Anal. Chem. 1997, 69, 1339-1346
Retention in Field-Flow Fractionation with a Moderate Nonuniformity in the Field Force Michel Martin,* Charles Van Batten, and Mauricio Hoyos
Ecole Supe´ rieure de Physique et Chimie Industrielles, Laboratoire de Physique et Me´ canique des Milieux He´ te´ roge` nes (URA CNRS 857), 10, rue Vauquelin, 75231 Paris Cedex 05, France
In some field-flow fractionation (FFF) techniques, the basic analyte-field interaction parameter, λ, is not constant but varies within the channel cross section as a result of the nonuniformity of the force exerted by the field on the analyte. This is the case, for instance, in thermal FFF, because of the temperature dependence of the relevant physicochemical transport parameters. To account for this effect, a new FFF retention model is developed, allowing a linear variation of λ from the accumulation to the depletion wall, which is assumed to describe correctly moderate nonuniformity in λ in the vicinity of the accumulation wall. A methodology for sample characterization on the basis of this model is proposed. It associates λapp, the apparent λ value derived from the retention ratio by means of the classical retention model, with a specific distance from the accumulation wall. An empirical relationship between that distance and λapp is derived. In the high retention limit, it is found that this specific distance is not equal, as sometimes intuitively believed, to the mean distance of the molecule or particle cloud to the accumulation wall but is approximately equal to twice this mean distance. The validity of the proposed approach is checked. The whole family of field-flow fractionation (FFF) methods of separation and characterization covers a very broad range of applications extending from macromolecules with a molecular mass of a few thousand daltons to particulate materials of a size of almost 100 µm.1 In FFF, components of a supramolecular sample are differentially transported along a ribbonlike channel by means of the flow of a carrier fluid under the influence of an external field applied perpendicularly to the flow axis, in the direction of the small dimension of the channel cross section. In well-operated FFF systems, the component migration occurs in quasi-equilibrium conditions, and the mean retention time of an analyte depends on the equilibrium distribution of its concentration in the field direction as well as on the flow velocity profile. The most common retention mechanism for macromolecules and colloids of submicrometer size is the Brownian (or normal) mode of retention. In this mode, the accumulation of an analyte near one of the channel main walls under the applied field is counteracted by the effect of its Brownian motion. This leads to a decrease of the analyte concentration with increasing x, the distance from the accumulation wall. In isothermal conditions, when the force exerted by the field on the analyte molecules or particles is constant, the concentration profile, c(x), decreases (1) Giddings, J. C. Science 1993, 260, 1456-1465. S0003-2700(96)00530-6 CCC: $14.00
© 1997 American Chemical Society
exponentially with increasing x, according to the Boltzmann distribution law.2 Then, the relative concentration profile, c(x)/ co, where co is the analyte concentration at the accumulation wall, depends on a single dimensionless parameter, λ, which characterizes the interaction of the analyte with the applied field:
x/w c ) exp co λ
(
)
(1)
where w is the channel thickness (distance between the two channel walls). The classical retention theory, established under the assumptions of an exponential analyte concentration profile and a parabolic velocity profile (Poiseuille flow), indicates that the retention ratio, R, depends only and monotonously on λ, according to eq 2:
[ (2λ1 ) - 2λ]
R ) 6λ coth
(2)
Hence, the experimental measurement of R allows the λ parameter to be determined and the analyte to be characterized. The accuracy of the resulting determinations depends on the degree to which the assumptions underlying eq 2 are fulfilled in practical situations. Deviations of the velocity profile from the parabolic shape as a result of the temperature dependence of the carrier viscosity have been long recognized in thermal FFF.3 They are conveniently taken into account by approximating the true velocity profile by a third-degree polynomial velocity profile containing a parameter, ν, adjusted in such a way that the slopes of these two profiles at the accumulation wall (generally the cold wall) are equal.4-6 The rationale behind this definition of ν arises from the fact that the migration of an analyte is mostly controlled by the velocity profile in the vicinity of the accumulation wall. The parameter ν depends on the carrier, the temperature drop between the two walls, and the cold wall temperature. Methods for estimating ν are available.6 In sedimentation FFF, deviations from the parabolic flow profile are due to the curvature of the flow (2) Martin, M.; Williams, P. S. In Theoretical Advancement in Chromatography and Related Separation Techniques; Dondi, F., Guiochon, G., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1992; pp 513-580. (3) Hovingh, M. E.; Thompson, G. H.; Giddings, J. C. Anal. Chem. 1970, 42, 195-203. (4) Gunderson, J. J.; Caldwell, K. D.; Giddings, J. C. Sep. Sci. Technol. 1984, 19, 667-683. (5) 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. (6) Belgaied, J. E.; Hoyos, M. H.; Martin, M. J. Chromatogr. A 1994, 678, 8596.
