Experimental and theoretical investigation of high- concentration elution bands in ion-pair chromatography

The effective separation of many solutes, including pharmaceuticals, can be performed using an ion-pair reagent (IPR) in the mobile phase. However, chromatographic separation and mathematical modelling are a challenge in ionpair chromatography (IPC), especially in preparative mode, due to the complicated chromatographic process.

Journal of Chromatography A 1656 (2021) 462541 Contents lists available at ScienceDirect Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma Experimental and theoretical investigation of high- concentration elution bands in ion-pair chromatography Marek Les´ ko a, Jörgen Samuelsson a, Krzysztof Kaczmarski b,∗, Torgny Fornstedt a,∗ a b Department of Engineering and Chemical Sciences, Karlstad University, SE, 651 88 Karlstad, Sweden Department of Chemical Engineering, Rzeszów University of Technology, PL, 35 959 Rzeszów, Poland a r t i c l e i n f o Article history: Received 16 April 2021 Revised September 2021 Accepted September 2021 Available online 10 September 2021 Keywords: Ion pair chromatography Overloaded profiles Multilayer adsorption model Implicit model Electrostatic theory a b s t r a c t The effective separation of many solutes, including pharmaceuticals, can be performed using an ion-pair reagent (IPR) in the mobile phase However, chromatographic separation and mathematical modelling are a challenge in ionpair chromatography (IPC), especially in preparative mode, due to the complicated chromatographic process In this study, we present a retention mechanism and a mathematical model that predict overloaded concentration profiles in IPC using a system with X-Bridge C18 as stationary phase and tetrabutylammonium bromide in the - 15 mM concentration range as the IPR Two different mobile phases were used: (i) 15/85 [v/v] acetonitrile/water, (ii) 25/75 methanol/water The model compounds were sodium salts of organic compounds with sulfonic acid functions The analytical and preparative elution profiles were obtained for specified conditions The analytical data were utilized to calculate the difference in electrical potential between the surface and bulk solution using firm electrostatic theory In the preparative mode in a certain range of IPR concentrations, complicated U-shaped overloaded profiles were observed In the other considered cases, Langmuir overloaded elution profiles were recorded A multilayer adsorption model was derived, which is consistent with the dynamic ion exchange models The model assumes that lipophilic IPR adsorbs on the stationary phase, creating charged active sites that serve as exchange sites for the solutes The molecules of the solute can adsorb on the already formed IPR layer It was also assumed that a subsequent layer of solute can form on the formed layer of complexes due to interactions between the solute molecules The model takes into account the electrostatic attraction and repulsion of the molecules, depending on the considered situation The proposed model allowed prediction of the overloaded concentration profiles with very good agreement for the model solute and followed the progression from Langmuirian, through U-shaped, to again Langmuirian profiles © 2021 The Author(s) Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Introduction Ion-pair chromatography (IPC) was developed for almost 40 years ago for the purpose of separating polar charged compounds that otherwise are not retained on the hydrophobic RPLC surfaces In ion-pair chromatography (IPC) a lipophilic ion-pair reagent (IPR) is added to the mobile phase for separation of polar/charged solutes in high resolution reversed-phase liquid chromatography (RPLC) systems [1] The analytical mode of IPC quickly became popular, but the preparative mode of IPC (prep-IPC) has not gained as much popularity, one of the reasons being that the collected fractions will contain large amounts of IPR However, if the prepIPC method is the best alternative, as example for the next genera- ∗ Corresponding authors E-mail addresses: kkaczmarski@prz.edu.pl (K Kaczmarski), Torgny.Fornstedt @kau.se (T Fornstedt) tion drugs molecule class oligonucleotides [2-4] it is worth adding a final polishing step for removing the IPR with a final purification step IPC is frequently used for separating inorganic and organic ions, peptides as well as oligonucleotides, for both purification and analytical purposes, and the retention mechanism of IPC has been discussed for years [1, 5] Generally, two different underlying mechanisms are considered: (i) ion-pair formation occurs in the mobilephase eluent, binding the complex to the non-polar stationary phase [5, 6]; (ii) ion-pair formation occurs between the solute and lipophilic IPR bonded to the stationary phase [7] The simpler ionpair adsorption model developed by Schill et al also assumes that the ion pairs are adsorbed to the stationary phase but leaves open whether the ion pairs are formed in the mobile phase or during the adsorption process [8] To describe these mechanisms, stoichiometric models, electrostatic models, or combined stoichiometric– electrostatic models have been proposed [9-12] https://doi.org/10.1016/j.chroma.2021.462541 0021-9673/© 2021 The Author(s) Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) M Le´sko, J Samuelsson, K Kaczmarski et al Journal of Chromatography A 1656 (2021) 462541 Stoichiometric models use simple ideal equilibrium theory to estimate the formation of ion pairs in solution or on the stationary phase [8, 9] Even though stoichiometric models lack firm foundations in physical chemistry, they can still be used to predict the retention rather well The non-stoichiometric models adopt different versions of the Stern–Gouy–Chapman theory regarding