Anal. Chem. 2007, 79, 9066-9073
Taylor Dispersion Analysis of Mixtures Herve´ Cottet,*,† Jean-Philippe Biron,† and Michel Martin*,‡
Institut des Biomole´ cules Max Mousseron, (UMR 5247 CNRSsUniversite´ de Montpellier 1-Universite´ de Montpellier 2), 2 place Euge` ne Bataillon CC 017, 34095 Montpellier Cedex 5, France, and Ecole Supe´ rieure de Physique et de Chimie Industrielles, Laboratoire de Physique et Me´ canique des Milieux He´ te´ roge` nes (PMMH-UMR 7636 CNRS-ESPCI-Universite´ Paris 6-Universite´ Paris 7), 10 rue Vauquelin, 75231 Paris Cedex 05, France
Taylor dispersion analysis (TDA) is a fast and simple method for determining hydrodynamic radii. In the case of sample mixtures, TDA, as the other nonseparative methods, leads to an average diffusion coefficient on the different molecules constituting the mixture. We set in this work the equations giving, on a consistent basis, the average values obtained by TDA with detectors with linear response functions. These equations confronted TDA experiments of sample mixtures containing different proportions of a small molecule and a polymer standard. Very good agreement between theory and experiment was obtained. In a second part of this work, on the basis of monomodal or bimodal molar mass distributions of polymers, the different average diffusion coefficients corresponding to TDA were compared to the z-average diffusion coefficient (Dz) obtained from dynamic light scattering (DLS) experiments and to the weight average diffusion coefficient (Dw). This latter value is sometimes considered as the most representative of the sample mixture. From these results, it appears that, for monomodal distribution and relatively low polydispersity (I ) 1.15), the average diffusion coefficient generally derived from TDA is very close to Dw. However, for highly polydisperse samples (e.g., bimodal polydisperse distributions), important differences could be obtained (up to 35% between TDA and Dw). In all the cases, the average diffusion coefficient obtained by TDA for a mass concentration detector was closer to the Dw value than the z-average obtained by DLS. Different techniques can be used for the measurement of diffusion coefficients (D). One can distinguish methods that allow determining an average diffusion coefficient of the entire sample from separation techniques possibly coupled to multiple detections that allow determining the diffusion coefficient of each separated zone. In the first category, average diffusion coefficients can be obtained by free diffusion,1 sedimentation,2 dynamic light scat* To whom correspondence should be addressed. Tel: +33 4 6714 3427. Fax: +33 4 6763 1046. E-mail:
[email protected].. Tel: +33 1 4079 4707. Fax +33 1 4079 4523. E-mail:
[email protected]. † Universite ´ de Montpellier. ‡ Universite ´ Paris. (1) Schachman, H. K. In Methods in Enzymology; Colowick, S. P., Kaplan, N. O., Eds.; Academic Press: New York, 1957; Vol. 4. (2) Van, Holde, K. E.; Baldwin, R. L. J. Phys. Chem. 1958, 62, 734-743.
9066 Analytical Chemistry, Vol. 79, No. 23, December 1, 2007
tering (DLS),3 pulsed NMR techniques,4 and Taylor dispersion (or diffusion) analysis (TDA). In the second category, one can cite separation methods for which retention is related to the analyte diffusion coefficient in the carrier fluid, such as hydrodynamic chromatography (HDC),5 size exclusion chromatography (SEC),6 or flow field-flow fractionation,7 or for which a stopped migration procedure provides access to D, such as arrested flow chromatography8 or “stop flow” capillary electrophoresis.9 TDA is an absolute, simple, and rapid method for determining average D values. TDA is based on the seminal work of Taylor,10 later extended by Aris,11 who computed the dispersion coefficient of a solute plug in an open tube under Poiseuille laminar flow conditions. Due to the parabolic velocity profile, molecules injected in a narrow band at the inlet end of the tube move with different velocities depending on their positions in the tube cross section. The dispersion of the solute plug depends on the molecular diffusion that redistributes the molecules over the cross section of the tube. While the method developed by Taylor required the measurement, at a given time, of the solute concentration at different positions along the length of the tube, the development, a few years after the work of Taylor, of fast flow-through chromatographic detectors allowed the measurement of the dispersion coefficient from the distribution of solute concentration at a given position along the tube (the tube outlet) as a function of time. This method, which is the basis of modern TDA, was first applied to the determination of gaseous diffusion coefficients12 and then to liquid diffusion coefficients.13-15 More recently, the instrument for capillary zone electrophoresis, which allows the detection at a given location inside the capillary, was shown to be particularly well suited for performing TDA.16 (3) Berne, B. J.; Pecora, R. Dynamic Light Scattering; Wiley-Interscience: New York, 1976. (4) Stejskal, E. O.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288-292. (5) Bos, J.; Tijssen, R. In Chromatography in the Petroleum Industry; Adlard, E. R., Ed.; Journal of Chromatography Library, Vol. 56; Elsevier: Amsterdam, 1995. (6) van Asten, A. C.; van Dam, R. J.; Kok, W. Th.; Tijssen, R.; Poppe, H. J. Chromatogr., A 1995, 703, 245-263. (7) Giddings, J. C.; Yang, F. J.; Myers, M. N. Science 1976, 193, 1244-1245. (8) Knox, J. H.; McLaren, L. Anal. Chem. 1964, 36, 1477-1482. (9) Walbroehl, Y.; Jorgenson, J. W. J. Microcolumn Sep. 1989, 1, 41-45. (10) Taylor, G. Proc. R. Soc., London A 1953, 219, 186-203. (11) Aris, R. Proc. R. Soc., London A 1956, 235, 67-77. (12) Giddings, J. C.; Seager, S. L. J. Chem. Phys. 1960, 33, 1579-1580. (13) Ouano, A. C. Ind. Eng. Chem. Fundam. 1972, 11, 268-271. (14) Pratt, K. C.; Wakeham, W. A. Proc. R. Soc., London A 1974, 393-406. (15) Grushka, E.; Kikta, E. J. J. Phys. Chem. 1974, 78, 2297-2301. (16) Bello, M. S.; Rezzonico, R.; Righetti, P. G. Science 1994, 266, 773-776. 10.1021/ac071018w CCC: $37.00
© 2007 American Chemical Society Published on Web 10/25/2007
TDA is applicable on (macro)molecules and particles of virtually any molar mass. Since it is absolute, no calibration is required and the knowledge of the sample concentration is not needed provided that this concentration is low enough for the diffusion coefficient to correspond to its high dilution limit. Despite the advantages of this method, there are relatively few works dealing with the use of TDA for mixtures of solutes and, more specifically, for polymer analysis. The theory pertaining for threecomponent systems has been developed by Price.17 It is somewhat complex as it involves diffusion cross-terms. Still, if the analyzed sample is a mixture of highly diluted solutes in the carrier solvent, these cross-terms can be neglected. Barooah et al. measured the diffusion coefficient of polystyrene in cyclohexane18 and in dioxane,19 at infinite dilution. Boyle et al.20 determined diffusion coefficients of pauci- and polydisperse poly(styrenesulfonate) samples by studying the variation of the peak width with the carrier velocity, by a method they called flow injection analysis. However, since they operated in conditions where the peaks eluted from the tube in a time smaller than the characteristic time of diffusion across the tube radius, their approach does not rely on Taylor’s analysis of dispersion and does not correspond to TDA. Mes et al.21 reported a comparison of different methods (DLS, TDA, HDC, SEC) for the determination of diffusion coefficients of synthetic polymers (styrene acrylonitrile copolymers). The TDA method relies on the expression of the dispersion coefficient given by Taylor10 and extended by Aris.11 There are several possible approaches for exploiting the time distribution of the sample concentration recorded at the capillary tube outlet or at a given position along the capillary by a suitable detector. A first method consists in fitting the experimental data to a theoretical expression of the concentration distribution as a function of time.22 A second approach relies on the measurement of the peak area and peak height, with the assumption of a Gaussian peak shape for a pulse injection.18 A third, more common, approach relies on the experimental determination of the temporal variance of the sample peak.12-16,21 The theory of the latter approach has been developed,23 and corrections to the variance arising from instrumental contributions have been studied in detail.24 In the case of binary systems (for instance, solutesolvent samples), all these approaches are expected to provide the same value of the diffusion coefficient. However, in the case of three or more component systems, two or more mutual diffusion coefficients are involved, even at infinite dilution. The fitting approach requires the knowledge of the exact number of components, its accuracy decreases with increasing sample complexity, and this approach becomes hardly applicable for more than three-component systems. The two other approaches provide an (17) Price, W. E. J. Chem. Soc., Faraday Trans. 1 1988, 2431-2439. (18) Barooah, A.; Chen, S. H. J. Polym. Sci., Part B: Polym. Phys. 1985, 23, 2457-2468. (19) Barooah, A.; Sun, C. K. J.; Chen, S. H. J. Polym. Sci., Part B: Polym. Phys. 1986, 24, 817-825. (20) Boyle, W. A.; Buchholz, R. F.; Neal, J. A.; McCarthy, J. L. J. Appl. Polym. Sci. 1991, 42, 1969-1977. (21) Mes, E. P. C.; Kok, W. Th.; Poppe, H.; Tijssen, R. J. Polym Sci., Part B: Polym Phys. 1999, 37, 593-603. (22) Umecky, T.; Kuga, T.; Funazukuri, T. J. Chem. Eng. Data 2006, 51, 17051710. (23) Alizadeh, A.; De Castro, C. A.; Wakeham, W. A. Int. J. Thermophys. 1980, 1, 243-284. (24) Sharma, U.; Gleason, N. J.; Carbeck, J. D. Anal. Chem. 2005, 77, 806813.
