Separation Speed and Power in Isocratic Liquid Chromatography

Article pubs.acs.org/ac

Separation Speed and Power in Isocratic Liquid Chromatography: Loss in Performance of Poppe vs Knox-Saleem Optimization Adam J. Matula and Peter W. Carr* Department of Chemistry, University of Minnesota, 207 Pleasant Street SE, Minneapolis, Minnesota 55455, United States S Supporting Information *

ABSTRACT: The best separation possible at a given analysis time and maximum system pressure is achieved by simultaneously optimizing column length, eluent velocity, and particle size. However, this threeparameter optimization is rarely practicable because only a few commercially available particle sizes exist. Practical optimization for systems described by the van Deemter equation therefore proceeds by first selecting an available particle size and then optimizing eluent velocity and column length. This two parameter (“Poppe”) optimization must result in poorer performance with respect to both speed and efficiency because one fewer degree of freedom is used. A deeper analysis identifies a distinct point on each pair of “Poppe” curves beyond which the more efficient (and faster) separation is maintained by changing from smaller to larger particles. Here, we present simple equations identifying these “crossover points” in terms of analysis time and plate count thereby allowing a practitioner to rapidly identify the correct particle size for use in tackling a particular separation problem. Additionally, we can now quantitatively compare two-parameter and three-parameter optimization. Surprisingly, we find that for systems well-described by the van Deemter equation there is little separating power lost (only about 11% in the worst case) as a result of the limited availability of different particle sizes in using two-parameter optimization when compared to the ideal three-parameter optimization so long as one changes particle size at the prescribed crossover points. If these crossover times are not used, a great deal of separating power will be needlessly lost.

T

resulting two-parameter optimization necessarily yields plate counts lower than the maximum achievable at a given analysis time because one fewer degree of freedom is available for optimization. Guiochon has proven mathematically that only when particle size is co-optimized along with velocity and length are the velocity and plate height equal to that corresponding to those predicted by the optimum in the van Deemter plot.2 Clearly, any other particle size must give poorer performance in terms of time or plate count and Poppe curves must lie above the Knox-Saleem line. Therefore, it is correct to say that an increase in the number and variety of available particle sizes will allow for improved separations. However, the potential amount of improvement has not been quantified. Further, when performing two-parameter optimizations, the selection of the best particle size for the separation is computationally fatiguing. Poppe’s numerical solution to this issue involves calculating the plate count (N) as a function of analysis times (gauged by the column dead time, t0) and creating so-called “Poppe plots” for the various available particle sizes, noting that for any given analysis time there is a particle size that maximizes theoretical

here have been many excellent studies of the problem of optimization in chromatography dating back to the work of Giddings and many others.1−6 Historically, the guiding principles of isocratic high-performance liquid chromatography (HPLC) indicate that the best separation possible at a given analysis time and maximum system pressure drop is achieved by simultaneously optimizing column length, eluent velocity, and particle size.1,2 This three-parameter optimization, frequently referred to as the “Knox-Saleem limit”, is only practicable when the optimized quantities can be achieved experimentally,7,8 that is, if the correct particle sizes and column lengths are available and the pumping system can deliver the velocity desired. Modern LC systems are able to accommodate many, if not all, desirable velocities. Further, column lengths can be varied in about 1 cm increments from 5 cm upward either directly by using a different length of column or by combining smaller columns in tandem to create a larger column. However, as only a few particle sizes are commercially available, it is impractical to treat particle size as a continuously variable quantity in a separation optimization scheme. The question of the best particle size has been studied extensively from an experimental perspective.9−13 This limitation necessitates a different formulation of the separation optimization that treats particle size as the discrete quantity that it really is. Thus, in practice, it is generally held constant as column lengths and velocities are optimized as espoused by Poppe.14 The © 2015 American Chemical Society