electrical double layers formed by a surface excess of adsorbed IPR ions on the stationary phase The solute is attracted by the charged interface, in that way increasing the retention [10 - 12] Purely electrostatic models disregard the ion-pairing process in the bulk eluent; these models are also often referred to as dynamic ion-exchange models A universal retention mechanism of IPC that can explain all phenomena taking place during the chromatographic process would be highly desirable Unfortunately, due to the complexity of the chromatographic process, such a model has not yet been proposed Most IPC studies concern the analytical mode of IPC, but only a few describe the overloaded or preparative mode As far as we know, there is a particular lack of studies addressing the modeling of overloaded concentration profiles Such models would be highly desirable because they could be used in the rationalization and optimization of chromatographic resolutions IPC has become increasingly important in recent years because of the trend toward biological drugs in the pharmaceutical area, and much larger complex molecules often require the use of one or another IPR in the eluent It has previously been shown how additive components, if more or less strongly adsorbed, can generate the strangest peak deformations in analytical and especially preparative liquid chromatography [13] These effects are difficult to predict even when neutral components have been separated and a rule of thumb was developed to avoid the effects in separations of neutral components in liquid chromatography [13] as well as in the more complex mode, supercritical fluid chromatography [14] There are strong indications these very strange deformations are exacerbated in IPC [13] and with fundamental studies such as this one, in the future we might be able to give guidance on avoiding such peak deformations in IPC The unusual band shapes were previously observed in the adsorption of Tröger’s base enantiomers on the chiral stationary phases in HPLC [15] and in TLC for organic acids [16] The results of these studies were successfully interpreted by a multilayer adsorption model There are indications that the strange band shapes with round U shapes observed in IPC might also result from multilayer adsorption For overloaded cases, as in IPC used in production, the simple models generally used often neglect all types of ion-pair formation [17] Recently, overloaded elution profiles for peptide separation were successfully described by also considering ion-pair formation in the mobile phase [18] However, the model could only handle single charge-peptides; for three-charged peptides, the problem become too complicated The aim of this study is to develop a model for predicting overloaded concentration profiles in IPC For this purpose, a model chromatographic system was used to separate organic sulfonic acids using as tetrabutylammonium bromide as the mobile phase IPR The study consists of an analytical and overloaded section; in the former section the solutes retention as well as the columns surface potentials are investigated as a function of the IPR In the overloaded section, models for the prediction of the overloaded concentration profiles was developed and validated mathematical models allowing prediction of the retention factor [5-12] These models have their sources in physical chemistry and usually use the concept of the difference in electrical potential between the surface and bulk solution In preparative mode, in addition to predicting the retention, it is also very important to predict the shape of the overloaded concentration profiles, because optimal separation conditions are achieved when band profiles overlap each other to some degree [19] Thus, modeling the preparative mode is much more difficult than modeling analytical chromatography due to the nonlinear relationship between the solute adsorbed and dissolved in the mobile phase and other competition occurring at high solute concentrations Moreover, unlike analytical chromatography, in which the appropriate equation for calculating the retention factor is usually derived, in this mode, a dynamic column simulation model must be connected with an appropriate description of the adsorption/desorption process This is especially challenging in IPC, whose multicomponent system, usually with a complex mechanism, requires a sophisticated mathematical description Section 2.1 briefly describes one of the most popular electrostatic theories, based on the Gouy–Chapman theory, for the interpretation of analytical retention times in IPC [20] Section 2.2 deals with the basic part of the study: predicting the overloaded concentration profiles in IPC; the assumptions, derivation, and mathematical model of the preparative mode will be presented there in detail 2.1 Analytical ion-pair chromatography According to the electrostatic model of Ståhlberg et al [11, 20], based on the Gouy–Chapman theory, the retention factor of a charged solute, E, in the presence of an ion-pair reagent, H, can be described using this relationship: k = φ exp − G0E + zE F ψ RT (1) where φ is the phase ratio, zE F ψ the electrostatic energy, G0E the free energy of adsorption without the electrostatic energy, T the temperature, R the gas constant, F Faraday’s constant, zE the charge of the solute, E, and k the retention factor of compound E The general idea of this model is that the adsorption of the charged solute on the stationary phase depends on two factors: (1) the electrostatic attraction to or repulsion from the surface of the stationary phase, which is governed by the electrostatic potential of the surface and the charge of the ionic form of the solute; and (2) the free energy of adsorption when the electrostatic potential equals zero Thus, using