Anal. Chem. 2004, 76, 6665-6680
Programmed Field Decay Thermal Field Flow Fractionation of Polymers: A Calibration Method Luisa Pasti,† Filippo Bedani,† Catia Contado,† Ines Mingozzi,‡ and Francesco Dondi*,†
Department of Chemistry, University of Ferrara, Ferrara, Italy, and Basell Polyolefins, Ferrara, Italy
Thermal field flow fractionation (ThFFF) is one of the techniques belonging to the field flow fractionation (FFF) family, where separation occurs as result of coupling a flow field,1-4 i.e., a spatial configuration of different flow rates, with a force field
orthogonal to the flow field. In ThFFF, the force field is a thermal gradient inducing thermodiffusion; it is particularly suited to the separation of macromolecules.2,4 From the very outset, thanks to several specific features, ThFFF was considered competitive versus size exclusion chromatography (SEC).4 The ThFFF separation system is, in fact, very simple: it is made up of a flat ribbonlike channel, free of any packing material, obtained by placing a trimming spacer between two flat bars kept at different temperatures; on the inside, the carrier solvent flow follows a well-defined flow profile. The ThFFF system does not significantly affect the sample: it does not produce tangential stresses or pore entrapment on separated macromolecules. ThFFF is thus particularly suitable for “difficult” samples such as those including ultrahigh molecular weight or microgel components. Since the only temperature limit is the thermal stability of the spacer, the ThFFF separation system does not have severe temperature limits as usually imposed in SEC by the thermal stability of the packing. The ThFFF system is easy to maintain since the channel can be opened for internal inspection and cleaning. This makes sample adsorption controls immediate. The ThFFF separation system is stable practically forever. Once the operating variables are fixed, ThFFF retention depends on both the size and composition of the macromolecule and on the type of solvent.2,6-9 However, unlike SEC, ThFFF retention can be expressed as a function of operating variables such as the temperatures of the hot and cold channel walls, Th and Tc, respectively, and thus on their difference ∆T ) Th - Tc, as well as of other physicochemical solvent and apparatus constants.5 This last feature makes system calibration in ThFFFs the transformation of the retention time axis into a molecular weight (M)s“universal” in type:6 once it has been determined for a specific solvent-polymer system, it is “universally” valid; i.e., it is instrument transferable. In SEC, any new column must be calibrated and periodically checked. Finally, ThFFF can even be performed under thermal field programming (TFP) conditions10-12 by varying the applied thermal field during the separation process, i.e., ∆T, in a way similar to
* To whom correspondence should be addressed. E-mail: F.Dondi@ unife.it. † University of Ferrara. ‡ Polyolefins. (1) Giddings, J. C. Unified Separation Science; Wiley: New York, 1991. (2) Schimpf, M. In Field-Flow Fractionation Handbook; Schimpf, M.; Caldwell, K., Giddings, J. C., Eds.; Wiley-Interscience: New York, 2000; pp 239-256. (3) Martin, M. In Advances in Chromatography; Brown, J. C., Grushka, E., Eds.; Marcel Dekker: New York, 1998; Vol. 39, pp 1-138. (4) Giddings, J. C In Size Exclusion Chromatography; Hunt, J. C., Holding, S., Eds.; Blackie and Son: Glasgow, 1989; pp 191-216.
(5) Cao, W.-J.; Williams, P. S.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1999, 71, 1597-1609. (6) Giddings, J. C. Anal. Chem. 1994, 66, 2783-2787. (7) Melucci, D.; Contado, C.; Mingozzi, I.; Hoyos, M.; Martin, M.; Dondi, F. J. Liq. Chromatogr., Relat. Technol. 2000, 23, 2067-2082. (8) Pasti, L.; Roccasalvo, S.; Dondi, F.; Reschiglian, P. J. Polym. Sci. Part B 1995, 33, 1225-1234. (9) Pasti, L.; Melucci, D.; Contado, C.; Dondi, F.; Mingozzi. I. J. Sep. Sci. 2002, 25, 691-702. (10) Giddings, J. C.; Smith, L. K.; Myers, M. N. Anal. Chem. 1976, 48, 15871592.
The universal calibration procedure typical of thermal field flow fractionation (ThFFF) under constant thermal field operation was extended to thermal field programming (TFP) operation. The method requires knowledge of the following: (a) the programming function, which only depends on the thermal field decay function, (b) the physicochemical properties of the solvent, and (c) the calibration plot under varying channel cold wall temperatures (Tc). Two field flow fractionation field programming conditions, with either a constant or a variable in time carrier flow velocity, are exploited. The method is based on determination, for each retention time position, of the average λ retention value typical of TFP ThFFF. This parameter is then used to obtain the calibration plot (i.e., the molecular weight of the species as a function of the retention time position) by using the programming function and the calibration plot under varying Tc values. The procedure approximation errors are also derived as a function of the programming type and solute-solvent system. To properly test the procedure, the calibration plot for the system constituted by polystyrene (PS) in cistrans Decalin was determined, under varying conditions Tc and thermal gradients, by using a set of monodisperse PS standards of different molecular weights (M). The procedure was first validated by simulation under two typical cases of TFP ThFFF operation. The approximation errors were found acceptable (in the worse cases, the accuracy in M prediction was 3%) and are in agreement with the theory. The procedure was then experimentally validated under varying programming decay function conditions. The reproducibility and accuracy of the M determination are both better than 2%.
10.1021/ac049399q CCC: $27.50 Published on Web 10/19/2004
© 2004 American Chemical Society
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the well-known temperature programming in gas chromatography or programmed carrier composition in high-performance liquid chromatography (HPLC). The advantages of operating under TFP are well known: the mass-based fractionation power (i.e., mass selectivity)12 is kept uniform over the retention time axis; separation speed increases; species detectability is kept uniform. The TFP method is, in general, more robust versus sample overloading effects than the constant field method. The only drawback is that, to date, no extended work has been performed to exploit the potential of TFP ThFFF. In particular, no way to extend “universal” calibration features to TFP ThFFF operations has yet been exploited, even though ThFFF programming instrumentation is now commercially available. The papers dealing with general and theoretical field FFF programming aspects are few but significant.10-12 The most extensive work has been devoted to exploiting field programming in sedimentation FFF (SedFFF).12-13 In this case, a numerical methodsand specific softwaresfor transforming the retention time axis into a species mass axis was developed. In this method, the specific sedimentation program computes the instantaneous retention ratio (see below) for a defined species mass value as a function of the applied instantaneous sedimentation field. The progress of the species inside the channel is monitored all the way to the end, and the species retention time is computed. The retention axis is thus fully calibrated, by repeating the retention time evaluation as a function of the species mass. The permitted range of sedimentation field species mass values must exclude steric retention effects.13 This numerical procedure could be extended to TFP ThFFF although there are serious complications. First, computation of the retention ratio as a function of the instantaneous applied field is not as straightforward in ThFFF as it is in SedFFF. In ThFFF, one must know the calibration curves as a function of both Tc and ∆T values, in the range spanned during TFP. This information is not generally available and must be experimentally determined. Moreover, in TFP ThFFF, the number of time-dependent variables to be monitored and computed during elution is significantly greater than in SedFFF. For example, in addition to changes in Tc and ∆T, one must also account for the flow profile time change due to the temperature dependence of the solvent physicochemical parameters (thermal conductivity, κ, and viscosity, η). The consequence is that program complexity and the computational time can be expected to increase significantly. Under such conditions, one can consider the timeliness of developing an approximate method. It is worth pointing out that, in computation, approximation is not an exception but a rule. In fact, one must remember that a lot of functions (e.g., the exponential function or the coth function entering the FFF retention expression) are based on truncated series expansions. Approximation is introduced in computation in two typical cases: when the computation complexity is too high or when ultimate precision is not required (e.g., when other relevant error sources come into play and overcome the approximation error). This is indeed the present case when comput(11) Williams, P. S. In Field-Flow Fractionation Handbook; Schimpf, M.; Caldwell, K.; Giddings, J. C., Eds.; Wiley-Interscience: New York, 2000; pp 145-165. (12) Williams, P. S.; Giddings, J. C. Anal. Chem. 1987, 59, 2038-2044. (13) Williams, P. S.; Giddings, M. C.; Giddings J. C. Anal. Chem. 2001, 73, 42024211.