Analytical Chemistry, Vol. 69, No. 7, April 1, 1997 1339
streamlines. It can be shown that the ν parameter of the approximate third-degree polynomial velocity profile is equal to plus or minus one-third of the curvature ratio, i.e., of the ratio of w to the average radius of curvature of the two channel walls, Rc, the sign depending on whether the accumulation wall is the inner or outer wall.7 In present-day sedimentation FFF systems, the curvature ratio is very small, and the deviation of the flow profile is not significant. Whatever the value of ν, the retention ratio for a third-degree polynomial velocity profile and an exponential concentration profile of the analyte is given by8,9
{
[ (2λ1 ) - 2λ]}
R ) 6λ ν + (1 - 6λν) coth
(3)
Unless ν is smaller than -1, this leads to a monotonous variation of R with λ; hence, there is no problem in the determination of λ from experimental R values once ν is known or estimated. The determination problem becomes more complex when the concentration profile of the analyte ceases to be exponential, and hence the force exerted by the field on the analyte, F, is no longer constant and becomes dependent on x. As this force can be expressed10 as F ) Sφ, i.e., as the product of the field strength, S, and the analyte-field interaction parameter, φ, which generally encompasses the analyte relevant transport parameters, this may happen when either S or φ (or both) depends on x. The former case is encountered, for instance, in sedimentation FFF since the field strength, which is then the centrifugal acceleration responsible for the migration of a non-neutrally buoyant analyte toward one of the two channel walls, increases with increasing distances from the axis of rotation, although the effect on the concentration profile is related to the curvature ratio and is likely to be small in usual systems. The latter case is found in thermal FFF, since the nonuniformity of the temperature across the channel thickness leads to a dependence of the relevant physicochemical parameter (Soret coefficient), which enters in the expression of φ, on x. In addition, the field strength itself, which is then the thermal gradient responsible for the migration of the analyte toward one of the two channel walls, is not constant due to the temperature dependence of the thermal conductivity of the carrier liquid. As a result of these combined effects, the concentration profile, c(x), is not exponential, and eqs 2 and 3 are no longer correct, although they have been and are still used for retention data interpretation in thermal FFF. An attempt to take into account this effect was made in thermal FFF. It amounted to making an assumption about the form of the functional dependence of the concentration profile on x, by means of a three-parameter relationship, and to determining these parameters by an appropriate fit of retention data obtained for a given analyte under different experimental conditions (in fact, with different cold wall temperatures) using a modified Simplex optimization procedure.5 Although it is a definitive improvement over classical methods, which fail to recognize that the concentration distribution is not exponential, this approach is relatively complex and time-consuming and cannot be applied to retention data of a single FFF experiment. (7) Martin, M. J. High Resolut. Chromatogr. 1996, 19, 481-484. (8) Giddings, J. C.; Martin, M.; Myers, M. N. Sep. Sci. Technol. 1979, 14, 611643. (9) Martin, M.; Giddings, J. C. J. Phys. Chem. 1981, 85, 727-733. (10) Giddings, J. C.; Caldwell, K. D. In Physical Methods of Chemistry, Vol. 3B; Rossiter, B. W., Hamilton, J. F., Eds.; John Wiley & Sons: New York, 1989; pp 867-938.
1340
Analytical Chemistry, Vol. 69, No. 7, April 1, 1997
There is a need for a practical methodology for handling nonexponential concentrations profiles in the Brownian retention mode of FFF. In the investigation of the thermal diffusion behavior of polymers by thermal FFF experiments, this need appeared to be crucial and motivated the present study. But since a field force nonuniformity leading to a nonexponential analyte concentration profile may be encountered, under some circumstances, in other FFF subtechniques, the methodological approach for handling such situations is, in the following, addressed in a general way applicable to any FFF subtechnique. THEORY Let us define, for simplifying notations in the following, the relative distance from the accumulation wall as
s ≡ x/w
(4)
The equilibrium concentration profile in FFF results from the balance of the convective flux of the analyte toward the accumulation wall under the applied field and of the diffusive flux, which counteracts its accumulation near this wall. Locally, this takes the following form:2
|U| ds dc )dx ) c D λ
(5)
with
λ)
kT D ) |U|w |F|w
(6)