average value of the diffusion coefficient. Because these approaches are based on measurements of different characteristics of the peak profile, they provide different average diffusion coefficients. Barooah and Chen18 have shown that the approach based on the determination of peak height and area leads to an average diffusion coefficient equal to the square of the weightaverage value of the square roots of the diffusion coefficients of the individual sample components. This was later noticed by Kelly and Leaist.25 The classical TDA approach, based on the measurement of two first moments of the elution peak (elution time and variance), does not require the assumption that the peak is Gaussian or symmetrical. However, the question arises of which kind of average diffusion coefficient is provided by this approach. Furthermore, in the case of a mixture of components having some common response property for the detector at hand (for instance, polymeric samples with molar mass-sensitive detector), one wishes to know how the average D depends on this common response property. Therefore, the main purpose of this work was to establish the equations giving the average diffusion coefficients in the case of TDA of sample mixtures and to verify these equations on experimental mixtures of polymeric and small molecules. Next, we found it interesting to compare the experimental average diffusion coefficients derived from TDA to the z-average diffusion coefficients (that can be derived from DLS experiments26) for polymeric samples. THEORY When the sample injected in a Taylor dispersion experiment is not a pure analyte, but a mixture of analytes (e.g., a polymeric sample with molar mass distribution), the diffusion coefficient derived from the TDA by means of the determination of the peak variance is an average value of the individual diffusion coefficients. The objective of this theoretical part is to specify which kind of average diffusion coefficient is obtained by TDA. Variance of the Sample Peak. The temporal variance, σs2, of the sample peak recorded by the detector is equal to
∫y(t - t ) ) ∫y dt
2
2
σs
R
dt (1)
where y is the detector signal, t the time elapsed since sample injection, and tR is the mean elution time (first moment of the elution peak). The subscript s refers to the sample peak, and the integration extends all over the elution profile. Similarly, the temporal variance of the peak of an individual analyte, i, is given by
∫y (t - t ) ∫y dt
2
σi2 )
R
i
dt (2)
i
yi being the detector signal that would be obtained by injecting the individual analyte i at the concentration it has in the sample. We note that, in Taylor dispersion analysis, the mean elution time, (25) Kelly, B.; Leaist, D. G. Phys. Chem. Chem. Phys. 2004, 6, 5523-5530. (26) Brown, J. C.; Pusey, P. N. J. Phys. D: Appl. Phys. 1974, 7, L31-L35.