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Analytical Chemistry, Vol. 76, No. 22, November 15, 2004
ing theoretical retention under TFP ThFFF. In fact, in field FFF programming, complex nonequilibrium phenomena (the so-called secondary relaxation; see below) can come into play and are intrinsically difficult to account for in a theoretical model.11 Moreover, in ThFFF, a non-negligible source of imprecision derives from its own instrumental setup, where the Th and Tc constancy over the entire length of the channel is not better than (1 °C. The present paper considers the general problems involved in accounting for retention under different constraints inherent to field programming conditions, and an approximate method for extending the universal calibration procedure under TFP ThFFF is presented and exploited. The approximation errors are theoretically derived and estimated for typical cases of FFP ThFFF operation. This will allow us to define the “a priori” validity and limits of the proposed procedure. The method is then applied in practice: TFP ThFFF of polystyrene (PS) in Decalin. To do this, the pertinent calibration curves under different conditions of Tc and ∆T are experimentally determined together with their confidence intervals. Then monodisperse PS samples are analyzed under varying TFP ThFFF conditions, and a comparison is made between certified and experimentally determined M values. This approach can also provide a basis for developing optimal ThFFF separation procedures. THEORY General Aspects. FFF separation methods belong to the general F(+)1 separation methods where the flow and gradient of the overall chemical potential (representing the sum of the external field effects and the internal molecular interactions) are perpendicular. Separation here is the result of coupling the local linear flow velocity of the carrier fluid, v, with the local crosssectional concentration c. The average cross-sectional mean velocity of the analyte zone in the direction of flow is
V)
∫
Af
cv dA/
∫
Af
c dA
(1)
where Af is the cross-sectional area occupied by the fluid. The downstream migration of the single analyte is characterized by the retention ratio R, a dimensionless parameter defined as
R ) V/〈v〉
(2)
where 〈v〉 is the cross-sectional average of v:
〈v〉 ) (1/Af)
∫vdA
(3)
V is the average cross-sectional mean velocity of the analyte also defined as
V ) dl/dt
(4)
where l is the axial position of the analyte zone inside the separation system and in the flow direction, and t the time. In
general, V is dependent on both l and t. By combining eqs 1-4, one obtains the general differential migration equation:
dl ) R(t,l) × [〈v〉(l,t)] dt
(5)
where both R and 〈v〉 are allowed to be dependent on both l and t. The text that follows advances three conditions defining the constraints of the differential migration equation. In all cases considered, both R and 〈v〉 are assumed to be independent of l. The simplest case is the one where both R and 〈v〉 are constant, i.e., independent of both l and t, here indicated as C1:
C1 w R and 〈v〉,
constant,
i.e., independent of l,t (6a)
The C1 condition is usually met in SedFFF and ThFFF techniques under constant applied field conditions. A second condition is that R is dependent on t and 〈v〉 is constant, here indicated as C2:
C2 w R dependent on t, independent of l; 〈v〉 constant, i.e., independent of l and t (6b) The third condition is that both R and 〈v〉 are dependent on t, here indicated as C3:
C3 w R and 〈v〉 dependent on t;
both independent of l (6c)
C1 and C2 are typical field program FFF conditions. We observe that eqs 5 and 6a-c, and the other equations derived in the present paper, do not account for zone broadening effects and describe the local migration of the first moment of the analyte zone.1,3 For the C3 fractionation condition (eq 6b), the differential migration equation becomes
dl ) R(t) × [〈v〉(t)] dt
C3
(7a)
The integrated form of eq 7a is
L)
∫
L
0
dl )
∫
tR
0
R(t) × [〈v〉(t)] dt
C3
(7b)
which defines tR, the retention time i.e., the mean analyte residence time in the separation system of length L. Equation 7b is an integral equation where tR is the unknown. For R ) 1, one defines the void time t0, which is the mean residence time of the carrier fluid in the separation system:
L)
∫
t0
0
[〈v〉(t)] dt
C3
(7c)
Equations 7b and 7c are still integral equations in the unknown tR and t0, respectively, and solution requires knowledge of both functions R(t) and [〈v〉(t)]. Under the C1 condition (see eq 6a), the solution of eq 5 is trivial:
L ) R〈v〉tR
C1
(8)
and, for R ) 1, one has
L ) 〈v〉t0
C1
(9)
C1
(10)
By combining eqs 8 and 9 one has
R ) t0/tR
In eq 10, t0 and tR are, in practice, evaluated from the experimental fractogram as the first moment of the unretained compound and analyte peak profiles, indicated as ht0 and htR, respectively (corrected for the connection dead times). The experimental quantity Rexp is
Rexp ) ht0/thR
(C1)
(11)
For straight conduits, as those employed in ThFFF, a correspondence can be advanced3 between R in eq 10 and Rexp in eq 11. Moreover, we will assume that the sample is injected as a narrow pulse (Dirac pulse injection), in a small quantity, and that, before starting the fractionation process, the sample is properly relaxed.3 In practice, when referring to the injected quantity, the term “small” means that the overloading effect causing peak distortion has been excluded.2 Consequently one can put
R ) Rexp ) ht0/thR
(C1)
(12)
The FFF techniques under constant field conditions met the C1 condition. In fact, the requisite of 〈v〉 is independent of l and t is assured as long as the channel section is constant along l, since the pressure drop along channel length is on the order of 0.1 bar for a flow rate of 1 mL min-1, and under these conditions, the liquid carrier is practically incompressible. The other condition required for R to be independent of l and t is assured, in FFF techniques, by keeping the separation field constant along the entire length of the channel. In ThFFF, this requires that the carrier composition be constant with respect to l and t, which in turn, requires the invariability of both Tc and ∆T at the crosssectional width of the separation channel, w. “Universal” Calibration Features of ThFFF under Constant Thermal Field Conditions. Among the various F(+) separation techniques, FFF techniques have unique features since they are known to exhibit, under certain conditions, either “absolute” or “universal” calibration capabilities.1-6 Thanks to welldefined dependency between physicochemical properties of the separated species and applied force, both these capabilities rely on the theoretical tractability of eqs 1 and 3 with given channel geometry and under specific conditions.14 These relationships make it possible to solve the integrals in eqs 3 and 1, respectively.1-3 In the classical FFF retention theory, which holds true, for example, for SedFFF under well-defined conditions (normal or (14) Dondi, F.; Martin, M. In Field-Flow Fractionation Handbook; Schimpf, M.; Caldwell, K.; Giddings, J. C., Eds.; Wiley-Interscience: New York, 2000; pp 103-132.
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“Brownian” mode3), the c profile is exponential:
c ) c0 exp(- x/l)
(13a)
c ) c0 exp(- x/λw)
(13b)
solving the integral in eq 1 can be theoretically handled. The ThFFF retention equation is
R(λ, ν) ) 6λ(ν + (1 - 6λν)(coth(1/2λ) - 2λ)) (18a)
or
In eqs 13a and b, x is the distance from the accumulation wall (x ) 0), where c ) c0, and l is the x value in which c ) c0/e. λ in eq 13b is the l value normalized with respect to w. In ribbonlike channels, the velocity profile in the cross section is parabolic:
v(x) ) 6〈v〉
[wx - (wx ) ] 2
(14)
From eqs 1-3, 13a, b, and 14, one obtains the classical FFF retention equation
R(λ) ) 6λ[coth(1/2λ) - 2λ]
(15)
By combining eq 12 with eq 15 one obtains
ht 0/thR - 6λ[coth(1/2λ) - 2λ] ) 0
C1
(16)
and this permits us to determine λ, which is related to the cross section equilibrium-stationary conditions:1
λ ) D/|u|w
(th0/thR) - 6λ(ν + (1 - 6λν)(coth(1/2λ) - 2λ)) ) 0 C1 (18b) Equation 18b is similar to eq 16 since it defines a ThFFF λ parameter. However, in this case, λ is conceptually different from what appears in eq 16 since, here, λ is no longer related to a real exponential distribution of the analyte in the channel cross section (see eqs 13a,b). Instead, the ThFFF λ parameter refers to a hypothetical exponential distribution giving the same observed ThFFF Rexp value (see eq 12), under defined conditions.17 For this reason, λ in eqs 18a,b is called an “apparent” λ parameter. Below, only the λ parameter related to eqs 18a,b and other equations related to eqs 18a,b will be considered and the specification “apparent” will be omitted. In ThFFF, the separation driving the physical process is thermodiffusion,1,15 where |u| is proportional to the temperature gradient,15 dT/dx, along the cross section of the separation channel:
|u| ) DT(dT/dx)
(19)
(17a) where DT is the thermal diffusion coefficient or more correctly the thermophoretic mobility. By combining eqs 17a and 19, one has
or
λ ) kT/|Ff|w
(17b)
where D is the analyte diffusion coefficient, |u| the velocity induced by the field on the analyte, |Ff| the field force, k the Boltzmann constant, and T the absolute temperature. By introducing the dependence of either |u| or |Ff| on the analyte property holding true for the specific FFF subtechnique,1,3,14 λ can be expressed (in a specific range) as a monotonic function of the analyte property, together with other measurable quantities (carrier physical constants, parameters of the instrumentation, working parameters, etc.). If the analyte property is the masssas in SedFFFsthe technique is said to be an “absolute” technique, since the determination of the analyte property can be obtained directly from the λ measurement. This is the case, for example, of SedFFF working under well-defined conditions (the so-called normal or Brownian mode and constant spinning rate).3 The classical FFF retention equation (eq 16) does not apply to ThFFF, since relevant physicochemical parameters, affecting both flow profile and analyte concentration distribution in channel cross section, are temperature dependent and thus not constant in the channel cross-sectional area. Under these conditions, the concentration profile differs from the exponential profile (see eqs 13a,b) and the velocity profile is strongly distorted with respect to the parabolic profile (see eq 14). Nonetheless, the problem of 6668
where ν is a computable parameter,3 related to the changes in η and κ in channel cross section and accounting for distortion of the ideal parabolic flow (eq 16) profile in the channel. By combining eqs 16 and 18a, one obtains3,17
Analytical Chemistry, Vol. 76, No. 22, November 15, 2004
λ)
( )
D 1 DT (dT/dx)w
(20)
Equations 19 and 20 hold true in an infinitesimal layer thickness, dx, located at a value x. The temperature does not change significantly within this dx layer; D and DT are thus constant and referred to this temperature value. In this sense, the λ parameter associated with eqs 18a,b is referred to a given local point. In fact, by differentiating eq 13b one has
dc 1 x )- d c λ w
()
(21)
which gives a precise meaning to λ. According to eq 21, the ThFFF λ parameter of eq 18b is thus associated with a well-defined value of x, the so-called equivalent position xeq, where the quantities (D/ DT), the local temperature T, and the local gradient (dT/dx) assume well-defined values, (D/DT)eq, Teq, and (dT/dx)eq, respectively.5,17 (15) De Groot, S. R. Thermodynamics of Irreversible Processes; North-Holland Publishing Co.: Amsterdam, 1963. (16) Martin, M.; Garcia-Martin, S.; Hoyos, M. J. Chromatogr., A 2002, 960, 165174. (17) Martin, M.; Van Batten, C.; Hoyos, M. Anal. Chem. 1997, 69, 1339-1346.