D is the diffusion coefficient, U and F the velocity impelled and the force induced by the field on each analyte molecule or particle, k the Boltzmann constant, and T the absolute temperature. λ is the basic FFF retention parameter involved in the classical FFF retention theory. When the applied field force is uniform, λ is constant, and integration of eq 5 yields the exponential concentration profile given by eq 1. Equation 5 shows that the hypothesis of an exponential concentration profile, which leads to eqs 2 and 3 for R, is associated with a constancy of the parameter λ, i.e., of D/U (or, which is equivalent, of F/kT), whatever the position in the FFF channel. Equation 6 is the key expression that allows the analyte to be characterized by relating FFF retention to the relevant analyte physicochemical parameter expressed by the field (i.e., effective mass in sedimentation FFF, Soret coefficient in thermal FFF, or Stokes diameter in flow FFF). When the applied field force is not uniform, i.e., when λ is not constant but varies continuously from the accumulation wall to the depletion wall, the concentration profile is not exponential. It then becomes difficult to characterize the analyte since the retention in the FFF channel is associated not with a single λ value from which the relevant physicochemical parameter can be determined but, more generally, with a continuous range of λ values. Concentration Profile. To take into account the nonconstancy of λ, which leads to a nonexponential concentration profile, one makes the assumption that the variation of λ with x is moderate and can be modeled by a linear relationship,
λ ) λo + δs
(7)
or, equivalently,
λ ) λo(1 + βs)
v ) 6[(1 + ν)s - (1 + 3ν)s2 + 2νs3] 〈v〉
(8)
λo is the value of λ at the accumulation wall. δ is the variation of λ across the channel thickness (δ ) λ1 - λo, where λ1 is the value at the depletion wall), and β is the relative change of λ across the channel (β ) δ/λo). In fact, it is unlikely that the actual variation of λ be linear over the whole channel thickness. But, in practice, one needs to describe as correctly as possible the variation of λ in the vicinity of the accumulation wall where the analyte is concentrated. Accordingly, if the field nonuniformity is not too strong, one can expect that eq 7 or 8 provides a satisfactory approximation to the actual λ profile and, in any case, an improvement over the classical constant λ retention theory. The concentration profile is obtained by combination of eqs 5 and 7 or 8 and integration, which gives, for β and δ * 0,
c ) (1 + βs)-1/βλo ) (1 + βs)-1/δ co
When ν is zero, one retrieves the classical parabolic profile. The retention ratio for the concentration profile given by eq 9, which is refered to as the linear λ concentration profile in the following, and for the third-degree polynomial velocity profile is obtained by integration of eq 11 with eqs 12 and 13, for δ * 0 and * 1: 3
∑ a 1 - (i + 1)δ
R)6
i
[ (
)]
(10)
One notes that, in practice, β may not be small, so that the approximate eq 10 becomes valid only for very small values of s. It shows that, as expected, the smaller is β, the closer are the profiles given by eqs 1 and 9. Retention. Whatever the concentration and flow velocity profiles, the application to FFF of the general definition of the retention ratio for separation methods with flow perpendicular to chemical potential gradient gives11
R)
∫
1
0
c v ds 〈c 〉 〈v 〉
(14)
1 - (1 + β)(δ-1)/δ
i)0
with
(
1 + ν 1 + 3ν 2ν + + 3 β β2 β
)
(15a)
1 + ν 2(1 + 3ν) 6ν + + 3 β β2 β
(15b)
a0 ) -
a1 )
c s 1 ) exp 1 - βs + O(βs)2 co λo 2
1 - (1 + β)[(i+1)δ - 1]/δ
1-δ
(9)
Clearly, this profile differs from the classical exponential profile of eq 1. The comparison between the two profiles can be better perceived by rewriting eq 9 in an exponential form and taking the limiting expression for small values of β:
(13)
(
a2 ) -
)
1 + 3ν 6ν + 3 β2 β
a3 )
(15c)
2ν β3
(15d)
As can be seen from eqs 14 and 15, R depends now on three parameters, β, δ, and ν, instead of two (λ and ν) in eq 3. This obviously arises from the fact that two parameters are now necessary to describe the λ profile and, hence, the concentration profile. While the two parameters of the λ profile appearing in eqs 14 and 15 are β and δ, it is a simple matter to express R as a function of λo and β, or of λo and δ, by noting that δ ) λoβ, or β ) δ/λo. The expression for R can be greatly simplified if one notes that, whatever i,
(11) 1 - (1 + β)[(i+1) δ - 1]/δ
where v(s) is the velocity profile and the symbol 〈 〉 represents a mean value in the channel thickness. The relative concentration profile, c/〈c〉, corresponding to eq 9 is given, for δ ) βλo * 0 and * 1, by
1 - (1 + β)(δ-1)/δ
≈1
for i ) 1-3
(16)
To compute the retention factor, one needs to specify the relative velocity profile, v/〈v〉. Since it is likely that, when the applied field is moderately nonuniform, the velocity profile differs somewhat from the parabolic shape, as discussed in the introduction section, one assumes that the relative velocity profile is described by a third-degree polynomial in s containing the adjustable parameter ν, which gives8,9
as long as λo and β are not simultaneously too large (for i ) 0, this expression is identically equal to 1). This is most often the case in practice, since λo should not exceed about 0.2 if the analyte peak is to be satisfyingly distinguished from the void peak of an unretained species. If it were not so, the field strength would likely be relatively low, and λ would not vary significantly within the channel thickness. Also, optimized FFF separations are linked to relatively low λo values.12 Thus, if λo ) 0.1 and β ) 1 (i.e., a 100% variation of λ across the channel thickness), the error made in the approximate eq 16 varies from 0.2% to 1.4% as i goes from 1 to 3. It is significant for λo larger than about 0.1 (depending on β) but decreases rapidly with decreasing λo. With the help of eq 16, eq 14 is greatly simplified and becomes, after combination with eq 14 and rearrangement, but no further approximation,
(11) Giddings, J. C. Unified Separation Science; Wiley: New York, 1991; p 192.
(12) Giddings, J. C. Sep. Sci. 1973, 8, 567-575.
