Modern analytical applications of liquid chromatography require more and more efficient columns. In this work, the possibility of utilizing particle size gradient in the chromatographic column was studied by a theoretical approach. In the course of our work three different scenarios of particle size gradients were considered with different shapes (linear, convex and concave).
Journal of Chromatography A 1651 (2021) 462331 Contents lists available at ScienceDirect Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma Theoretical study of the efficiency of liquid chromatography columns with particle size gradient Szabolcs Horváth a, Fabrice Gritti b, Krisztián Horváth a,∗ a b Research Group of Analytical Chemistry, University of Pannonia, Egyetem utca 10, Veszprém H-8200, Hungary Waters Corporation, 34 Maple Street, Milford, MA 01757, United States a r t i c l e i n f o Article history: Received 24 July 2020 Revised June 2021 Accepted June 2021 Available online June 2021 Keywords: Particle size gradient Physical bandwidth Efficiency Isocratic Gradient elution modes Stationary phase design a b s t r a c t Modern analytical applications of liquid chromatography require more and more efficient columns In this work, the possibility of utilizing particle size gradient in the chromatographic column was studied by a theoretical approach In the course of our work three different scenarios of particle size gradients were considered with different shapes (linear, convex and concave) The evolution of bandwidth inside the column was plotted for each scenario As a reference point, the bandwidth of the uniform column was used, which had the same pressure drop as the non-uniform column According to our calculations, in isocratic elution mode, the non-uniform column does not offer any advantage compared to the uniform column, regardless the type of the particle size gradient In gradient elution mode, however, extra band compression occurs was found For negative particle size gradients, the final physical bandwidth was found to be approximately 1–4 % smaller than for uniform columns This slight gain in efficiency in terms of bandwidth compression can be expanded to 5–8 % by the optimization of the limiting particle sizes These optimized results are obtained when the final particle size is approximately 40% of the initial particle diameter © 2021 The Authors Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Introduction For complex samples, single column liquid chromatography often does not provide separations with adequate resolution and analysis time [1] There are several approaches that combine stationary phases to improve chromatographic performance The use of parallel columns is the simplest approach: the analysis is performed in two runs The second run has the aim to resolve the overlapped compounds in the first run A rather sophisticated software is often required to find true complementary situations that give adequate resolution of all analytes [2,3] Stationary phases of mixed-mode columns exhibiting intermediate selectivity can be prepared by combining different separation mechanisms, such as reversed phase or size exclusion, and cation or anion exchange Mixed-mode columns manifest the properties of each constituent stationary phase By changing their ratio, the selectivity can be adjusted [4–6] However, mixed-mode column approach might have some difficulties First, dedicated columns are needed Second, the interaction of different types of stationary phases with each other can give rise to complex non-linear retention behaviours [7] ∗ Corresponding author Tandem columns in a multi-dimensional configuration can be an alternative to mixed-mode columns In this approach, the elution of the sample goes through two or more tandem (i.e., subsequent) columns, each one packed with a specific stationary phase The separation in the second dimension can involve all the fractions collected in the first dimension (comprehensive approach) [8–10], or only selected fractions (heart-cutting approaches) [11– 14] Along with the advantage of combined utilization of different retention mechanisms, two-dimensional LC offers the opportunity to manipulate independently the experimental conditions (eluent composition, gradient program, flow rate, and temperature), for each particular column Main disadvantages are the considerable sample dilution that may occur and the longer analysis times In addition, optimization in two-dimensional LC is not straightforward due to a large number of interdependent factors[1] A technically more simple design is the tandem columns in a serial configuration The idea is to connecting directly two or more