Chromatography and Separation Chemistry Downloaded from pubs.acs.org by TEXAS STATE UNIV SAN MARCOS on 11/21/18. For personal use only.
11 Calculation of Retention for Complex Gradient Elution High-Performance Liquid Chromatographic Experiments: A Universal Approach Sterling A. Tomellini , Shih-Hsien Hsu , R . A. Hartwick , and Hugh B . Woodruff 1,3
1
1
2
Department of Chemistry, Rutgers, The State University of New Jersey, New Brunswick, NJ 08903 Merck Sharp & Dohme Research Laboratories, Rahway, NJ 07065
1
2
A numerical method for the calculation of retention under complex gradient HPLC elution conditions is presented. The approach is applicable to virtually any gradient or solvent-solute relationship. Examples include binary, ternary, quaternary, linear and multiple-linear solvent gradients as well as for cases involving stationary phase programming (coupled column gradient elution experiments). The chromatographic experiment i s somewhat unique i n that more time i s normally spent designing the experiment than interpreting the resulting data. It i s for this reason that much of the current work i n l i q u i d chromatography i s directed at helping the researcher determine which experimental conditions w i l l produce an adequate separation. A number of papers have been published which demonstrate various approaches for the "optimization" of l i q u i d chromatographic parameters. The researcher should, however, be conscious of two questions before choosing to use any "optimization" strategy. F i r s t , one should ask when an optimization strategy should be used. The second question the researcher should address i s what i s meant by an "optimum" i n regard to any LC separation and i n particular to the separation which i s presently under consideration. It i s obvious that the only reason to use an optimization strategy i s that the response surfaces are unknown and would require too much effort to evaluate d i r e c t l y . Liquid chromatography has an added complication i n that the chromatographic response may be time dependent (due to changes i n the column, e t c . ) . The problem with a l l such strategies i s , however, defining what i s meant by the "optimum" separation. The problems associated with such indicators are well known ( 1 ) · DeGalan (2) has demonstrated that the chromatographic response surface actually consists of two independent surfaces (time and resolution). The "optimum" separation w i l l , therefore, most 3
Current address: Department of Chemistry, University of New Hampshire, Durham, N H 03824 0097-6156/ 86/ 0297-0188506.50/ 0 © 1986 American Chemical Society
11.
T O M E L L I N I ET A L .
189
Calculation of Retention
probably be a compromise between time and resolution. Furthermore, i t should be emphasized that the information desired from a chromatographic experiment must also be considered when determining the "optimum" separation. Thus, the "optimum" separation can vary considerably among various researchers even for a single sample. An alternative approach for determining which conditions produce adequate separations i s to take a limited amount of chromatographic data and use available knowledge to predict expected responses. Such an approach can be termed a " c a l c u l a t i o n a l " approach and can generally be broken into two parts. The available data must f i r s t be f i t t e d to an expected retention vs. solvent concentration curve. Once the c o e f f i c i e n t s for this curve are known, calculations can be made for both i s o c r a t i c and gradient conditions. The gradient equations can be solved using either numerical integration techniques or exact solutions. Normally exact calculations can be used to predict elution times and widths for i s o c r a t i c conditions. Exact solutions are also available for some gradient conditions (3-21); the solutions, however, are dependent on gradient shape, number of solvent components and the relationship between solvent components and retention data. Often exact solutions are not available for complex gradient conditions which leads to the use of more f l e x i b l e techniques employing numerical integration (22)* Furthermore, use of numerical methods to treat gradient conditions allows for independence between the f i t t i n g of retention to solvent concentration data and the gradient c a l c u l a t i o n s . Also, numerical integration i s generally more computer amenable as programming complexity i s s i g n i f i c a n t l y reduced i f only one c a l c u l a t i o n a l algorithm i s required. The development of computer programs capable of using the c a l c u l a t i o n approach to a s s i s t the researcher i n determining which chromatographic conditions are useful w i l l be presented. The theory, assumptions, advantages and disadvantages of the calculations on which the programs are based w i l l also be presented. Examples w i l l be given for separations of test compounds using binary, ternary and quanternary, linear and multiple-linear gradients. Calculation of retention times for combined stationary and mobile phase programming w i l l also be presented. Theory f