(
)
o (1 + βs) 1 c ) βλo [(1 + β)1-1/βλo - 1] 〈c 〉
-1/βλ
(12)
Analytical Chemistry, Vol. 69, No. 7, April 1, 1997
1341
R ) 6λo
1+ν 1 + 3ν - 12λo2 + 1 - 2δ (1 - 2δ)(1 - 3δ) 72λo3
ν (17) (1 - 2δ)(1 - 3δ)(1 - 4δ)
This expression admits some familiar limits. For instance, when δ becomes vanishingly small, one obtains the low λ approximation of R for an exponential concentration profile and a third-degree polynomial velocity profile (see eq 55 of ref 9). When both δ and ν are negligible, one gets the well-known low λ approximation of eq 2, R ) 6λ - 12λ2, for a parabolic flow profile and an exponential concentration profile.13 Special Cases. Equation 9 as well as eqs 12 and 14 does not apply to the case β ) δ ) 0, although the limit of eq 9 for small values of βs (eq 10) does. It is obvious that this case corresponds to the classical constant λ situation for which the concentration profile is given by eq 1 and the retention ratio by eqs 2 and 3 for second-degree and third-degree polynomial velocity profiles, respectively. Equations 12 and 14 apply for δ * 1. When δ ) 1, which corresponds to a large and rather unlikely variation of λ across the channel thickness, these equations become, respectively,
c 1 1 ) 〈c〉 ln(1 + 1/λo) λo + s
for δ ) 1
(18)
and
{
R ) 6 a0 +
1 × λo ln(1 + 1/λo)
[ ( ) ( a1 +
)]}
a2 a3 1 3 1 2+ + 3+ + 2 2 λo 3 λo λ o
(19)
where the a0 - a3 coefficients are still defined by eqs 15, with, in this case, 1/β ) λo. In addition, one notes that, when δ is equal to 1/(i + 1), for i ) 1, 2, or 3, the corresponding i term in the summation of eq 14 has to be replaced by ai ln(1 + β). In these cases, eq 17 does not hold. RESULTS AND DISCUSSION The differences between the concentration profiles corresponding to constant λ and linear λ situations are better seen when ln c/co is plotted vs s, since the former is then a straight line according to eq 1. Such a plot is shown in Figure 1, where the exponential profile corresponding to λ ) 0.05 is compared to the linear λ profile with λo ) 0.05 and β ) 0.5. This latter value can commonly be encountered in thermal FFF with a relatively large temperature difference between the two channel walls.14 As expected, the two profiles converge as one approaches the accumulation wall (s ) 0). The quadratic-like shape of the curve corresponding to the linear λ profile indicates that eq 10 provides a reasonable approximation to the true profile given by eq 9. Influence of the Relative Variation of λ across the Channel Thickness on Retention. The variations of the retention ratio as a function of the λ value at the accumulation wall, λo, are shown (13) Giddings, J. C. J. Chem. Educ. 1973, 50, 667-669. (14) Van Batten, C.; Hoyos, M.; Martin, M., to be published
1342
Figure 1. Comparison of the concentration profiles (plotted as ln c/co vs s ) x/w) corresponding to constant λ and linear λ situations. The straight line (constant λ) corresponds to eq 1 with λo ) 0.05, the curved line (linear λ) to eq 9 with λo ) 0.05 and β ) 0.5 (δ ) 0.025).
Analytical Chemistry, Vol. 69, No. 7, April 1, 1997
Figure 2. Variations of the retention ratio, R, vs the λ parameter at the accumulation wall, λo, for a given value of the third-degree polynomial velocity profile parameter, ν ) -0.1, and various values of the relative rate of variation of λ across the channel thickness, β. The dashed curve represents the constant λ situation (β ) 0). From lower to upper curve, β ) -0.5, -0.25, 0; 0.25, 0.5, 0.75, and 1.
in Figure 2 for different values of β and a fixed value of ν (ν ) -0.1). Whatever β, the retention ratio is seen to increase steadily with λo, as expected. For the sake of comparison, the dashed curve in Figure 2 corresponds to the classical constant λ situation (i.e., β ) 0) expressed by eq 3. Curves for negative values of β are also shown in Figure 2. They correspond to situations in which λ is decreasing from the accumulation wall to the depletion wall. In these cases, the value of β can never be lower than -1. Indeed, if it were so, λ would reach a zero value at some position within the cross section, which is impossible since this would correspond to either a zero diffusion coefficient or an infinite fieldinduced velocity, an unrealistic situation in both cases. As can be seen in Figure 2, for any value of λo, R increases steadily with increasing β. Under given field conditions, when λ decreases with increasing distance from the accumulation wall (β < 0), the cloud of analyte molecules or particles does not extend as far from the accumulation wall as when λ is constant. Then, the molecules or particles move along the channel, as an average, at a lower velocity, and R is lower than when λ is constant. The opposite is true when λ increases with increasing distance from the accumulation wall (β > 0), the more so when β is larger. For some particular values of β and ν, there might be some finer effects on the retention curve which are not apparent on Figure 2. For instance, even for an exponential concentration profile (constant λ), an inflection point is found in the R vs λ curve when the third-degree velocity profile parameter, ν, is lower than
Figure 3. Variations of the retention ratio, R, vs the λ parameter at the accumulation wall, λo (upper curve), and the λ parameter at the depletion wall, λ1 (lower curve), for a given value of the third-degree polynomial velocity profile parameter, ν ) -0.1, and of the relative rate of variation of λ across the channel thickness, β ) 0.5. These curves are obtained from eqs 14 and 15 by noting that λo ) δ/β, and λ1 ) λo + δ. The intermediate curve represents the constant λ situation (β ) 0). The bottom abscissa axis shows the relationship between seq, λapp, λo, and δ () λ1 - λo).