columns in series and flowing the same mobile phase (or gradient) through all columns With a given serially coupled column combination, a quasi-new column is obtained, which in some instances have much better performance, than the individual columns [3] The idea of coupling the columns already arose at the dawn of the history of chromatography when analyzing complex sam- https://doi.org/10.1016/j.chroma.2021.462331 0021-9673/© 2021 The Authors Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) S Horváth, F Gritti and K Horváth Journal of Chromatography A 1651 (2021) 462331 of bands of compounds inside the column, σz2 , can also be determined by solving the following ordinary differential equation [33]: ples [15–18] In the case of column coupling, on the one hand, the kinetic performance can be increased by increasing the column length, and on the other hand, the selectivity can be adjusted by combining the stationary phases with different chemistry [19] Some reports [18,20] also highlighted the increased number of drugs that could be monitored using coupled columns Besides the development of analytical procedures, an attempt was made to use serially coupled columns in preparative chromatography for separating multi-component mixtures [21] The authors reported an enhanced production by coupling two columns in series It should be finally commented that the analysis of samples containing mixtures of chiral and achiral compounds, by coupling chiral and achiral stationary phases, is another active field [22–26] The foundations of the kinetic theory of separation in an arbitrary linear medium (time-varying and / or non-uniform) were explained in details by Giddings [27,28] and Blumberg [29,30] In these studies, a detailed model for calculating the theoretical plate height in non-uniform medium is provided These theories also extend to the case of coupled columns and gas chromatography, taking into account the gas compression effects as well Fekete et al [19] developed a theory for both isocratic and gradient modes, to predict the peak widths for coupled column systems They concluded that in isocratic mode, the plate numbers are not additive when the columns possess different plate count, and kinetic performance has a limiting value which depends on the efficiency and length of the individual columns In gradient elution mode, the column order is no longer indifferent The observed apparent efficiency significantly depends on the column order In combinations, when the latter column has higher efficiency, a decrease in the peak width is predicted Therefore, the columns should be placed in order of increasing efficiency As previously seen, decreasing particle size along columns (in a coupled system) can result in extra peak sharpening effect in addition to the common gradient band compression [19,31] In a recent work, Codesido et al [32] studied the potential of columns packed with particles of decreasing size (particle size gradient) They reported that in isocratic elution mode, varying particle size does not present any advantage However, in gradient elution mode, with steep mobile phase gradient, the potential benefit of a particle size gradient might become important In the best scenario, about 15– 20% increase in efficiency can be expected at a given retention time when utilizing a particle size gradient, compared to constant particle size The purpose of this work is to study the evolution of the bandwidth in non-uniform columns by a theoretical approach considering different scenarios for particle size gradients dσz2 kz + kz = Hz dz (2) where Hz and kz are the local height equivalent to a theoretical plate, HETP, and the local retention factor Hz can be calculated by the Knox equation in the knowledge of the local particle size: Hz = A νz1/3 + B νz + C νz d p ( z ) (3) where νz is the reduced linear mobile phase velocity: νz = u0 d p ( z ) Dm (4) and Dm is the diffusion coefficient of the solute molecules Note, that by using this approach, the same u0 might be optimal for a given particle diameter and not optimal for any other particle sizes The physical bandwidth, 4σz , is the spatial width of a concentration profile of a solute inside the chromatographic column In reversed phase gradient elutions, the local retention factor of the compound is determined as kz = kϕ0 (5) ϕ /tG (z/u0 ) + kϕ0 S where kϕ0 is the retention factor of solute at the beginning of analysis (ϕ = ϕ0 ): kϕ0 = k0 exp(−S ϕ ) (6) ϕ is the volume fraction of the stronger eluent modifier, S is a practical measure of the retention sensitivity of a compound toward the change of eluent composition (-S is the slope of ln k[ϕ ] vs ϕ plot), ϕ is the change of stronger eluent component in tG gradient time and k0 is the retention factor of the compounds for ϕ = 2.2 Equivalent particle diameter Pressure drop of a column in which particle diameter varies axially can be calculated by Darcy’s law: dP φ u0 η =− dz d p (z ) (7) where z is the spatial variable, d p (z ) the particle diameter that is a function of z, η the dynamic viscosity of the eluent, and φ the column resistance factor which is in the range of 50 0–10 0 (10 0 is assumed in this work) By neglecting the effect of viscous heating on u0 and η, Eq (7) can be solved as Theory L P = φ u0 η 2.1 Efficiencies of non-uniform columns dz d2p (z ) (8) where L is the column length If the particle diameter is constant throughout the column, Eq (8) simplifies into Kozeny-Carman equation, that is a well known relation used in fluid dynamics to calculate the pressure drop of a fluid flowing through a packed bed of solids, and is only valid for laminar flow [34,35] Equivalent particle diameter, d p,e , can be defined as the diameter of particles packed in a non-gradient, uniform column which generates the same pressure, P , as a column with particle-size gradient For a chromatographic column with a gradient in particle size, the differential mass balance equation of equilibrium dispersive model can be written as: u0 ∂ c ∂c ∂ ∂c =− + D ∂t 1+k∂z 1+k∂z ∂z kz +1 kz (1) where t is the time variable, c is the concentration of the compound in the mobile phase and D is the longitudinal dispersion coefficient Since the particle diameter changes throughout the column axis, value of D depends on the spatial variable, z The efficiency of the chromatographic column with a gradient in particle size can be determined by calculating the retention time and peak variance from the solution of Eq (1) Assuming that zones of analytes are so narrow that particle sizes can be considered nearly constant within the zone, variances d p,e = L L d2p (z ) − 12 dz = φ u0 η L P (9) The column packed with the equivalent particle diameter acts as a reference, whose efficiency can be compared to the different particle-size-gradient columns S Horváth, F Gritti and K Horváth Journal of Chromatography A 1651 (2021) 462331 Table The values of the numerical parameters necessary for the numerical calculations Parameter Value column length (L) parameters of the Knox equation (A, B, C) molecular diffusion coefficient (Dm ) column resistance factor (φ ) linear mobile phase velocity (u0 ) eluent viscosity (η ) initial molar fraction of the stronger eluent (ϕ0 ) change of the eluent composition ( ϕ ) retention factor for ϕ = (k0 ) retention sensitivity (S) gradient time (tG ) 0.05 m 1, 1.5, 0.05 10−9 m2 /s 1000 0.0111 m/s 10−3 Pas 0.05 0.7 104 30 600 s (b) negative gradient ⎡ d p (z ) = d p,0 + d p,L ⎣1 − ( Lz ) i d p,0 d p,L ⎤ ⎦ (15) where i is a positive integer (i = or in this work) where d p,0 and d p,L are particle size diameters at the beginning and end of the column, respectively The shape of the different particle size gradients are illustrated in Fig in case of positive gradients Results and discussions 4.1 Determination of equivalent particle diameters Methods In order to be able to compare the columns with different particle size gradients with uniform columns in respect of efficiency, the column pressure was used as a reference point Pressure drops of the non-uniform columns were calculated by substituting the different d p (z ) scenarios (Eqs (10)–(15)) into Eq (7) With the knowledge of the pressure drops of the non-uniform columns, the equivalent particle diameter can be determined by inserting the calculated P values to Eq (9) for each scenario When d p,e is determined for a given particle size gradient scenario, a theoretical reference column will be obtained, so it can be decided whether the non-uniform column has better or worse separation power compared to this reference column In Fig 2, the pressure profile of the non-uniform and reference columns are plotted for the linear and convex particle size gradients It is apparent that the pressure drop curve of the positive gradient is a concave function, while the negative gradient has a convex pressure profile It is also apparent that these two curves are point symmetric and the center of the symmetry is the middle of the column Obviously, the pressure change is more significant in the region of the column where small particles are located That is, in the case of a negative particle size gradient, the pressure changes more strongly at the end of the column, while in the case of a positive particle size gradient, the pressure changes more strongly at the column head The calculated