Anal. Chem. 2000, 72, 1555-1563
Theoretical Investigation of the Spatial Progression of Temporal Statistical Moments in Linear Chromatography Kevin Lan and James W. Jorgenson*
Department of Chemistry, Venable Hall, CB 3290, The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3290
Migration and dispersion in chromatography are modeled by analogy to an effective eddy diffusion process. On the basis of this model, the spatial rates of temporal statistical moment change are derived for general chromatography in linear media. In most practical cases, these equations can be simplified so that temporal statistical moments can be calculated by solving a system of ordinary differential equations that depend only on the local HETP, solute velocity, and initial values of the temporal statistical moments. The calculations of temporal centroid, temporal variance, temporal skew, and temporal excess are demonstrated for the case of linear solvent strength gradients. It is shown for the case of temporally invariant separation environments, such as isocratic liquid chromatographic systems and isothermal gas chromatographic systems, that temporal variance contributions are spatially additive and that the temporal third normalized central moment is unaffected by spatial variations in the medium. A refined explanation is given for how peak symmetry is improved in gradient forms of chromatography. Modern fundamental theories of chromatographic behavior describe peak characteristics in terms of spatial quantities, such as spatial centroid and spatial variance. Unfortunately, virtually all modern chromatographic detectors acquire their measurements temporally; i.e., a signal is acquired at a specific point in space over a duration of time. Accordingly, such detectors are incapable of directly measuring the spatial quantities given by modern theory. This inherent incompatibility between theory and experiment leads to some confusion in the prediction of chromatographic behavior for all but the simplest of systems. To enhance the predictive ability of chromatographic theory and the general understanding of chromatographic behavior, fundamental theories should be redeveloped so that they directly describe the measured temporal quantities. Conventional Theory. One of the most fundamental models of chromatographic behavior is based on a one-dimensional system where migration is modeled as a bulk displacement and dispersion is modeled as an effective diffusion process.1 The basic assumption governing this model is that (1) Giddings, J. C. Unified Separation Science; John Wiley & Sons: New York, 1991; pp 37-54. 10.1021/ac990533u CCC: $19.00 Published on Web 03/03/2000
© 2000 American Chemical Society
j(x, s) ) v(x, s) c(x, s) - D(x, s)
∂c (x, s) ∂x
(1)
where j is flux, v is the displacement velocity, c is concentration, D is the effective Fick diffusion coefficient, x is space, and s is time. This equation is essentially Fick’s first law with an additional term to account for the bulk displacement of solute. Applying the equation of continuity to (1) gives the partial differential equation1 that is frequently used to model chromatography:2-5
∂c ∂ ∂c ∂vc (x, s) ) D(x, s) (x, s) (x, s) ∂s ∂x ∂x ∂x
(2)
On the basis of this equation, Blumberg has been able to describe generally the spatial variance of a peak migrating through a linear medium under temporally and spatially changing conditions.4 This theory gives the progression of spatial peak variance as the solution to the following ordinary differential equation:
dσ2 ∂ ln u ∂ ln u ∂H ) H(z, t) + 2σ2 (z, t) + σ2 (z, t) (z, t) (3) dz ∂x ∂x ∂x where σ2 is the spatial peak variance, z is the peak centroid, t is the time required for the centroid to reach position z, H is the local height equivalent of a theoretical plate (HETP),4 and u is the net solute velocity due to displacement and any spatial variation of diffusivity.3 Less general forms of (3) have also been derived,2,3,6,7 and these equations have been used successfully to predict peak variances in linear solvent strength gradient liquid chromatography2,8 and reversed-phase alternate-pumping recycle liquid chromatography.7,9,10 Practical Limitations of Conventional Theory. The major practical restriction in the derivation of (3) is that the chromatographic system must be spatially moderate (i.e., there must be (2) Poppe, H.; Paanakker, J.; Bronckhorst, M. J. Chromatogr. 1981, 204, 7784. (3) Blumberg, L. M.; Berger, T. A. J. Chromatogr. 1992, 596, 1-13. (4) Blumberg, L. M. J. Chromatogr. 1993, 637, 119-128. (5) Miyabe, K.; Guiochon, G. J. Chromatogr., A 1999, 830, 263-274. (6) Rubey, W. A. J. High Resolut. Chromatogr. 1991, 14, 542-548. (7) Lan, K.; Jorgenson, J. W. Anal. Chem. 1998, 70, 2773-2782. (8) Snyder, L. R.; Dolan, J. W. Adv. Chromatogr. 1998, 38, 115-187. (9) Biesenberger, J. A.; Tan, M.; Duvdevani, I.; Maurer, T. Polym. Lett. 1971, 9, 353-357. (10) Duvdevani, I.; Biesenberger, J. A.; Tan, M. Polym. Lett. 1971, 9, 429-434.