-0.333. This can be noticed on curves previously plotted for some specific negative ν values.8 Then, the lower ν, the larger the λ value at the inflection point. In the case of a linear λ concentration profile, whatever the ν value, even positive, the inflection point can be observed for sufficiently large β values. For instance, the critical β value for observing the inflection point in the R vs λo curve under relatively moderate retention conditions (R larger than 0.1) is equal to 0.35, 0.67, 0.95, 1.19, 1.40, and 1.55 for ν equal respectively to -0.3, -0.2, -0.1, 0, 0.1, 0.2. It can be shown that, whatever λo, R varies nearly linearly with β in the 0-1 range of β. The slope of the R vs β curves can be estimated, for relatively small β and λo, by derivation of eq 17:
∂R (for β and λo small) ) ∂β
depletion wall, λo and λ1, respectively, for given values of β and ν. For a given experiment leading to a given R value, the range of λ values is relatively large since, from one wall to the other, λ changes by a relative amount equal, of course, to β. In Figure 3, ν ) -0.1 and β ) 0.5. These values are typical in thermal FFF.14 The fact that λ varies in a relatively large range for a given analyte in a given experiment poses the problem of the proper strategy for sample characterization, the more so as one does not know what is, in practice, the value of β. This problem is addressed in the following section. Methodology for Retention Data Interpretation and Sample Characterization. In practice, the retention ratio, R, is determined from the retention time. The parameter ν, which controls the velocity profile, can be found from the values of the operating parameters (for instance, cold wall temperature, temperature drop across the channel, and nature of the liquid carrier in thermal FFF). One can, therefore, determine what would be the λ value of a hypothetical exponential concentration profile which would give the same retention ratio as the actual one. Let λapp (apparent λ) be this value. This is the value of λ for which the intermediate curve of Figure 3 gives the actual retention ratio (assuming that the ν value for which Figure 3 is constructed corresponds to the actual one). It is easily determined from eq 3, knowing R and ν. As shown in Figure 3, λapp falls between λo and λ1. Therefore, there is a particular distance from the channel wall for which the actual value of λ is equal to λapp. Let xeq be this distance. If xeq was known, it would be possible to characterize the analyte by associating its λapp value with the value of the field force at the position xeq. (For instance, in thermal FFF, λapp would be associated with the temperature T(xeq) in the thermal gradient.) Defining seq as
seq ≡ xeq/w
(21)
one gets, from eq 7,
seq )
(
)
λapp - λo 1 λapp ) -1 δ β λo
(22)
12λo2[(1 + ν) - 5(1 + 3ν)λo + 54νλo2] (20)
It is noticeable that the slope is approximately proportional to λo2. It can be negative when ν is lower than -1. The combination of eqs 17 and 20 indicates that, for small β and λo, the relative rate of variation of R with β is approximately equal to 2λoβ. The error in R made by assuming λ to be constant across the channel thickness may not be large, especially when both λo and β are small. Nevertheless, erroneous conclusions can be drawn about fine effects leading to relatively small retention differences when data interpretation is performed under the constant λ assumption while it does not apply. The characterization of an analyte from FFF retention is generally performed by relating the retention ratio determined from the experimental measurement of the retention time to the basic FFF parameter λ using eq 3 once ν is known. When λ is not constant across the channel thickness, eq 3 does not apply, and one cannot associate a single λ value with a particular retention ratio. This can be seen in Figure 3, where the retention ratio is plotted vs the λ values at the accumulation wall and at the
Clearly, seq cannot be directly determined since both λo and δ (or β) are unknown. If they were known, seq would be easily graphically determined, as shown on the bottom abscissa axis of Figure 3. When λo and/or β (or δ) are changed, for ν fixed, R is modified. So are λapp and seq. The variations of seq with λapp (determined from eqs 3, 14, and 22) are shown in Figure 4 for all combinations of 19 values of λo (from 0.02 to 0.20 by increments of 0.01) and eight values of β (0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, and 1). Since seq and λapp both depend on independent variables λo and β, the (seq,λapp) points can be scattered in the whole first quadrant of the (seq,λapp) plan. However, it is quite noticeable that, in Figure 4, there is very little scattering of these points around some master curve, shown in Figure 4, especially when λo is small. For other ν values, one gets very similar plots. This remarkable propertysthe high correlation between seq and λappssuggests that one uses an empirical relationship between these variables, so that seq can be estimated from a known λapp value. A third-degree polynomial fit was searched for, containing only three coefficients since the zero-order coefficient is obviously Analytical Chemistry, Vol. 69, No. 7, April 1, 1997
1343
accumulation wall in the channel). Then, sample characterization is performed by associating λapp to the field strength at this position. Discussion on the Low λapp Limit of seq. The remarkable result expressed by eq 25, in the high retention limit, may appear surprising and merits discussion. In the traditional approach of handling retention data in thermal FFF, the λ value obtained from the classical retention equation for an exponential concentration profile, i.e., λapp, is associated with the temperature (and field strength) at the position of the center of gravity of this profile, xcg, i.e., at a distance from the accumulation wall equal to the mean layer thickness. Whatever the concentration profile, scg ) xcg/w is defined as Figure 4. Variations of seq vs λapp for 19 λo values (from 0.02 to 0.20 by increments of 0.01) and eight β values (0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, and 1). ν ) -0.1. The solid curve corresponds to eq 26.