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9067
tR, is the same for all analytes and is equal to L/u, where L is the capillary length to the detector and u is the average velocity of the eluent. We assume in the following that the detector is linear, i.e., first, that it provides a signal linearly proportional to the analyte or sample concentration and, second, that it is operated in its linearity domain (for sufficiently diluted solutions). Most chromatographic detectors are linear, but a few others are intrinsically nonlinear (such as the evaporative light scattering detector or the flame photometric detector in the sulfur mode). In these conditions, the detector signal becomes
∑y
y)
(3)
i
i
∫∑ 2
σs )
i
∑ ∫ i
)
∫∑ (
)
∑∫
yi) dt
i
yi dt
∑φ σ
2
i i
i
∑ ∫y dt
)
Dav,1
Dav,2 )
with
∫y dt
i
R2u
φi
∑ 24 D
+ u
i
(9)
i
It thus appears that, for a mixture, the plate height curve, Hs(u), is the sum of an hyperbolic term and of a linear term, as for a single analyte, but that the average diffusion coefficients appearing in the expressions of these two terms are not the same, in contradiction with what happens for a single analyte. Hence, the average diffusion coefficient, Dav,1, obtained from the slope, dHs/ du, of the ascending branch of the Hs versus u curve is equal to
(4)
i
i
φi )
i
i
24 dHs
)
R2 du
φi
∑D i
(at large u)
(10)
i
while that, Dav,2, obtained from the slope, dHs/d(1/u), of the linear branch of the Hs versus 1/u curve is
σi2 yi dt
yi)(t - tR)2 dt
∑φ D
Hs )
1
the summation extending for all analytes present in the sample. Combining eqs 1-3, one gets
(
2
(5)
1 dHs
)
2 d(1/u)
∑φ D i
(at low u)
i
(11)
i
Hence Dav,1 is a harmonic average of the individual Di weighted by their fractional response area, while Dav,2 is an arithmetic average. Average Frictional Coefficients. The diffusion coefficient, Di, of a species is related to its frictional coefficient, fi, as
i
i
Di ) kBT/fi
φi is the fractional peak area of peak i, i.e., the ratio of the peak area of the analyte i to the total peak area of the sample. Average Diffusion Coefficients. In chromatography, the peak dispersion is commonly measured by the plate height, H, of the sample peak. The dispersion coefficient is equal to Hu/2. H is related to the two first moments of the elution profile:
Hs ) L(σs2/tR2)
(6)
Combining eqs 4 and 6, one gets
Hs )
∑φ H i
i
where kB is the Boltzmann constant and T the absolute temperature.28 Determining a diffusion coefficient is equivalent to determining a frictional coefficient. In some instances, the knowledge of f is of more direct interest than that of D since, for analytes of a size significantly larger than that of the solvent molecules, f is proportional to the hydrodynamic radius of the species (Stokes law). The average frictional coefficient, fav,1, derived from Dav,1, with the help of eqs 9 and 11, as
fav,1 ) (7)
(12)
kBT Dav,1
) kBT
φi
∑D ) ∑φ f
ii
i
(13)
i
i
i
is an arithmetic average, while, fav,2, derived from Dav,2, as The plate height, Hi, of an unretained analyte in a capillary column of radius R is related to its molecular diffusion coefficient Di and to the average eluent velocity according to10,11,27
2Di R2u + Hi ) u 24Di
kBT Dav,2
)
kBT
∑
φiDi
i
(8)
Combining eqs 7 and 8, one gets (27) Golay, M. J. E. In Gas Chromatography; Desty, D. H., Ed.; Butterworths: London, 1958; pp 36-53.
9068
fav,2 )
Analytical Chemistry, Vol. 79, No. 23, December 1, 2007
1
)
(14)
φi
∑f i
i
is a harmonic average. Similar expressions are obtained for the average hydrodynamic radius, Rh,av of the individual radii Rh,i, by replacing fav and fi by Rh,av and Rh,i, respectively, in eqs 13 and 14. (28) Einstein, A. Ann. Phys. 1906, 19, 371-381.
It can be shown that a harmonic average is always smaller than an arithmetic one. Hence, one always has: Dav,2 > Dav,1, and fav,1 > fav,2. Detector Response. As assumed above, the detector signal for analyte i is proportional to the mass concentration, ci, of that analyte in the detector cell:
yi ) kici
(15)
of their molar mass. The case of r ) +1 is typical of light scattering detectors for which the response is proportional to ciMi, whatever Mi, at least when the virial terms are negligible in the detector response function. For all these detectors, the λr coefficient being the same for all macromolecular analytes, it disappears from the expression of the average diffusion coefficients, which take then some particular forms. Combining eqs 10 and 11 for Dav, eqs 13 and 14 for fav(or Rh,av) with eqs 18 and 19, one gets
where ki is the response factor for analyte i. The contribution of analyte i to the sample peak area is equal to
∫
yi dt ) ki
∫
kimi,inj Q
ci dt )
Dav,1 ) (16)
kimi,inj
∑k m i
kici,inj
)
∑k c
i,inj
∑
Dav,2 )
i
i
∑
(21) Mi1+rNi
∑M
1+r
Nifi
i
(18) i
i
Most often, the detectors used in polymer analysis have a response factor that is a simple power law function of the molar mass and can be expressed as
ki ) λrMir
NiDi
i
i
(17)
