho State Board of Education and grant (2-3272 from Research Corp. M.E.S. is also pleased to announce continued support of this work by the NSF-Idaho EPSCoR Program and National Science Foundation Grant OSR-9350539. Received for review May 11, 1994. Accepted August 1, 1994.” Abstract published in Aduance ACS Abstracts, October 1 , 1994.