Eq (1), it is possible to define the retention factor, k0 , at a reference composition of the IPR in the mobile phase, with the difference in electrostatic potential between the mobile and stationary phases being set to zero In this condition, the retention factor, k0 , is equal to φ exp(− G0E /RT ) Eq (1) can be rearranged so that it is possible to estimate the electrostatic potential as a function of the retention factors of solutes: = RT k0 ln zE F k (2) In the above discussed model the formation of ion-pairs in solution is not considered and according to Ståhlberg and Häglund this is a valid simplification when using tetrabutylammonium ion as IPR [12] Theory 2.2 Overloaded ion-pair chromatography In analytical mode, the most important thing is to predict the retention times of the eluted solutes There are several theories concerning the interpretation of the mechanism of IPC and the For correct modeling of the overloaded concentration profiles in the IPC case, an appropriate model is required As in classical M Le´sko, J Samuelsson, K Kaczmarski et al Journal of Chromatography A 1656 (2021) 462541 Fig Schematic illustration of the equilibria (a) considered when modeling the adsorption mechanism of SBS in the presence of the ion-pairing reagent (IPR) according to the multilayer adsorption model (b) The IPR, H, adsorbs to the surface of the stationary phase L, while the compounds of the solute, E, adsorb to already adsorbed H or is the ratio of the adsorption sites form the next layer of E The excess of anions over IPR, X, can adsorb to the layer of H K denotes the association equilibrium constant; occupied by an appropriate combination of compounds to the saturation capacity nonlinear chromatography, phenomena generally neglected in linear chromatography can be important, phenomena such as lateral interaction between adsorbed solutes or multilayer adsorption [16, 19] In addition, in IPC, complexes between the IPR and solute can form in the mobile phase and then adsorb We tested several models and found the best agreement between experiment and theory when multilayer adsorption was assumed However, including in the model the possibility of complex formation between IPR and solute did not improve the model’s accuracy On the other hand, a model that neglected multilayer adsorption and accepted the complex formation failed to correctly model overloaded IPC in our case Here, a new 3-layer ion pair model is proposed The model is based on several surface adsorption-desorption reactions, see Fig 1a In formulating the equations of the model, we assume that the process of the ion-pairing takes place on the stationary phase and it is responsible for observed shapes of the overloaded concentration profiles This is consistent with postulated so-called dynamic ion exchange models [6, 7] which assume that the lipophilic ions first adsorb to the stationary phase and thus dynamically create charged active sites These sites serve as exchange sites for the solute ions In the following equation H, E and X are the IPR, the solute, and the excess of anions as compared to the IPR coming from the inorganic salt as the counter-ion for the IPR, respectively The following reactions are considered (see Fig 1a): (1) adsorption of H to the stationary phase L, (2) accumulation of E to the already established layer of H on the column, (3) the accumulation of X to the already established layer of H on the column and finally (4) the accumulation of E to the already established layer of HE on the column In other words, we have up to three layers on the surface of the stationary phase: first layer of H second layer of E on the H-layer and X on H-layer and third layer of E on the HE-layer The schematic illustration of the model is presented in Fig 1b Moreover, on the base of the electrostatic theory, it was assumed, that affinity of adsorption of the H is decreasing with the increase of its concentration in the stationary phase It is caused by electrostatic repulsion of the positively charged compound off the positively charged layer Similarly, the model assumes that the affinity of the adsorption of the E and X is increasing with the increase of the H concentration on the stationary phase It is due to the electrostatic attraction of oppositely charged ions of solute to the dynamically formed ion pair layer on the stationary phase Under the above assumption the ratio of the free adsorption sites to the saturation capacity can be expressed by the following equation: =1− H − HE − HE2 − HX (3a) where: H , HE , HE2 and HX are the ratios of the sites occupied by the: H, first layer of E on the H-layer, two layers of the E on the H-layer, and X on the layer of the H-layer to the saturation capacity, respectively Here, the electrostatic contribution is handled by modifying the equilibrium constants and introducing a M Le´sko, J Samuelsson, K Kaczmarski et al Journal of Chromatography A 1656 (2021) 462541 new electrostatic modified equilibrium constant Ti TH = KH exp(−S1 H ) (3b) THE = KHE exp(S2 H ) (3c) THX = KHX exp(S3 H ) d qE d = qs ( dt dt d qX d = qs dt dt (3d) THE2 = KHE2 The constants Ki are the equilibrium constants (see Fig 1a) of the adsorption processes for each considered cases to model the repulsion constants S1 , S2, and S3 are introduced these expressing the strength of the repulsion or attraction of the IPR, solute, and anions adsorption, respectively Observe that the adsorption constant THE2 does not contain any exponential factor, because the complex HEL is uncharged and would not repell futher adsorption of the the solute E The retention mechanism for multi-layer adsorption using the definitions described using Eqs (3) can be described as mass balances for a specific layer In all equations additional layers are going to be described using