- Email: [email protected]
Comparing two models of gradient elution in counter-current chromatography
Journal of Chromatography A, 1274 (2013) 77–81
Contents lists available at SciVerse ScienceDirect
Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Comparing two models of gradient elution in counter-current chromatography夽 Shihua Wu a , Junling Liang a , Alain Berthod b,∗ a b
Research Center of Siyuan Natural Pharmacy and Biotoxicology, College of Life Sciences, Zhejiang University, Hangzhou 310058, Zhejiang Province, China Institut des Sciences Analytiques, Université de Lyon, CNRS, 69100 Villeurbanne, France
a r t i c l e
i n f o
Article history: Received 1 October 2012 Received in revised form 20 November 2012 Accepted 28 November 2012 Available online 19 December 2012 Keywords: Counter-current chromatography Linear gradient Step-gradient Modeling
a b s t r a c t Gradient change of mobile phase composition is commonly used in liquid chromatography to shorten analysis duration. Gradient elution is possible in counter-current chromatography if it is demonstrated that the mobile phase composition changes are not associated with liquid stationary phase composition changes. Also, the solute distribution ratios in initial (1) and final (2) eluting systems must be known. If the solute distribution ratios during the changing mobile phase composition are modeled, the full gradient separation can be modeled using classical equations of liquid chromatography. Another approach allowed modeling empirically the gradient step. It considered the decreasing volume of mobile phase 1 and the increasing volume of mobile phase 2 used at each gradient time to derive a very simple equation. The two equations were compared computing the retention volumes of five test solutes. A remarkable agreement was observed considering the very different mathematical expressions of the two models. The stepgradient method was also modeled and its results compared to those of the linear gradient method. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Counter-current chromatography (CCC) is the separation technique that uses two liquid phases to separate solutes by partitioning only [1–3]. The support-free liquid stationary phase is obtained using centrifugal fields. The CCC columns are actually machines containing motor and rotating parts required to generate a centrifugal field able to hold the liquid stationary phase [3,4]. CCC is mainly used for preparative purification of delicate compounds [5]. In any chromatographic methods there are solutes highly retained. Gradient methods were developed to speed up the retention of slowly eluting compounds. The principle is simple: the elution conditions are gradually modified to increase the elution strength of the mobile phase. This poses little problem in liquid chromatography: the mobile phase composition is changed with no or minimum changes of the solid stationary phase. Some reequilibration time is needed after the gradient run. Gradients are much more difficult in CCC where the liquid mobile phase is in equilibrium with the liquid stationary phase. Any composition change in one phase may induce a composition change in the other phase [2].
Many CCC gradient separations were performed successfully separating complex mixtures [6–10]. They were either linear composition change gradients [6,7] or step-gradient switching directly and abruptly from one mobile phase composition to another one [8–10]. The chromatograms are shown in the articles, but the results are always experimental observations optimized on a caseby-case basis. In recent works, we demonstrated that it is possible to model the solute band position inside the CCC column [11,12]. Since the CCC retention mechanism is exclusively liquid–liquid partitioning, the knowledge of the column characteristics, mobile and stationary phase volumes, flow rate and solute distribution ratios are the parameters needed to model the peak positions and widths [11,12]. It was also shown that particular liquid system can have composition changes in one liquid phase associated with little or no change in the other phase [13]. In this work, two different ways to model gradient composition changes in CCC are presented and compared.
2. Solute band motion inside the CCC column 2.1. Modeling band position
夽 Presented at the 7th International Conference on Countercurrent Chromatography, Hangzhou, China, 6–8 August 2012. ∗ Corresponding author. Fax: +33 472 448 319. E-mail address: [email protected] (A. Berthod). 0021-9673/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chroma.2012.11.078
Assuming that the CCC column has a length L, all solutes move inside the column at different speeds depending on the mobile phase flow rate and the affinity for the stationary phase, expressed by the distribution ratio (KCi values) of Solute i. At any time, after the
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S. Wu et al. / J. Chromatogr. A 1274 (2013) 77–81
volume V of mobile phase has be pumped into the column, Solute i is located at the distance Xi from the column entrance: Xi =
LV VRi
dxi =
(2)
in which F is the constant mobile phase flow rate and VRi (t) is the solute isocratic retention volume at time t with KCi (t), its distribution ratio at this time. Integrating dxi over the [0,tG ] period of time corresponding to the gradient step, xG (t), the solute band motion inside the CCC column, can be expressed as [13]:
in which the subscripts R, M and S stand for retention, mobile and stationary phase volumes, respectively [1–3]. The distance X is comprised between X = 0 at the column entrance (injection) and X = L, at the column exit (V = VRi , solute elutes). To simplify the writing, the dimensionless relative distances: xi =
Xi V = L VRi
(3)
will be used with the condition: 0 ≤ xi ≤ 1.