where ci,inj is the mass concentration of analyte i in the injected sample. Detector Response for Polymeric Samples. When the sample is a mixture of macromolecules of various molar masses made of a common repeating unit, the average diffusion coefficients determined from TDA can take specific expressions. Let Ni be the number of moles of macromolecules of molar mass Mi in the injected sample. Noting that mi,inj ) Mi Ni, one gets, from eq 17
∑k M N
1+r
i
fav,1 )
φi )
(20)
{Mi1+rNi}/{Di}
∑M
i
kiMiNi
Ni
i
i i,inj
i
1+r i
i
where mi,inj is the mass of analyte i in the injected sample and Q the eluent flow rate. The fractional peak area of analyte i becomes
φi )
∑M
(19)
λr being a constant independent of the molar mass for a given polymer-solvent system, and r an exponent taking generally values of -1, 0, or +1. Detectors corresponding to r ) 0 are called mass concentration-sensitive detectors since, as seen from eqs 15 and 19, their response is proportional to the mass concentration of the macromolecules, whatever their molar mass. This is generally the case of differential refractometers or of UV detectors when the chromophores belong to the repeating units of the macromolecules. Detectors for which r ) -1 are molar concentration-sensitive detectors, since their response is proportional to the molar concentration, ci/Mi, of the macromolecules. This is the case of UV detectors when the chromophore(s) is(are) beared by the end group(s) of the macromolecules, so that all macromolecules have the same number of chromophores, irrespective
i
∑
(22) Mi1+rNi
i
∑M fav,2 )
1+r i
Ni
i
∑
(23)
{Mi1+rNi}/{fi}
i
For molar concentration-sensitive detectors (r ) -1), the average diffusion coefficient obtained from the linear part of the Hs ) f(u) plot (see eq 9) is a number-average harmonic value, while the average friction factor and hydrodynamic radius are numberaverage values. For mass concentration-sensitive detectors (r ) 0), they become a weight-average harmonic value and weightaverage values, respectively. For a light scattering detector (r ) +1), the linear part of the Hs ) f(u) plot would provide a z-average harmonic diffusion coefficient and z-average values of the friction factor and of the hydrodynamic radius. From the low-velocity part of the plate height curve, i.e., from the linear part of Hs ) f(1/u) plot, one gets number-average, weight-average, and z-average diffusion coefficients for the r ) -1, r ) 0, and r ) +1 detectors, respectively. Then the friction factors and hydrodynamic radii are number-average, weightaverage, and z-average harmonic values, respectively. EXPERIMENTAL SECTION Reagents. Borax (disodium tetraborate decahydrate) was purchased from Prolabo (Paris, France). Mesityl oxide, phthalic acid, sodium dihydrogenophosphate, and disodium hydrogenophosphate were obtained from Aldrich (Milwaukee, WI). Sodium hydroxide was from Merck (Darmstadt, Germany). The water used to prepare all buffers was further purified with a Milli-Q system from Millipore (Molsheim, France). The borate buffers were directly prepared by dissolving the appropriate amount of borax in water. Analytical Chemistry, Vol. 79, No. 23, December 1, 2007
9069
Standard of poly(styrenesulfonate) (PSS; weight-average molecular mass Mw 1.45 × 105 g/mol) was purchased from Polymer Standards Service (Mainz, Germany). The polydispersity index of the PSS is below 1.2. The degree of sulfonation of the PSS is higher than 90%. Taylor Dispersion Analysis. TDA experiments were performed on a PACE MDQ Beckman Coulter (Fullerton, CA) apparatus. Capillaries were prepared from bare silica tubing purchased from Composite Metal Services (Worcester, United Kingdom). Capillary dimensions were 60 cm (50 cm to the detector) × 50 µm i.d. New capillaries were conditioned with the following flushes: 1 M NaOH for 30 min, 0.1 M NaOH for 30 min, and water for 10 min. Before sample injection, the capillary was filled with the buffer (80 mM borate buffer, pH 9.2). Phthalate and PSS samples were dissolved in the buffer at 0.5 g/L. Mixtures of these two samples were prepared with the following volumetric proportions (25/75; 50/50; 75/25). Sample injection was performed hydrodynamically on the inlet side of the capillary (1.5 psi, 9 s). Mobilization pressures of 0.5, 1, 2, 4, and 6 psi were applied with buffer vials at both ends of the capillary. Between two TDA analyses, the capillary was successively flushed with the following: (i) water (50 psi, 1 min); (ii) 1M NaOH (50 psi, 2 min), and (iii) buffer (50 psi, 3 min). Since the solutes are highly anionic, no specific capillary coating was required for performing TDA on fused-silica capillary. Solutes were monitored by UV absorbance at 200 nm. The temperature of the capillary cartridge was set at 25 °C. The elution time was corrected from the delay in the application of the pressure using the following equation (in min):
tR ) tR,obs - 0.125
Figure 1. Overlay of the TDA signals obtained for phthalate, a standard of PSS (Mr 1.45 × 105), and different mixtures of both. Volumetric proportions of the mixtures are given on the graph. Fusedsilica capillary, 60 cm (50 cm to the detector) × 50 µm i.d. All experiments were performed in 80 mM borate buffer, pH 9.2. Mobilization pressure, 2 psi. Samples at 0.5 g/L in buffer. Hydrodynamic injection, 1.5 psi, 9 s. UV detection at 200 nm. Temperature, 25 °C.