P and d p,e values are summarized in Table Note, that the equivalent particle diameters are presented in the Table in two decimal precision Obviously, it is not possible to prepare particles with that precision However, this precision was necessary due to the sensitivity of pressure drop toward the particle size variations The data prensented in Table shows that convex scenarios produce larger pressure drops than linear and concave particle size gradients Note that the particle diameters varied between 2.0 and 10.0 μm in each case 3.1 Parameters of calculations The numerical calculations were carried out with a software written in house in Python programming language (v 3.6, Anaconda Python Distribution), using the NumPy and SciPy packages The values of the numerical parameters necessary for the numerical calculations, such as the column and the molecular parameters are listed in Table Note that the retention sensitivity parameter, S, has a quite high value That large value was deliberately chosen in order to represent large polymers and proteins Since S has an impact on the overall evolution of bandwidth, solute size might have a minor impact on the efficiency of columns possessing particle size gradients 3.2 Particle size gradients In the course of our work, three scenarios with different stationary phase particle size gradients were studied For each scenario both the positive and negative gradients were examined, and the calculated efficiencies (physical bandwidth at the end of the column) of these non-uniform columns were compared to the efficiency of the uniform column (the column with the same pressure drop as the non-uniform column, packed with the particles of the equivalent diameter) These three scenarios were the following: Linear gradient (a) positive gradient d p (z ) = d p,L − d p,0 z + d p,0 L (10) (b) negative gradient d p (z ) = d p,0 − d p,L z + d p,L L (11) 4.2 Isocratic elution mode Convex gradient (a) positive gradient d p (z ) = d p,0 Based on the determination of equivalent particle diameters, efficiencies of non-uniform (particle-size-gradient) columns can be compared with reference uniform columns for both isocratic and gradient elution modes If the physical bandwidth at the end of the ( ) z L d p,L d p,0 i (12) (b) negative gradient Table Pressure drop and equivalent particle diameter values for the five scenarios ( L−zL ) i d p (z ) = d p,L d p,0 d p,L Concave gradient (a) positive gradient (13) Scenario ⎡ d p (z ) = d p,L + d p,0 ⎣1 − ( L−zL ) i d p,L d p,0 Linear Convex Convex quinteda Concave Concave quinted ⎤ ⎦ (14) a P (bar) 277.5 413.8 1005.5 192.0 88.6 d p,e (μm) 4.47 3.66 2.35 5.38 7.91 quinted means: “to the fifth power” S Horváth, F Gritti and K Horváth Journal of Chromatography A 1651 (2021) 462331 Fig Variation of the particle diameter along the column length in case of positive gradients a b C Fig Pressure drop curves of the non-uniform columns for positive (blue) and negative (orange) particle size gradients plotted together with the pressure drops of the reference columns (green) for (a) linear, (b) convex, and (c) convex-quinted particle distribution scenarios (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) non-uniform column is smaller compared to the bandwidth at the end of the reference column, then it offers better separation power For the calculations of physical bandwidths, Eq (2) was used Since in isocratic elution mode the retention factor, k, does not change with the spatial variable, z, Eq (2) can be simplified as: dσz2 = Hz dz columns and of the uniform columns are concave functions The bandwidth curves of the positive gradient columns have inflection points This is most significant in case of convex quinted scenario In case of positive particle size gradient, the widths of chromatographic bands are always bigger all along its migration through the columns compared to the uniform column In case of negative particle size gradient, however, the bandwidth is smaller in the beginning compared to the reference column, since the initial particle sizes are smaller than that of the uniform column As the particle sizes increase, the bandwidths increase rapidly Finally, at the end of the column, the bandwidths are the same for positive and negative particle size gradients as well Accordingly, in case of iso- (16) In Fig 3, the evolution of the width of bands of compounds inside the column (4σz ) can be seen for non-uniform columns and the corresponding reference columns in case of isocratic elution mode The bandwidth curves of the negative particle-size-gradient S Horváth, F Gritti and K Horváth Journal of Chromatography A 1651 (2021) 462331 a b C Fig