xG (t) =
2.2. Modeling linear gradient elution The change of mobile phase composition to perform gradient elution is most often started after an initial isocratic elution. After this first step, a linear change of mobile phase composition is programmed between initial mobile phase of composition 1 and final mobile phase of composition 2 for a period of time tG . When a given mobile phase composition is prepared in the pumping system and introduced in the CCC equipment, the composition change is not seen by all solutes at the same time. The new mobile phase composition progresses in the CCC column crossing first the injection valve and connecting tubing (dwell volume). Next there is a sweep volume inside the CCC column needed for the changed mobile phase composition to reach the solutes that are not all located at the same position since the separation started. As presented in our previous work, the modeling of the solute retention volume under linear gradient conditions, VRi , must be done in four steps expressed by [13]: (4)
Step I: isocratic initial solute elution with VCM volume of mobile phase 1, Step II: dwell volume, Vdw , and sweep elution volume, Vsw,i , solute depending, Step III: gradient composition change volume, VG , Step IV: isocratic final solute elution, ViIV , with mobile phase 2. The problem is to relate VG to xGi , the solute i displacement inside the CCC column. Two different approaches were developed to model the solute motion under linear gradient change of mobile phase composition: a classical chromatographic model and an empirical “volumic” model. 2.2.1. Classical chromatographic model Gradient elution is classically modeled assuming that the mobile phase elution strength increases linearly with the composition change [14]. In our recent work, we demonstrated that the solute i distribution ratio, KCi , was related to the mobile phase water content, %W: ln KCi = A %W + B
(5)
with A and B being constants. In the gradient step, the mobile phase water content is linearly decreased. Since the solute distribution ratios are known for all changing mobile phase composition, it is
F VM
t−
(6)
1 ln AG
V + K (t)V M S Ci VR1i
(7)
in which V with the subscripts M, S and R1i are respectively the mobile and stationary phase volume inside the CCC column and the solute i retention volume under elution with mobile phase composition 1. A is the slope of Eq. (5) and G is the gradient slope defined as: G=
VRi = VCM + Vdw + Vsw,i + VG + ViIV
F dt dV = VRi (t) VRi (t)
(1)
in which VRi is the solute retention volume given by: VRi = VM + KCi VS
possible to express for each solute, its solute displacement, dxi , as [13]:
%W2 − %W1 tG
(8)
For all solutes still inside the column after the gradient step, the relative displacement, xG (tG ), can be expressed using Eqs. (5), (7) and (8) as: xG (tG ) =
VG VM
1−
ln(VR2 /VR1 ) ln(KC2 /KC1 )
(9)
In a real analysis, some solutes elute during the initial isocratic step with mobile phase 1, others during the gradient step and the remaining solutes elute during the last isocratic step with the high elution strength mobile phase 2 as fully previously considered [13]. 2.2.2. Empirical volume model Another approach to model the solute band motion during the gradient step is to consider the experimental conditions used to perform the mobile phase composition changes. Two pumps are used; the first pump contains mobile phase 1 with %W1 water content, the second pump contains the higher elution strength mobile phase 2 with %W2 water content. To perform the gradient for a time tG and slope G (Eq. (8)), the flow rate of pump 1 is linearly decreased from F to 0 mL/min while, simultaneously, the pump 2 flow rate is linearly increased from 0 to F. Considering the respective mobile phase volumes, it is assumed that all solute bands, reached by the gradient mobile phase composition change, are moved independently by a volume V1 (t) of mobile phase 1 and a volume V2 (t) of mobile phase 2. The volume of mobile phase 1 decreases linearly and that of mobile phase 2 increases linearly as the gradient time increases. The flow rate, F, is constant. Each solute band moves by a distance xG (t) expressed by: xG (t) =
V1 (t) V2 (t) + VR1i VR2i
(10)
After the gradient step of duration tG , a volume VG of mobile phase has been flown, volume equally made by mobile phases 1 and 2. Eq. (10) gives easily xG (tG ), the gradient step solute displacement after a volume VG /2 of mobile phase 1 and 2 as been used during time tG : xG (tG ) =
VG 2
1
VR1i
+
1 VR2i
(11)
It is very interesting to compare the results obtained using the two models that give equations mathematically very different. Fig. 1A compares the gradient band displacement inside the CCC column for two compounds used in [13] dihydrotanshinone and cryptotanshinone using the classical model (Eq. (7) and full lines) and the volume model (Eq. (10) and dashed lines). The agreement
S. Wu et al. / J. Chromatogr. A 1274 (2013) 77–81
79
Step II: dwell volume, Vdw , and sweep elution volume, Vsw,i , solute depending, Step III: isocratic final solute elution, ViIII , with mobile phase 2. As previously demonstrated [13], after step II, the solutes are located inside the column at relative distances xiII : xiII =
VCM + Vdw KC1i VS
(12)
with xiII < 1 otherwise the solute eluted in an isocratic way with mobile phase 1. The volume ViIII , of mobile phase 2 needed to elute the solute depends on the remaining distance to the column outlet: ViIII = (1 − xiII )VR2i
(13)
Combining all equations, the step gradient elution volume, VRi , of solute i is: VRi = VCM + Vdw + Vsw,i + ViIII
(14)
rearranged as: VRi =
(VCM + Vdw ) × (KC1i − KC2i ) + VR2i KC1i
(15)
with the condition: VRi < VR1 ; the calculated step gradient solute retention volume must be lower that the solute retention volume under isocratic elution with mobile phase 1. If the calculated step gradient VRi is larger than VR1 it simply means that the solute eluted before the start of the step change of mobile phase composition. 3. Comparing gradient modes
Fig. 1. Comparing models: classical gradient model in full lines; empirical volume model in dashed lines. (A) Dihydrotanshinone (KC1 = 0.78; KC2 = 0.22) and cryptotanshinone (KC1 = 2.06; KC2 = 0.50) displacements inside a 200 mL CCC column during a 30 min gradient between mobile phase 1 of the system hexane/ethyl acetate/ethanol/water 8:2:5:5 (v/v) and mobile phase 2 of the system 8:2:7:3 (v/v). Flow rate: 2 mL/min, VG = 60 mL, experimental data from [13]. (B) Calculated band displacement for a 25 min gradient (VG = 50 mL) and a distribution ratio change as KC1 /KC2 = 4 (Eqs. (9) and (11)).