equal to zero. The first-order coefficient in λapp, which is the slope of the seq vs λapp curve at the origin, is easily obtained by comparing the polynomial development of R, for the linear λ profile, in terms of λo (i.e., eq 17), with that of the R vs λapp (i.e., the polynomial development of eq 3 with λ replaced by λapp). This latter has been shown to be given by9
R ) 6(1 + ν)λapp - 12(1 + 3ν)λapp2 + 72νλapp3
(23)
Comparison of eqs 17 and 23, limited to the first order in λo and λapp, respectively, gives
λapp )
λo 1 - 2δ
in the low λo limit
(24)
and the combination with eq 22 gives the low λapp limit of seq:
seq ) 2λapp
for low λapp
(25)
As observed in seq vs λapp plots like those of Figure 4, the slope at the origin is equal to 2 and does not depend on specific values of λo, δ, or ν. A refined relationship between seq and λapp can be obtained from a combination of eqs 17, 22, and 23. Since the result depends slightly on ν, the second and third coefficients are set proportional to their first-order polynomial development in ν, equal to (1 + 2ν) and (2 - ν), respectively. Then, these coefficients are obtained from a least-mean-square linear regression of {(seq - 2 λapp)/[(1 + 2ν)λapp2]} vs [(2 - ν)λapp/(1 + 2ν)] for all (seq,λapp) data points resulting from the combination of 12 λo values (0.005, 0.01, 0.02, 0.04, 0.06, 0.08, 0.10, 0.12, 0.14, 0.16, 0.18, and 0.20), eight β values (0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, and 1) and five ν values (0, -0.05, -0.1, -0.15, and -0.2). This gives
seq ) 2λapp - 2.1365(1 + 2ν)λapp2 - 6.1678(2 - ν)λapp3 (26) Equation 26 provides the desired relationship for correlating a given λapp value obtained from R by means of eq 3, when ν is known or estimated, to a given position (distance from the 1344
Analytical Chemistry, Vol. 69, No. 7, April 1, 1997
scg )
∫
1
0
c s ds 〈c〉
(27)
For an exponential concentration profile, this gives2
scg ) λapp -
e-1/λapp 1 - e-1/λapp
(28)
(constant λ concentration profile)
which, in the high retention (low λapp) limit, becomes scg ) λapp. (One notes, in passing, that this property has sometimes led some FFF researchers to define λ as the mean layer thickness divided by the channel thickness. This is erroneous. λ must fundamentally and unambiguously be defined by eq 6. In other words, one must not confuse an approximation with a definition and certainly must not extend the approximation to a domain in which it is not valid.) Thus, the traditional approach associates, in the high retention limit, λapp with the field parameters at the position seq ) scg ) λapp.15-20 However, as indicated by eq 25 and illustrated in Figure 5, this is not correct, since λapp must be associated, in the high retention limit, with the position seq ) 2λapp. For the linear λ concentration profile, the center of gravity is given, from eqs 27 and 12, as
scg )
[
(2δ-1)/δ
]
1 1 - δ 1 - (1 + β) -1 β 1 - 2δ 1 - (1 + β)(δ-1)/δ
(29)
(linear λ concentration profile)
In the high retention (low λo) limit, this gives, using approximation 16 for i ) 2 and replacing β by δ/λo, (15) Giddings, J. C.; Caldwell, K. D.; Myers, M. N. Macromolecules 1976, 9, 106-112. (16) Brimhall, S. L.; Myers, M. N.; Caldwell, K. D.; Giddings, J. C. J. Polym. Phys., Polym. Phys. Ed. 1985, 23, 2443-2456. (17) Schimpf, M. E.; Giddings, J. C. J. Polym. Phys., Polym. Phys. Ed. 1989, 27, 1317-1332. (18) Van Asten, A. C.; Venema, E.; Kok, W. Th.; Poppe, H. J. Chromatogr. 1993, 644, 83-94. (19) Sisson, R. M.; Giddings, J. C. Anal. Chem. 1994, 66, 4043-4053. (20) Van Asten, A. C.; Kok, W. Th.; Tijssen, R.; Poppe, H. J. Chromatogr. A 1994, 676, 361-373.
Figure 5. Illustration of the relationship between scg, seq, and λapp in the high retention limit. Variations of s ) x/w vs λ, in a narrow s domain, for λo ) 0.02 and δ ) 0.2. The first diagonal represents the scg ) λapp relationship.
scg )
λo 1 - 2δ
Figure 6. Variations of λo vs δ at constant λapp. ν ) -0.1. From lower to upper curve λapp ) 0.02, 0.05, 0.75, 0.1, and 0.15. Corresponding R values are respectively equal to 0.105, 0.248, 0.355, 0.449, and 0.599.