Langmuiur adsorption model or electrostatic modified Langmuir adsorption isotherm, that is using the electrostatic equilibrium constants described in Eqs (3b-3d) For – only IPR can adsorb due to hydrophobic interaction − k−H H ) = −kH CH H − H H = (kHCH dt − k−H −(kHXCX H − k−HX −kHE CE H − H ) − (kHECE HX HE ) = kH CH − kHX CX THE H − k−HE − H − HE HX H HE HE dt = (kHECE = kHE CE for d HE HE2 dt and H H − k−HE HE − HX HE ) − (kHE2 CE − kHE2 CE THE HE − k−HE2 HE − HE2 HE2 ) HE − k−HE2 HE2 ) = kHE2 CE HE − HE2 THE2 HX dt = kHX CX H − HX (7) H+ HE + HE2 + HX ) = CX THX (15) HE (16) H = HE2 =0 (17) = + CH, H = H, HX + CX, inlet TH follows from the solution of the set inlet THXCH,inlet TH −1 (18) 0CH,inlet TH (19) = HCX,inlet THX (20) H + HX ) qE (t = ) = In the above equations CH , CE, and CX are the concentration of the H, E and X in the mobile phase, respectively In Fig 1a the adsorption equilibrium constants (KH , KHE , …) for all the reactions are presented Similary we could define the rate constant for the adsorption process is ki (kH , kHE , …) and the rate constant for the desorption process is k-i (k-H , k-HE , …) Please, observe that Ki = ki /k-i The concentration change of the adsorbed compounds (qi ) are: d qH d = qs ( dt dt = CE THE2 qH (t = ) = qs ( (8) THX (14) H where CH, inlet and CX, inlet are the concentration of H and X introduced with the mobile phase into the column inlet, respectively Finely the initial conditions are: and d = CE THE HX – adsorption/desorption of solute = (kHE2 CE (12) and (6) THE2 −1 (13) Initial value of , of tree equation: (5) for the HE , – on adsorbed in second layer solute the next solute particle can adsorb due to hydrophobic interaction d (11) It should be noticed that above set of equations are implicit where H is a function of H , see Eq (3b) In others words, the amount of adsorbed IPR depends on the IPR already adsorbed on the surface of the stationary phase due to repulsion from the charged layer The model requires to define the initial condition For t = 0, it was assumed equilibrium: TH HX (10) HX = CH TH HE2 ) THX ) ) + TH cH + TH THE cH cE + TH THX cH cX + TH TE THE2 cH cE2 = HE for: H – we must consider H, E and X adsorb due to ion-ion interaction d HE2 The expression for each case of the occupying active places by the considered components are: (4) TH +2 Assuming infinitely fast surface pseudo-reaction process, aka the steady state approximation, the left-hand sides of Eqns (4 - 8) are equal to zero From these equations connected with Eq (3a), after simple algebraic manipulations we obtain: (3e) d = −(kHCH dt HE qX (t = ) = qs ( (21) (22) HX ) (23) The kinetic adsorption model was solved with transport dispersive model [19]: ∂ Ci u ∂ Ci − εt ∂ qi ∂ 2Ci + + = DL ∂ t εt ∂ t εt ∂ t ∂x (24) where u (m s−1 ) is the superficial velocity, εt is the total porosity, DL (m2 s−1 ) is the axial dispersion coefficient (9) M Le´sko, J Samuelsson, K Kaczmarski et al Journal of Chromatography A 1656 (2021) 462541 Experimental enough adsorption rates, i.e., kH , kHE , kHX , and kHE2 , is equivalent to solving the ED model using the implicit isotherm In our case, for adsorption rates greater than about 10 0, the solutions of Eqs (24) and (9)–(11) were almost the same It should be also noted that the typical initial and boundary conditions used in simulating the column dynamics had to be modified in our case It was assumed that the concentration of E in the mobile phase equals zero at the initial time, so the surface concentrations HE and HE2 also equal zero at the initial moment The values of the surface concentrations , H , and HX at t = followed from the concentration of H and X in the mobile phase in the steady state The presented model contains eight parameters (qs , KH , KHE , KHE2 , KHX , S1 , S2 , S3 ) estimated based on four or five overloaded concentration profiles depending on the considered case The concentration profiles chosen during the estimation covered different IPR concentrations and peak shapes so that the estimations always included Langmuir-type peaks obtained at the lowest IPR concentration in the mobile phase, U-shaped peaks, and two or three Langmuir-type peaks obtained at the higher concentration of IPR The estimation was done based on concentration profiles obtained at the highest injection volume The parameters of the model were estimated with the inverse method using the stochastic simulated annealing method [23] with the minimum value of temperature established on 0.001 and accuracy of optimum calculation at 10−5 In the considered cases, due to the many existing local minimums, faster deterministic methods such as the Levenberg–Marquardt algorithm not give satisfactory results; therefore, the stochastic method was found to be more effective in this case 3.1 Chemicals and column The mobile phase consisted of gradient-grade MeOH purchased from Fisher Scientific (Loughborough, UK) or gradient-grade MeCN purchased from VWR International (Radnor, PA, USA) and deionized water with a resistivity of 18.2 M cm from a Milli-QPlus 185 water purification system (Merck Millipore, Darmstadt, Germany) As solutes, we used sodium salts of the sulfonic acids: sodium benzenesulfonate (SBS), p-toluenosulfonic acid monohydrate (SPTS), sodium 2-naphtalenesulfonate (S2NS), 1,5-naphthalenedisulfonic acid disodium salt (S15NS), and 2,6-naphthalenedisulfonic acid disodium salt (S26NS) The IPR used in this study was tetrabutylammonium bromide (TBuABr) The constant ionic strength of the mobile phase was maintained by adding sodium chloride All solutes, the IPR and salts were purchased from Sigma-Aldrich (St Louis, MO, USA) The column was a 100 × 2.1 mm XBridge C18 from Waters Corporation (Milford, MA, USA with an average particle diameter of 5.0 μm According to the vender, in the