From eq 20, one sees that the λ parameter in ThFFF is related to (D/DT), but not specifically to a conventional analyte property, such as molar mass M. However, since (D/DT) depends on M,1,3,8,18 it also mediates the dependence between λ and M. For this reason, ThFFF cannot be defined as an “absolute” calibration technique, since the relationship between(D/DT) and M is not known a priori. Nonetheless, at constant Tc conditions, the relationship among λ, (D/DT), M, and ∆T can be experimentally exploited by using monodisperse or polydisperse9 standard polymers for a given specific polymer-solvent system. Once this relationship is obtained, it is “universal”.6 A typical calibration function of λ versus M is
log (λ∆T) ) a0 + a1 log M
(22)
where a0 and a1 are suitable calibration constants. Under nonconstant Tc conditions, the dependence of the λ parameter on M, ∆T, and Tc or Teq can be modeled according to any one of the following equations,5 depending on user preference and the desired accuracy:
log (λ∆T) ) a0 + a1log M + a2 log(Tc/298.15) (23a) log (λθc∆T) ) a0 + a1log M + a2 log(Tc/298.15) (23b) log (λθc∆T) ) a0 + a1log M + a2 log(Teq/298.15) (23c)
Since these models relate λ to ∆T and Tc (or Teq), whose values change throughout the fractionation process, they will form the basis for extending the “universal” calibration features of the ThFFF under constant thermal field to TFP conditions. The quantities a0, a1, and a2 in eqs 23a-c are typical of the chosen calibration function and characteristic of the specific polymersolvent system, i.e., of the composition and structure of a polymer in a given solvent, but not its M value. The θc parameter in eqs 23b and c is
[ ( ) ]
θc ) 1 +
1 dκ ∆T κc dT c 2
(24)
where κc is the thermal conductivity of the solvent at Tc. θc accounts for the thermal gradient deviation from linearity across the channel cross section. Once the specific form of the calibration function is selected and its coefficients are experimentally determined, it can be used to characterize an unknown monodisperse or polydisperse sample from the experimental determination of λ under given experimental conditions. For example, if eq 23b is employed, the species M value will be
log(M) )
(
log(λθc∆T) - a0 - a2 log a1
Tc 298.15
)
calibration features of ThFFF from constant to TFP operation lies in the complexity in finding a solution for eq 7b in terms of species property versus tR. In fact, if we introduce the ThFFF retention (see eq 18a) in eq 7b, we obtain an integral equation of tR versus λ and ν, both functions of time. Then, if one of the calibration eqs 23a-c is used to replace λ, we obtain an integral equation on tR, which depends on M and on other system (a0, a1, a2) and operating time-dependent variables (v, ∆T, Tc, θc). Moreover, to this last set of time-dependent variables, one should also add the time variation of 〈v〉, since changing Tc and ∆T produces a variation in ht0.16 In terms of tR, the final solution for each M value and the given programming conditions can be obtained numerically by following the approach already developed by Williams et al. for SedFFF.13 However, when applied to the TFP ThFFF case, such an approach appears from the very outset to be much more complex than the SedFFF analogue. Thus, a simplified method is quite appealing. Below, an approximate method is developed based on the computation of a unique time-dependent programming function. Before describing the method, some aspects of TFP will be focused on. In FFF, programming can be performed in a wide variety of ways:11 during 0 e t e tR, one can impose a specific change to some specific “primary” variables, i.e., to those variables over which the operator has direct control. This list includes the following: (1) flow rate, (2) carrier composition, (3) applied force field or a combination thereof. Only force field programming is considered here. However, besides the primary variable (the force field, here), which is intentionally changed by the set program, one must also carefully consider the effect on other variables that are subject to variation as a consequence of the primary variable variation. In TFP ThFFF, the primary variable programmed on time is ∆T. However, at the same time, one cannot avoid the simultaneous variation in Tc, since the present state of ThFFF instrumentation does not permit its simultaneous control. There is, however, experimental evidence that both ∆T and Tc are practically constant over 0 e l e L at a given t: under these conditions, it can be said that the thermal field is constant over 0 e l e L. On the other hand, even if 〈v〉 is practically homogeneous over 0 e l e L at constant t, it does change in time. Several factors are involved in such changes: carrier fluid η variations induced by changes in both ∆T and Tc and variations in void volume.16 Under these conditions, TFP ThFFF operations are performed under C3 fractionation conditions (see eq 6c). This condition is nevertheless significant since, at a given t value, both the λ quantity (related to eqs 20 and 21) for a given position x in the channel cross section and R (see eq 18a) will be constant over the entire channel length, i.e., in 0 e l e L. Note that, since eqs 20 and 21 are assumed, secondary relaxation effects are neglected. With reference to one of the calibration equations, e.g., to eq 23b, under TFP THFFF, for a given analyte one has
(25)
( )
∆T0 Tc,0 ∆Tt Tc,t
λt ) λ0
-a2
θc,0 M given, 0 e t e tR, θc,t
C3
(26)
where the subscripts 0 and t refer to two distinct time values in Extending “Universal Calibration” of ThFFF to TFP Operation. The main difficulty in extending the “universal”
(18) Tyrrel, H. J. V. Diffusion and Heat Flow in Liquids; Butterworth & Co.: London, 1961; pp 230-245.
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6669
the 0 e t e tR range, ∆T0, ∆Tt and Tc,0, Tc,t are their associated ∆T and Tc values during the experimental TFP operation. For the sake of simplicity, the “0” reference is chosen at t ) 0, i.e., at the beginning of the program. Equation 26 can be rewritten as
λ(t)|M ) λ0|M pr(t)
0 e t e t R,
C3
(27)
where
pr(t) )
∆T0
( )
-a2
Tc,0
∆T(t) Tc(t)
θc,0
C3
θc(t)
(28)
is the time programming function; i.e., it is a function of the field program run time. Note that pr(t) depends on the polymersolvent systems (through a2, see eqs 23a-c), but not on the species M value. The dependence of M in eq 27 is instead assumed by the quantity λ0. A time average operator is applied to eq 27 in the 0-tR range, and the following expression is obtained:
〈λ|M〉tR ) λ0|M〈pr〉tR
C3
(29)
where
1 tR
〈λ|M〉tR )
∫
tR
0
λ(t)|M dt
C3
(30)
is the time average value of the λ parameter, experienced by the species M during the programmed fractionation conditions and
1 tR
〈pr〉tR )
∫
tR
0
pr(t) dt
C3
(31)
or
log Tc(t) ) log (Tc,0) + p2 log
∆T ) ∆T0
0 e t e t1
( )
t1 - t a ∆T(t) ) ∆T0 t - ta
p1
∫
t g t1 > ta
(32b)
where t1, ta, and p1 are suitable parameters. Equations 32a-c describe the so-called power program.11,12 Normally, in ThFFF, p1 is chosen as equal to 2, but the actual p1 value can be different from the set one. When ∆T is changed according to the above time variation, Tc too undergoes to a time variation and can be modeled as
0 e t e t1
( )
Tc(t) ) Tc,0 6670
t g t1 > t a
)}
2λ(t)
( )
p2
tR
R(t) dt
0
(33c)
C2; M
(34)
( [ ]
{
R[λ(t),ν(t)] ) 6λ(t) ν(t) + (1 - 6λ(t)ν(t)) coth
t1 - t a log ∆T(t) ) log (∆T0) + p1 log t g t1 > ta (32c) t - ta
t1 - t a t - ta
t g t1 > ta
If secondary relaxation11 is neglected during the fractionation process, R(t) in eq 34 can be modeled by
(32a)
or
Tc ) Tc,0
t1 - t a t - ta
From eq 29, one can see that, for a given species, λ0|M can be determined at the “0” condition if one has an experimental estimate of 〈λ|M〉tR relative to the species eluting at tR. In fact, 〈pr〉tR can be independently evaluated (see eqs 28 and 31). The M value, i.e., the searched solution, can be obtained from λ0|M (e.g., from eq 25) by entering λ ) λ0|M, ∆T ) ∆T0, θc ) θc,0, and Tc ) Tc,0 into the equation. In this way, each tR value of the fractogram is associated with the corresponding M value; in other words, the time axis is calibrated in terms of species molecular weight. How to obtain a good estimate of 〈λ|M〉tR is discussed in the following section. Average λ Estimate. While the C3 fractionating condition that holds true for TFP ThFFF operations allows one to define properties of λ and of its time averages (eqs 26-31), the same condition does not allow us to solve the integral migration equation (see eq 7b) because 〈v〉 is time-dependent and remains under the integral. However, there is experimental evidence that the temporal variation of 〈v〉 is moderate.11 Consequently, we will first exploit the condition of 〈v〉 constant over both l, t () C2 fractionation condition) and then extend the solution to the C3 condition, i.e., by including the time variation of 〈v〉. This approach allows us to understand the relevance of the temporal variation in 〈v〉 during thermal decay and to estimate the differences between C2 (〈v〉 constant) and C3 (〈v〉 variable) fractionation conditions. Under C2 condition eq 7b becomes
L ) 〈v〉 is the time average programming function. The time dependence of ∆T is usually modeled according to the following functions:
( )
1 2λ(t)
(secondary relaxation neglected) (35)
Throughout the present handling, this assumption will be implicitly assumed to hold true and will not be further specified below. From eq 7c
L ) 〈v〉
∫
t0
0
dt ) 〈v〉t0
C2; M
(36a)
C2; M
(36b)
By combining eqs 34 and 36a, one has
L ) t0 ) 〈v〉
∫
tR
0
R(t) dt
(33a) (33b)
Analytical Chemistry, Vol. 76, No. 22, November 15, 2004
which is an integral equation with respect to unknown tR. From eqs 36a-b, one can see that the retention time of the species, tR, under C2 condition, is only an average effect of the instantaneous
retention ratio R(t). It is useful to introduce the retention time average 〈R(t)〉, for a given species M:
〈R|M〉tR )
∫
tR
0
R(t) dt/tR
(37)
By combining eqs 36b and 37 one has
〈R|M〉tR ) t0/tR
C2
(38)
As for constant field operation conditions (see eqs 8-10 and 11), t0 and tR of eq 38 can in practice be evaluated by ht0 and htR, respectively.