Received: January 26, 2015 Accepted: June 12, 2015 Published: June 12, 2015 6578

DOI: 10.1021/acs.analchem.5b00329 Anal. Chem. 2015, 87, 6578−6583

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parameter optimization.3,4 Extending this idea, by plotting “Poppe curves” for different particle sizes ad inf initum (i.e., where the difference between two adjacent particle sizes approaches zero) and connecting the tangent points, one recovers the “Knox-Saleem line”. Because the number of available particle sizes is small, it is important to select the particle size (dp) such that it differs minimally from KnoxSaleem optimized particle size at the operating velocity and column length (see Figure 2A). Noting that there is a point of intersection between each set of two sequential particle sizes’ “Poppe curves” at which the best separation changes from occurring at a smaller particle size to a larger one (see Figure 2B), it is clear that by switching particle sizes at the appropriate intersection time/plate count the loss in speed and efficiency will be minimized. One of the major objectives of this paper is to find an analytical solution to the calculation of these intersection points, termed “crossover points”. Furthermore, the solution to the above problem leads to an equation which allows the simple computation of the loss in plate count resulting from the use of two-parameter instead of three-parameter optimization. In doing so, an equation for the intersection between two-parameter optimization (Poppe optimization) and three-parameter optimization (Knox-Saleem limit) in terms of the crossing point time is provided. Finally, a few reference tables and a more comprehensive web-based approach are presented which allow the rapid determination of the crossover points between different particle sizes at various common separation conditions.20

plates relative to analysis time. Poppe’s approach has been expanded upon, simplified, and further developed by Desmet as well as others who recast “Poppe plots” into a family of “kinetic plots”.15−18 These optimization calculations can be implemented via a system of linear equations that have recently been further simplified.10,15 These computations can be done using a web-based approach.19 Simultaneously plotting an isocratic “Poppe curve” at a particular particle size and the “Knox-Saleem line” for the same separation conditions shows that the “Knox-Saleem line” lies exactly tangent to the “Poppe curve” only when the particle size given by the Knox-Saleem optimization is equal to the particle size chosen for the Poppe optimization (see Figure 1). This

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Figure 1. Isocratic “Poppe” curve for 1.8 μm particles with its KnoxSaleem line tangent. Conditions: Pressure = 1000 bar, T = 40 °C, ϕ = 500, ACN = 40% (v/v), A = 1.0, B = 5, C = 0.05, and Dm = 10−5 cm2/ s. The diagonal lines indicate lines of constant column dead time. The shaded region represents potential efficiency lost from using twoparameter instead of three-parameter optimization, the distance between the “Poppe” curve and the Knox-Saleem limit for a given analysis time.

RESULTS AND DISCUSSION

Calculation of the Crossover Point Time and Plate Number. By definition, at the crossover point of Poppe curves for any two particle sizes, dp1 (the particle size being transitioned from) and dp2 (the particle size being transitioned toward), both curves must have the same plate count. Thus,

N1 = N2 = N ⧧

point of tangency occurs where the “Poppe optimized” eluent velocity (u*e ) coincides with the eluent velocity at the minimum of the van Deemter curve and should be understood as the point which must exist for each particle size where the twoparameter optimization yields the same results as the three-

(1)

and t0,1 = t0,2 = t0⧧

(2)

Figure 2. (A) Isocratic Poppe curves for three different particle sizes. Indicated points represent “crossover times” between adjacent particle sizes. Conditions: Pressure = 1000 bar, T = 40 °C, ϕ = 500, ACN = 40% (v/v), A = 1.0, B = 5, C = 0.05, and Dm = 10−5 cm2/s. With the above parameters, we see that 1.8 μm particles are optimal for dead times less than about 90 s and 3.5 μm particles are best up to about dead times of 700 s. (B) Panel A under the same conditions but enlarged to demonstrate that the crossover points, when obeyed, represent the maximum deviation from the KnoxSaleem limit. This maximum decreases with the ratio of the particle sizes. 6579

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Figure 3. Crossover times (A) and crossover plate counts (B) between 1.8 and 3.0 μm particles at various system pressures at 40 and 80 °C. All conditions not specified are as in Figure 1.

where Dm is the diffusion coefficient of the analyte in the eluent and ue* is the value of the eluent velocity that maximizes the plate count at a given t0, it is easily seen that at crossover conditions:

where the double dagger denotes the value of the property at crossover. By definition:1 L d ph

N=

(3)

⎛ d p1 ⎞2 ⎟⎟ v1 = v2⎜⎜ ⎝ d p2 ⎠

Since the crossover point is occurring at a point along a twoparameter optimized “Poppe curve”, the equations fundamental to the Poppe model hold7 Ψd p Pmax dp = t0 ϕηλt0

ue* =

Substitution into the reduced van Deemter equation and cancellation of like terms shows that

(4)

v1 = Pmaxλt0 d p = Ψλd p t0 ϕη

Le* =

Ψ= λ=

Pmax Φηλ εe εtot

(5)

(6)

(9)

(10)

Noting that reduced velocity (v) is defined such that v=

ue* dp Dm

(13)

Dm d p2

B C

(14)

(15)