unbound phase the average pore diameter was 147 A˚ and the surface area was 192 m2 g–1 3.2 Instrumentation and procedure Experiments were performed on an Acquity H-Class Bio System from Waters (Milford, MA, USA) with a quaternary solvent manager, an autosampler with a 50-μL flow-through needle, a stillair column oven equipped with a mobile phase pre-heater, and a PDA detector with a 500-nL flow cell The extra volume from the autosampler to the detector was 23 μL Two different mobile phases were used: 1) 15/85 [v/v] MeCN/water, and 2) 25/75 [v/v] MeOH/water Tetrabutylammonium bromide was chosen as the IPR The concentrations of the IPR in both mobile phases were: 0, 1, 1.5, 2, 3, 5, 7.5, 10, 12.5, and 15 mM The constant ionic strength of the mobile phases was maintained at 15 mM by adding an appropriate amount of sodium chloride The final prepared mobile phase was filtered through a 0.20-μm filter before use The solutes were dissolved in the mobile phase at concentrations of 0.0010 g L–1 and 1.0 or 1.1 g L–1 for analytical and preparative injections, respectively The injection volume was μL for analytical injections; for the preparative investigations, the injected volumes were 10, 20, and 30 μL The signal was detected at 210 and 250 nm At least two replicates were always made The mobile flow rate in all experiments was 0.25 mL min–1 The temperature of the column oven and the heat exchanger prior the column inlet was 25°C Results and discussion The aim of this study is to develop theory and methods allowing prediction of the retention and shape of the overloaded concentration profiles in IPC For this purpose, a model chromatographic system was considered in which several sodium salts of the sulfonic acids were eluted, with the mobile phases containing tetrabutylammonium bromide as the IPR The analytical and overloaded concentration profiles were obtained for the studied chromatographic system The study has three parts: analytical and overloaded Section 4.1 presents the analytical mode, in which the surface potential for a considered experimental condition is estimated In Section 4.2, the overloaded concentration profiles are presented at various IPR concentrations In Section 4.3, the results for the overloaded concentration profiles predicted using the mathematical model are presented in the SBS case and the profiles shapes are investigated The particular aspects and challenges of modeling preparative ion pair chromatography are also discussed there 3.3 Method of estimating the model parameters and simulation The model for predicting the overloaded concentration profiles was solved using the method of orthogonal collocation on the finite element described by Kaczmarski et al [21] As discussed in Section 2.2, it should be noted that the proposed model is implicit because the amount of adsorbed H depends on the amount of H already adsorbed Thus, the model requires application of the appropriate procedure for solving the equations Detailed information about solving such problems can be found in Kaczmarski et al [22] To simulate the column dynamics, Eq (24) (see Section 2.2) must be coupled with an appropriate equation that describes the equilibrium between the concentrations of solute in the mobile and stationary phases; this equation can also be the implicit equation If the rate of the adsorption–desorption process is infinitely fast, then the model is known as the equilibrium–dispersive (ED) model It is worth emphasizing that solving the ED model using implicit isotherm equations is not easy However, the solution of Eq (24) using adsorption kinetics, i.e., Eqs (9)–(11), for large 4.1 Analytical investigation The variation in the retention factor of the studied solutes with the amount tetrabutylammonium bromide (TBuABr) in the eluent is presented in Fig 2a for MeOH containing eluent (25/75 [v/v] MeOH/water) and in Fig 2c for MeCN containing eluent (15/75 [v/v] MeCN/water) NaCl was added to the eluents in order to keep the ionic strength constand in all experiments, see Section 3.2 for details As can be seen, in both Fig 2a and c, the retention increased with increasing the IPR in the mobile phase according to a parabolic curvature, as expected The increase in solute retention depends on the solute’s characteristics, mainly on the solute lipophilicity and the number of charges that the solute have in ionized form The smallest retention was observed in the SBS case Increasing the solute lipophilicity increased the retention SPTS had M Le´sko, J Samuelsson, K Kaczmarski et al Journal of Chromatography A 1656 (2021) 462541 Fig The dependence of the: a) retention factor and b) surface potential as function of concentration of IPR in the mobile phase The mobile phase was 25/75 [v/v] MeOH/water Subplots c) and d) are the same as those in a), and b) except the mobile phase was 15/85 [v/v] MeCN/water higher retention than did SBS, while S2NS was the strongest retained because it had the highest lipophilicity of the solutes considered here S15NS and S26NS were similar, displaying almost the same increase in retention factor as when increasing the IPR in the mobile phase S15NS and S26NS differed from S2NS only in having one more sulfone group, which significantly reduced their lipophilicity Moreover, SPTS and especially S2NS displayed simultaneous adsorption onto the stationary phase together with the IPR The other studied compounds were very weakly retained, using only the mobile phase without the IPR Using Eq (2) we can estimate the surface potential for all these experiments Note, that the retention of some solutes at zero IPR concentration cannot be used, due to the weak adsorption of these solutes to the phase without IPR A small error in the reading of k0 will manifest itself in large errors in the estimated surface potential On the