Rexp,pr ) ht 0/thR
〈ν〉tR )
1 tR
∫
tR
0
ν(t) dt
(43)
is the time average of the ν quantity. ν and thus 〈v〉tR can be computed for any tR value as long as the temperature dependence of both η, R, and κ are known and the time dependence of both Tc and ∆T are established. By combining eqs 40, 41, and 42a,b one can write
(
( ( ) ))
〈R(λ|)M,ν〉tR - 6〈λ〉t/R 〈ν〉tR + (1 - 6〈λ〉t/R〈ν〉tR) coth 2〈λ〉t/R
1 2〈λ〉t/R
) 0 (44a)
(39) By combining eqs 40, 41, and 42a,b one obtains
Under C2, condition 〈v〉 is assumed constant (see eq 6b) and ht0 is independent of tR. Consequently eqs 38 and 39 can be equated obtaining
〈R|M〉tR ) ht 0/thR
C2
( (
6[λ|M(t)]ν(t)) coth
[∫
tR
0
(
6[λ|M(t)] ν(t) + (1 -
)
1 - 2[λ|M(t)] 2[λ|M(t)]
)) ]
dt /tR (41)
In association with eq 40, eq 41 is the basis for calibrating the retention time axis in TFP ThFFF under C2 conditions in terms of M. Although integration of eq 41 appears complex, it can be simplified if eq 41 is viewed as a case where the average of a function of a random variable(s) is computed and handled using statistical mathematics.19 In particular, it is known that the average of the random variable function can be approximated by the function of random variable average. We will proceed in this way. In Appendix I (see Supporting Information (SI)), the problem is discussed and the final solution is derived together with the approximation errors. Only the results are reported here:
〈R(λ|M,ν)〉tR ≈ R[〈l|M〉tR, 〈ν〉tR]
0 e t e tR
(42a)
where
(
ht R
(
( ( ) ))
- 6〈λ〉t/R 〈ν〉tR + (1 - 6〈λ〉t/R〈ν〉tR) coth
2〈λ〉t/R
(40)
〈R|M〉tR can be theoretically evaluated by introducing eq 35 into eq 37:
〈R|M〉tR ≡ 〈R(λ|M(t),ν(t))〉tR )
ht 0
1 2〈λ〉t/R )0
C2 (44b)
where the value of 〈λ〉t/R is the unknown, obtained by numerically solving eq 44a or b. The difference between these two equations lies in the fact that eq 44a includes the average retention ratio that can be theoretically computed using eq 41. In the case of eq 44b, the experimental retention ratio is included for the determination of the unknown 〈λ〉t/R. The symbol * in 〈λ〉t/R of both eqs 44a and b indicates that approximations were introduced, and, more specifically, (1) the theoretical average 〈R(λ,ν)〉 function was approximated by R(〈λ〉,〈ν〉) and (2) secondary relaxation was neglected. The 〈λ〉t/R approximation errors will be considered later. Let us now consider the C3 conditions which, for TFP ThFFF, are more appropriate than C2 since 〈v〉 is not independent of t. The migration equation is now eq 7b, which replaces eq 34. The following time average is defined:
∫
tR
〈R × 〈v〉〉tR )
0
R(t) × [〈v〉(t)] dt tR
C3, M given (45)
where R(t) is the instantaneous retention ratio and [〈v〉(t)] is the instantaneous average cross-sectional linear carrier velocity (assumed to be independent of l). By combining eqs 41 and eq 7b, one has
〈R × 〈v〉〉tR ) L/tR
C3, M given
(46)
R[〈λ|M〉tR,〈ν〉tR] ≡ 6〈λ|M〉tR 〈ν〉tR +
( (
(1 - 6〈λ|M〉tR〈ν〉tR) coth
)
))
1 - 2〈λ|M〉tR 2〈λ|M〉tR
(42b)
and (19) Hamilton, W. C. Statistics in Physical Science; The Ronald Press Co.: New York, 1964.
From eqs 45 and 46, one sees that the effective average species M migration velocity, L/tR, is, under C3 condition, the result of the average effect of the product of two instantaneous quantities, R(t) and [〈v〉(t)], which both change with t. The instantaneous velocity can be evaluated if an experimentally determined function of ht0(Tav) is available, e.g.:16
ht 0(Tav) ) b1 + b2Tav Analytical Chemistry, Vol. 76, No. 22, November 15, 2004
(47a) 6671
〈th0〉t/R
where
Tav ) Tc + (∆T/2)
(47b)
ht R
(
( ( ) ))
- 6〈λ〉t//R 〈ν〉tR + (1 - 6〈λ〉t//R 〈ν〉tR) coth 2〈λ〉t//R
and if the time dependence of Tav, Tav(t), is known. In fact, by using eqs 47a,b one can put
[〈v〉(t)] ) L/th0[Tav(t)]
(47c)
i.e., the instantaneous carrier velocity can also be evaluated at any point in the program as long as ∆T(t) and Tc(t) are known. The position is advanced that the average of a product of two varying quantities is approximated by the product of the average of the those two quantities19 (see Appendix I, SI):
〈R × 〈v〉〉tR ≈ 〈R〉tR × 〈〈v〉〉tR
C3, M given
(48)
where 〈R〉tR can be evaluated from eq 41. In eq 48, 〈〈v〉〉tR is the 〈v〉 time average, within the time range 0 e t e tR:
∫
tR
〈〈v〉〉tR )
0
[〈v〉(t)] dt C3, M given
tR
(49a)
By combining eq 49a and 47c, one has
∫
〈〈v〉〉tR )
{
tR
0
}
L dt ht 0[Tav(t)] tR
C3, M given (49b)
1 2〈λ〉t//R
)0
C3 (51)
the solution of which gives 〈λ〉t//R . Equation 51 is derived by following the same procedure employed for deriving eq 44b, holding true for C2 conditions. The symbol ** was used to indicate that an additional approximation was introduced: the changing value of the carrier velocity during the fractionation process. This additional error is evaluated in Appendix I of SI. At the end of the derivation, one sees that the final expressions (eq 44b or 51)sallowing us to obtain a 〈λ〉 parameter estimation from the void and retention times under TFP operationssare formally similar to the one employed in ThFFF under constant thermal field operations to derive a λ parameter value (cf. eq 18b) with the average quantities 〈 〉 replacing the corresponding constant quantities. The accuracy of the 〈λ〉 estimation is considered in the following section. Approximation Errors in the λ Average Method. Equations 44b and 51 give approximate estimates of 〈λ〉tR,〈λ〉t/R, and 〈λ〉t//R , under C2 and C3 conditions, respectively. The approximation errors are derived in Appendix I of SI. The approximation errors related to eq 44b are evaluated from the two successive neglected terms of the Taylor series expansion employed to derive eq 42a. They are
R′′〈λ〉t 1 R 2 ∆1〈λ〉tR ≈ [λ0]2 σ 2 R′〈λ〉t pr,tR
(52)
R
If the two instantaneous quantities R(t) and [〈v〉(t)] are stochastically independent of one another, eq 48 is exact instead of being approximate.19 This is not the case since, during TFP, R(t) of a given species increases with time, whereas[〈v〉(t)] decreases (see Discussion). The approximation error will be considered later. By combining eqs 48 and 46 one has
〈R〉tR ≈ L/tR〈〈v〉〉tR
C3, M given
(50a)
which can be rewritten as
〈R〉tR ≈ 〈th0〉t/R/tR
C3, M given
(50b)
if one defines the apparent void time 〈th〉t/R as
〈th0〉t/R
≡ L/〈〈v〉〉tR
C3, M given
C3, M given
R
R′〈λ〉t
F〈λ〉t
,〈v〉t
R
σpr,tR
(53)
R
R
where R′′ and R′ are the first and second derivatives, respectively, computed with respect to the quantities reported in the subscript, σ the standard deviations, and F the correlation coefficients referred the subscript quantities in the time range 0 e t e tR. With reference to eqs 52 and 53, since approximations were introduced in obtaining these final expressions (see Appendix I of SI), to recall that approximation was introduced, the symbol ≈ was kept instead of the ) previously used. This notation is employed in the following. Consequently, one has
C2 (54)
(50c)
(50d)
Equation 50d is the analogue of eq 50b with tR evaluated by htR. Finally if eqs 50b and d are equated and combined with eq 44a, one has, for C3 condition, the following equation: 6672
,〈v〉t
R
∆〈λ〉t/R ≡ 〈λ〉tR - 〈λ〉t/R ≈ ∆1〈λ〉tR + ∆2〈λ〉tR
The experimental retention ratio under C3 condition is defined:
R* exp,pr ) 〈th0〉t/R/thR
R′′〈λ〉t
∆2〈λ〉tR ≈ [λ0]
Analytical Chemistry, Vol. 76, No. 22, November 15, 2004
In using eq 51, which is suitable for the C3 conditionsi.e., for the case of time variation of the cross-sectional carrier velocity during the programsone makes an additional error since the program correlates the time variation of [〈v〉(t)] toR(t). The resulting error is
∆3〈λ〉tR ≈
[FR,〈v〉t ]σR,tR σ〈t 〉,t 0
R
R′〈λ〉t
R
R
〈th0〉tR
C3
(55)
Table 1. Calibration Equations of ThFFF Retention Data of PS in Decalin in Isocratic Condition at Various Tc and ∆T Values model
ref eq (see text)
a0
a1
a2
RMSPE
(I) log(λ∆T) ) ao + a1 log(M) + a2 log(Tc/298.15) (II) log(λθc∆T) ) ao + a1 log(M) + a2 log(Tc/298.15) (III) log(λθc∆T) ) ao + a1 log(M) + a2 log(Teq/298.15)
23a 23b 23c
3.35 ( 0.11 3.34 ( 0.11 3.21 ( 0.12
-0.536 ( 0.013 -0.533 ( 0.012 -0.513 ( 0.014
3.28 ( 0.24 3.12 ( 0.23 3.02 ( 0.26
0.0724 0.0720 0.0760
Consequently, under C3 conditionscsi.e., the proper case for TFP ThFFFsone makes the three types of approximation errors described by eqs 52-54. Thus
∆〈λ〉t//R ≡ 〈λ〉tR - 〈λ〉t//R ≈ ∆1〈λ〉tR + ∆2〈λ〉tR + ∆3〈λ〉tR
C3 (56)
It is useful to relate the percent relative error in M, (E(M) %), to the error in 〈λ〉, ∆〈λ〉. If one considers eqs 23b, 29, and 56 and remembers that dl ln x ) dx/x, one has
[E(M)%]theor )
∆〈λ〉t//R 100 〈λ〉tR
a1
7725i Rheodyne (Cotati, CA) valve. Solutions containing a single monodisperse PS standard were injected separately to determine the retention calibration plot. The detector was a UV Model 106 (Linear Instrument Corp., Reno, NV) used at a fixed wavelength, 254 nm. Data handling of both ThFFF output and ThFFF temperature control were performed by NOVAFFF TF3 Control version 2.0 (Postnova Analytics GmbH, Landsberg, Germany) run on a PC equipped with an I/O acquisition board model NI-6034E (National Instruments, Austin, TX). The maximum of 1200 data points was obtained as a product of data rate and run time. The experimental fractograms and the relative Tc and ∆T values were acquired at a data rate of 10 points min-1.