(24)
Since the diffusion coefficient was determined from the slope of the Hs ) f(u) curve, the correction from the finite injection plug was not considered since this contribution only translates the experimental Hs points by a constant value. Moreover, the correction due to the finite injection plug on the observed elution time remains small (∼1%). RESULTS AND DISCUSSION Taylor Dispersion Analysis of a Small Molecule and Polymer Mixtures. The TDA was performed using a 80 mM borate buffer (pH 9.2) as mobile phase and a 60 cm long fusedsilica capillary (50 cm to the detector) with internal diameter of 50 µm. A superposition of the elution profiles obtained for phthalate, PSS (Mw ) 1.45 × 105 g/mol), and mixtures of both in different volumetric proportions is given in Figure 1 for a mobilization pressure of 2 psi (∼143 mbar). As expected, the elution profile of the phthlate is narrower than the one corresponding to the polymer sample. This is in accordance with the decrease of the second term of eq 8 for high diffusion coefficients. Indeed, in our conditions of mobilization pressures, the second term of eq 8 is preponderant. The condition for eq 8 to be valid (tR . R2/D see, e.g., ref 23) was satisfied for almost all the experimental data. It is worth noting that the experimental UV traces obtained for the mixtures correspond to the addition of the two individual signals weighted by the volumetric proportions. This is demonstrated in Figure 2 where the experimental trace 9070
Analytical Chemistry, Vol. 79, No. 23, December 1, 2007
Figure 2. Comparaison of the experimental and calculated UV traces obtained for a 75/25 v/v PSS/phthalate mixture. The calculated UV trace of the mixture has been obtained by adding the experimental TDA UV traces of the two individual components weighted by the volumetric proportions.
corresponding to the 75/25 v/v PSS/phthalate mixture is superimposed on the arithmetic sum of the individual signals weighted by the volumetric proportions. Next, the plate heights of the experimental elution profiles were calculated using eq 6 and were plotted as a function of the linear velocity of the eluent, u, in Figure 3. Straight lines were obtained and the average diffusion coefficients were derived from the slopes (R2/24D) of these lines. For the PSS sample, the experimental diffusion coefficient was found to be (2.42 ( 0.27) × 10-11 m2 s-1 while for the phthalate D ) (7.93 ( 0.26) × 10-10 m2 s-1. The experimental D values obtained for PSS/phthalate mixtures can be compared to the theoretical Dav,1 values given by eq 10 in combination with eq 17. Actually, the sample mixtures were made from stock solutions of the individual analytes at concentration ci,0 in the carrier liquid (i ) 1
Figure 3. Hs ) f(u) plots obtained from the TDA of the individual samples and the mixtures of both in different proportions. Other conditions as in Figure 1.
Figure 4. Experimental and calculated diffusion coefficients as a function of the mass fraction of PSS in the binary mixture. Experimental values (Dav,1) are derived from Figure 3 using eq 10. Calculated values were obtained using eq 26.
and 2, for the binary PSS/phthalate mixture in the carrier) by adjusting the volumetric fractions, Fi (with F1 + F2 ) 1). The concentration of analyte i in the sample mixture becomes then ci,inj ) Fi ci,0. Then, according to eqs 10 and 17, Dav,1 becomes
Dav,1 )
k1F1c1,0 + k2F2c2,0 k1F1c1,0 k2F2c2,0 + D1 D2
(25)
Since, in the present study, c1,0 ) c2,0 ) 0.5 g/L, the average diffusion coefficient is then given by
Dav,1 )
F1 + F2 F1 F2 + D1 D2
(26)
with ) k2/k1 being the ratio of the response factors (in this work ) kPSS/kphthalate ) 0.736 ( 0.005). Figure 4 shows the comparison between the calculated Dav,1 values using eq 26 (closed circles) to the experimental values (open squares). A very good agreement between these values was obtained demonstrating the validity of the equations set in the theoretical section. Comparison of Average Diffusion Coefficients for Polymer Samples. To get a better insight in the differences of average diffusion coefficients that can be obtained by TDA or by DLS, we found it useful to compare these average values to the weightaverage diffusion coefficient Dw for typical polymeric distributions. The weight-average diffusion coefficient Dw is sometimes considered as the most representative D value of the polymer sample since each diffusion coefficient is weighted by the mass proportion of the corresponding polymer. Let consider three polymer samples, each having a Gaussian distribution in molar mass, centered at Mw equal to 14.5 × 103, 145 × 103, and 1450 × 103 g/mol, respectively, and with polydispersity indexes of 1.15. In other words, for each polymer sample, the molar mass distribution obtained with a mass concentration-sensitive detector (detector response proportional to NiMi) leads to a Gaussian curve (see,
Figure 5. Molar mass and diffusion coefficient distributions of a typical polymeric sample (Mw ) 145 × 103 g/mol PSS sample) used in this work for comparison of the average diffusion coefficients.