Accordingly, we see that the crossover time increases with both particle sizes, as the C/B ratio gets bigger, as the solute’s diffusion coefficient gets smaller, and as Pmax increases. The dependence on viscosity is implicit in that both Ψ and Dm vary with viscosity (see eq 6). Combining Ψ and using the WilkeChang dependence of Dm on viscosity indicates that as the viscosity decreases so will the crossover time. Therefore, increasing the temperature will decrease the crossover time (see Figure 3). Some values of the crossover time as a function of the maximum pressure and column temperature in 40% (v/v) acetonitrile−water for a species that has a diffusion coefficient of 1 × 10−5 cm2/s at 40 °C in this eluent are given in Figure 3A and Table 1. Results at other compositions as well as those having a peptide-like diffusion coefficient (2 × 10−6 cm2/s) under the same conditions are given in the Supporting Information. Figure 3 shows that the crossover time for the 1.8/3.0 μm particle pairing increases linearly with the pressure maximum. This means that as the available system pressure increases the time at which one should change to the larger particle increases.

Resultantly, we see that under the crossover conditions both particle sizes have the same reduced plate height (h):

h1 = h2 = h⧧

B C

⎛ Ψd p1d p2 ⎞2 C t0⧧ = ⎜ ⎟ ⎝ Dm ⎠ B

where L is the column length, h is the reduced plate height, Pmax is the maximum system pressure, ϕ is the dimensionless column flow resistance parameter (about 500), η is the eluent viscosity, and εe and εtot are the interstitial and total porosities of the column. Application of the crossover point conditions N⧧ and t⧧0 to eq 3 shows that

* d p2 Le,2 = * d p1 Le,1

d p2

Applying this equation to eq 4 yields an analytical solution for t⧧0 , the crossover time:

(7)

(8)

d p1

Consequently, by eq 11: *= ue1

L1 L2 = d p1h1 d p2h2

(12)

(11) 6580

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increased, t⧧0 decreases substantially whereas N⧧ decreases only slightly, leading to a considerable increase in the ratio of N⧧ to t⧧0 . Thus, temperature has a qualitatively different and beneficial effect on the speed of LC in comparison to the effect of increased pressure. The Supporting Information give additional results on the performance at additional eluent compositions (0 to 100% ACN (v/v) in 20% increments), for solutes with smaller diffusion coefficients (5 × 10−6 and 2 × 10−6 cm2/s representative of larger molecules and peptides). In addition, results are given for different values of the column parameters B and C as they impact the crossing time (see eq 15) whereas the A term only effects the plate count at crossover (see eq 17). Loss in Plate Count of Two-Parameter Relative to Three-Parameter Optimization. We turn now to the second major part of this work, namely, estimation of the loss in separating power due to use of two-parameter in contrast to the ideal three-parameter optimization. It is important to explore how closely two-parameter optimization results approximate those obtained from three-parameter optimization in order to understand the impact of particle size limitations on separation efficiency and speed. Examination of Figure 2 shows that, provided one changes the particle size at the prescribed crossover time for the conditions of the relevant separation, the crossover point itself is the point of maximum loss in N/t0 for twovs three-parameter optimization and thus the largest loss in efficiency or speed resulting from the limited availability of particle sizes. The plate number from three-parameter optimization (N3P) is7

Table 1. Crossover Times for Various Particle Size, Temperature, and Pressure Combinations at ACN = 40% (v/ v) t0 (s) T = 40 °C P (Barr)

1.8/3.0 μm

400 1000 1200

53 133 159

P (Barr)

1.8/3.0 μm

400 1000 1200

25 62 74

P (Barr)

1.8/3.0 μm

400 1000 1200

13 33 39

1.8/3.5 μm

3.0/5.0 μm

5.0/10 μm

409 1023 1227

4547 11 367 13 640

3.0/5.0 μm

5.0/10 μm

190 476 571

2113 5284 6341

1.8/3.5 μm

3.0/5.0 μm

5.0/10 μm

18 45 53

101 252 303

1122 2805 3366

72 180 216 t0 (s) T = 80 °C 1.8/3.5 μm 34 84 101 t0 (s) T = 120 °C

The same holds true for all particle size pairings (see Table 1). Note that for low molecular weight solutes 1.8 μm particles will give better results than 3.0 μm particles up to dead times of about 2.5 min when Pmax is 1200 bar and T is 40 °C. When a maximum retention factor of about 10 is assumed, it follows that for any “fast” LC one should always use 1.8 μm particles provided that the higher pressure (1200 bar) is available. Increasing temperature has the opposite effect. That is, the crossing time decreases (see Figure 3A and Table 1). This may seem to be an inconvenience or even deleterious but what it means is that larger particles become more useful than the smaller particles at shorter times. An increase from 40 to 80 °C virtually halves the crossover time. Note that in this calculation as temperature is increased viscosity decreases in accord with the Chen-Horvath equation and diffusion coefficients increase as per the Wilke-Chang equation.21,22 Given t⧧0 , computation of the column length follows from substitution into eq 5: *= Le1