between the two models is surprisingly good considering the different mathematical expression of the two equations. Fig. 1B compares the band displacement xG (tG ) for hypothetical compounds with distribution ratios in mobile phase 1 between 0.1 and 10. Here again, the curves computed with Eq. (9) involving logarithms (full line) and Eq. (11) with simple quotients (dashed line) are extremely close. The empirical volume model produces results so close to those of the exact classical model that it would be difficult to experimentally detect the small volume difference. The empirical model is much easier to use that the classical model, needing only the solute retention volumes in the initial and final mobile phases. 2.3. Modeling step-gradient elution The step-gradient experiment consists in switching directly from mobile phase 1 to mobile phase 2 with stronger elution strength. The solute distribution ratios, KC2 , in the final mobile phase 2 are obviously smaller that the respective KC1 in the initial mobile phase 1. The modeling of step-gradient elution is much simpler that that of the linear gradient elution since it can be considered that there is only three steps: Step I: isocratic initial solute elution with VCM volume of mobile phase 1,
Linear gradient and step gradient elution modes were compared using the developed models. The experimental validation of the classical linear model (Eqs. (4), (7) and (9)) was done with a tanshinone natural mixture and a gradient composition change of the hexane/ethyl acetate/ethanol/water 8:2:x:y liquid system [13]. The empirical model of linear gradient elution (Eqs. (4), (10) and (11)) gives results so close to the classical model (Fig. 1) that there is no further experimental validation needed. Linear and step gradient elution of a hypothetical mixture of solutes were compared calculating the solute retention volumes with Eqs. (4), (7) and (9) for the linear gradient elution and Eq. (15) for the step-gradient elution. Table 1 lists the distribution ratios of the five hypothetical solutes 1–5 in the two liquid systems 1 and 2 as well as the respective isocratic retention volumes with mobile phase 1 and 2 and the retention volumes under linear and step-gradient conditions. Fig. 2 compares all chromatograms. The isocratic separation of the 5 compounds with mobile phase 1 not only last for 11 h but compounds 4 and 5 coelute. Using the more eluting mobile phase 2 allows for a separation lasting only 4 h but compounds 1 and 2 and compounds 3 and 4 coelute (Fig. 2 top chromatograms). This set of compounds and liquid systems was selected to present an obvious case of valuable use of gradient elution. Both ways: linear and step-gradient elutions are compared. An initial isocratic elution of 30 min (VCM = 60 mL) with mobile phase 1 is needed to develop the compound 1 and 2 separation. Then, the linear or step-gradient elution is started. The dwell volume was taken as 20 mL. These initial conditions were used for all three modeling. A single chromatogram is presented for the linear gradient experiment (Fig. 2, bottom left) because there was no visible difference between the chromatogram modeled using the classical model (Eqs. (4), (7) and (9)) and the one modeled with the volume model (Eqs. (4), (10) and (11)). Table 1 lists the computed retention volumes using the two sets of equations. The higher relative difference is 1.4% seen on compounds 4 returning a retention volume of
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S. Wu et al. / J. Chromatogr. A 1274 (2013) 77–81
Table 1 Distribution ratio and retention volumes of the test solutes of Fig. 2 computed using the different models presented. Compound
KC1
KC2
VR1 isocratic 1 (mL)
VR2 isocratic 2 (mL)
VR linear gradient (mL) Eqs. (4), (7) and (9)
VR linear gradient (mL) Eqs. (4), (10) and (11)
VR step gradient (mL) Eq. (15)
1 2 3 4 5
0.2 0.6 1.5 6 7
0.1 0.2 0.8 1 2
80 140 275 950 1100
65 80 170 200 350
80 136 200 286 423
80 136 199 282 420
80 133 207 267 407
VC = 200 mL; VM = 50 mL; VS = 150 mL; Sf = 75%; F = 2 mL/min; N = 200 plates. Linear gradient retention volumes obtained with the classical model (Eqs. (4), (7) and (9)) and empirical volume model (Eqs. (4), (10) and (11)), both with an initial isocratic elution with mobile phase 1, VCM = 60 mL and a dwell volume Vdw = 20 mL, tG = 25 min; VG = 50 mL. Step gradient (Eq. (15)) with an initial isocratic elution with mobile phase 1, VCM = 60 mL and a dwell volume Vdw = 20 mL.