Single Column. It i s well known that the capacity factor, k , for a compound often varies i n a predictable manner with respect to the concentration of organic modifier for aqueous solutions under reverse phase conditions. I t has been shown for a single organic modifier the relationship i s : Ln(k») - AC
X
+ DCx
2
+ Β
(1)
where A, B, and D are constants and C^ i s the concentration of organic modifier which i s generally given i n volume percent. Additional modifiers are expected to affect the capacity factor of a compound i n a similar manner. It i s expected, therefore, that
C H R O M A T O G R A P H Y A N D SEPARATION CHEMISTRY
190
for an aqueous solution containing two organic modifiers the relationship w i l l be: f
2
2
Ln(k ) - ACi + B C i + DC + E C + FCiC + G (2) where A, B, D, E, F and G are constants. Notice that a terra has been included which varies as a function of both and C Using similar mathematical f i t t i n g any number of organic modifiers can be used. Once the expected relationship between k' and organic modifer i s known, the values of the constant coefficients must be determined. Using a limited amount of experimentally determined i s o c r a t i c data the c o e f f i c i e n t s can be calculated by using a linear least squares f i t algorithm. The minimum number of data points required varies with the number of terms i n the equation being f i t . The data for mobile phases containing one organic modifier are f i t to Equation 1 and require at least 3 experimentally determined data points. Likewise, 6 data points are required for aqueous mobile phases having two organic modifiers (e.g. aqueous solution containing methanol and a c e t o n i t r i l e as modifiers). Using similar f i t t i n g equations any number of organic modifiers may be employed. Liquid chromatography i s often thought of as a volume problem but i s i n fact fundamentally a length problem since the column length i s a l i m i t i n g parameter. A l l separations must be completed by the time the compounds have traveled the length of the column. Thus, one of the fundamental equations of LC i s : 2
2
2
2
Length =* Velocity χ Time
(3)
In the i s o c r a t i c case the velocity i s constant and Equation 3 becomes : L
col
55
fc
x
R
u
4
band
( )
Where L ^ i s the column length, t i s the retention time for a given solute and U ^ j i s the linear velocity of that solute band. If the v e l o c i t y i s not constant, as i s generally the case i n gradient elution, the i n t e g r a l solution for Equation 4 must be used which i s given as: t col / band.inst' () 0 c o
R
an(
R
L
s
u
dt
5
where U ^ j ^ i s the instantaneous velocity of the solute band and the integration l i m i t s are from time 0 to time = t . The v e l o c i t y of a band can be found for the i s o c r a t i c case by f i r s t rearranging the often used expression for k : a t l (
e
n s t
s
R
f
to get t
R
= t ( l + k») M
(7)
11.
Calculation of Retention
TOMELLINI E T A L .
191
where i s the elution time of an unretained solute. U i for elution under i s o c r a t i c conditions can be calculated by dividing the column length by t (as given i n Equation 7) resulting i n : D a n c
R
"band - W / ( t ( l M
+
k
'>)
W
For the gradient case where k' i s generally changing throughout, the instantaneous band v e l o c i t y i s of interest. I t follows from Equation 8 that Ub i. can be found i n terms of the instantaneous k , k i , such that: an(
f
e
i
n
s
t
f
n s t
u
band.inst
Thus the fundamental
L
col - I
s
L
c o l
/(t (l + ^ M
1
η
β
ϋ
))
(9)
equation for gradient elution becomes:
W / U M U
f
d
+ k inst))' t
(10)
0 Obviously, Equation 10 reduces to Equation 4 for i s o c r a t i c cases where k £ Is constant over time. Using Equation 10 i t i s , i n theory, possible to solve exactly for the retention time of any band knowing the relationship between the solute's instantaneous k and time. I f , however, the instantaneous k vs time relationship i s mathematically complex, as can be the case i n gradient elution, then an exact solution for t w i l l be d i f f i c u l t i f not impossible to determine. There are a number of factors a f f e c t i n g the complexity of the k ^ vs. time relationship, among these are: f
n s t
f
f
R
,
n s t
1.
2. 3.
4.