(57)
EXPERIMENTAL SECTION The ThFFF system was a model TF3 polymer fractionator (Postnova Analytics GmbH, Landsberg, Germany). The coolant liquid, tap water or a mixture of water and glycerine (Carlo Erba, Milan, Italy), was circulated by means of a Haake N3-B thermostat (Haake Mess-Technik GmbH, Karlsruhe, Germany). Channel dimensions were as follows: length 45.6 cm tip to tip, width 1.9 cm, and thickness 0.0127 cm obtained from a polyimide sheet, sandwiched between two chrome-plated copper bars clamped together. The void volume at room temperature was calculated from the retention time of an unretained solute (BHT, 2,6diterbutyl-p-cresol, Fluka Chemie, Buchs, Switzerland) and was 1.08 ( 0.04 cm3 (30 data points): this value agrees with the geometrical value of 1.05 cm3. The carrier flow rates were measured at room temperature (23-25 °C). The relationship relating the dead time to the average temperature inside the separation channel (see eq 47a) was obtained by measuring the void time (retention time of an unretained sample, BHT) ht0 at a flow rate of 0.204 cm3 min-1 and at seven different values of Tc in the 295.15-349.15 K range and ∆T in the 22-80 K range. Three repeated injections were performed for each condition. Carrier flow was generated by a model 420 pump (Kontron Instruments S.p.A.) operating at a flow rate of 0.204 cm3 min-1. The carrier was HPLC-grade cis- + trans-Decalin (Fluka Chemie, Buchs, Switzerland) (51% cis+ 49% trans, determined by means of gas chromatographic analysis). The monodisperse polystyrene standards were provided by the manufacturer (Polymer Laboratories Ltd.). The concentration of the solutions injected in the ThFFF system was 0.05% w/v in cis + trans-Decalin and 0.01% w/v in BHT as antioxidant. The samples were prepared at room temperature and filtered on steel filters (10 µm) before injection. Injected quantities were 50 and 10 µg for molecular weights higher than 106 u. A 20-µL loop was used to inject the standards into the ThFFF system by means of a model
COMPUTATION The computations were performed with Matlab version 5.3 using our own programs. The computations involving the dependence of solvent physical-chemical properties ((1/η) ) d0 + d1T + d2T2 + d3T3, d0 ) 5.67 cP-1; d1 ) 5.45 × 10-2 cP-1 K-1; d2 ) 1.61 × 10-4 cP-1 K-2; d3 ) 1.25 × 10-7 cP-1 K-3, thermal conductivity3,7 κ ) q0 + q1(T - TC), q0 ) 1.40 × 104 erg cm-1 s-1 K-1; q1 ) - 9 erg cm-1 s-1 K-2) on temperature were performed to obtain the ν values for any given Tc and ∆T values by using the algorithms reported in the literature.3 Recorded fractograms were used to evaluate the retention time from the peak barycenter. The λ values were obtained from the retention and dead time data using a routine Matlab FZERO to find the root of a function (see eqs 18b, 44a, and 51). θc was computed according to eq 24. In isocratic mode under different Tc and ∆T conditions, the calibration plots were obtained by multiple linear regression according to the model (see eqs 23a-c). Teq in eq 23c was evaluated as described in refs 9 and 17. The best fitted parameters are reported in Table 1. The predictive ability of the model was quantified by the “leave-one-out” root-mean-squared prediction error,21 RMSPE ) ((∑(yˆi - yi)2/n)1/2, where n is the number of the observation and yi and yˆi, respectively, the experimental y-values and those predicted from the model built without the ith observation, i.e., log(λθc∆T). Parts a and b of Figure 1 provide examples of calibration plots under constant and varying Tc and thermal gradient conditions, respectively. During the time programming, Tc(t) and ∆T(t) values (see, for example, Figure 2a) were fitted according to models eqs 33c and 32c, respectively, by using the linear least-squares model. One fitting example is reported in Figure 2b. (20) Vandeginste, B. M. G.; Massart, D. L.; Buydens, L. M. C.; De Jong, S.; Lewi, P. J.; Smeyers-Verbeke, J. Handbook of Chemometrics and Qualimetrics: Part A; Elsevier: Amsterdam, 1998. (21) Vandeginste, B. M. G.; Massart, D. L.; Buydens, L. M. C.; De Jong, S.; Lewi, P. J.; Smeyers-Verbeke, J. Handbook of Chemometrics and Qualimetrics: Part B; Elsevier: Amsterdam, 1998; pp 228-229.
Analytical Chemistry, Vol. 76, No. 22, November 15, 2004
6673
Figure 1. ThFFF calibration plot in isocratic mode; PS standards in cis-trans-Decalin (51:49%); flow rate 0.204 mL min-1. (a) Calibration plot; log(λθc∆T) vs log M. Tc ) 357 K, ∆T ) 39 K: experimental values (b); regression line (s); 95% confidence limits (- - -) 95% confidence limits in prediction (‚ ‚ ‚). (b) Multilinear regression calibration plot under varying conditions of ∆T and Tc: log(λθc∆T)p predicted by eq 23b, vs experimental log(λθc∆T)exp (s); confidence limits in prediction (‚ ‚ ‚).
Figure 3. Exploitation of various quantities during ThFFF programming. (a-d) (A) ∆T0 ) 80 K, Tc,0 ) 303.15 K, t1 ) 10 min, ta ) -20 min, p1 ) 1.71, p2 ) 0.031. (a′-d′) (B) ∆T0 ) 80 K, Tc,0 ) 303.15 K, t1 ) 10 min, ta ) 0 min p1 ) 1.71, p2 ) 0.031. (a) and (a′) The decay functions Tc/Tc,0 (- - -); and ∆T/∆T0 (‚ ‚ ‚) vs elution time specify the programs A and B, respectively. Simulated fractograms (s); (b) and (b′) instantaneous flow distortion parameter ν(t) (- ‚ -); time-averaged flow distortion parameter 〈ν〉tR (s); and cross-sectional average carrier velocity 〈υ〉tR (- - -) vs elution time. (c) and (c′) instantaneous retention parameter λ(t) (- ‚ -); time-averaged retention parameter 〈λ〉tR (s); and thermal gradient correction parameter θc(t) (- - -) vs elution time. (d) and (d′) instantaneous R(t) (- ‚ -); and averaged 〈R〉tR (s) retention ratio vs elution time. 1-3 correspond to M ) 105, 106, and 107 u fractionation parameters, respectively.
Figure 2. Temperature decay in a power programmed run. Selected parameter set t1 ) 20 min, ta ) -40 min, ∆T0 ) 80 °C; Tc,0 ) 303.5 K; (a) experimental data tc °C, (- -) and ∆T (‚ ‚ ‚) versus time. (b) Linear fitting of log(tc) and log(∆T) vs the time function log((t - ta)/(t1 - ta)): experimental data (- - -); linear fitting (s); tc (°C) ) Tc (K) -273.15.