Fig. 2. Modeled chromatograms of Table 1 compounds. VC = 200 mL; VM = 50 mL; VS = 150 mL; Sf = 75%; F = 2 mL/min. Linear gradient duration tG = 25 min. For both linear and step gradient the classical mode (CM) isocratic elution with mobile phase 1 was VCM = 60 mL and the dwell volume Vdw = 20 mL. Red line: mobile phase 1; blue line: mobile phase 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
286 mL with the classical model and only 282 mL with the empirical volume model. A linear gradient with tG = 25 min (VG = 50 mL) from mobile phase 1 to mobile phase 2 composition followed by an isocratic mobile phase 2 separation produced a full separation of all five compounds. The step-gradient experiment was modeled with the same initial conditions (VCM = 60 mL and Vdw = 20 mL) i.e., the separation is simply performed in two steps: (1) isocratic elution for 30 min at 2 mL/min with mobile phase 1 followed by (2) isocratic elution with mobile phase 2 till the full chromatogram developed (Fig. 2, bottom right). A significant overlap of compounds 3 and 4 peaks is seen. It is pointed out that the chromatogram comparison was done maintaining the same first step for both linear and step gradient Fig. 2 runs. This is doing the step change too early. The 3 and 4 overlap problem was mostly solved by doubling the initial isocratic step with mobile phase 1 using VCM = 120 mL. It obviously increased the analysis duration by 30 min (chromatogram not shown) but would allow for a complete separation of the 5 compounds with a much simpler experimental setting. The red lines in Fig. 2 (dark gray) correspond to mobile phase 1 and the blue lines (light gray) to mobile
phase 2. Taking in account the 20 mL dwell volume, the actual isocratic initial step lasts for 40 min in both gradient experiments. 4. Conclusion The first major conclusion of this work is that the countercurrent chromatography technique, with its support-free liquid stationary phase and immiscible liquid mobile phase, can be modeled using the long established chromatographic theory if the chemistry and chemical physics of biphasic liquid systems is known and understood. Another interesting conclusion is that solutions for the CCC gradient modeling developed independently with two different approaches produced mathematically two very different equations that turned out to both model the problem very correctly. The last conclusion is that the popular step-gradient method is also easily modeled and can produce very acceptable separations using an experimental setting simpler (i.e., a single pump) than that needed for linear gradient experiments. In all cases, the bottom line is that the biphasic liquid system used for gradient separations in CCC must be able to tolerate significant polarity changes in one
S. Wu et al. / J. Chromatogr. A 1274 (2013) 77–81
phase, the mobile phase, associated with minimum changes in the other phase, the stationary phase [13]. Acknowledgements SW thanks the National Science Foundation of China for Grant #20972136, Zhejiang Province for Grant #Y4080353 and the Fundamental Research Funds for the Central Universities #2010 QNA6006. AB thanks the French National Center for Scientific Research (CNRS) and University of Lyon 1 for continuous support with UMR5280 ISA (P. Toulhoat and P. Lanteri). References [1] W.D. Conway, Countercurrent Chromatography: Apparatus, Theory & Applications, VCH Publishers, Weinheim, Germany, 1990.