(30)
(linear λ concentration profile in high retention limit)
which, as can be seen from eq 24, corresponds to scg ) λapp. Hence, whether the concentration profile is associated with a constant λ profile or a linear λ profile, the center of gravity corresponding to a given value of R (hence of λapp) is the same and such that scg ) λapp. Therefore, in Figure 5, on the vertical line corresponding to λapp, the distance between the abscissa axis and the first diagonal (such that s ) λ) is equal to the distance between this diagonal and the line of the equation given by eq 7. It can be noted that, as shown in Figure 5 for the linear λ profile in the high retention limit, λ(scg) ) λ(λapp) is equal not to λapp but, according to eqs 7 and 24, to (1 - δ)λapp. In fact, the equivalency of the positions of the centers of gravity for the two concentration profiles giving the same retention ratio in the high retention limit is a general property which can be shown to apply whatever the form of the concentration profile. Indeed, in the high retention limit, the cloud of analyte particles or molecules is located in the vicinity of the accumulation wall where the velocity profile is nearly linear, expressed as v(s)/〈v〉 ) ks (for the velocity profile given by eq 13, k is equal to 6(1 + ν)). Then, the comparison of eqs 11 and 27 shows that, whatever the concentration profile, R becomes equal to
R ) v(scg)/〈v〉 ) kscg (for a linear velocity profile) (31)
Accordingly, the scg values of all concentration profiles giving the same retention ratio are identical in the high retention limit. Remarks about the Quasi-Unicity of the seq vs λapp Curve. Since two independent parameters (λo and δ, or λo and β) describe the interaction of the analyte with the nonuniform field, one would expect that two independent informations on the analyte migration process in the FFF channel are required to allow the extraction of these two parameters. One may then be surprised that the methodology described above, based on only the measurement of the retention time, allows an analyte to be characterized. It must, however, be emphasized that this methodology allows only the determination of λ at a specified position, but not of λo and δ
separately. Nevertheless, this apparent paradox, which is related to the quasi-unicity of the seq vs λapp curve, can be shown to have its origin in some peculiar properties of the relationships between seq, λapp, λo, and δ. When ν is fixed, the combination of eqs 3 (with λ replaced by λapp), 14, and 15 indicates that λapp is a function of λo and δ, λapp ) f1(λo,δ). So is seq, seq ) f2(λo,δ), if the seq vs λapp curve is unique, i.e., if there is well-defined dependence of seq on λapp, seq ) g(λapp). In the three-dimensional space (λapp,λo,δ), the intersection of the surface λapp ) f1(λo,δ) with a plane at λapp ) constant gives a twodimensional curve in the (λo,δ) plane. Its equation is derived from eq 22:
λo ) λapp - seqδ ) λapp - g(λapp)δ
(32)
It appears that, as λapp and, hence, g(λapp) are constant, this curve is a straight line. So is the curve resulting from the intersection of the surface seq ) f2(λo,δ) with a plane at seq ) constant. Reciprocally, one can similarly demonstrate that, if the λo vs δ curves, at constant λapp, are straight lines, there is a unique relationship between seq and λapp. One can check the validity of these findings in Figure 6, showing the curves of variation of λo vs δ for various constant λapp values. Since λapp is constant for points belonging to a given curve, all sets of (λo,δ) values for such points provide the same retention ratio. It is clearly apparent in Figure 6 that these curves are straight lines. Linear regression analysis of the (λo,δ) data gives in all cases a correlation coefficient exceeding 0.999 for λapp up to 0.15 and δ up to 0.25. This high linearity, which results from the peculiar dependences between R and λapp, on one hand, and R and λo and δ on the other hand, explains why a nearly unique relationship can be found between seq and λapp. Furthermore, eq 32 shows that, when g(λapp) ) 2λapp according to eq 24, in the case of very small values of λapp, the straight lines λo vs δ, for various λapp, must converge to the point (δ ) 0.5;λo ) 0). Such a convergence is not found for the curves of Figure 6 because they are obtained for λapp values that exceed the domain for which eq 24 is valid. Nevertheless, a convergence area is obtained for these curves in the vicinity of the low λapp convergence point. Validity of the Proposed Methodology. One can test the ability of the methodological approach proposed above to provide Analytical Chemistry, Vol. 69, No. 7, April 1, 1997
1345
Table 1. Relative Errors, E, in the Estimation of the Distance from the Accumulation Wall for Which the Value of λ in a Linear λ Profile is Equal to λapp, Determined from Eq 3a
R λapp seq,ex seq,est 1 scg 2
λo ) 0.05 δ ) 0.005
λo ) 0.05 δ ) 0.015
λo ) 0.05 δ ) 0.025
λo ) 0.05 δ ) 0.05
λo ) 0.05 δ ) 0.1
λo ) 0.1 δ ) 0.005
λo ) 0.1 δ ) 0.015
λo ) 0.1 δ ) 0.025
λo ) 0.1 δ ) 0.05
λo ) 0.1 δ ) 0.1
0.250 0.050 0.096 0.095 -0.6% 0.050 -47.2%
0.255 0.051 0.097 0.097 -0.6% 0.051 -46.7%
0.259 0.052 0.099 0.098 -0.5% 0.052 -46.9%
0.271 0.055 0.103 0.103 - 0.2% 0.055 -46.5%
0.297 0.061 0.112 0.113 0.8% 0.061 -45.4%
0.452 0.101 0.172 0.171 - 0.3% 0.101 -41.1%
0.458 0.103 0.173 0.173 0.0% 0.103 -40.8%
0.464 0.104 0.175 0.175 0.4% 0.104 -40.3%
0.479 0.109 0.178 0.181 1.3% 0.109 -39.0%
0.510 0.119 0.185 0.191 3.4% 0.118 -36.1%
a ν ) -0.1. , relative error on the equivalent position made by using the present approach (eq 26). , relative error on the equivalent 1 2 position made by using the classical approach based on the center of gravity (eq 28).