N3P =

Dm

hmin = A + 2 BC

N =

λ Ψ t0⧧ h⧧

N2P = (16)

Dm

B C

⎡ ⎢⎣A +

⎛ d p1 BC ⎜ d + ⎝ p2

Ψλ t0 hopt

hopt = A +

d p2 ⎞⎤ d p1 ⎠⎥ ⎦

(19)

(20)

where hopt is the two-parameter optimized reduced plate height and hmin is the three-parameter optimized reduced plate count. These are not the same quantity as hopt is constrained by the preselected particle size while the “true” minimum (hmin) is not and is accordingly smaller. It is given by7

Ψ2λd p1d p2

=

(18)

The general two-parameter optimization plate number (N2P) is7

Equations corresponding to eqs 12−16 exist for the second particle size and are given in the Supporting Information. Finally, computation of N⧧, the crossover plate count, is achieved by using eq 3 giving: ⧧

hmin

where

λ Ψ2d p12d p2 B C

Ψλ t0 Pmaxλt0 = hmin ϕη

1

BDm t0 Ψd p2

+

C Ψd p2 Dm t0

(21)

⎟

Dividing N2P by N3P and canceling like terms, it becomes apparent that the relationship between two- and threeparameter optimization depends only upon the difference in the practically optimized reduced plate heights and the absolute minimum reduced plate height.

(17)

Here, we see that the column efficiency at crossover has a complicated dependence on particle size, increasing as Pmax increases and increasing as the solute diffusion coefficient decreases. When the Wilke-Chang relationship between diffusion coefficient and viscosity is taken into account, the column efficiency at crossover is independent of eluent viscosity. Using the column parameters employed in Figure 1, the dependence of crossover plate count on particle size is shown in Figure 3B. It is interesting to note that increasing Pmax has no effect at all on the ratio of N⧧ to t⧧0 . In contradistinction as temperature is

N2P h = min N3P hopt

(22)

N2P A + 2 BC = C Ψd p2 BD t N3P A + m 20 + D t Ψd p

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m

0

(23) DOI: 10.1021/acs.analchem.5b00329 Anal. Chem. 2015, 87, 6578−6583

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Analytical Chemistry Substituting the crossover time for t0 gives ⧧ N2P = N3P

A BC A BC

+

+2 d p2 d p1

+

d p1 d p2

(24)

Thus, the relative efficiencies of two-parameter and threeparameter optimizations depends only on the ratio of the two particle sizes being considered and the fundamental column parameters A, B, and C. The relative loss in eff iciency does not depend on the maximum operating pressure, column temperature, the eluent viscosity, or the solute’s dif f usivity. This equation represents a dramatic simplification over eq 23, but because it is not explicitly dependent on the analysis time, it does not provide information about any condition (analysis time) other than that at the crossover time. To remedy this, a dummy time variable τ is defined such that any analysis time can be treated in terms relative to the crossover time. τ≡

(25)

Figure 4. Loss in plate count due to use of two-parameter and not three-parameter optimization vs particle ratio provided that the correct crossover time is correctly employed. The particle size must be changed at the correct target time to observe such results. For the transition from 5 μm particles to 10 μm particles, the largest “gap” between commercially available particle sizes on the market, the loss in plate count is only about 11%. All other conditions are as in Figure 1.

(26)

The performance loss is quite modest provided that the correct crossover time is used; however, it is important to evaluate the magnitude of additional losses which will certainly take place when the proper crossover time is not used. This is achieved via eq 26. Figure 5 shows that the performance loss is