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[2] A. Berthod, in: D. Barcelo (Ed.), Countercurrent Chromatography, the Supportfree Liquid Stationary Phase, Comprehensive Analytical Chemistry, vol. 38, Elsevier, Amsterdam, 2002. [3] N.B. Mandava, Y. Ito, Countercurrent Chromatography, Theory and Practice, in: Chromatographic Science Series, vol. 44, M. Dekker, NY, 1988. [4] A. Berthod, T. Maryutina, B. Spivakov, O. Shpigun, I.A. Sutherland, Pure Appl. Chem. 81 (2009) 355. [5] G.F. Pauli, S.M. Pro, J.B. Friesen, J. Nat. Prod. 71 (2008) 1489. [6] M.J. vanBuel, L.A.M. vanderWielen, K. Luyben, J. Chromatogr. A 773 (1997) 13. [7] P. Duret, M.A. Fakhfakh, C. Herrenknecht, A. Fournet, X. Franck, B. Figadere, R. Hocquemiller, J. Chromatogr. A 1011 (2003) 55. [8] G.G. Leitao, P.A. de Souza, A.A. Moraes, L. Brown, J. Liq. Chromatogr. Relat. Technol. 28 (2005) 2053. [9] G.G. Leitao, S.S. El-Adji, W.A.L. de Melo, S.G. Leitao, L. Brown, J. Liq. Chromatogr. Relat. Technol. 28 (2005) 2041. [10] F.D. Costa, G.G. Leitao, J. Sep. Sci. 33 (2010) 336. [11] A. Berthod, J. Chromatogr. A 1126 (2006) 347. [12] A. Berthod, J.B. Freisen, T. Inui, G.F. Pauli, Anal. Chem. 79 (2007) 3371. [13] S. Wu, D. Wu, J. Liang, A. Berthod, J. Sep. Sci. 35 (2012) 964. [14] P. Jandera, J. Churáˇcek, J. Chromatogr. Libr. 31 (1985).
Contents lists available at SciVerse ScienceDirect
Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma
Comparing two models of gradient elution in counter-current chromatography夽 Shihua Wu a , Junling Liang a , Alain Berthod b,∗ a b
Research Center of Siyuan Natural Pharmacy and Biotoxicology, College of Life Sciences, Zhejiang University, Hangzhou 310058, Zhejiang Province, China Institut des Sciences Analytiques, Université de Lyon, CNRS, 69100 Villeurbanne, France
a r t i c l e
i n f o
Article history: Received 1 October 2012 Received in revised form 20 November 2012 Accepted 28 November 2012 Available online 19 December 2012 Keywords: Counter-current chromatography Linear gradient Step-gradient Modeling
a b s t r a c t Gradient change of mobile phase composition is commonly used in liquid chromatography to shorten analysis duration. Gradient elution is possible in counter-current chromatography if it is demonstrated that the mobile phase composition changes are not associated with liquid stationary phase composition changes. Also, the solute distribution ratios in initial (1) and final (2) eluting systems must be known. If the solute distribution ratios during the changing mobile phase composition are modeled, the full gradient separation can be modeled using classical equations of liquid chromatography. Another approach allowed modeling empirically the gradient step. It considered the decreasing volume of mobile phase 1 and the increasing volume of mobile phase 2 used at each gradient time to derive a very simple equation. The two equations were compared computing the retention volumes of five test solutes. A remarkable agreement was observed considering the very different mathematical expressions of the two models. The stepgradient method was also modeled and its results compared to those of the linear gradient method. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Counter-current chromatography (CCC) is the separation technique that uses two liquid phases to separate solutes by partitioning only [1–3]. The support-free liquid stationary phase is obtained using centrifugal fields. The CCC columns are actually machines containing motor and rotating parts required to generate a centrifugal field able to hold the liquid stationary phase [3,4]. CCC is mainly used for preparative purification of delicate compounds [5]. In any chromatographic methods there are solutes highly retained. Gradient methods were developed to speed up the retention of slowly eluting compounds. The principle is simple: the elution conditions are gradually modified to increase the elution strength of the mobile phase. This poses little problem in liquid chromatography: the mobile phase composition is changed with no or minimum changes of the solid stationary phase. Some reequilibration time is needed after the gradient run. Gradients are much more difficult in CCC where the liquid mobile phase is in equilibrium with the liquid stationary phase. Any composition change in one phase may induce a composition change in the other phase [2].
Many CCC gradient separations were performed successfully separating complex mixtures [6–10]. They were either linear composition change gradients [6,7] or step-gradient switching directly and abruptly from one mobile phase composition to another one [8–10]. The chromatograms are shown in the articles, but the results are always experimental observations optimized on a caseby-case basis. In recent works, we demonstrated that it is possible to model the solute band position inside the CCC column [11,12]. Since the CCC retention mechanism is exclusively liquid–liquid partitioning, the knowledge of the column characteristics, mobile and stationary phase volumes, flow rate and solute distribution ratios are the parameters needed to model the peak positions and widths [11,12]. It was also shown that particular liquid system can have composition changes in one liquid phase associated with little or no change in the other phase [13]. In this work, two different ways to model gradient composition changes in CCC are presented and compared.