e.g., Figure 5 for the Mw ) 145 × 103 g/mol sample). Obviously, for a molar concentration-sensitive detector (detector response proportional to Ni), the molar mass distribution is no longer Gaussian. For each molar mass, the corresponding diffusion coefficient was calculated using the following equation:29
Di )
(
kBT 10πNA 6πη 3[ηi]Mi
)
1/3
(27)
where [ηi] is the intrinsic viscosity, NA is the Avogadro number, η is the solvent viscosity, and with the reasonable assumption that the Stokes radius (from diffusion) and the viscosity radius are identical for statistical chains. The intrinsic viscosity was calculated according to [ηi] ) 1.39 × 10-4 Mi0.72, [ηi] in dL/g, for a PSS in water at 0.05 M ionic strength.30 As an example, Figure 5 displays the distribution in molar mass and in diffusion coefficient for a (29) Rudin, A.; Johnston, H. K. J. Polym. Sci., Part B: Polym. Lett. 1971, 9, 5560. (30) Tricot, M. Macromolecules 1984, 17, 1698-1704.
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Table 1. Comparison of the Average Diffusion Coefficients That Can Be Obtained by TDA (Mass Concentration- or Molar Concentration-Sensitive Detector) or by DLS for Four Fictive PSS Samplesa TDA (mass concn, r ) 0)
PSS 14.5 k I ) 1.15 PSS 145 k I ) 1.15 PSS 1 450 k I ) 1.15 (50 /50 w/w) PSS 14.5 k + PSS 145 k
TDA (molar concn, r ) -1)
DLS (r ) +1)
Dav,1
Dav,2 ) Dw
Dav,1
Dav,2
Dz
7.79 × 10-11 (-4.1%) 2.08 × 10-11 (-4.1%) 5.55 × 10-12 (-4.0%) 3.28 × 10-11 (-36%)
8.13 × 10-11
8.53 × 10-11 (+4.9%) 2.28 × 10-11 (+5.1%) 6.10 × 10-12 (+5.5%) 6.83 × 10-11 (+33%)
9.56 × 10-11 (+17%) 2.89 × 10-11 (+33%) 1.06 × 10-11 (+83%) 8.96 × 10-11 (+74%)
7.59 × 10-11 (-6.6%) 2.03 × 10-11 (-6.5%) 5.42 × 10-12 (-6.2%) 2.53 × 10-11 (-51%)
2.17 × 10-11 5.78 × 10-12 5.15 × 10-11
a All the average diffusion coefficients were calculated and not experimentally determined. Average diffusion coefficient values were calculated using eqs 20 and 21 with r ) -1 and r ) 0, and eq 28 by integrating on the entire polymeric distribution. For that, the diffusion coefficient corresponding to each molar mass was calculated by eq 27 using viscosity data from the literature.30 For each average value, the relative difference from Dw is given in parentheses.
