811
Anal. Chem. 1985, 57,811-816
Computer-Based Numerical Integration for the Calculation of Retention Times in Gradient High-Performance Liquid Chromatography Sterling A. Tomellini and Richard A. Hartwick*
Department of Chemistry, Rutgers University, P.O. Box 939, Piscataway, New Jersey 08854 Hugh B. Woodruff
Merck Sharp & Dohme Research Laboratories, P.0. Box 2000, Rahway, New Jersey 07065
An approach to the calculatlon of solute retentlon under gradient elution conditions using numerlcal approximations on a DEC VAX 11/780 computer Is presented. Thls approach ellminates the need for exact solutlons of the Inherently complex gradient Integrals. The programs developed can be used for any solvent composltlon vs. solute capacity factor or solvent composttlon vs. tlme relatlonshlps and are, therefore, universal in nature. Gradient mlcrobore HPLC in the reversed-phase mode was used to demonstrate the accuracy of the approach. The assumptlon of a quadratic dependency between In k’ and solvent composltlon allowed for the least-squares fittlng of solute lsocratlc retentlon wtth a minimal number of experiments. By use of thls approach and the corresponding FORTRAN program, predlction of solute retention under complex multiple-llnear gradient shapes was achleved wlth an accuracy of better than 5 % .
The chromatographic experiment is somewhat unique in that more time and effort are normally invested in determining the required experimental conditions than in interpreting the resulting data. Research has thus been directed at finding efficient ways to “optimize” an HPLC separation. Drouen et al. (1)have characterized the two major optimization schemes in HPLC as being either of the sequential simplex or of the lattice type design. While automated optimization programs of either design can be useful in finding adequate separation conditions, it must be realized that a separation can be “optimized” according to many different criteria. Thus, an optimization scheme should be flexible to the specific demands of a particular separation problem and not be user independent, contrary to the philosophy being adopted by numerous instrument companies in automated methods development. An alternative to the completely automated methods development approach is to permit the scientist to use the computer as an interactive tool, without the restriction of a single optimization algorithm. In order to permit this, versatile calculation procedures must be available. Jandera and Churacek (2-111, Schoenmakers et al. (12-14), Snyder (15)) and others (16-18) have demonstrated that exact mathematical solutions can be derived for certain gradient conditions and that these solutions can predict the retention time of a compound with good accuracy. Such mathematical solutions can then form the basis for subsequent optimization procedures. The problem with using exact mathematical solutions is that the form of the solution depends upon a number of factors, including the gradient shape and number of components in the mobile phase, the function describing the capacity factor ( k ? and mobile phase composition for each solute and the 0003-2700/85/0357-08 1 1801.50/0
instrumental delay volume, As the complexity of these factors increases, exact solutions become progressively more difficult to implement. In addition, programming effort becomes excessive when writing algorithms to handle any of a variety of potential conditions within a single computer program. Due to the inflexibility of the existing solutions and the mathematically intractable nature of other solutions for more complex cases, it was decided to develop an approach and corresponding computer programs which could more easily handle the various conditions often used by the scientist attempting to develop a separation. The strength of the approach presented arises by the separation of the solute-solvent composition relationship from the mathematics of the gradient integral. Complex equations of this type are often most easily solved by using numerical integration techniques. Such approximate solutions require numerous calculations and therefore must run on relatively fast computers. The reward for using approximate, solutions and larger computers is that a single program can handle complex solute-solvent relationships as well as complex isocratic and gradient elution conditions with reasonable calculation time. This paper will report on the results obtained with a program written to determine retention times for binary, ternary, quaternary, etc. gradient elution conditions with single or multiple linear gradient segments based on limited isocratic chromatographic data. Multiple linear gradient segments were chosen since these are widely used on commercial HPLC instruments.
THEORY Isocratic Data. The first step in deriving solutions for the prediction of retention times for solutes eluted under gradient conditions is to know the capacity factor, k’, for each solute as a function of solvent composition. For the reversed-phase mode of HPLC, using a binary solvent, a logarithmic function of the following form can usually be assumed (18) In k’ = AC1
+ BC12+ D
(1)
where A , B , and D are constants and C, is the concentration of organic modifier which is generally given in volume percent. For an aqueous solution containing two organic modifiers, the relationship of solvent composition to k’ can be expected to be in the form In k’ = A C ,
+ BC12 + DCZ + EC2’ + FClCz + G
(2)
where A , B , D,E , F, and G are constants. A cross term has been included which varies as a function of both C1 and Co. By use of similar mathematical fitting, any number of organic modifiers can be used. Also, if a different function is followed, as for example a reciprocal function as is often observed in ion exchange, similar equations of the same general type can be derived. 0 1985 American Chemical Society
812
ANALYTICAL CHEMISTRY, VOL. 57, NO. 4, APRIL 1985
Once the expected relationship between k ’ and organic modifier is known, the values of the constant coefficients must be determined by using a linear least-squares fit. The minimum number of data points required varies with the number of terms in the equation being fit. A binary mobile phase (single organic modifier) fit to eq 1 requires a t least three experimentally determined data points. Likewise, six and ten data points are required for ternary and quaternary mobile phases, respectively. Calculation of Retention Times. Liquid chromatography is often treated as a volume problem but is in fact fundamentally a length problem since the column length, Lco1,is a limiting parameter. All separations must be completed by the time the compounds have traveled the length of the column. Thus
Lcol= velocity of solute
X retention time
(3)
In the isocratic case the velocity may be assumed to be constant and eq 3 becomes Lcol
=
UbandtR
the initial isocratic conditions before being overtaken by the gradient front must be calculated. This delay has two sources. First, due to a physical instrumental delay time, t D , the solute band will move down the column a t a velocity inversely proportional to (1 + k ? (eq 7) while the gradient front travels to the column head. Secondly, the solute band continues to advance at the same rate until overtaken by the gradient traveling at a rate equal to the mobile phase velocity, u, Le., Lcol/tm.Knowing the instrumental time delay of the solute band allows for the calculation of a time correction, t,,,,. Simply stated the distance traveled by the solute band during the instrumental delay plus the time until overtaken by the gradient front in the column must be equal to the distance traveled by the gradient front a t the time the two are coincidental, or (Lcol/(tM(l
tR
=
tM(1
+ k?
(6)
where t M is the elution time of an unretained solute. for elution under isocratic conditions can be calculated by dividing the column length by tR (as given in eq 6) resulting in Uband
= LcoI/(tM(1
+ k?)
(7)
Under gradient conditions where k’ is changing throughout, the instantaneous band velocity is of interest. From eq 7, U ’ h d can be found in terms of the instantaneous k’, k’bt, such that Uband
= Lcol/(tM(1
+ k