2. Solute band motion inside the CCC column 2.1. Modeling band position
夽 Presented at the 7th International Conference on Countercurrent Chromatography, Hangzhou, China, 6–8 August 2012. ∗ Corresponding author. Fax: +33 472 448 319. E-mail address: [email protected] (A. Berthod). 0021-9673/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.chroma.2012.11.078
Assuming that the CCC column has a length L, all solutes move inside the column at different speeds depending on the mobile phase flow rate and the affinity for the stationary phase, expressed by the distribution ratio (KCi values) of Solute i. At any time, after the
78
S. Wu et al. / J. Chromatogr. A 1274 (2013) 77–81
volume V of mobile phase has be pumped into the column, Solute i is located at the distance Xi from the column entrance: Xi =
LV VRi
dxi =
(2)
in which F is the constant mobile phase flow rate and VRi (t) is the solute isocratic retention volume at time t with KCi (t), its distribution ratio at this time. Integrating dxi over the [0,tG ] period of time corresponding to the gradient step, xG (t), the solute band motion inside the CCC column, can be expressed as [13]:
in which the subscripts R, M and S stand for retention, mobile and stationary phase volumes, respectively [1–3]. The distance X is comprised between X = 0 at the column entrance (injection) and X = L, at the column exit (V = VRi , solute elutes). To simplify the writing, the dimensionless relative distances: xi =
Xi V = L VRi
(3)
will be used with the condition: 0 ≤ xi ≤ 1.
xG (t) =
2.2. Modeling linear gradient elution The change of mobile phase composition to perform gradient elution is most often started after an initial isocratic elution. After this first step, a linear change of mobile phase composition is programmed between initial mobile phase of composition 1 and final mobile phase of composition 2 for a period of time tG . When a given mobile phase composition is prepared in the pumping system and introduced in the CCC equipment, the composition change is not seen by all solutes at the same time. The new mobile phase composition progresses in the CCC column crossing first the injection valve and connecting tubing (dwell volume). Next there is a sweep volume inside the CCC column needed for the changed mobile phase composition to reach the solutes that are not all located at the same position since the separation started. As presented in our previous work, the modeling of the solute retention volume under linear gradient conditions, VRi , must be done in four steps expressed by [13]: (4)
Step I: isocratic initial solute elution with VCM volume of mobile phase 1, Step II: dwell volume, Vdw , and sweep elution volume, Vsw,i , solute depending, Step III: gradient composition change volume, VG , Step IV: isocratic final solute elution, ViIV , with mobile phase 2. The problem is to relate VG to xGi , the solute i displacement inside the CCC column. Two different approaches were developed to model the solute motion under linear gradient change of mobile phase composition: a classical chromatographic model and an empirical “volumic” model. 2.2.1. Classical chromatographic model Gradient elution is classically modeled assuming that the mobile phase elution strength increases linearly with the composition change [14]. In our recent work, we demonstrated that the solute i distribution ratio, KCi , was related to the mobile phase water content, %W: ln KCi = A %W + B
(5)
with A and B being constants. In the gradient step, the mobile phase water content is linearly decreased. Since the solute distribution ratios are known for all changing mobile phase composition, it is
F VM
t−
(6)
1 ln AG
V + K (t)V M S Ci VR1i
(7)
in which V with the subscripts M, S and R1i are respectively the mobile and stationary phase volume inside the CCC column and the solute i retention volume under elution with mobile phase composition 1. A is the slope of Eq. (5) and G is the gradient slope defined as: G=
VRi = VCM + Vdw + Vsw,i + VG + ViIV
F dt dV = VRi (t) VRi (t)
(1)
in which VRi is the solute retention volume given by: VRi = VM + KCi VS
possible to express for each solute, its solute displacement, dxi , as [13]:
%W2 − %W1 tG
(8)
For all solutes still inside the column after the gradient step, the relative displacement, xG (tG ), can be expressed using Eqs. (5), (7) and (8) as: xG (tG ) =
VG VM
1−
ln(VR2 /VR1 ) ln(KC2 /KC1 )
(9)
In a real analysis, some solutes elute during the initial isocratic step with mobile phase 1, others during the gradient step and the remaining solutes elute during the last isocratic step with the high elution strength mobile phase 2 as fully previously considered [13]. 2.2.2. Empirical volume model Another approach to model the solute band motion during the gradient step is to consider the experimental conditions used to perform the mobile phase composition changes. Two pumps are used; the first pump contains mobile phase 1 with %W1 water content, the second pump contains the higher elution strength mobile phase 2 with %W2 water content. To perform the gradient for a time tG and slope G (Eq. (8)), the flow rate of pump 1 is linearly decreased from F to 0 mL/min while, simultaneously, the pump 2 flow rate is linearly increased from 0 to F. Considering the respective mobile phase volumes, it is assumed that all solute bands, reached by the gradient mobile phase composition change, are moved independently by a volume V1 (t) of mobile phase 1 and a volume V2 (t) of mobile phase 2. The volume of mobile phase 1 decreases linearly and that of mobile phase 2 increases linearly as the gradient time increases. The flow rate, F, is constant. Each solute band moves by a distance xG (t) expressed by: xG (t) =