The simulated fractograms reported in Figure 3 were obtained by using the following temperature programs: (A) ∆T0 ) 80 K, Tc,0 ) 303.15 K, t1 ) 10 min, and ta)-20 min; (B) ∆T0 ) 80 K, Tc,0 ) 303.15 K, t1 ) 10 min, and ta ) 0 min. The decay functions of ∆T and of Tc were calculated, respectively, according to eq 32b with p1 ) 1.71 and eq 33b with p2 ) 0.031. These are close to those found in practice (see Figure 2). All simulated fractograms and molecular weight distributions contained 1000 points in a time 6674 Analytical Chemistry, Vol. 76, No. 22, November 15, 2004
interval of 100 min, which corresponds to a data rate of 10 points min-1. The computations used to build Figure 3 and to evaluate the data reported in Table 2 were performed following the sequence reported below: 1. The chosen ∆T(t) and Tc(t) decay plots were modeled by eqs 32a,b and 33a,b. The decay functions for case A and B are traced in Figure 3a and a′, respectively. 2. The instantaneous ν(t) values (see above), and their progressive time average values 〈ν〉tR (see eq 43), are reported for the two programming cases in Figure 3b, b′, respectively, as a dotted line and solid line (left-hand axis). 3. The instantaneous carrier velocity,[〈v〉(t)] (see eq 47c), and the parameter θc(t) (see eq 24) are reported, respectively, in Figure 3b,b′ and 3c, c′ (right-hand axis). 4. The instantaneous λ(t) values for three M values (105, 106,and 107 u) are computed using the model in eq 23b of Table 1 together with the two decay functions of ∆T(t), Tc(t) (see Figure
Figure 4. Different steps and functions through which the experimental fractogram is calibrated in terms of M values for the simulated fractogram / (s); approximated averaged λ parameter (see eq 51), 〈λ〉t// , (- - -). of Figure 3a. (a) Averaged void volume 〈ht0〉tR (- - -); retention ratio Rexp,pr R (b) pr(t) function (s); decay function ∆T/∆T0(Tc/Tc,0)-a2 (-‚-) left-hand axis; relative thermal gradient correction parameter θc,0/θc (‚‚‚) righthand axis. (c) time averaged 〈pr〉 function (s) and λ0 (- - -). (d) time axis calibration plot log(M)exp,pr; (s) and confidence limits (- - -) for program A. (B) Calibration plot (- ‚‚ -) for program B: 1-3 are the retention time positions of M ) 105, 106, and 107 u, respectively. Table 2. λ Estimation from Simulated Fractograms
M (u)
/ 〈λ〉t//R tR|M 〈th0〉tR R/exp,pr (min) (min) (eq 47) (eq 51)
Mexp,pr (u)
[E(M)%]exp,pr
Aa
Program 1.00 × 105 17.72 5.264 0.2958 0.0750 1.00 × 105 6 1.00 × 10 35.74 5.338 0.1468 0.0336 9.92 × 105 1.00 × 107 63.35 5.393 0.0827 0.0175 9.81 × 106
0.80 1.9
Program Ba 1.00 × 105 15.99 5.218 0.3261 0.0828 1.02 × 105 1.00 × 106 26.36 5.272 0.2000 0.0459 9.92 × 105 1.00 × 107 40.55 5.318 0.1310 0.0274 9.71 × 107
2.0 0.82 2.9
a
See under Computation.
3a,a′), and θc(t) (see Figure 3c,c′, right-hand axis). These are reported in Figure 3c, c′ (see dotted lines, left-hand axis), 5. The instantaneous values of R(t) are the values R[λ(t),ν(t)] computed according to eq 35 for the various M values and decay programs. The λ(t) values are calculated as reported under (4) (see Figure 3c,c′, left-hand axis). The instantaneous ν(t) values were calculated as reported under (3) (see Figure 3b, b′). 6. [〈λ(t)〉]t and 〈R(t)〉t computed respectively using eqs 30 and 37 (or 41) are reported in Figure 3c,c′ and d, d′, respectively (solid lines, left-hand axis). 7. tR|M of the various species (see the vertical lines in Figure 3 and Table 2) are computed as numerical solution of the integral equation eq 7b, where R(t) and [〈v〉(t)] are the values calculated and reported in Figure 3d,d′ and b,b′, respectively as explained above. The tR|M values are the time at which species M reaches the end of the channel (L ) 45.2 cm); i.e., they are the simulated retention times under C3 condition. 8. The 〈th0〉t/R values are computed by using eqs 50c, 49b, and 47a and reported in Table 2 for the different species M and programs A and B. The fitting degree of ht0 experimental values to eq 47a was expressed by the coefficient of determination20 r2 (see below, under Discussion).
/ of Table 2 and Figure 4a are computed by using eq 9. Rexp,pr 50d. Finally, the fractograms are traced in Figure 3a,a′: these fractograms are obtained by convoluting at the three tR|M values, Gaussian peak shapes simulating the effect of a polydispersity µ ≈ 1.03. Column band broadening is not accounted for in this handling. Nonetheless, the fractograms reported in Figure 3a,a′ roughly correspond to those of the three standards at µ ≈ 1.03 under the specific TFP ThFFF conditions, since the polydispersity band-broadening effects overcome column band-broadening effects. The procedure described in ref 14 was followed in obtaining M distribution of Figures 3 and 5.
DISCUSSION Experimental Data for Exploiting TFP ThFFF. To exploit and apply the universal calibration ThFFF method under TFP operations, the following experimental data must be available: (1) the calibration plot, for the specific polymer-solvent system, at the various cold wall temperature conditions in ∆T and the Tc ranges exploited during the TFP (see eqs 23a-c); 2) the ∆T(t) and Tc(t) values, necessary to evaluate the programming function and its average (see eqs 28 and 31); (3) the ht0(Tav) function (see eq 47a) allowing one to determine 〈th0〉t/R (see eq 50c), necessary / for the species eluting at tR (see eq 50d) and to compute Rexp,pr to solve eq 51 in terms of 〈λ〉t//R . Experimental Determination of ThFFF Calibration Curve of PS in cis-trans-Decalin Extended to Different Cold Wall Temperature Conditions. In the literature, only one study is reported concerning full ∆T and Tc calibration modeling in ThFFF.5 For the purposes of this paper, λ data for the PS were measured in the cis-trans-Decalin system at varying Tc (307333 K) and ∆T (75-20 K) values. The data are reported in Figure 1a,b. Figure 1a presents an example of an experimental calibration plot at Tc ) 347 K for the PS-cis-trans-Decalin system. The scattering observed in this plot is close to that observed for similar Analytical Chemistry, Vol. 76, No. 22, November 15, 2004
6675
Figure 5. (a) Experimental fractogram (s); and ∆T (‚‚‚) vs time plot. Sample, PS, Mp) 2 .310 × 106 u, µ ) 1.04; solvent, cis-transDecalin (51:49%); flow rate, 0.204 mL min-1; Tc,0 ) 300 K, ∆T0) 51 K, t1 ) 20 min, and ta ) -40 min. (a′) Molar mass distribution obtained from the fractogram reported in (a). (b) Experimental fractogram of a synthetic mixture (s); and ∆T (- - -) vs time plot. Sample, PS, Mp ) 1.02 × 106, 6.5 × 106 u, µ )1.05 and 1.12, respectively; solvent, cis-trans-Decalin (51:49%); flow rate, 0.204 mL min-1; Tc,0 ) 335 K, ∆T0 ) 51 K, t1 ) 20 min, and ta ) -40 min, (b′) Molar mass distribution obtained from the fractogram reported in (b). (c) Method reproducibility under two different program conditions. (a) Fractogram PS, Mp ) 1.02 × 106 u, µ ) 1.05; solvent, cis-trans-Decalin (51: 49%); flow rate, 0.204 mL min-1; t1 ) 20 min, ta ) -40 min, Tc,0 ) 303.5 K, ∆T0) 76 K (s); Tc,0 ) 305 K, ∆T0 ) 68 K (- - -). (b) Molar mass distribution obtained from the fractograms of the sample reported in (a) at Tc,0 ) 303.5 K and ∆T0 ) 76 K (s); Tc,0 ) 305 K and ∆T0 ) 68 K (- - -).
cases.5 In this plot, two types of confidence intervals are reported: the inner one is the confidence interval for calibration at 95% probability, whereas the external one is the confidence interval “in prediction”;20 it refers to the uncertainty involved when the calibration line is employed to determine the log M for an unknown species from a single experimental determination of log(λθc∆T). One can see that the log M 95% confidence interval is on the order of ∼0.1 unit, which corresponds roughly to an error of ∼(20% in M determination. To improve the precision in prediction, one must either improve calibration plot precision, e.g., by making replicates or increasing the number of points in the calibration interval or improve experimental precision, e.g., by improving control over experimental variables such as flow rate, ∆T, and Tc. Table 1 presents the ThFFF “universal” PS-cis-trans-Decalin calibration plots under varying conditions of Tc, modeled according to eqs 23a-c. From Table 1 one can see that, for the three different models, the fitting degrees, expressed by the RMSPE (see Computation), show reciprocal statistical equivalence. Figure 6676
Analytical Chemistry, Vol. 76, No. 22, November 15, 2004
1b reports the fitting by model eq 23b. One can see that this model represents the experimental points fairly well, since the observed experimental data scattering versus the ideal line at zero intercept and unit slope does not significantly depend on log(λ∆Tθc). Moreover, the 95% confidence interval in prediction on log(λ∆Tθc) is on the order of 0.1 unit, i.e., the same order of magnitude observed at constant Tc (see Figure 1b): apparently increasing the number of points in Figure 1b versus those in Figure 1a does not reduce calibration plot uncertainty, as one would expect. This may be due to the increased number of experimental factors (∆T and Tc) undergoing variation or the lack of model fitting (eq 23). Full exploitation of these aspects lies beyond the aims of the present paper. The conclusion is that, at variable Tc, the determined calibration plot should permit predictive experimental precision for M determination similar to what is obtained at constant Tc. Equation 23b has been chosen as the basis for modeling TFP ThFFF. Operative Variables in TFP ThFFF. The primary variable that undergoes variation in thermal TFP ThFFF is ∆T, modeled according to eq 33b with p1 ) 2, while Tc should remain constant: in fact, these conditions correspond to optimal separation conditions.22 However, with the available instrumentation (see under Experimental Section), not only is the actual ∆T(t) function different from the set one, but Tc, too, does not remain constant. Parts a and b of Figure 2 report an example of the experimental ∆T(t) and tc(t) (with tc(°C) ) Tc(K) - 273.15) programming decay function together with their modeling according to eqs 32c and 33c, respectively. One can see that the chosen models fit the experimental trends of both ∆T and tc fairly well. It is worth noting that the t1 and ta values in the models in eqs 32c and 33c correspond to the set ones (see Figure 2 caption) and did not undergo minimization. The other parameters ∆T0 and p1 in eq 32c and Tc,0 and p2 in eq 33c did undergo minimization. On the other hand, the best fitting p1 ()1.71) and p2 ()0.031, which correspond to p2 calculated on tc ) 0.47) values are significantly different from the set ones (2 and 0, respectively, see Figure 2b). Consequently, in computing pr(t) and its average values 〈pr〉tR (see eq 28 and 31, respectively), one must employ either real ∆T and Tc data sets or the best fitting equations (see Figure 2). We chose the first option. Void Time Variation under ∆T and Tc Decay. The void timesevaluated by using the first moment of an inert tracer, ht0, (corrected for extracolumn contributions)sis not constant during TFP but is known to vary with variations in ∆T, Tc, or both.16 This has significant consequences in both the modeling type (C2 vs C3) and in the proper equations used to derive the λ parameter (eq 44b vs eq 51). The ht0 variation versus Tav (see eq 49) in the 306-360 K range was measured and modeled according to eq 47a, with the following result: ht0 ) 6.9((0.12) - 0.0041((0.00036)Tav; r2 ) 0.9774 (see Computation). This model is satisfactory for the subsequent applications since it explains ∼98% of the variability. (22) Giddings, J. C.; Kumar, V.; Williams, P. S.; Mayer, M. N. In Polymer Characterization, Physical Property, Spectroscopic and Chromatografic Methods; Craver, C. D.; Provder, T., Eds.; Advances in Chemistry Series 227, American Chemical Society: Washington, DC, 1990; pp 1-21. (23) Crame´r, H. Mathematical Methods of Statistics; Princeton University Press: Princeton, NJ, 1974; pp 62-88.