The number of modifiers i n the mobile phase and the relationship between k* and concentration for these modifiers. The gradient shape ( i . e . , l i n e a r , multiple l i n e a r , complex). The instrumental delay time which causes most solutes to t r a v e l i s o c r a t i c a l l y before being overtaken by the gradient. A time correction which i s necessary since the gradient front after overtaking the solute band generally moves at a faster v e l o c i t y than the solute band.
While exact mathematical solutions for retention times are preferable (when available) for manual calculations, numerical integration has two major advantages when the computing power of a large computer i s a v a i l a b l e . F i r s t , solutions are not always a v a i l a b l e . Second, the v e r s a t i l i t y of numerical integration reduces the amount of programming e f f o r t necessary and also the size of the required programs i f multiple experimental conditions are to be allowed. As when using exact solutions, the f i r s t step when using solutions employing numerical integration i s to determine the relationship between k* and solvent composition for each solute using limited i s o c r a t i c data. Next the retention times of each solute must be calculated i n turn. F i r s t , the time spent by the
C H R O M A T O G R A P H Y A N D SEPARATION CHEMISTRY
192
solute traveling under the i n i t i a l i s o c r a t i c conditions caused by the instrumental delay time, t , must be calculated. Knowing the actual time delay before the gradient front reaches the solute band and the k' of the solute band during this time, the distance traveled by the solute under i s o c r a t i c conditions can be determined. Notice that this problem i s e s s e n t i a l l y a related rate problem of the type normally presented i n beginning calculus courses. I t i s easiest to v i s u a l i z e the problem by thinking of the solute band as having an e f f e c t i v e head start i n distance equal to the measured gradient time delay multiplied by the linear velocity of the gradient front down the column which i s equal to t i i / t ^ . If the gradient front travels for a time, t at a velocity equal to l i i / t j 4 and the solute band travels f o r the same time at a v e l o c i t y given by Equation 8, L i / ( t j 4 ( l + k ' ) ) , then knowing the e f f e c t i v e head start of the solute band allows for the calculation of t . Simply stated the distance traveled by the solute band plus the head start must be equal to the distance traveled by the gradient front at the time the two are coincidental, or: D
C O
c
o
r
r
CO
c o
c o r r
1
(Lcol/tMC + k')) x t
c o r r
) + ((L
Solving Equation (11) f o r t tcorr
β
c
o
r
c o l
/t ) χ t M
D
« L
c o
i/t
M
χ tc
o r r
(11)
gives:
r
,
f
((1 + k ) / k ) x t„
(12)
The distance traveled by the solute band i s o c r a t i c a l l y , l>± can be found by substituting Equation 12 into Equation 8 and rearranging to get: 30
L
i s o - ((Lcol * t ) / ( t D
1
M
χ k ))
(13)
Knowing the time spent by the solute band traveling i s o c r a t i c a l l y and i t s position i n the column when the gradient takes effect allows the calculation of the distance over which the gradient w i l l affect the solute since: L
This, L
g r a (
iso
+
L
L
14
grad • c o l
j can be substituted into Equation 10 for L
( > c o
i to become:
t i R
L
grad
β
k ,
/
" / ( i n s t + 1>·
0 where tR
s
t
c
o
r
r
+ t» R
(16)
and u i s the linear velocity which i s equal to Ι ^ Ο Ι ^ Μ · Notice that while the problems previously noted as being associated with the instrumental delay have been overcome, i t i s s t i l l often not possible to solve Equation 15 exactly. I t i s easiest, therefore, to evaluate the i n t e g r a l i n a stepwise manner by simply incrementing the time by some small step, calculating the corresponding k' f o r the resulting time and then calculating the
11.
Calculation of Retention
TOMELLINI E T A L .
193
length traveled during the time i n t e r v a l . The sum of the time steps w i l l equal t , when the sum of the lengths traveled i s equal to L g j . A continuous correction must be made, however, since the actual time spent traveling and the time corresponding to the gradient concentration which the solute band encounters are generally not the same. The reason for this i s that the gradient concentration seen by the band i s not only dependent on the time after the gradient has started but also on the position of the band i n the column. The simplest way to eliminate length from the problem i s to calculate a "gradient" time as well as an actual time. While the actual time i n t e r v a l i s specified, the "gradient" time i n t e r v a l , t ^ ^ i i can be calculated using the r e l a t i o n s h i p : R
r a