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no sudden spatial variations in the system).2-4 (Specifically, the spatial derivatives of the local HETP and net velocity functions must be nearly constant across the width of the peak.) Although most practical chromatographic systems meet this requirement, there are a few practical systems that are not spatially moderate, such as two different columns connected in series. The requirement is also not met when the chromatographic system is considered in its entirety: Junctions, such as those between connective tubing and the column, are not spatially moderate. Theoretical Limitations of Conventional Theory. A theoretical difficulty in using (3) is that it gives spatial peak variance, not temporal peak variance, as its solution. Equation 3 thus depends on the approximate conversion4 of spatial variances into temporal variances τ2:
τ2 ≈
σ2 u2(z, t)
dτ )
H(z) u2(z)
dz
(-1)nµ jn vn(z, t)
(6)
THEORY Cumulative Amount. Consider a one-dimensional chromatographic system containing a single solute. Let us define the spatially cumulative amount n-(x, s) as the amount of solute at
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Analytical Chemistry, Vol. 72, No. 7, April 1, 2000
∂n(x, s) ∂x
(8)
The rate at which the temporally cumulative amount increases with respect to time is the one-dimensional flux j:
j(x, s) ≡
∂n+ (x, s) ∂s
(9)
(5)
Unfortunately, this conversion is very unreliable for higher order moments (n > 2) when the chromatographic medium has spatial or temporal variations (Appendix, part 1). Other methods of obtaining temporal statistical moments also assume that the medium has no spatial or temporal variations.12,13
(11) Giddings, J. C. Anal. Chem. 1963, 35, 353-356. (12) Kucera, E. J. Chromatogr. 1965, 19, 237-248. (13) Grushka, E. J. Phys. Chem. 1972, 76, 2586-2593.
(7)
Amount Density. The rate at which the spatially cumulative amount increases with respect to space is the one-dimensional concentration c:
c(x, s) ≡
By use of (4) to convert the spatial variance in (3) to a temporal variance, it can be derived4 that the validity of (5) also relies on the additional condition that the local HETP must be uniform in space. Indeed, if the definition of temporal variance were given by (4), this conclusion would be correct. However, (4) is an approximation, so it is likely that the conditions surrounding the validity of (5) may be different from what is implied by (3) and (4). The complications associated with converting spatial variances into temporal variances are exacerbated when higher order statistical moments are converted. For example, spatial statistical moments µj n at high efficiencies are most commonly converted into temporal statistical moments M h n by
M hn≈
ntot ≡ n-(x, s) + n+(x, s)
(4)
Although using this conversion does not seriously limit the practical accuracy of (3), it does have some theoretical implications. For example, Giddings proposed that temporal variance contributions are spatially additive in chromatographic systems in which variations occur in space but not in time:11 2
positions less than x at time s. Similarly, let us define the temporally cumulative amount n+(x, s) as the amount of solute at positions greater than x at time s. (All solute that has passed a given point x by time s must be at positions greater than x; hence, n+ is temporally cumulative.) Let us assume that both cumulative amount functions are continuously differentiable. The sum of the spatially cumulative amount and the temporally cumulative amount must be the total amount of solute in the system ntot, which we assume to be a constant:
Note that concentration in one dimension has units of amount per distance, whereas the conventional definition of concentration in three dimensions has units of amount per volume. Similarly, flux in one dimension has units of amount per time, whereas the conventional definition of flux in three dimensions has units of amount per area time. All references to concentration and flux in the remainder of this work are in the context of one dimension. Let us assume that concentration and flux are continuously differentiable functions. To relate concentration and flux quantities, it is convenient to express concentration and flux as derivatives of the same function. Rearranging the spatial derivative of (7) gives the concentration in terms of the temporally cumulative amount:
∂n∂n+ (x, s) ) (x, s) ) c(x, s) ∂x ∂x
(10)