t0 t0⧧

Using this definition, eq 23 is rewritten as N2P = N3P

A BC A BC

+

d p2 d p1

+2 τ +

d p1 d p2

1 τ

This equation allows the determination of the maximum loss in performance of two- vs three-parameter optimization at any desired analysis time. That is, it allows estimation of the loss in plates because the ideal particle size as dictated by the threeparameter (Knox-Saleem) optimization is not available. To maximize separation performance, it is vital to select the correct particle size for the intended analysis time. In a series of “Poppe curves”, the intersections between each pair of curves indicate the analysis time at which one should change the particle size to optimize a given separation. Previously, these intersection points were calculated by generating entire “Poppe curves” and numerically searching for the intersection. Here, we developed the analytical solution for the crossover time given a set of separation conditions and two particle sizes (eq 15). Because the locus of best possible separations (i.e., the KnoxSaleem line) is tangent to the two intersecting Poppe curves, the crossover point is the point (time and efficiency) at which the largest loss in performance is observed due to the limited number of particle sizes. Extending upon this, eq 24 quantifies this performance loss as the ratio of the plate counts under two- and three-parameter optimization. Most interestingly, this result does not depend on any dynamic separation conditions (e.g., temperature, pressure, or analyte diffusion coefficient); only the ratio of the particle sizes and the fundamental column parameters influence the plate count ratio. Figure 4, which gives the performance loss as a function of the ratio of particle sizes for a standard column, makes it obvious that this loss is really quite small ( 1. This is because the definition of dp2 refers to the particle size being transitioned “toward”, and the case of τ > 1 describes a situation where the transition should have happened but has not (dp2 is the larger of the two sizes) while τ < 1 indicates that the transition has already taken place but ideally would not have (dp2 is the smaller of the two sizes).

nearly doubled when the crossover time is exceeded by 50%. This performance loss is unnecessary and can be readily avoided simply by observing the proper crossover time. This analysis also assumes that in missing the crossover time between the two analyzed particle sizes the practitioner has not missed a second particle size crossover, which would have the effect of sharply increasing the wasted performance. Finally, the authors present a web-based calculator that rapidly yields the crossover parameters (plate count, crossover 6582

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Analytical Chemistry time, etc.) for an entered set separation conditions.20 This determine the correct particle problem with ease, minimizing power.

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of column parameters and allows the practitioner to size for a given separation needless loss of separating

AUTHOR INFORMATION

Corresponding Author

*Phone: (612) 624-0253. Fax: (612) 626-7541. E-mail: [email protected]. Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

CONCLUSIONS

Current physical limitations on particle size force chromatographers to consider two-parameter optimization instead of three-parameter optimization. For each pairing of particle sizes, there is an analysis time after which one particle size always out performs the other. These “crossover points” can be readily calculated given a set of separation conditions. Two-parameter optimization necessarily results in a loss of performance when compared to three-parameter optimization because one less degree of freedom is available. This loss in plate count is at worst only about 11% provided the crossover time is observed. However, failing to change particle size at the crossover time can cause this loss to double. It is remarkable that the fractional loss in plate count of Poppe (2-parameter) vs Knox-Saleem (3parameter) optimization is dependent only on the column parameters (A, B, and C) and the ratio of particle diameters. It is independent of pressure, temperature, solute diffusion coefficient and column porosity parameters, and permeability. To select a particle size to optimize a separation in HPLC and UPLC, a chromatographer should calculate the crossover time between each two particle sizes and chose the particle size which gives the best performance under conditions which can be practically achieved. If the desired time of analysis changes, it is important to reevaluate the size of particles being used to avoid unnecessary loss in performance. Because the crossover time depends strongly on the diffusion coefficient of the analyte in the solvent used, these calculations must be repeated for different separations (e.g., small molecules vs peptides). Given this information, there is little incentive for manufacturers to diversify particle sizes further, particularly in cases where a new particle size would not significantly decrease the ratio of particle sizes between it and the nearest size above and below it. It is important to note that the optimal separation for each particle size will occur with a different eluent flow rate and column length. Because the maximum system pressure is treated as an input in this analysis, all calculated eluent velocities are achievable. Though technically column length is nondiscrete like particle size, it is effectively variable in practice and many length combinations are available. This analysis is performed using the van Deemter equation to describe theoretical plate heights. Some systems may demonstrate plate height curves which deviate from this equation and are better described by other plate height equations.23 However, the velocity near the crossing-point is never very far from the velocity at which the plate height is minimized and thus the data fitting equation hardly matters.

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Article

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the contributions of Prof. Dwight Stoll (Gustavus Adolphus College) in the creation of the webbased calculator for crossover point times and plate counts as well as the generous financial support of the University of Minnesota Undergraduate Research Opportunities Program (UROP) and a National Science Foundation Grant 1213561.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Crossover times and associated plate counts for a variety of relevant separation conditions. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.analchem.5b00329. 6583

DOI: 10.1021/acs.analchem.5b00329 Anal. Chem. 2015, 87, 6578−6583