Liquid chromatography (LC) is one of the most commonly used methods for the analysis of both small and macromolecules, such as synthetic polymers and proteins. Due to its widespread use, an increase in separation efficiency is always desirable, whether to increase throughput or enable separations that were previously not feasible or practical to perform. Most analytical and preparative-scale LC separations are performed using packed-bed columns. The dispersion or band broadening that occurs in these columns depends primarily on the packing quality and the diffusion that occurs in different parts of the packed bed [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Typically, all band broadening resulting from diffusion is quantified using the molecular (Dm) and effective diffusion coefficient (Deff). The former is the diffusion rate in the unobstructed mobile phase, while the latter is an obstructed diffusion rate, hindered by the presence of the impermeable parts of the particles and the retention experienced by the analytes. Hence, Deff is the convoluted effect of the diffusion (Dm) that occurs outside the particles and the one (Dpart) occurring within the particles. For a core-shell particle, Dpart encompasses both the effect of the solid core and the diffusion (Dpz) in the mesoporous material of the shell. This Dpz then in turns depends on the diffusion (γmpDm) in the mobile phase part of the mesopores and that experienced by the analyte when retained within the stationary phase volume (γsDs). An explanatory schematic illustrating the different zones and the respective diffusion coefficients for a column packed with core-shell particles is included in the supplementary material (SM, Fig. S1).

Knowledge of the diffusion coefficients is vital for understanding what mechanisms contribute most to band broadening in a given packed bed [16], [17], [18], [19], [20], [21], [22]. Within a packed bed, it is generally assumed there are four different contributions to a band's width; namely, i) eddy dispersion (hA, or hinhom), ii) longitudinal or axial diffusion (hB), iii) mobile-phase mass-transfer resistance (hC,m), and iv) stationary-phase mass-transfer resistance (hC,s). For all but the first contribution, analytical expressions exist that allow modelling their impact on the overall (reduced) plate height (h) . These expressions are shown below in Eqs. (1)–(4) in reduced coordinates, using ν0=u0dpDm or νi=uidpDm, with u0 and ui the linear velocity and interstitial velocity, respectively and dp the particle diameter.h=hA+hB+hC,m+hC,shB=2γeff(1+k′)ν0=2γeff(1+k″)νi=BνihC,m=2α1Shmk″2(1+k″)2εe1−εeνi=CmνihC,s=2α1Shpartk″(1+k″)2DmDpzνi=Csνiwhere γeff is the reduced effective diffusion coefficient (γeff=DeffDm), often also referred to as an “obstruction” parameter [23], [24], [25], [26], [27], which is always smaller than one, k′ is the phase-based retention factor, and k″ the zone-based retention factor [28]. The other parameters are εe, the external or interstitial porosity defined as the ratio of interstitial to geometrical volume (usually between 0.36–0.40 for a well-packed bed of spherical particles), α a geometrical constant (6 for spherical particles), and Shm and Shpart the Sherwood numbers for the mobile and stationary-phase mass transfer resistance. Usually Shm is between 10 and 20 [29], while Shpart may be estimated using Eq. (5) [30].hpart=431−ρ3215−23ρ3+65ρ5−23ρ6where ρ=dcoredp, indicates the solid, non-porous fraction of the particle. For fully porous particles ρ=0 and 2α1Shpart=130, while for non-porous particles ρ=1 and 2α1Shpart≈0, and the Cs-term turns to zero. Except for some uncertainty regarding the exact expression for Shm (it depends on the (unknown) packing structure) these are well-established equations, allowing one to estimate hA by process of elimination [11,31,32].

The B- and C-terms depend on the diffusion in different parts of the bed: Deff, Dm, and Dpz. While Dm is an intrinsic property of the employed combination of analyte and mobile phase, Deff and Dpz also depend on the intra-particle microstructure and composition. It is hence interesting to investigate how these vary between packings and different particle types. For example, such an approach has previously been used to compare the obtainable efficiency of small molecules on packed beds of fully porous particles and core-shell particles [33]. This study illustrated that for small compounds the observed reduction in plate height when going from fully porous to core-shell particles is primarily a result of smaller A and B-term contributions. The latter partially being a result of the solid core blocking diffusion but being primarily thanks to differences (mainly in porosity of the mesoporous zone) between the porous material of core-shell and fully porous particles [33,34].

Among the different diffusion coefficients, only Dm and Deff can be directly measured, while Dpz and Dpart can be estimated from Deff via models or approximations. To measure Dm, a variety of methods can be used [35], [36], [37]. Of these, the most straightforward is the measurement of peak broadening in an empty tube or capillary, otherwise known as a Taylor–Aris (TA) experiment. Since the underlying theory has been described extensively in literature, it is not repeated here. Instead, the reader is referred to several previous works [37], [38], [39], [40], [41], [42].