It is important to note that nearly all chromatographic detectors measure either concentration or flux as their signals. A detector that measures flux, such as a flame ionization detector, is usually called a “mass-sensitive detector” because its temporally integrated signal is proportional to the total mass of solute. Stepwise Model of Eddy Diffusion. The progression of peak shape in chromatography is usually modeled as an effective Fick diffusion process combined with bulk displacement. Let us instead model the progression of peak shape as an effective eddy (multipath) diffusion process. Consider a simple model of eddy diffusion where the column consists of evenly spaced cells (A, B, C, etc.) that are connected in series by identical pairs of incongruent paths (Figure 1). Let us denote each pair of paths as a step. When solute is introduced at cell A (Figure 1A), it is divided between the two paths of the step as it travels to cell B. Since the two paths are incongruent, the solute will arrive at cell B at two
between these two equations. In both (1) and (11), the first term on the right-hand side (rhs) of the equation accounts for the bulk displacement of solute and the second term diminishes the variations of the corresponding amount density profile. In Fick diffusion, variations of concentration across space are diminished by the progression of time. In eddy diffusion, variations of flux across time are diminished by progression through space. Equation of Continuity. The mixed second partial derivatives of a continuous function are the same regardless of the order in which the derivatives are taken. The temporally cumulative amount function is continuous, so
∂ ∂n+ ∂ ∂n+ (x, s) ) (x, s) ∂x ∂s ∂s ∂x
(12)
Substitution of (9) and (10) into this equation gives Figure 1. Stepwise model of eddy (multipath) diffusion. Cells A-E (hatched squares) are connected in series by short (S) and long (L) paths. See text for explanation. In this example, the solute is evenly divided between short and long paths at each step.
different times (Figure 1B). For each arrival of solute at cell B, the solute is again divided between two paths as it travels to cell C (Figure 1C), and this process is repeated for each step between cells (Figure 1D). In this system, the distribution of times required for solute to reach a given cell is binomial, and such distribution can be closely approximated by a Gaussian function after it has traversed a sufficiently large number of steps (Figure 1E, central limit theorem). Moreover, the mean time and the temporal variance of this distribution increase linearly with the number of steps (Supporting Information, part 1). The process of eddy diffusion after a large number of steps can be described generally by making the fundamental assumption that
c(x, s) ) λ(x, s) j(x, s) - δ(x, s)
∂j (x, s) ∂s
(11)
where λ and δ are parameters that we denote as the lenticity and eddy diffusion coefficient, respectively. When the lenticity and eddy diffusion coefficient are constants, the lenticity can be interpreted as the average amount of time required for the solute to traverse a unit of space; i.e., lenticity is the reciprocal of velocity, and the eddy diffusion coefficient can be interpreted as half the rate at which the temporal variance increases with respect to space (Supporting Information, part 2). (The Fick diffusion coefficient D is half the rate at which the spatial variance increases with respect to time.) The reciprocal of velocity has been used as a key variable in other works, and it has been referred to as the delay rate,3 residency,14 and retention profile.7 In this work, the reciprocal of velocity is referred to as lenticity (lentic, adj., relating to still waters; lento- slow; veloc- quick). Note that the symbol for lenticity, λ, appears similar to an inverted vee, v. Equation 11 is the eddy diffusion analogue of Fick’s first law with bulk displacement (eq 1). Note the structural similarity (14) Blumberg, L. M. J. High Resolut. Chromatogr. 1993, 16, 31-38.
∂j ∂c (x, s) ) - (x, s) ∂x ∂s
(13)
which is the one-dimensional form of the equation of continuity.1 Applying this equation to (11) yields
∂j ∂ ∂j ∂λj (x, s) ) δ(x, s) (x, s) (x, s) ∂x ∂s ∂s ∂s
(14)
This partial differential equation is the eddy diffusion analogue of (2). The Gaussian distribution that arises from the stepwise eddy diffusion model is a solution to this equation (Supporting Information, part 2). Statistical Moments. The statistical moments considered in this work are determined by measuring flux at a hypothetical detection point z, which can exist anywhere in the system. Note that all statistical moments are implicit functions of the detection point. Zeroth Moment. The zeroeth moment M0 is given by15
M0 ≡
∫
∞
-∞
j(z,s) ds
(15)
which can be interpreted as peak area or the total amount of a solute that eventually traverses the detection point in the positive spatial direction:
∫
∞
-∞
∂n+ (z, s) ds ) ∂s
∫
∞
-∞
dn+(z, s) s)∞ ds ) n+(z, s)|s)-∞ ds
(16)
Let us assume that the zeroth moment is the total amount of solute. First Normalized Moment. The first normalized moment M1 is given by15
M1 ≡
1 M0
∫
∞
-∞
sj(z, s) ds
(17)
(15) Grushka, E.; Myers, M. N.; Schettler, P. D.; Giddings, J. C. Anal. Chem. 1969, 7, 889-892.