Variation of the physical bandwidth inside the column for the non-uniform columns with positive (blue) and negative (orange) particle size gradients plotted together with the bandwidth curve of the reference column (green) in case of (a) linear, (b) convex and (c) convex-quinted particle size gradient scenarios (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table Bandwidth values (mm) at the end of the column for the five scenarios in isocratic elution mode Scenario Linear Convex Convex quinted Concave Concave quinted Table Bandwidth values (mm) at the end of the column for the five scenarios in gradient elution mode Physical bandwidth (mm) Scenario positive gradient negative gradient uniform column 3.50 3.12 2.78 3.86 4.56 3.50 3.12 2.78 3.86 4.56 2.84 2.52 1.98 3.19 2.50 Linear Convex Convex quinted Concave Concave quinted cratic elution mode, non-uniform columns are identical regarding their efficiencies The direction of the particle size gradient does not affect the overall separation power The calculated physical bandwidth values are summarized in Table for each scenarios The main conclusion of Fig and Table is that uniform columns always provide higher efficiencies than non-uniform columns in case of isocratic elution mode The same conclusion was drawn by studying serially coupled columns under isocratic conditions [31] Physical bandwidth (mm) positive gradient negative gradient uniform column 2.49 2.28 1.66 2.67 2.91 1.70 1.47 1.15 1.94 2.55 1.72 1.53 1.20 1.93 2.50 column In case of the convex quinted scenario, the bandwidths are constant in 80% of the column As a result of the band compression, in case of negative particle size gradients, the final physical bandwidths may be smaller than for uniform columns When the particle size gradient was linear, the advantage of non-uniform column was ∼ 1% For convex and convex quinted scenarios the bandwidths was ∼ 4% thinner than in case of uniform column Accordingly, the steepness of the decrease of the particle diameter towards the end of the column is a decisive factor regarding the evolution of the physical bandwidth The concave gradient did not offer better separation power compared to the uniform column Amongst the non-uniform columns Scenario with negative particle size gradient offered the highest efficiency but it also had the highest pressure drop (smallest d p,e ) The calculated physical bandwidth values are summarized in Table Note that, since the gain of efficiency is characterized by the physical widths of solute bands at the end of the column, this gain in efficiency can be converted to resolution or peak capacity improvement directly 4.3 Gradient elution mode The same plots were generated for gradient elution mode as in the previous section, only this time Eq (2) was used in its full form – since in this case, k, changes with, z, – to determine the shape of the bandwidth functions The plots presented in Fig are very similar to the isocratic ones (Fig 3), but there are some important differences to note First, that columns with positive and negative particle size gradients cannot be considered identical regarding their efficiencies Negative particle size gradients offer significantly better separation power than columns with positive particle size gradients In the latter case, none of the scenarios offered better efficiencies compared to the uniform column In case of negative particle size gradient, however, band compression can be observed For the linear and convex scenarios, even the narrowing of bandwidths can be observed in the last section of the 4.4 Optimization of limiting particle sizes The important conclusion of the previous section, that by delicately choosing the shape and type of particle size gradient, application of non-uniform columns with negative particle size gra5 S Horváth, F Gritti and K Horváth Journal of Chromatography A 1651 (2021) 462331 a b C Fig Variation of the physical bandwidth along the column for the non-uniform columns with positive (blue) and negative (orange) particle size gradients plotted together with the bandwidth curves of the reference, uniform columns (green) in case of (a) linear, (b) convex and (c) convex-quinted particle size gradient scenarios (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) a b c Fig Ratios of the uniform and non-uniform column bandwidths as a function of particle diameter at the end of the column in case of (a) linear, (b) convex and (c) concave particle size gradient scenarios dients can be a potential minor improvement to achieve more efficient separations The column efficiency can be further optimized by the variation of limiting particle sizes (d p,0 , d p,L ) In Fig 5, the physical bandwidths of sample compounds at the end of nonuniform columns relative to the uniform case as a function of the final particle diameter, d p,L , can be seen at three different initial particle sizes (d p,0 = 5, 10, 20 μm) When the ratio is below one, the non-uniform column has better efficiency than the uniform one Fig shows clearly that the minimal (final) particle diame- ter has a significant effect on the overall separation efficiency The optimum of the curves is reached when the final particle size is approximately 40% of the initial particle diameter In case of linear and convex scenarios (Fig 5a and b) the efficiency gain is close to 6–8% in terms of band width reduction Note, that even in case of the concave scenario (Fig 5c), the column efficiency can be improved by 5% in spite of the suggestions of the results of previous sections Fig emphasizes the importance of in-silico optimization of novel stationary phase structures S Horváth, F Gritti and K Horváth Journal of Chromatography A 1651 (2021) 462331 Conclusions [6] R Nogueira, M Lämmerhofer, W Lindner, Alternative high-performance liquid chromatographic peptide separation and purification concept using a new mixed-mode reversed-phase/weak anion-exchange type stationary phase, J Chromatogr A 1089 (2005) 158–169 [7] E.Y Ordoñez, J.B Quintana, R Rodil, R Cela, Computer assisted optimization of liquid chromatographic separations of small molecules using mixed-mode stationary phases, J Chromatogr A 1238 (2012) 91–104 [8] S.P Dixon, I.D Pitfield, D Perrett, Comprehensive 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studied by a theoretical approach According to our results, in isocratic elution mode the uniform column always has the better efficiency regardless of the parameters of the particle size gradient As for gradient elution mode, extra band compression can be achieved in some cases For linear particle size gradient with decreasing diameter, the gain in efficiency was ∼ % and for convex and convex quinted negative particle size gradient the advantage was ∼ % Optimizing the decrease of the size of the particles between the two ends of the column, the advantage can be expanded to ∼ 5–8 % that still does not result in a significant difference in peak shapes Former studies [32] may have been reported more significant gains, but it is important to note that in this work, the reference column was chosen to have the same apparent permeability as the non-uniform column By delicately choosing the shape and type of particle size gradient, application of non-uniform columns with negative particle size gradients can be a potential minor improvement to achieve more efficient separations Even if it is not possible to pack columns to produce particle-size gradients, at the dawn of 3D printing in HPLC [36–40], a legitimate prediction may be that it will be possible to produce columns with efficiency gradients The results also emphasize the importance of in-silico optimization of novel stationary phase structures 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 Szabolcs Horváth: Methodology, Software, Visualization, Formal analysis, Writing - original draft Fabrice Gritti: Conceptualization Krisztián Horváth: Conceptualization, Funding acquisition, Project administration, Software, Formal analysis, Validation, Writing - review & editing Acknowledgment Financial support of the Hungarian National Research, Development and Innovation Fund (NKFIH FK128350) is greatly acknowledged This work was also supported by the TKP2020-IKA07 project financed under the 2020-4.1.1-TKP2020 Thematic Excellence Programme by the National Research, Development and Innovation Fund of Hungary References [1] T Alvarez-Segura, J.R Torres-Lapasió, C Ortiz-Bolsico, M.C García-Alvarez– Coque, Stationary phase modulation in liquid chromatography through the serial coupling of columns: a review, Anal Chim Acta 923 (2016) 1–23 [2] C Ortiz-Bolsico, J.R Torres-Lapasió, M.C García-Alvarez-Coque, Approaches to find complementary separation conditions for resolving complex mixtures by high-performance liquid chromatography, J Chromatogr A 1229 (2012) 180–189 [3] T Alvarez-Segura, C Ortiz-Bolsico, J.R Torres-Lapasió, M.C García-Alvarez– Coque, Serial versus parallel columns using isocratic elution: a comparison of multi-column approaches in mono-dimensional liquid chromatography, J Chromatogr A 1390 (2015) 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Accordingly, the steepness of the decrease of the particle diameter towards the end of the column is a decisive factor regarding the evolution of the physical bandwidth The concave gradient did not offer