Mw of 145 × 103 g/mol PSS sample. The average diffusion coefficients Dav,1 and Dav,2 that can be potentially derived from TDA were calculated according to eqs 20 and 21 for a molar concentration-sensitive detector (r ) -1) and for a mass concentrationsensitive detector (r ) 0). The corresponding values are given in Table 1 and were compared, on one hand, to the z-average diffusion coefficient corresponding to DLS measurements and to TDA at low velocities with a light scattering detector (Dav,2 for r ) +1), and on the other hand, to the weight average diffusion coefficient Dw. It is worth noting that Dw indeed corresponds to Dav,2 in the case of a mass concentration-sensitive detector (third column in Table 1). However, TDA generally leads to Dav,1 values since the second term in eq 9 is preponderant. The determination of Dav,2 requires operating at low eluent velocities and is therefore less convenient to perform experimentally. From Table 1, it appears that, depending on the detector, Dav,1 can be higher (for the molar concentration-sensitive detector) or lower (for the mass concentration-sensitive detector) than Dw. However, for the three polymer samples (with I ) 1.15), deviations between Dav,1 and Dw remains moderate (within 6%). Lower differences were obtained with the mass concentration-sensitive detector (-4%) than for the molar concentration-sensitive detector (+5 to +6%). The difference in the Dav,1 values obtained with these two detectors is ∼10% whatever the molar mass of the sample. On the contrary, the corresponding difference on Dav,2 depends on the molar mass and could be very high (up to 83% for Mw1450 × 103 g/mol sample). The z-average values obtained by DLS according to the following equation:
∑ NM i
Dz )
2
i
∑ NM i
∑N M R i
mass
Rh,av,1 )
Di
i h,i
i
∑
(29) NiMi
i
(28) 2
i
are about -6 to -7% below Dw. Whatever the detector and the sample, Dz was always lower than Dav,1, and Dav,1 was always lower than Dav,2. This means that the z-average value gives larger weight to the smaller diffusion coefficients (larger hydrodynamic radii and larger molar masses) than number or weight harmonic 9072
average values (Dav,1). Similarly, harmonic average values (Dav,1) are always smaller than arithmetic average values (Dav,2). As for Dav,1, the mass concentration-sensitive detector leads to smaller Dav,2 values than the molar concentration-sensitive detector. Regarding the case of a bimodal polymer sample (50/50 w/w mixture of Mw 14.5 × 103 and 145 × 103 g/mol aforementioned samples), similar trends are observed except that the differences between Dav,1 and Dw, on one hand, and Dz and Dw, on the other hand, drastically increase. Dav,1 is ∼36% lower (mass concentrationsensitive detector) or 33% higher (molar concentration-sensitive detector) than Dw, while Dz is ∼51% lower than Dw. It is worth noting that, in the case of highly disperse samples, the average values Dav,1 (mass concentration-sensitive detector), Dw, and Dz are very different from each other. Dav,1 (mass concentrationsensitive detector) is however closer to Dw than Dz. It is also interesting to notice that, whatever the polymer distribution (mono or bimodal), Dw is close to the arithmetic average value of the two Dav,1 diffusion coefficients. Since the determination of diffusion coefficients is generally performed to get information on the size of the (macro)molecules, it seems interesting to discuss the results in terms of average hydrodynamic radii. As a general rule, TDA with a mass concentration-sensitive detector leads to the weight-average hydrodynamic radius (see eq 22 with r ) 0):
Analytical Chemistry, Vol. 79, No. 23, December 1, 2007
TDA with a molar concentration-sensitive detector leads to the number-average hydrodynamic radius (see eq 22 with r ) -1):
∑N R
i h,i
molar
Rh,av,1 )
i
∑ i
(30) Ni
Finally, DLS leads to an harmonic z-average value according to the following equation:
∑N M i
DLS Rh, av )
2 i
i
NiMi2
∑R i
(31)
h,i
CONCLUSION This work sets the equations giving the average diffusion coefficients that can be measured by the classical TDA approach based on the determination of the variance of the sample peak recorded by a detector located at the capillary outlet or at a fixed position along the capillary. The average D value experimentally obtained by TDA depends on the nature of the detector (mass concentration- or molar concentration-sensitive detector), but also on the part of the Hs ) f(u) plot that is considered. Four different mass molar average values can be obtained by TDA (namely Dav,1 , Dav,1 , mass molar Dav,2 , and Dav,2 where the superscript refers to the type of
detector). For the sake of completion, the average diffusion coefficients that would be obtained by means of a light scattering detector are also provided. Generally, the average diffusion coefficient obtained by TDA corresponds to the ascending portion of the Hs ) f(u) curve and therefore, the corresponding average mass mole and Dav,1 depending on the nature of the D values are Dav,1 mass mole detector. Dav,1 is a weight-average harmonic value, while Dav,1 is a number-average harmonic value. The average hydrodynamic radii that are derived from these values from the Stokes-Einstein relationship are weight-average and number-average arithmetic values, respectively. From numerical calculations using the previously established equations, it appears that, for samples with monomodal and relatively low polydispersity (I ) 1.15), the average diffusion coefficients generally measured by TDA (Dav,1) are very close to Dw (∼4-5% differences). For highly polydisperse samples (e.g., bimodal polydisperse distributions), important differences could be obtained between TDA, Dw, and DLS (Dz). Received for review May 18, 2007. Accepted September 17, 2007. AC071018W
Analytical Chemistry, Vol. 79, No. 23, December 1, 2007
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