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which can be interpreted as the average amount of time required for the solute to reach the detection point. Let us also denote the first normalized moment as the temporal centroid t:
t ≡ M1
(18)
where τ2 is the temporal variance of a peak if it were located at position x after time s. Since the eddy diffusion coefficient is half of the spatial rate at which the temporal variance increases, we have
δ(x, s) ) The temporal centroid measured at the end of the column is nearly equal to the retention time in almost all practical cases. (Retention time is usually defined as the time corresponding to the peak’s maximum signal at the end of the column.) Let us assume that the temporal centroid is bounded. Normalized Central Moments. The nth normalized central moment M h n is given by15
M hn≡
∫
1 M0
∞
-∞
(s - t)nj(z, s) ds
(19)
Let us assume that all normalized central moments are bounded (Supporting Information, part 3). Note that the zeroth- and firstorder normalized central moments are constants:
M h0)1
(20)
M h1)0
(21)
In accordance with convention, we may denote the second normalized central moment as the temporal peak variance τ2:
τ2 ≡ M h2
(22)
Relationship between the Eddy Diffusion Coefficient and the Local HETP. A definition of local HETP that is free of internal inconsistencies has been given by Blumberg:4
dσ2 σ2f0 dz
H(x,s) ≡ lim
(23)
where σ2 is the spatial variance of a peak if it were located at position x after time s. The local HETP can be interpreted as the sum of all band-broadening contributions other than that arising from the spatial variation of retention.7 Spatial variances can be converted approximately into temporal variances by applying (6) for n ) 2: 2
2
τ ≈ λ (z, t)σ
2
(24)
This equation is inexact because of the potential for the migration and dispersion rates to vary in space or time. If the peak is infinitesimally wide, however, no such variation can exist. The relationship is thus exact for infinitesimally wide peaks:
lim τ2 ) λ2(z, t) lim σ2
τ2f0
σ2f0
(25)
H(x, s) λ2(x, s) ) lim
τ2f0
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dτ2 dz
Analytical Chemistry, Vol. 72, No. 7, April 1, 2000
(26)
(27)
Let us assume that both the local HETP and lenticity are known functions, so that the eddy diffusion coefficient is also a known function. Let us also assume that the eddy diffusion coefficient is positive, bounded, continuously infinitely differentiable in time (i.e., everywhere differentiable at any order with respect to time), and independent of amount densities (i.e., the chromatographic medium is linear). Net Rate of Migration. Blumberg has shown that a spatial gradient in the Fick diffusion coefficient contributes to the net migration of solute.3 A similar effect arises when the eddy diffusion coefficient varies in time. Consider a peak as it moves past a given detection point. If the level of eddy diffusion remains constant in time, the broadening incurred by the peak is symmetric. Thus, the average rate of solute migration for the peak is that of the bulk displacement. However, if the level of eddy diffusion changes as the peak migrates past the detection point, one side of the peak will broaden more than the other. The average rate of solute migration for the peak is thus affected. The contribution of a dynamically changing eddy diffusion coefficient to the net migration rate of solute can be combined with the lenticity of displacement to give the net lenticity Λ:
Λ(x, s) ≡ λ(x, s) +
∂δ 1 ∂Hλ2 (x, s) ) λ(x, s) + (x, s) ∂s 2 ∂s
(28)
Since the local HETP and lenticity are known functions, the net lenticity is also a known function. Let us assume that the net lenticity is positive, bounded, continuously infinitely differentiable in time, and independent of amount densities. Incorporating (28) into (14) yields (Supporting Information, part 4)
∂j ∂2δj ∂Λj (x, s) ) 2 (x, s) (x, s) ∂x ∂s ∂s
(29)
Spatial Rates of Statistical Moment Change. The progression of a statistical moment is completely specified by the initial value of the moment and the rate at which the moment changes as the peak travels through the column. Finding convenient expressions for the spatial rates of statistical moment change thus provides a foundation for the prediction of statistical moments. First Normalized Moment. The rate at which the temporal centroid increases with detection point is equivalent to the fluxweighted average of net lenticities (Supporting Information, part 5):
dt 1 ) dz M0
Equations 23 and 25 imply that
H(x, s) λ2(x, s) 2
∫
∞
-∞
Λ(z, s) j(z, s) ds
(30)
If the function Λ can be accurately expanded about s ) t as a
Taylor series, (30) can be restated in terms of normalized central moments (Supporting Information, part 6):
dt dz
M h2∂ Λ M hm∂ Λ (z, t) ) Λ(z, t) + 0 + (z, t) + 2 ∂s2 m)0 ∂sm M h 3 ∂3Λ (z, t) + ... (31) 6 ∂s3 ∞
≈
∑ m!