Retention and Retention Parameter Variation under TFP ThFFF. Under Computation, a numerical procedure is described for full simulation of the TFP ThFFF under C3 conditions, i.e., under conditions of nonconstant carrier velocity. Figure 3 and Table 2 present the results of two full simulation examples. The aim of Figure 3 is to describe the whole fractionation process including its hidden features, i.e., those not accessible to the experimenter. Two series of different programming conditions (A and B; see under Computation and Figure 3a,a′) and three different samplessrespectively, of 105, 106, and 107 us were considered along with an assumed polydispersity of µ ) 1.03 (see below). The type A program (left-hand side of Figure 3) is an optimum program, whereas program B (right-hand side of Figure 3) exhibits a fast decay at the beginning and is not optimized.22 In Figure 3b,b′, one can see how the progressive thermal field release (reported in Figure 3a and a′) affects the parabolic distortion parameter ν and its average value 〈ν〉tR: when ∆T ) 0, ν is zero, and the progressive time average values 〈ν〉tR increase with tR. Likewise θc(t) in Figure 3c,c′, right-hand axis, converges to 1 when ∆T f 0 (see eq 24). The instantaneous carrier velocity,[〈v〉(t)], reported in Figure 3b,b′ (right-hand axis), corresponds to the above-described trend in ht0 with Tav, exhibiting an increasing trend as t increases. The tR|M of the various species (see the vertical lines in Figure 3 and Table 2) are the times at which species M reaches the end of the channel (L ) 45.2 cm), i.e., the simulated retention times under the C3 condition. This parameter is the solution of eq 7b and reflects the combined effect of both R(t) and [〈v〉(t)]. Fractograms are traced in Figure 3a and a′: they were obtained by convoluting at the three tR|M values, Gaussian peak shapes simulating the effect of a polydispersity µ ≈ 1.03 (see Computation). Figure 3c,c′ and d,d′ provide a detailed picture of the species “history” inside the channel during the fractionation process. In particular, plots of Figure 3c,c′ describe how the three different species (105, 106, and 107) are constantly held at different fixed relative λ positions during the first period of programming when both Tc and ∆T are kept constant. It can be seen that when M increases, the corresponding λ value decreases (see either Figure 3c or c′), and when the ∆T value decreases, the corresponding λ value increases (cf. the λ(t) values in Figure 3c and c′ for the same M value). Moreover, the effect of decreasing ∆T prevails over the decrease in Tc and this determines the observed increase in the λ parameter (cf. eq 23b). Likewise one can observe the instantaneous values of the retention ratio R(t) reported in Figure 3d,d′: its behavior is closely related to λ(t) and expresses the instantaneous relative migration velocity of the band (see eqs 2 and 18a). Note, however, that the effective, instantaneous band migration velocity is given by (〈v〉(t)) × R(t) (see eq 5). Time Axis Calibration by the λ Average Method. If Figure 3 was built to clarify the fractionation process under TFP ThFFF conditions once all the features of the experiment are perfectly known (experimental parameters, calibration functions, and M values of the species), Figure 4 was built to shed light on the different steps and functions for calibration of the experimental fractogram in terms of M. The example worked out mainly refers to the A type program in Figure 3a-d. However, now the M value associated with a given tR is the unknown. The different steps to be used are as follows:
1. 〈th〉t/R for each tR value of the fractogram time axis is computed by using eq 50c together with eqs 49b and 47a,b (see Figure 4a, right-hand axis). / 2. Rexp,pr for each tR value of the fractogram time axis is computed by using eq 50d (with tR ≡ htR) (see Figure 4a, righthand axis). 3. 〈λ〉t//R values versus tR (see Figure 4a) are determined by using eq 51, with tR ≡ htR and 〈ν〉tR read in Figure 3b, for each tR value (full line, left-hand axis). The 〈λ〉t//R function is reported in Figure 4a, left-hand axis. 4. The ∆T0/∆T/(Tc,0/Tc)a2 and θc,0/θc (see eq 24) are calculated from the ∆T(t) and Tc(t) decay plots (see in Figure 3a), and from them, the pr function is obtained (see eq 28). All these functions are reported in Figure 4b. 5. The 〈pr〉tR function is computed from eq 31. The λ0** values for each tR are computed from 〈λ〉t//R (reported in Figure 4a) and 〈pr〉tR (reported in Figure 4c) by using eq 29. 6. Using eq 25, the log Mexp,pr versus tRswith the confidence intervals in “prediction”20sare computed from the λ0** versus tR plot in Figure 4c and reported in Figure 4d. The same figure also reports the final calibration plot for the case B program (see Figure 3a′). Note that the subscript exp, pr has been used since the M values refer to the experimental calibration under programming condition. In Figure 4b, one can see that pr(t) is almost coincident with ∆T0/∆T/(Tc,0/Tc)a2, since the θc,0/θc varies between 1 (at the beginning of the program) and 0.97 at the end of the program (tR ) 120 min). Thus, in practice, the effect of θc time variation is marginal. In Figure 4d, the two programs A and B are compared through their calibration plots. One can see that, with program A, the different species are fractionated over a wider tR range than with program B and this makes program A more efficient than program B. As an example of the procedure, the M values for the three standards at M values of 105, 106, and 107 u under the programs A and B were determined by using the setup procedure (see Table 2 and Figure 4d). In fact, for these species, the exact tR|M values (see third column of Table 2 and arrows in Figure 4a) were computed as solution of eq 7a (see under Computation), and thus, these values play a role as the experimental quantities. One can see that the true M values (second column) and the “experimental” ones (seventh column, Mexp,pr), obtained by following the above-described computational sequence, are close to each other. Column 8 reports the M relative errors ((M Mexp,pr)/M × 100 ) [E(M)%]exp,pr). The maximum observed absolute error is 3% for program B (see Table 2), significantly lower than the error determined by the uncertainty of the calibration curve when employed in prediction ((10%, see above and Figure 1). In the case of program A, the “experimental” Mexp,pr values lie within the confidence interval “in prediction” (see Figure 4d) Degree of Accuracy for the λ Average Method. The good agreement between true and determined M values indicates that, in general, the assumed approximations are good, at least for the two simulated program cases. In Table 3, a quantitative study of the accuracy is reported by comparing the found “experimental” errors on 〈λ〉 with those expected from eqs 54 and 56, which account for the first-order Taylor Series expansion of the R(λ,ν) function and of the correlation between R(t) and [〈v〉(t)], respecAnalytical Chemistry, Vol. 76, No. 22, November 15, 2004
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Table 3. Errors in λ Estimation from Simulated Fractograms
M (u)
〈λ|M〉tR (eq 30)
〈λ〉t/R (eq 44a)
〈λ〉t//R (eq 51)
〈λ〉t/R (eq 54)
〈λ〉t//R (eq 56)
∆1〈λ〉tR + ∆2〈λ〉tR (eq 54)
∆1〈λ〉tR + ∆2〈λ〉tR + ∆3〈λ〉tR (eq 56)
[E(M)%]theor (eq 57)
1.00 × 105 1.00 × 106 1.00 × 107
0.0751 0.0334 0.0173
0.0750 0.0333 0.0172
0.0750 0.0336 0.0175
-0.0001 -0.0001 -0.0001
Program Aa -0.0001 0.0002 0.0002
-0.00009 -0.00012 -0.00007
-0.00006 0.0002 0.00027
0.15 1.1 2.7
1.00 × 105 1.00 × 106 1.00 × 107
0.0832 0.0457 0.0269
0.0826 0.0454 0.0267
0.0828 0.0459 0.0274
-0.0008 -0.0003 -0.0002
Program Ba -0.0005 0.0002 0.0005
-0.0007 -0.0005 -0.0002
-0.0005 0.0002 0.0006
1.1 0.83 4.2
a
See under Computation.