The other measurable quantity (Deff) can be determined directly by peak-parking or stop-flow experiments [8,13,23,43,44]. In such experiments, the analyte is moved approximately halfway into the column after which the flow is stopped, allowing the analyte to diffuse for a given period of time (tpark). According to the Einstein-Smoluchowski law, the broadening of the peak is linearly proportional to the time it is left to diffuse in the column. Hence, the obstructed diffusion rate, Deff, is the slope or increase of the peak's axial variance (Δσx2) over the parking duration, i.e.Δσx2=2Defftparkwhere σx2 can be calculated from the measured temporal variance as σx2=σt2.(LtR)2and tR is the retention time obtained without additional parking time, and after subtracting the system time.

To obtain a correct measurement of σt2, several conditions must be met. Firstly, adverse effects on σt2 coming from valve switches or turning on and off the flow must be eliminated by correcting for them. For example, by subtracting the measured σx2 with a very short (e.g. 1 min or shorter) parking duration from all other measured σx2. Secondly, care should be taken to not stop the peak at the beginning or end of the column due to small differences in packed bed structure that can exist within these parts of the column [45], [46], [47]. Thirdly, and finally, when possible, σt2 should be determined by a method that is appropriate for the obtained peak shape and potential background artifacts, e.g. using peak moments, peak width at half-height, or an asymmetric distribution function such as the exponentially modified Gaussian [48]. When working with slowly diffusing analytes, it should be considered that part of the peak skewness also originates from radial column heterogeneities [6].

After obtaining Dm and Deff, one can estimate Dpart and Dpz. In literature, both the residence-time-weighted (RTW)/parallel-zone model [2,15,49] and the effective medium theory (EMT) [16,33,[50], [51], [52]] have been used for this purpose. Because the former is based on the assumption that diffusion in- and outside the bed occur in parallel, which is likely more akin to what happens in a porous-layer open-tubular column or a coated capillary, it can lead to large errors when used for packed bed columns [50]. For such columns, it is hence better to use EMT-derived expressions. The EMT is a well-established theory used in many fields of science and technology to describe certain properties (e.g. the thermal or electrical conductivity, or the dielectric constant/permittivity) of composite materials, by assuming that these materials are homogenous on the macroscopic-scale. The EMT approach has been introduced into the area of chromatography [50] and adapted for cases with retention more than a decade ago and has since then been used many times to model the diffusion inside a packed bed [16,33,51,52]. The simplest EMT approximation, Maxwell's model [53], adapted for use in liquid chromatography is given by:DeffDm=11+k′1εt1+2β1(1−εe)1−β1(1−εe)where εt is the total porosity and β1 is the so-called particle polarizability constant. While the Maxwell model has been derived for dilute suspensions of spheres, it is often sufficiently accurate to estimate Dpz. Even more accurate solutions can be obtained using an extension of the model by Torquato [54,55]:DeffDm=11+k′1εt1+2β1(1−εe)−2εeξ2β121−β1(1−εe)−2εeξ2β12where ξ2 is the three-point parameter, the value of which depends on the packing orientation of the spheres, typically in the range of 0.2–0.3 for packed bed columns [56,57]. Irrespective of the model used, for spherical particles β1 is given byβ1=αpart−1αpart+2 with αpart,αpart=(1+k′)εt−εe1−εeDpartDmFor core-shell particles, the diffusion inside the particles is hindered by the core, leading to shorter diffusion paths. This means that the diffusion inside the porous zone, Dpz is actually slightly larger than the estimated Dpart. This can be accounted for by correcting Dpart usingDpz=2+ρ32Dpart

Of course, Dpart=Dpz for fully porous particles, while Dpart<Dpz for core-shell particles. The largest increase from Dpart to Dpz is observed for the smallest relative porous shell thickness.