2
m
This equation cannot be solved alone because the normalized central moments are unknown functions of the detection point. Normalized Central Moments. The rate at which a normalized central moment changes with respect to detection point is (Supporting Information, part 7)
dM hn dt ) -nM h n-1 + dz dz n(n - 1) ∞ (s - t)n-2 H(z, s) λ2(z, s) j(z, s) ds + -∞ 2M0
∫
∫
n M0
∞
-∞
(s - t)n-1 Λ(z, s) j(z, s) ds (32)
Lnk ) Ank + Bnk + Cnk
dt -n k - n ) -1 dz (37) 0 k - n * -1 1 n(n - 1) ∂(k-n+2)Hλ2 (z, t) k - n g -2 Bnk ) 2 (k - n + 2)! ∂s(k-n+2) (38) 0 k - n < -2 n ∂(k-n+1)Λ (z, t) k - n g -1 (39) Cnk ) (k - n + 1)! ∂s(k-n+1) 0 k - n < -1
{
Ank )
{
Equations 37-39, respectively, represent the first, second, and third terms on the rhs of (33). Note that the second row (n ) 1) of matrix L contains nonzero elements but the dot product of this row and the vector M h is always zero (Supporting Information, part 9). We may thus replace all elements in the second row with zeros to achieve the same result. Similarly, the second column (k ) 1) contains nonzero elements but these elements are multiplied by the first normalized central moment, which is zero (eq 21), when the dot product is taken. We may thus also replace the elements in the second column with zeros:
Lnk ) Since a factor of n exists in each term on the rhs, we find that the zeroth normalized central moment is unchanged as the peak progresses through the column, which is consistent with (20). For the first normalized central moment, a factor of n - 1 eliminates the second term and the first and third terms cancel via (30). Thus, the first normalized central moment also remains unchanged as the peak progresses through the column, which is consistent with (21). If the functions Hλ2 and Λ can be accurately expanded about s ) t into a Taylor series, (32) can be restated in terms of normalized central moments (Supporting Information, part 8):
dM hn
h m+n-2 ∂mHλ2 dt n(n - 1) ∞ M ≈ -nM h n-1 + (z, t) + dz dz 2 m! m)0 ∂sm
∑
M h m+n-1 ∂mΛ (z, t) (33) m! m)0 ∂sm ∞
n
∑
The rate of change in the nth normalized central moment can thus be written for any given n. The collection of all such equations (0 e n < ∞) can be written in matrix form as
( )(
dM h 0/dz L00 L10 dM h 1/dz dM h 2/dz ) L20 ·· ·· · ·
L01 L11 L21 ·· ·
L02 L12 L22 ·· ·
··· ··· ··· ·· ·
)( ) M h0 M h1 M h2 ·· ·
(34)
or simply
M h ′ ) LM h
where
(35)
{
(36)
{
Ank + Bnk + Cnk n * 1 and k * 1 0 n ) 1 or k ) 1
(40)
For the remainder of this work, we will use this equation instead of (36) to calculate elements Lnk. System of Ordinary Differential Equations. Despite the use of matrix notation, (34) is not a mathematically linear system of differential equations because the matrix has two parameters: the temporal centroid t and the detection point z. The progression of the temporal centroid with detection point is given by (31), so the combination of (31) and (34) constitutes a complete system of ordinary differential equations that describe the progression of all statistical moments. This system cannot be solved directly because of its infinite size, but there are simplifications that allow the system to be approximately solved. Simplifications Based on Temporal Moderation. The basic strategies of simplifying (31) and (34) rely on the assumption of temporal moderation; i.e., any temporal variation in either Hλ2 or Λ is assumed to be sufficiently slow that the sum of low-order terms in the Taylor series accurately represents the original function in the domain about the peak. This assumption does not seriously restrict the applicability of the theory because the vast majority of real chromatographic systems are completely temporally moderate from the instant after injection. An important exception is step