Presenters: Raffi Manjikian, Nicole Charles, Antonio Macaluso

Department of Chemistry and Biochemistry

Seton Hall University

South Orange, NJ 07079

AbstractWe have developed novel finite difference software for modeling chromatographic peak shape when both partition and adsorption simultaneously control the distribution on the column. This software uses four dimensionless input parameters (the mobile phase fraction, the surface area constant, the adsorption constant, and the amount of theoretical plates) to compute the numerical representation of the peak. Analysis performed upon each peak yields its retention factor and its USP tailing factor (defined as the ratio of two extensions from the maximum point of a peak measured at 5% of the peak height). Variation of these input parameters allows the output to be varied between partition and adsorption and any combination between these extremes. This presentation focuses on the finite difference simulation both in the partition extreme and under adsorptive control when high values of the adsorptive mass balance parameter, ΓoA/CoVm, produce peak shapes indistinguishable from partition peaks. The results of this work help to confirm the validity of the original simulation and assist in the identification of the experimental conditions that require both partition and adsorption considerations to describe peak shape more accurately while also improving our understanding of various column materials.

IntroductionWe have recently described novel finite difference software to model adsorption effects in partition chromatography. Written as an Excel® VBA, this program predicts behavior when both partition between mobile and stationary phases and adsorption on the stationary surface control the chromatographic distribution.

Using 4 dimensionless input parameters representing

(1) the mobile phase partitioning fraction (X);(2) the adsorption equilibrium constant (KadCo); (3) the area-volume ratio constant (ΓoA)/(CoVm); and (4) the number of theoretical plates (No);

the VBA generates a numerical representation of each peak and computes its chromatographic characteristics such as tR, σ2, k’, and the USP peak tailing factor.

Simulation Input / Output

Partition – Adsorption Mechanism

Separation & Transport in the Partition-Adsorption Model

Computing the Mobile Phase Fraction x(n,r)x(n,r) = ½ { ξ + * ξ2 + 4X/(KadCo)∙ft(n,r)]½ } where

ξ = X – 1/(KadCo)ft(n, r) – (ΓoA/CoVm)∙X/ft(n,r)

X = [1 + Kp(Vs/Vm)]-1 = fixed partitioning fraction

(KadCo) = dimensionless adsorption constant

(ΓoA/CoVm) = area-volume ratio constant

This computation is performed at each plate during each iteration – in theory, (nmax)(rmax) times during the simulation of a single peak.

Computing the Mobile Phase Fraction x(n,r)This computation is performed at each plate during each

iteration – in theory, (nmax)(rmax) times during the simulation of a single peak.

In practice, the computation of x(n,r) is performed many fewer times than (nmax)(rmax) by using an algorithm that limits it to those cells where f(n,r) > 0.

Mass balance is performed at the conclusion of the simulation of each peak to assure that this algorithm has produced valid results.

Fundamental Plate Count ExpressionsN = tR

2/σ2 (1)

N’ = tR(tR-to)/σ2 (2)

No = defined number of theoretical plates (3)

In partition chromatography N’ agrees with No

while N does not.

Direct rearrangement of eq. 2 yields:

tR = N’(σ2/tR) + to (4)

Eq. 4 is of the form y = mx + b

Linear Regression Diagnostic: tR vs. (σ2/tR)tR = N’(σ2/tR) + to (4)

Eq. 4 is of the form y = mx + b with

y tR & x (σ2/tR) and m N’ & b to

so that a plot of tR vs (σ2/tR) will have a predicted slope N’ and intercept to.

This linear regression is employed diagnostically in all of the work that follows.

Other Regression DiagnosticsN’= tR(tR-to)/σ2 to give tR= N’(σ2/tR)+ to

For well behaved partition character, N’→No and to→No. Therefore, the slope is equal to N’ and the intercept is equal to to.

Since there are No transfers to get to the leading edge to the detector plate, No/b= to/b= 1.0 in well behaved region.

More Diagnostics for Partition ChromatographySince N/N’= (tR

2/σ2)/((tR2-tRto)/σ2)= 1/(1-(to/tR))

N/N’= (1-(to/tR))-1 = (1-(1/(k’+1)))-1

N/N’= (k’/(k’+1))-1 = (1+ k’-1)

Thus, (N/m)/(1+ k’-1) = 1 for well behaved partition chromatographs.

Typical Simulation OutputResults shown in the following figure were obtained by setting No = 1000; X = 0.5; ΓoA/CoVm = 1.0; and KadCo = 1.5, 4.0, 6.5 and 9.0, respectively.

The computed tailing factors of the resulting peaks were 1.150, 1.254, 1.304, and 1.335, respectively.

Linear regression (inset) on a plot of tR vs σ2/tR gave a slope m = 952 (corresponding to N’) and an intercept b = 1116 (corresponding to to) with R2 = 1.0.

Typical Partition - Adsorption Output

Achieving Partition BehaviorThe results shown above exhibit significant tailing, as expected when adsorption is operative.

We have discovered two ways to achieve partition-like behavior using this VBA:

(1) Set ΓoA/CoVm = 0 and allow the retention factors to be determined by the input values of X and No.

(2) Set X = 0.9999 and allow the retention factors to be determined by ΓoA/CoVm, KadCo, and No.

No=1000 X=0.9999 and ΓoA/CoVm=0

No=1000 X=0.9999 and ΓoA/CoVm=0.1

No=1000 X=0.9999 and ΓoA/CoVm=0.316

No=1000 X=0.9999 and ΓoA/CoVm=1

No=1000 X=0.9999 and ΓoA/CoVm=3.16

No=1000 X=0.9999 and ΓoA/CoVm=10

No=1000 X=0.9999 and ΓoA/CoVm=31.6

No=1000 X=0.9999 and ΓoA/CoVm=100

ConclusionsIn the course of this work, where partition was excluded by setting X = 0.9999 to prohibit extraction in the stationary phase, partition-like results were obtained by employing values of ΓoA/CoVm > 1.0.

In general, it was observed that larger values of ΓoA/CoVm produced adsorption peaks that were indistinguishable from partition peaks.

What does this finding mean?The results obtained by applying the partition

diagnostic to adsorption peaks clearly indicate

that when [ΓoA/CoVm] is large, adsorption is

indistinguishable from partition. These

observations suggest that the adsorption model

we have developed using these finite difference

methods may be generally more applicable to

understanding chromatographic processes than

traditional methodology based solely on partition.

Future WorkWe are developing a theoretical expression for retention factor (k’) in partition–adsorption chromatography.

k’ = ko’ + (KadCo) ∙ [(ΓoA)/(VmCo)] ∙ (1-θ)

where ko’ is the retention factor for partition in the absence of adsorption. Given k’ as simulation output for each peak and all else but θ as simulation input, this equation allows θ, the average fraction of occupied adsorption sites to be computed for each peak.

When X=0.9999, ko’=0 and as (ΓoA)/(VmCo) approaches infinity, θapproximately equals zero.

Therefore, k’= (KadCo) ∙ [(ΓoA)/(VmCo)] which leads to

k’= Kad(ΓoA)/Vm

References

1. Pittcon11, #860-6

2. EAS11, #294

3. Pittcon12, #810-5P