other hand, due to the nonlinear nature of the adsorption of IPR, selecting a reference state at a high IPR concentration would not properly represent the surface potential To solve these issues, we calculated the surface potential at low concentrations using a solute that has good retention, S2NS Then we used this surface potential in estimating k0 values for the rest of the solutes The estimated surface potential is plotted in Fig 2b and d for the mobile phase containing MeOH and MeCN, respectively For both the MeOH and MeCN cases, we can see that the surface potential is more or less the same for all solutes This indicates that the electrostatic model describes the retention well for the systems under investigation This also indicates that ion-pair formation in solution in our cases are not crucial to describe the retention of the solutes to approximately mM IPR in the mobile phase, the concentration profiles were U shaped Above this concentration, the profiles were again Langmuirian In the case of U-shaped profiles, the left part of the peak decreased while the right part increased with increasing IPR concentration in the mobile phase This phenomenon was observed for all solutes The only difference was in the IPR concentration range in which the U-shaped profiles appeared, and this was connected with the retention of the solutes For solutes more retained than SBS, for example, SPTS, the U-shaped profiles could be observed at lower concentrations of IPR, and then at still lower concentrations, they again became Langmuirian In the case of S2NS (not shown), characterized by very high retention in comparison with the other compounds, the shape was Langmuirian throughout almost the entire studied IPR concentration range Only at 0.5 mM IPR could a small distortion of the concentration profile be observed, with U-shaped peaks appearing between and 0.5 mM IPR in the mobile phase In the case of S15NS and S16NS, the concentration profile distortion was observed at a slightly higher IPR concentration than for SBS; in this case, the solute retention was lower than for SBS at a lower IPR concentration in the mobile phase 4.3 Prediction of overloaded profiles - multilayer adsorption model The interesting change in the profile shape was the basis for deriving and validating the mathematical model for predicting the overloaded concentration profiles The considerations were started from SBS which retention is negligible at mM concentration IPR.The main idea of the postulated model presented in Section 2.2 is the assumption that ion pairs are created on the surface of the stationary phase and further interact with the molecules of the solute, which leads to multilayer adsorption This is consistent with what is postulated in the dynamic exchange models in the literature [6, 7] dynamic exchange models We have also tested several different models and found that the best agreement between experiment and theory is obtained when multilayer adsorption is assumed, see Fig 1a The proposed model does not include ion pair formation in the mobile phase which indicate that accouting this phenomenon should not improve the prediction of the overloaded profiles The models that were investigated and failed to describe the elution profiles were: (1) adsorption of the IPR to the phase followed by adsorption of the solute 4.2 Overloaded concentration profiles Overloaded concentration profiles were obtained for all considered solutes and mobile phases The experiments were conducted at several IPR concentrations spanning from 0.5 to 15 mM with constant ionic strength of the mobile phase (see Section 3.2) Fig presents examples of the overloaded concentration profiles obtained for SBS, SPTS, and S15NS using a methanol/water mobile phase containing 0.5, 1.5, 2, and mM tetrabutylammonium bromide As can be seen, the shape of the overloaded concentration profiles changed with increasing IPR concentration in the mobile phase In the case of SBS with a small concentration of IPR (about 0.5 mM), the concentration profiles were Langmuirian in shape Up M Le´sko, J Samuelsson, K Kaczmarski et al Journal of Chromatography A 1656 (2021) 462541 Fig Overloaded concentration profiles of SBS in first row (a -d), SPTS, in second row (e-h), S15NS in third row (i –l), were obtained at 0.5, 1, and mM ion-pair reagent (IPR) The examples cover the different shapes of the obtained profiles, starting from Langmuirian profiles, through U-shaped profiles, to Langmuirian profiles once again with increasing IPR concentration in the mobile phase The injection volumes were 30, 20, and 10 μL; the mobile phase was 25/75 [v/v] MeOH/water Fig Experimental (solid lines) and calculated (dotted lines) concentration profiles of SBS eluted with 25/75 [v/v] MeOH/water containing: a) 0.5, b) 1, c) 1.5, d) 2, e) 5, f) 10 mM ion-pair reagent in the mobile phase The injection volume was 30 μL The parameters were estimated based on the concentration profiles obtained at IPR concentrations: 0.5, 2, 5, and 10 mM to the already adsorbed IPR (2) as (1) but also considering the adsorption of X to the established H-layer and (3) as (1) but with an additional layer of solute layer formed on the already established HE-layer on the column The lattar is without considering the adsorption of X to the established H-layer It could be concluded that witthout including the second solute layer as well as the adsorption of X to the established H-layer, it is impossible to predict correctly overloaded concentration profiles Fig presents the final results of predicting the overloaded concentration using the discussed model The figure shows overloaded concentration profiles of SBS obtained for MeOH eluent Even though the experiment was conducted using IPR concentrations up to 15 mM in the final estimations of the model