gradient elution liquid chromatography, where a sudden change in a mobile phase makes the system both temporally and spatially immoderate.2 Another exception is alternate-pumping recycle chromatography,9,10 where repeated pressure jumps affect solute retention.7 Finite-System Approximation. The system of equations can be reduced to a finite size by truncating references to normalized central moments of orders greater than a given value. This task is easily accomplished by limiting the dimensions of the matrix and vectors in (34) to a finite value r, so that the matrix has dimensions of r × r and the vectors have a dimension of r. Accordingly, (31) must also be truncated so that only the first r Analytical Chemistry, Vol. 72, No. 7, April 1, 2000
1559
terms (up to order r - 1) in the Taylor series are used. For example, at a dimension size of r ) 5, (31) and (34) become
M h 2 ∂ 2Λ M h 3 ∂3Λ M h 4 ∂4Λ dt ≈ Λ(z, t) + (z, t) + (z, t) + (z, t) 2 3 dz 2 ∂s 6 ∂s 24 ∂s4 (41)
( )
dM h 0/dz dM h 1/dz dM h 2/dz ≈ dM h 3/dz dM h 4/dz
(
0 0
0 0
()
Hλ2
0 0
0 0 2
1 ∂ Hλ ∂Λ +2 Hλ2 0 2 ∂s ∂s2 2 ∂Hλ dt 0 0 -3 + 3 + 3Λ dz ∂s 0
6Hλ2
0 0 3
2
2
1 ∂ Hλ ∂Λ + 2 6 ∂s3 ∂s ∂Λ 3 ∂2Hλ2 + 3 2 ∂s2 ∂s ∂Hλ2 dt + 4Λ -4 + 6 dz ∂s
4
2
3
1 ∂ Hλ 1∂Λ + 24 ∂s4 3 ∂s3 1 ∂3Hλ2 3 ∂2Λ + 2 ∂s3 2 ∂s2 2 2 ∂ Hλ ∂Λ 3 2 +4 ∂s ∂s
() M h0 M h1 M h2 M h3 M h4
×
dt ≈ Λ(z, t) dz
(43)
This equation indicates that the progression of the temporal centroid, under temporally moderate conditions, is relatively insensitive to the second-order and higher order normalized central moments. The solution to this ordinary differential equation gives an estimate of the temporal centroid as a function of detection point. When this function is used to relate the two parameters of matrix L, (34) becomes a mathematically linear system of differential equations. For example, (42) can be approximated as a mathematically linear system by using (43). Furthermore, substitution of (43) into Analytical Chemistry, Vol. 72, No. 7, April 1, 2000
0 0 0 0 ∂Λ 1 ∂2Hλ2 +2 0 2 ∂s2 ∂s
0
0 3∂Hλ ∂s
0
0 6Hλ2
(42)
In (42), the functions Hλ2, Λ, and their partial derivatives are evaluated at (z, t). A finite-system approximation eliminates the fewest possible terms to make the system theoretically solvable for a given dimension size r, so the solutions to these systems are presumably the most accurate for that dimension size. This accuracy, however, comes at the cost of complexity. Unless further simplifications are made, it is usually very difficult to solve these systems analytically. Nonetheless, numerical solutions to these systems can provide an important utility by serving as references to test the accuracy of further simplifications. Low-Order Approximation. Equations 31 and 34 may be further simplified by excluding terms that have derivatives above a certain order h. Let us denote the resulting equations as hth-order approximations. Mathematically Linear System Approximation. The first-order approximation of (31) is
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( )
dM h 0/dz dM h 1/dz dM h 2/dz ≈ dM h 3/dz dM h 4/dz 0 0
2
0
elements L32 (fourth row, third column) and L43 (fifth row, fourth column) of (42) gives a cancellation of terms:
2
)
0 0 0 0 1 ∂3Hλ2 ∂2Λ 1 ∂4Hλ2 1 ∂3Λ + 2 + 6 ∂s3 3 ∂s3 ∂s 24 ∂s4 × 2 2 3 2 3 ∂ Hλ ∂Λ 1 ∂ Hλ 3 ∂ 2Λ + 3 + 2 ∂s2 ∂s 2 ∂s3 2 ∂s2 2 2 2 ∂ Hλ ∂Λ ∂Hλ 3 2 +4 6 ∂s ∂s ∂s M h0 M h1 M h 2 (44) M h3 M h4
()
In general, these cancellations occur in elements Lnk, where k n ) -1. Triangular-Matrix Approximation. All elements above the main diagonal (k - n > 0) in matrix L contain only second-order or higher order derivatives of Hλ2 and Λ. When the chromatographic system is temporally moderate, these elements can be approximated as zero, resulting in a triangular matrix. For example, (44) becomes