V1 (t) V2 (t) + VR1i VR2i
(10)
After the gradient step of duration tG , a volume VG of mobile phase has been flown, volume equally made by mobile phases 1 and 2. Eq. (10) gives easily xG (tG ), the gradient step solute displacement after a volume VG /2 of mobile phase 1 and 2 as been used during time tG : xG (tG ) =
VG 2
1
VR1i
+
1 VR2i
(11)
It is very interesting to compare the results obtained using the two models that give equations mathematically very different. Fig. 1A compares the gradient band displacement inside the CCC column for two compounds used in [13] dihydrotanshinone and cryptotanshinone using the classical model (Eq. (7) and full lines) and the volume model (Eq. (10) and dashed lines). The agreement
S. Wu et al. / J. Chromatogr. A 1274 (2013) 77–81
79
Step II: dwell volume, Vdw , and sweep elution volume, Vsw,i , solute depending, Step III: isocratic final solute elution, ViIII , with mobile phase 2. As previously demonstrated [13], after step II, the solutes are located inside the column at relative distances xiII : xiII =
VCM + Vdw KC1i VS
(12)
with xiII < 1 otherwise the solute eluted in an isocratic way with mobile phase 1. The volume ViIII , of mobile phase 2 needed to elute the solute depends on the remaining distance to the column outlet: ViIII = (1 − xiII )VR2i
(13)
Combining all equations, the step gradient elution volume, VRi , of solute i is: VRi = VCM + Vdw + Vsw,i + ViIII
(14)
rearranged as: VRi =
(VCM + Vdw ) × (KC1i − KC2i ) + VR2i KC1i
(15)
with the condition: VRi < VR1 ; the calculated step gradient solute retention volume must be lower that the solute retention volume under isocratic elution with mobile phase 1. If the calculated step gradient VRi is larger than VR1 it simply means that the solute eluted before the start of the step change of mobile phase composition. 3. Comparing gradient modes
Fig. 1. Comparing models: classical gradient model in full lines; empirical volume model in dashed lines. (A) Dihydrotanshinone (KC1 = 0.78; KC2 = 0.22) and cryptotanshinone (KC1 = 2.06; KC2 = 0.50) displacements inside a 200 mL CCC column during a 30 min gradient between mobile phase 1 of the system hexane/ethyl acetate/ethanol/water 8:2:5:5 (v/v) and mobile phase 2 of the system 8:2:7:3 (v/v). Flow rate: 2 mL/min, VG = 60 mL, experimental data from [13]. (B) Calculated band displacement for a 25 min gradient (VG = 50 mL) and a distribution ratio change as KC1 /KC2 = 4 (Eqs. (9) and (11)).
between the two models is surprisingly good considering the different mathematical expression of the two equations. Fig. 1B compares the band displacement xG (tG ) for hypothetical compounds with distribution ratios in mobile phase 1 between 0.1 and 10. Here again, the curves computed with Eq. (9) involving logarithms (full line) and Eq. (11) with simple quotients (dashed line) are extremely close. The empirical volume model produces results so close to those of the exact classical model that it would be difficult to experimentally detect the small volume difference. The empirical model is much easier to use that the classical model, needing only the solute retention volumes in the initial and final mobile phases. 2.3. Modeling step-gradient elution The step-gradient experiment consists in switching directly from mobile phase 1 to mobile phase 2 with stronger elution strength. The solute distribution ratios, KC2 , in the final mobile phase 2 are obviously smaller that the respective KC1 in the initial mobile phase 1. The modeling of step-gradient elution is much simpler that that of the linear gradient elution since it can be considered that there is only three steps: Step I: isocratic initial solute elution with VCM volume of mobile phase 1,
Linear gradient and step gradient elution modes were compared using the developed models. The experimental validation of the classical linear model (Eqs. (4), (7) and (9)) was done with a tanshinone natural mixture and a gradient composition change of the hexane/ethyl acetate/ethanol/water 8:2:x:y liquid system [13]. The empirical model of linear gradient elution (Eqs. (4), (10) and (11)) gives results so close to the classical model (Fig. 1) that there is no further experimental validation needed. Linear and step gradient elution of a hypothetical mixture of solutes were compared calculating the solute retention volumes with Eqs. (4), (7) and (9) for the linear gradient elution and Eq. (15) for the step-gradient elution. Table 1 lists the distribution ratios of the five hypothetical solutes 1–5 in the two liquid systems 1 and 2 as well as the respective isocratic retention volumes with mobile phase 1 and 2 and the retention volumes under linear and step-gradient conditions. Fig. 2 compares all chromatograms. The isocratic separation of the 5 compounds with mobile phase 1 not only last for 11 h but compounds 4 and 5 coelute. Using the more eluting mobile phase 2 allows for a separation lasting only 4 h but compounds 1 and 2 and compounds 3 and 4 coelute (Fig. 2 top chromatograms). This