parameters, only concentration profiles obtained for IPR concentrations up to 10 mM were taken (see Table S1 in Supplementary mate- rials) In this range (0.5–10 mM), the model predicts overloaded concentration profiles of SBS that agree very well with experimental results The analogous results displaying the same agreement with experimental results in predicting the concentration profiles of SBS eluted with MeCN eluent can be seen in Fig S1; the model parameters are presented in Table S2 (see Supplementary materials) It should be noted, that the model predicts complicated U – shaped profiles (see Fig and Fig S1) and can follow from Langmuirian peaks to Langmuirian peaks through U – shaped peaks without any change in the mathematical equations and values of its parameters This very valuable property of the model emphasizes its correctness, in our opinion However, an unfavorable feature of the model is the lack of very good agreement or difficulties in finding model parameters when modeling include the highest M Le´sko, J Samuelsson, K Kaczmarski et al Journal of Chromatography A 1656 (2021) 462541 Fig The adsorbed concentration profiles of: a) ion-pair reagent (H), b) SBS (E), and c) X as a function of the H and E concentrations The mobile phase was 25/75 [v/v] MeOH/water IPR concentrations, i.e., 12.5 and 15 mM, see Fig S2 in Suplementary materials In general, the model cannot accurately describe the concentration profiles in the highest IPR concentration region and there is a deterioration in prediction in all regions of IPR when the consideration includes the H concentration from to 15 mM The reason for this is probably that existence of a secondary sites, i.e the adsorption of IPR on the C18 stationary phase is somewhat more complicated with heterogenous adsorption Here, the adsorption is derived using electrostatic modified Langmuir models, see Section 2.2 Previous result have shown that often the adsorption of a charged molecules is best described using a more heterogeneous model [24] It should be noted that tetrabutylammonium bromide has four alkyl chains The model does not recognize the way of the adsorption The IPR can adsorb on the C18 phase in several different ways, and this probably dominates more at higher IPR concentrations Based on the equations, the description appears complex, but it is nevertheless insufficient to model such complicated processes as IPC The proposed model predicts overloaded concentration profiles in the SBS case with acceptable agreement with experimental results and can be useful in modeling other single charged com- pounds which are unretained or only slightly retained without any IPR present in the eluent It should be noticed that analyzed problem in prediction of concentration profiles is difficult due to unusual shapes The proposed model should also give good agreement in the cases when usual shapes of the concentration profiles are observed In the case there would be problems with the accuracy of the prediction of concentration profiles, for examples when difficult shapes would be considered or wide range of IPR concentration will be included, the model can be used separately for different concentration of IPR In such way an almost the perfect agreement between experimental and simulated concentration profiles was obtained for SBS Thus, the correlation of the model parameters with concentration of IPR can be done and increase the applicability of the model The consideration was limited to the SBS case The other compounds (i.e., SPTS, S2NS, and S15NS) display retention without IPR in the mobile phase or are two changed, which entails changing the assumptions and rewriting the model; this case is not considered here Fig shows the corresponding adsorbed isotherm for the MeOH eluent Fig 5a) for H, Fig 5b) for SBS (E) and Fig 5c) for X is M Le´sko, J Samuelsson, K Kaczmarski et al Journal of Chromatography A 1656 (2021) 462541 plotted as a function of the concentration of H and E in the mobile phase The adsorption isotherm were calculated using the determined model parameters presented in Table S1 The analogical figure for the MeCN eluent is presented in Fig S3 (see Supplementary material) As can be seen, the calculated amount of adsorbed H is drastically increased with the increase of H in the mobile phase (see Fig 5a) It takes place up to approximately mM of the H in the eluent then beyond this range the increase is leveled out more This is caused by saturation of available active sites and the repulsion of H molecules from the adsorbed H layer The adsorption of H increase also with the increase of the E concentration in the mobile phase This is observed because an adsorption of E, will decrease the: H As mentioned before H is responsible for the repulsion, see Eq (3b) of H molecules thereby inhibiting the ability to H adsorption on the stationary phase Fig 5b shows the plot analogous to that in Fig 6a, but this time the adsorption of SBS is considered In this case, the amount of adsorbed solute increased with the increasing concentrations of IPR and solute in the mobile phase Here the explanation is rather obvious: the increasing amount of SBS in the mobile phase caused the increase in its adsorption, while the increasing number of H molecules in the mobile phase caused the increase in the H layer on which the solute could adsorb Fig 5c presents the adsorption of X As can be seen, the adsorption of X increased with the increasing IPR concentration due to the increasing H layer on the stationary phase; thus, more sites became available for X The drastic increase in X adsorption could be observed at the lower IPR concentration in the mobile phase and was connected with the abovediscussed H adsorption in this region The adsorption isotherm for X was more or less constant, and started