( )
dM h 0/dz dM h 1/dz dM h 2/dz ≈ dM h 3/dz dM h 4/dz
(
0 0
0 0 0 0 0 0 2 2 ∂ Hλ 1 ∂Λ +2 0 Hλ2 0 2 ∂s ∂s2 2 3 ∂2Hλ2 ∂Λ +3 0 0 3∂Hλ 2 ∂s2 ∂s ∂s 0
0 6Hλ2
∂Hλ2 ∂s
6
0 0 0 0 ∂2Hλ2 ∂Λ +4 2 ∂s ∂s
3
() M h0 M h1 M h2 M h3 M h4
)
×
(45)
A mathematically linear system of differential equations with a triangular matrix can be solved much more easily because the differential equations of the system are solved sequentially instead of simultaneously. Note that a first-order approximation of the matrix L always results in a triangular matrix. For example, the first-order approximation of (42) is
( )(
0 0
0 0 0 0
0 0
0 0
dM h 0/dz ∂Λ dM h 1/dz 0 0 Hλ2 0 2 ∂s dM h 2/dz ≈ 2 ∂Λ 0 0 3∂Hλ 3 0 dM h 3/dz ∂s ∂s dM h 4/dz 2 ∂Λ 0 0 6Hλ2 6∂Hλ 4 ∂s ∂s
)( ) M h0 M h1 M h2 M h3 M h4
(46)
Simplification of the Net Lenticity. In many instances, the net lenticity is nearly equal to the lenticity:
Λ(x, s) ≈ λ(x, s)
(47)
According to (28), this approximation holds if
|
|
1 ∂Hλ2 (x, s) , λ(x, s) 2 ∂s
(48)
This condition is frequently true, but not necessarily true, when the chromatographic system is temporally moderate. Temporally Invariant Systems. Systems used in techniques such as isocratic liquid chromatography and isothermal gas chromatography have separation environments that have reached a steady state. The local HETP and lenticity functions of such systems may vary in space, but they do not vary in time. Let us denote these systems as being temporally invariant. The theoretical treatment of temporally invariant systems is greatly simplified because the temporal derivatives of Hλ2 and Λ are zero. As a result, (43) and (47) are exact for these systems:
dt ) λ(z) dz
(49)
Equation 33 is simplified to (Supporting Information, part 10)
dM hn)
n(n - 1) M h n-2H(z) λ2(z) dz 2
(50)
Statistical moments can thus be calculated by integration. DISCUSSION Limiting Assumptions of Temporal Statistical Moment Theory. The limitations of temporal statistical moment theory are implicitly given by the assumptions used in its derivation. Since many of the assumptions used in the work are nearly always accurate in practical chromatography, only the assumptions that may have important practical implications are discussed below. Equation 11. The fundamental assumption behind all migration and dispersion processes considered in this work is given by (11). Although there is no process that strictly obeys (11), it is likely that chromatographic migration and dispersion obey this relationship very closely. The strongest evidence in support of this assertion is that the most common function used to describe the temporal profile of a peak is the Gaussian function and this function is a solution to the differential equation given in (14) (which is naturally derived from eq 11).
Please note that (1), which is the fundamental assumption behind conventional Fick-diffusion-based theory, is also an approximation. There is only a single dispersion process that actually obeys (1), and that process is longitudinal Fick diffusion. Absolutely no other sources of dispersion obey (1), but they appear to obey it approximately in a chromatographic medium, which is why the conventional model is so commonly used. We propose that (11) is a similar type of relationship: an approximate, yet accurate, descriptor of migration and dispersion in chromatography. High Efficiency. Equation 11 is applicable only after a sufficiently large number of steps such that the temporal distribution of solute is nearly Gaussian. Each step in the stepwise eddy diffusion model represents a random event, so a large number of steps naturally corresponds to a high efficiency. Since high efficiencies (>100) are achieved in relatively short distances (