set of compounds and liquid systems was selected to present an obvious case of valuable use of gradient elution. Both ways: linear and step-gradient elutions are compared. An initial isocratic elution of 30 min (VCM = 60 mL) with mobile phase 1 is needed to develop the compound 1 and 2 separation. Then, the linear or step-gradient elution is started. The dwell volume was taken as 20 mL. These initial conditions were used for all three modeling. A single chromatogram is presented for the linear gradient experiment (Fig. 2, bottom left) because there was no visible difference between the chromatogram modeled using the classical model (Eqs. (4), (7) and (9)) and the one modeled with the volume model (Eqs. (4), (10) and (11)). Table 1 lists the computed retention volumes using the two sets of equations. The higher relative difference is 1.4% seen on compounds 4 returning a retention volume of
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Table 1 Distribution ratio and retention volumes of the test solutes of Fig. 2 computed using the different models presented. Compound
KC1
KC2
VR1 isocratic 1 (mL)
VR2 isocratic 2 (mL)
VR linear gradient (mL) Eqs. (4), (7) and (9)
VR linear gradient (mL) Eqs. (4), (10) and (11)
VR step gradient (mL) Eq. (15)
1 2 3 4 5
0.2 0.6 1.5 6 7
0.1 0.2 0.8 1 2
80 140 275 950 1100
65 80 170 200 350
80 136 200 286 423
80 136 199 282 420
80 133 207 267 407
VC = 200 mL; VM = 50 mL; VS = 150 mL; Sf = 75%; F = 2 mL/min; N = 200 plates. Linear gradient retention volumes obtained with the classical model (Eqs. (4), (7) and (9)) and empirical volume model (Eqs. (4), (10) and (11)), both with an initial isocratic elution with mobile phase 1, VCM = 60 mL and a dwell volume Vdw = 20 mL, tG = 25 min; VG = 50 mL. Step gradient (Eq. (15)) with an initial isocratic elution with mobile phase 1, VCM = 60 mL and a dwell volume Vdw = 20 mL.
Fig. 2. Modeled chromatograms of Table 1 compounds. VC = 200 mL; VM = 50 mL; VS = 150 mL; Sf = 75%; F = 2 mL/min. Linear gradient duration tG = 25 min. For both linear and step gradient the classical mode (CM) isocratic elution with mobile phase 1 was VCM = 60 mL and the dwell volume Vdw = 20 mL. Red line: mobile phase 1; blue line: mobile phase 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
286 mL with the classical model and only 282 mL with the empirical volume model. A linear gradient with tG = 25 min (VG = 50 mL) from mobile phase 1 to mobile phase 2 composition followed by an isocratic mobile phase 2 separation produced a full separation of all five compounds. The step-gradient experiment was modeled with the same initial conditions (VCM = 60 mL and Vdw = 20 mL) i.e., the separation is simply performed in two steps: (1) isocratic elution for 30 min at 2 mL/min with mobile phase 1 followed by (2) isocratic elution with mobile phase 2 till the full chromatogram developed (Fig. 2, bottom right). A significant overlap of compounds 3 and 4 peaks is seen. It is pointed out that the chromatogram comparison was done maintaining the same first step for both linear and step gradient Fig. 2 runs. This is doing the step change too early. The 3 and 4 overlap problem was mostly solved by doubling the initial isocratic step with mobile phase 1 using VCM = 120 mL. It obviously increased the analysis duration by 30 min (chromatogram not shown) but would allow for a complete separation of the 5 compounds with a much simpler experimental setting. The red lines in Fig. 2 (dark gray) correspond to mobile phase 1 and the blue lines (light gray) to mobile
phase 2. Taking in account the 20 mL dwell volume, the actual isocratic initial step lasts for 40 min in both gradient experiments. 4. Conclusion The first major conclusion of this work is that the countercurrent chromatography technique, with its support-free liquid stationary phase and immiscible liquid mobile phase, can be modeled using the long established chromatographic theory if the chemistry and chemical physics of biphasic liquid systems is known and understood. Another interesting conclusion is that solutions for the CCC gradient modeling developed independently with two different approaches produced mathematically two very different equations that turned out to both model the problem very correctly. The last conclusion is that the popular step-gradient method is also easily modeled and can produce very acceptable separations using an experimental setting simpler (i.e., a single pump) than that needed for linear gradient experiments. In all cases, the bottom line is that the biphasic liquid system used for gradient separations in CCC must be able to tolerate significant polarity changes in one
S. Wu et al. / J. Chromatogr. A 1274 (2013) 77–81
phase, the mobile phase, associated with minimum changes in the other phase, the stationary phase [13]. Acknowledgements SW thanks the National Science Foundation of China for Grant #20972136, Zhejiang Province for Grant #Y4080353 and the Fundamental Research Funds for the Central Universities #2010 QNA6006. AB thanks the French National Center for Scientific Research (CNRS) and University of Lyon 1 for continuous support with UMR5280 ISA (P. Toulhoat and P. Lanteri). References [1] W.D. Conway, Countercurrent Chromatography: Apparatus, Theory & Applications, VCH Publishers, Weinheim, Germany, 1990.
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