the correct value of seq. Assuming that the λ profile in the channel is given by eq 7, with known values of λo, δ, and ν, the retention ratio is obtained from eqs 14 and 15. From R and ν, λapp is determined using eq 3, and the seq value estimated by means of eq 26, seq,est, can be compared to the exact value, seq,ex, given by eq 22. The relative error, 1 ) (seq,est - seq,ex)/seq,ex, resulting from the present approach can be compared with that, 2, made with the classical method by taking seq,est equal to the relative position, scg, of the center of gravity of the exponential profile, which is given by eq 28. The results are reported in Table 1 for two λo and five δ values. It is clear that the present approach provides a much better estimation of the true equivalent position than the classical approach, which as discussed above gives an error nearly equal to 50% for small λapp values. For most cases, the error of the present method is well within 1%, which is quite good. As can be seen in Figure 4, one expects the estimation of eq 26 to be less satisfying for large λo (and, to a lesser degree, large δ) values. Indeed, for β ) 1 and for λo ) 0.15 and 0.2, which lead to relatively large R values (0.674 and 0.773, respectively), the errors on seq arising from eq 26 are 2.8% and -17.4%, while those arising from the classical approach (eq 28) are -15.8% and 12.5%, respectively. Clearly, the present approach, based on eq 26, must not be used for λapp larger than about 0.15-0.2. This limit corresponds to R values larger than about 0.7. However, one should emphasize that working in this R > 0.7 retention domain should be prohibited, since then the analyte peak is badly separated from the void peak, the fractionating power is poor, and the accuracy of the determination of the physicochemical parameters characterizing the analyte is low. Accordingly, the present approach is expected to work well in the whole domain of acceptable retention ratio values.
the present approach will correctly represent the behavior of an analyte if the λ profile can satisfyingly be approximated by a nearly linear profile in the thin layer where the analyte molecules or particles are compressed in the vicinity of the accumulation wall. The model described here is restricted not to small values of β (or δ) but to a nearly uniform rate of decrease of λ with the distance from the accumulation wall as this distance increases. It is, however, believed that such a nearly linear λ behavior will, in practice, be observed for relatively moderate variations of λ, under conditions such that the migration occurs in the Brownian mode, i.e., under conditions for which Brownian motion is the dominating mechanism that allows the analyte molecules or particles to escape deposition on the accumulation wall. Such a retention mode is characterized by a monotonous decrease of λ with the distance from the accumulation wall. The implementation of the proposed approach is relatively simple and is summarized as follows. First, the flow distortion parameter, ν, is computed or estimated (see refs 6 and 7 for estimation of ν in thermal FFF and sedimentation FFF, respectively). With this value of ν and the experimentally determined retention ratio, R, λapp is computed from eq 3. Then, the equivalent position, seq, is estimated using eq 26. The characterization of the analyte can then be performed by associating λapp with the value of the field strength and of the relevant operating parameters at the position seq. For instance, in the particular case of thermal FFF where λ depends on temperature, this λapp value is that λ value of the analyte at the temperature, Teq, at the position seq. Teq can be determined from the values of the hot wall and cold wall temperatures, knowing the temperature dependence of the thermal conductivity of the carrier liquid.15 Applications of this methodology to thermal FFF will be presented in a forthcoming publication.
CONCLUSION These results give us confidence in the proposed methodology for interpreting retention data. This approach is quite general and is not restricted to a particular FFF subtechnique. It can be applied each time the force, F, exerted by the field on the analyte molecules or particles (or more exactly, according to eq 6, the ratio, F/T, of the force to the absolute temperature) is expected to vary moderately within the channel thickness. Although the term “moderately” is rather vague, it is apparent, from eq 7, that
ACKNOWLEDGMENT Fruitful discussions with Franc¸ ois Martin about the quasiunicity of the seq vs λapp curve are gratefully acknowledged.
1346 Analytical Chemistry, Vol. 69, No. 7, April 1, 1997
Received for review May 30, 1996. Accepted January 21, 1997.X AC9605307 X
Abstract published in Advance ACS Abstracts, March 1, 1997.