tively (see Appendix I, SI). One can see that the agreement is remarkably good in all cases, considering the unavoidable computational rounding-off errors. In fact, in Table 3, one must compare ∆〈λ〉t/R ≡ 〈λ〉tR - 〈λ〉t/R, in column 6 with ∆1〈λ〉tR + ∆2〈λ〉tR in column 8, which expresses the eq 54 approximation. Likewise, one must compare ∆〈λ〉t/R ≡ 〈λ〉tR - 〈λ〉t//R in column 7 with ∆1〈λ〉tR + ∆2〈λ〉tR + ∆3〈λ〉tR in column 9, which expresses the eq 56 approximation. Column 10 in Table 3 reports the relative errors in M indicated as theoretical since it is computed according to eq 57. These values agree with the “experimental” ones [E(M)%] exp,pr reported in Table 2, column 8. The setup procedure is thus fully validated by numerical simulation. In fact, the “experimental” accuracy in M determination is within 3% in the worse case and thus compatible with the expected “real” precision in “prediction” ((10%), but also the “experimental” errors fit the “theoretical” ones. In Appendix II of SI, the goodness of the method is accounted for in term of goodness of the first-order Taylor series expansion and the dependence of the various sources of approximation are investigated with respect to tR, M, and TFP type. Experimental Validation. Figure 5 reports three applications of the present method to real TFP ThFFF cases of monodisperse PS samples. The fractograms and the shape of the ∆T decay function are reported on the left-hand side of the figure whereas the M distribution, obtained by following the above-discussed procedure for each data point of the fractogram, is traced on the right-hand side. It must be remember that the main value to consider here is the M peak value, Mp, since as mentioned above, band-broadening effects have not been accounted for here. In case a, the elution of the sample was performed in the first-medium part of the ∆T decay, whereas in case b, the Mp ) 1.030 × 106 u and Mp ) 6.5 × 106 u standards were eluted in the beginning and in the middle of the ∆T decay program, respectively. One can see that, in all cases, the values of Mp are unbiased and lie within the 2% approximation range; these errors are compatible with the experimental precision obtained by employing isothermal calibration plots, as discussed above. Finally, the reproducibility of the method was evaluated by determining the M distribution of a given PS standard (Mp ) 1.030 × 106 u) obtained under two different programming conditions. One can see that the peak positions in the molecular weight axis (see Figure 5c′) are close to each other, the very minor differences in distribution arising from the effect program decay has on band broadening and then on the fractionation power of ThFFF.12 6678
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CONCLUSIONS The essence of the present approach was to extend the classical retention equation of ThFFF at constant field strength (see eq 18b) to condition of TFP. In writing, this issued the same equation with pertinent average quantities (see, for example, eq 51). The approach involved relating the obtained apparent λ average parameter to the conditions at the beginning of the temperature program, λ0. Specific hypotheses and approximations were assumed. The equations describing the approximation errors were derived. The numerical procedure was checked by simulation by showing that the equation describing the approximation errors has excellent predictive value. The method was also validated in practice. This result was attained by determining the experimental error in the calibration step and by comparing this error with the theoretical approximation errors. These two types of errors were comparable. In this way, both the validity of the assumed hypotheses and the approximations were tested. It seems that the secondary relaxation effects here neglected, even if present, are only second-order effects, at least under the conditions exploited here. Likewise, the starting hypotheses seem acceptable. The detailed procedure here developed can provide the basis for exploiting the influence of various experimental parameters for optimization studies. Likewise, the numerical code here developed for a full simulation of the TPF allows one to identify the relevance of the various effects on the fractionation result. It seems that the major effects are the variables ∆T, Tc, and ν, and their time variation, θc is of marginal influence. GLOSSARY a0
intercept of the ThFFF calibration equation (eqs 23a-c)
a1
slope of the ThFFF calibration equation vs log(M) (eqs 23ac)
a2
slope of the ThFFF calibration equation vs log(Tc/298.15) (eqs 23a-c)
Af
channel cross-sectional area
bi
ith coefficient of the linear relationships between the void time ht0 and average channel temperature (eq 47)
c
cross-sectional local concentration
c0
analyte concentration at the accumulation wall
coth hyperbolic cotangent di
ith coefficient of the third-degree expression of fluidity (1/ η) vs temperature
D
analyte diffusion coefficient
analyte thermodiffusion coefficient
ht0
experimentally determined void time
[E(M)%]exp,pr relative error on the molecular weight determined by calibration in programming conditions
〈th0〉t/R
apparent void time averaged in the time range 0 e t e tR
[E(M)%]theo
t1
predecay time
T
temperature
Tav
temperature averaged on the channel thickness
Tav(t)
instantaneous temperature averaged on the channel thickness
Tc
cold wall temperature
Tc,0
initial cold wall temperature
Tc,t
cold wall temperature at a given time t
DT
theoretical relative error on the molecular weight determined by calibration in programming conditions
FFF
field flow fractionation
|Ff|
force exerted by the external field on an individual analyte macromolecule
GC
gas chromatography
HPLC high performance liquid chromatography Boltzmann constant ) 1.380 662 × 10-23 J K-1 ) 1.380 662 × 10-16 erg K-1
k
Tc(t)
instantaneous cold wall temperature
Teq
temperature at the equivalent analyte position
Th
hot wall temperature
ThFFF
thermal field flow fractionation
l
analyte axial position inside the separation channel
l
space constant of the exponential transverse concentration distribution of the analyte (eq 13a)
L
separation medium length
TFP
thermal field programming
M
molecular weight
|u|
velocity induced by the force on the analyte
Mp
molecular weight at the peak maximum
u
atomic mass unit
Mw
weight-average molecular weight
v
local linear flow velocity of the carrier fluid
Mn
number-average molecular weight
〈v〉
cross-sectional average carrier flow velocity
MLR
multi linear regression
〈〈v〉〉tR
pr(t)
function of the field program run time
the time average linear cross-sectional average linear carrier velocity in the time range 0 e t etR
p1
power decay of the temperature gradient (eq 32b)
V
cross-sectional average axial velocity of an analyte zone
p2
power decay of the cold wall temperature (eq 33b)
x
axis coordinate of the channel thickness
xeq
equivalent analyte position
w
channel thickness
〈•〉
average operator applied to a function
〈•〉tR
time-average operator applied to a function of time, in the time range 0 e t e tR
〈•〉t/R
time-average operator applied to a function of time in the time range 0 e t e tR, obtained by introducing first type approximation
〈•〉t//R
time-average operator applied to a function of time in the time range 0 e t e tR, obtained by introducing first and second type approximations
ith coefficient of the first-degree expression of κc vs temperature
qi
retention ratio
R Rexp
experimentally determined retention ratio in isocratic elution
R|M
retention time of a given species characterized by molecular weight M
Rexp,pr
experimentally determined retention ratio in programmed elution
/ Rexp,pr
retention ratio in programmed elution determined by using experimental retention time and apparent void time
R′〈λ〉t
first derivative of the retention ratio with respect to the λ values averaged in the time range 0 e t e tR
R
R′′〈λ〉t
second derivative of the retention ratio with respect to λ values averaged in the time range 0 e t e tR
R′′〈λ〉t
second derivative of the retention ratio with respect to the λ and ν values averaged in the time range 0 e t etR
R
〈ν〉t
R
R
RMSPE root mean squared prediction error SEC
Greek Symbols ∆〈λ〉t/R
error on 〈λ〉tR estimation derived from truncating the Taylor series expansion at the first term
∆
error on 〈λ〉tR estimation derived from truncating the Taylor series expansion at the first term and neglecting the time variation of the cross-sectional carrier velocity during the program
〈λ〉t//R
∆1〈λ〉tR
theoretical error in λ estimation derived from neglecting the second term of the Taylor series expansion (second degree in λ)
∆2〈λ〉tR
theoretical error in λ estimation derived from neglecting the second term of the Taylor series expansion (first degree in both λ and ν)
∆3〈λ〉tR
theoretical error in λ estimation derived from time variation of the cross-sectional carrier velocity during the program
size exclusion chromatography
SedFF
sedimentation field flow fractionation
t
time
ta
program time constant
tR
retention time
htR
experimentally determined retention time
t0
void time
Analytical Chemistry, Vol. 76, No. 22, November 15, 2004
6679
∆M
error in M estimation derived from truncating the Taylor series expansion at the first term and neglecting the time variation of the cross-sectional carrier velocity during the program
∆T
temperature gradient
∆T0
initial temperature gradient
∆Tt
temperature gradient at a given time t
θc,t
thermal gradient correction parameter at a given time t
θc(t)
instantaneous thermal gradient correction parameter
µ
sample polydispersity
F〈λ〉tR,〈v〉tR correlation coefficient of the λ and ν quantities averaged, in the time range 0 e t e tR FR,〈ν〉tR
correlation coefficient of the R and ν quantities averaged, in the time range 0 e t e tR
σλ,tR
standard deviation of λ averaged in the time range 0 e t e tR
σ2pr,tR
variance of the program function averaged in the time range 0 e t e tR
∆T(t)
instantaneous temperature gradient
η
carrier viscosity
κ
carrier thermal conductivity
κc
carrier thermal conductivity at Tc
ν
distortion flow profile parameter in isocratic elution
σR,tR
ν(t)
instantaneous distortion flow profile in programmed elution
standard deviation of the retention ratio averaged in the time range 0 e t e tR
σ〈t0〉,tR
standard deviation of the void time averaged in the time range 0 e t e tR
〈v〉tR
distortion flow profile averaged in the time range 0 e t etR
λ
analyte FFF parameter
λ0
λ value at the beginning of the program
λt
λ value at a given time t
ACKNOWLEDGMENT This work was financially supported by The Italian Ministry of University and Scientific Research (Grant 2002038818_002) and by Basell Polyolefins Italia.
λ0|M
λ0 value referred to a given molecular weight M
λ(t)|M
instantaneous λ value referred to a given molecular weight M
〈λ〉tR
average of instantaneous λ values in the time range 0 e t e tR
〈λ〉t/R
average of instantaneous λ values in the time range 0 e t e tR, obtained by introducing an approximation
〈λ〉t//R
average of instantaneous λ values in the time range 0 e t e tR, obtained by introducing two approximations
θc
thermal gradient correction parameter
Received for review April 22, 2004. Accepted August 26, 2004.
θc,0
initial thermal gradient correction parameter
AC049399Q
6680
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SUPPORTING INFORMATION AVAILABLE Appendix I (derivation of theoretical expressions of the errors due to the approximations) and Appendix II (examples of estimation of the errors in the molecular weight determination of sample eluted in TFP ThFFF). This material is available free of charge via the Internet at http://pubs.acs.org.