Most of the work that involves estimating diffusion coefficients has been performed using small molecules [13,32,33]. This is probably partly because such analytes diffuse rapidly in both the axial and radial directions making peak parking measurements much more straightforward and significantly faster than for macromolecules. Furthermore, for large molecules, Deff and subsequently the B-term contribution to band broadening are small, meaning their impact under typical separation conditions (e.g. flow rates from 0.1 to 2.0 mL·min−1 with corresponding velocities well above the minimum of the van Deemter curve) will be rather limited. However, because Deff allows estimating Dpz, which determines the magnitude of the Cs-term, it remains important to determine Deff to estimate the relative contributions of the A, B, Cm and Cs-terms for separations that involve macromolecules. Gritti et al. previously applied the peak parking method to determine Deff for large molecules such as insulin and bradykinin [12]. The authors were, however, only able to obtain results under non-retained conditions. Under retained conditions, non-linear plots of σx2 versus tpark were obtained, and the retention times of the molecules were not constant when increasing the parking time. The authors explained this as the result of a continuous conformational change of the biomolecule during tpark, while in contact with the stationary phase at atmospheric pressure. As the flow rate and system pressure were resumed, it was assumed that the affinity of the biomolecules for the stationary phase differed from the one before stopping the flow, due to slow conformational changes. To obtain values for the Cs-term (4th term in Eq. (1)) under retained conditions, the authors therefore multiplied values of Dpz obtained under non-retained conditions with the k″-values of the compounds.

Another possible approach to obtain values for the individual plate height contributions is by fitting one of several plate height equations to experimentally obtained plate height data. However, the advantage of the diffusion-based approach is that it does not make any assumptions regarding the velocity dependence of the A-term, which is still unclear and subject to extensive research [58,59].

A change in pressure is accompanied by changes in solvent compressibility, viscosity, and small changes in column diameter, leading to changes in column porosity and permeability [60]. However, these changes are often small, or in case of the latter, insignificant over a typical pressure range of about 0 to 500 bar. Usually, separations will be more affected by the influence of pressure on retention [60], [61], [62], [63], [64], [65], [66]. This effect has also been observed for small dipolar molecules but is especially pronounced for large molecules, such as proteins. When the separation is dominated by one retention mechanism, the change in retention with pressure can be derived from basic thermodynamic equations and described by a log-linear equation of the form:ln[k′k0′]=−ΔVRTP+ln[ββ0]where ΔV is the change in partial molar volume of the analyte upon its transition from the mobile to the stationary phase, R is the universal gas constant, T the absolute temperature, P the pressure, and β the phase ratio of the column, in LC commonly defined as the ratio of the stationary-phase volume to the mobile-phase volume. The change in V is given with respect to reference conditions, e.g. atmospheric pressure, with k0′ and β0 the retention factor and phase-ratio at these conditions. Typically, β decreases slightly with pressure because the column will expand at higher pressures and the packing particles can be compressed. However, these changes are very small and are typically neglected. For proteins, the sign of ΔV is typically negative, and often increases with analyte size [65]. In an isocratic experiment, the pressure decreases linearly over the column length, which is a good approximation at conventional operating pressures [60]. This results in a linear decrease in lnk′ along the column. In turn, this leads to a difference in retention factor between the front and trailing edges of the peak zone inside the column (measured in distance units). This causes a velocity gradient over the peak zone that can lead to compression of the zone (negative velocity gradient, ΔV>0), or an expansion of the zone (positive velocity gradient, ΔV<0) [67], [68], [69]. Naturally, this effect should be larger when the pressure-drop over the column is steeper (i.e. when columns are used that generate higher back pressures) and when the absolute slope (given by −ΔVRT) in Eq. (12) becomes larger, i.e. for very large proteins.

Another aspect that can affect a peak parking experiment at elevated pressure is the lesser-known change in self-diffusion with pressure [60]. Based on existing data, the self-diffusion coefficient of some common solvents decreases with pressure. However, for the water-acetonitrile mixtures used in this work, the changes are expected to be small, although they may increase for large molecules. In this work it was chosen not to account for this effect.

We first set out to assess whether Deff could be measured at an elevated pressure, and whether this had a noticeable effect on the obtained Deff. To do so, a six-port two-position valve was used that allowed closing both ends of the column (for details see Section 3.1) to keep its pressure. This set-up was first used to determine Deff for several peptides and proteins; namely, bradykinin, insulin, lysozyme, β-lactoglobulin and carbonic anhydrase. The experiments were performed at an average pressure of about 105 bar as the resulting data was to be used in a related study where an extensive set of plate height measurements was performed. For the sake of comparison, additional “conventional” peak parking experiments were also performed on the same 400 Å-pore column for a set of small analytes (acetophenone, propiophenone, benzophenone, valerophenone and hexanophenone). When including the experiments performed for the small analytes in the general assessment of the diffusion behaviour, it could be concluded that there was a clear decrease of DpzDm with analyte size, expressed as hydrodynamic radius (RH). Our second objective was to investigate whether the diffusion of bradykinin and lysozyme would be affected by performing peak parking experiments at different elevated pressures. Since a higher pressure is expected to lead to an increase in retention, it was expected that the obtained Deff should scale inversely with pressure, which was indeed observed. However, an unexpected decrease in DpzDm with pressure was also observed.