decreasing with increasing IPR in the mobile phase This was due to the decreasing growth in available adsorption sites on the surface of adsorbed H molecules as well as the decreasing concentrations of X in the mobile phase Note that the model takes into account only the excess of negatively charged ions over the IPR concentration, so for a high concentration of IPR there is a low concentration of X As can also be seen, the number of adsorbed anions decreased with the increasing SBS concentration, due to increased solute adsorption and fewer available sites on the H layer It should be also emphasized that the analyzed problem is very complicated due to the solution of the model and estimation of its parameters As stated above, the mathematical model is implicit The amount of adsorbed IPR depends directly on its already adsorbed amount This requires the use of a special method for solving the model Moreover, estimating the model parameters requires the use of the stochastic simulated annealing method due to existence of many local minimums ized form We found that a simple electrostatic model based on the Gouy–Chapman theory that does not consider ion pair formation in solution could be used to describe the retention The determined electrostatic potential as a function of IPR concentration were found to be larger in the MeCN case, resulted in a more sensitive system to IPR changes especially for solutes with multiple charges In contrast to the analytical mode, in which typical and expected IPC results were obtained, the overloaded mode resulted in interesting overloaded concentration profiles The shapes of the overloaded concentration profiles changed with increasing IPR concentration in the mobile phase In the case of SBS with a low concentration of IPR (about 0.5 mM), the concentration profiles were Langmuirian in shape Then, up to approximately mM IPR in the mobile phase, the concentration profiles were U-shaped Above this concentration, the profiles were again Langmuirian This pattern was observed for all solutes, which differed only in the IPR concentration range in which the U-shaped profiles appeared; this range was directly connected with the retention of the solutes To predict the overloaded concentration profiles, a multilayer electrostatic modified adsorption model was derived The model assumed that lipophilic IPR would adsorb on the stationary phase, creating the charged active sites serving as exchange sites for the solute The solute would then adsorb on the already formed IPR layer It was also found that subsequent adsorption layer of solute could form on the already formed complex due to hydrophobic interaction between the solute molecules The excess of the anions over the IPR had to be included in the model equations The electrostatic contribution were account for by introducing electrostatic attraction and repulsion of the molecules, depending on the considered situation The prediction was limited to SBS, for which good agreement with experimental results was obtained Here, we have demonstrated that the applied methodology and mathematical model, including the proposed description of the adsorption/desorption process, led to good agreement between simulated and experimental results The model allowed us to predict overloaded concentration profiles for SBS with good accuracy over a wide range of IPR concentrations in the mobile phase in the studied cases Conclusion Marek Les´ ko: Conceptualization, Methodology, Investigation, Validation, Formal analysis, Resources, Writing – original draft, Writing – review & editing, Visualization Jörgen Samuelsson: Conceptualization, Methodology, Validation, Formal analysis, Writing – review & editing Krzysztof Kaczmarski: Conceptualization, Methodology, Investigation, Software, Validation, Formal analysis, Data curation, Writing – review & editing, Supervision Torgny Fornstedt: Conceptualization, Data curation, Writing – review & editing, Supervision, Project administration, Funding acquisition Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper CRediT authorship contribution statement In this study we considered ion-pair chromatography (IPC) Attention was focused on numerically modeling the overloaded concentration profiles to develop the models and tools needed for this type of chromatography The mobile phase consisted of two eluents (i.e., 25/75 [v/v] MeOH/water and 15/85 MeCN/water) containing the ion-pair reagent (IPR), which was tetrabutylammonium bromide The concentration of IPR in the mobile phase ranged from to 15 mM while the ionic strength was kept at a constant 15 mM by addition of NaCl to the eluents The model solutes were the sodium salts of sulfonic acids and a XBridge C18 column was used The retention factors of the eluted solutes increased with increasing the IPR in the mobile phase according to a parabolic curve, as expected The increased retention of the solute was dependent on the nature of the solute: i.e., on the solute’s lipophilicity and on the number of charges that the molecules had in ion- Acknowledgements This work was supported by the Swedish Knowledge Foundation via the KKS SYNERGY project “BIO-QC: Quality 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(14) H where CH, inlet and CX, inlet are the concentration of H and X introduced with the mobile phase into the column inlet, respectively Finely the initial conditions are: and d = CE THE HX... the amount of adsorbed solute increased with the increasing concentrations of IPR and solute in the mobile phase Here the explanation is rather obvious: the increasing amount of SBS in the mobile... theories concerning the interpretation of the mechanism of IPC and the For correct modeling of the overloaded concentration profiles in the IPC case, an appropriate model is required As in classical