Over a decade ago, micro-pillar array columns were introduced for pressure-driven liquid chromatography [1]. These columns were designed as a method to create chromatographic beds with significantly enhanced uniformity compared to the conventionally used randomly packed sphere beds. This leads to a reduced band broadening, which, together with the fact that the non-touching aspect of the pillar bed also leads to a lower flow resistance, makes them score very good on kinetic performance index scale [2], albeit that very small pillar sizes are needed to optimally benefit from this [3]. Due to their inherent suitability for miniaturization, micro-pillar array columns have since their inception [1,[4], [5], [6], [7], [8], [9]] been extensively employed to address considerable issues in proteomic analyses [10], [11], [12], [13], [14], [15], [16], [17], for instance, to enabling detection/exploration of low-abundance and targeted proteins existing in a cell. Additionally, micro-pillar array columns have gained substantial traction in metabolomics, leveraging on the very large plate numbers provided by the longest pillar arrays [18].

Micro-pillar array columns are produced using silicon micromachining [19,20]. First a 2D drawing of the channel's geometry is transmitted to a layer of photoresist using deep-UV photolithography to locally remove the photoresist. The thus bared regions are subsequently attacked using deep reactive ion etching, locally fretting away the silicon to produce 2D arrays of silicon pillars, embedded in a half open-channel with silicon bottom and side-walls. In the currently adopted fabrication strategy, the retention surface is subsequently enhanced using electrochemical anodization with methanol and HF to transform the outer regions of the silicon material into a mesoporous silicon shell [21,7]. As the anodization process cannot differentiate between the pillar and channel walls, the current generation of pillar array columns has a retentive layer which not only covers the cylindrical pillars, but also the bottom wall. As the anodization needs to proceed in an open liquid bath to provide a uniform electrical field, the columns can only be sealed afterwards. This is done using anodic bonding with a glass plate, a material selected for its superior bonding properties to the silicon. The top wall is therefore not covered with a mesoporous retentive layer, a situation which is further referred to as the Bottom Only (BO)-case (cf. Fig. 1a-top).

The absence of a retentive layer at the top wall creates an intrinsic asymmetry of retention along the z-axis (running perpendicular top to bottom) and it can be assumed this asymmetry constitutes an additional source of band broadening. It is hence fundamentally and technologically very interesting to explore what improvement can be expected if this symmetry could be restored, either by avoiding the presence of a retentive layer at the bottom-wall (NTB-case, see Fig. 1a-middle), or by having an identical retentive layer on both the top and bottom walls (TB-case, Fig. 1a-bottom). In practice, the TB-case could be obtained by applying a mesoporous layer using sol-gel deposition [22,23]. This could be done after the addition of the top wall. As the adherence of the deposited mesoporous silica is mainly dictated by the presence of hydroxyl-groups, and since these are present on both the silicon walls as well as on the glass top wall, it can be assumed all walls will be uniformly covered. The NTB-case could be created using sophisticated masking techniques to protect the bottom wall during the anodization process. This would add additional steps to the fabrication process but is nevertheless practically feasible.

In the present study, the answer to the above question is pursued using a recently introduced numerical method [24], [25], [26], [27], [28], [29], [30], [31] transforming the time-dependent advection-diffusion equation governing the species transport into two steady-state advection-diffusion equations that furthermore only need to be solved over an elementary unit cell of the column (cf. green coloured volume in Fig. 1a). This method first solves two equations to compute the so-called b-field (with the “b” named after Brenner [32,33] whose work was adapted to be applied in porous zone chromatography systems), one corresponding to the mobile zone (flow channel space between pillars) and another to the stationary zone (mesoporous layer zone):Dm∇2bm−u→·∇→bm=<ux>m1+k″−uxDs∇2bs=<ux>m1+k″(−Dm∇→bm+Dmex→)·n→|Γ=K(−Ds∇→bs+Dsex→)·n→|Γbm|Γ=bs|Γ(−Di∇→bi+Diex→)·n→=0,i=m,s

Eq. (1) describes the b-field equation within the mobile zone, featuring a source term that accounts for the local velocity variation relative to the effective average velocity <ux>m1+k″. Eq. (2) on the other hand, represents the b-field equation in the stationary zone, where the source term remains constant due to the suppression of the velocity field in the mesoporous zone. Eq. (3) enforces the flux boundary condition at the interface Γ between the mobile and stationary zones, while Eq. (4) ensures the continuity of the b field. Eq. (5) specifies the boundary condition applied at impermeable walls. In these equations, bm and bs denote the magnitude of the b-fields (dimension length) in the mobile and stationary zones, respectively. ∇→ represents the gradient operator, u→ is the velocity field vector, ux is the local x component of the velocity of the mobile zone and k’’ is the zone retention factor. The partition coefficient K defined as K=k″Vm/Vs, incorporates the total volume of the mobile zone Vm and the total volume of the stationary zone Vs.Dm and Ds are the diffusion coefficients in the mobile and stationary zones respectively. The vector n→ is the normal vector to the surface, pointing from the mobile zone to the stationary zone in Eq. (3) and from the stationary zone to the impermeable walls when i = s and from the mobile zone to the impermeable walls when i = m in Eq. (5). ex→ is the x-axis unit vector and the operator 〈〉i is the volume average operator over zone i (i = m for the mobile zone and = s for the stationary zone).

Subsequently, the b-field is employed to calculate the axial dispersion coefficient Dax leading to the determination of the reduced plate height h via Eqs. (6) and (7) with dp chosen as a reference length. In this notation, dp is the pillar diameter in case of a pillar array column and the wall width in case of a parallel plate column [34], [35], [36]:Dax=Dm1+k″<∥∇→bm−ex→||2>m+Dsk″1+k″<∥∇→bs−ex→||2>sh=Hdp=2Dax(uxm1+k″)dp=2Pe·(DaxDm)·(1+k″)

As it only requires a steady-state solution over a small unit cell, this method, referred to as the two-zone moment analysis (TZMA) method, can be pushed to very high accuracies. The method is in terms of performance equivalent to the one proposed by Yan et al. [37], [38], [39] and is much faster and accurate than the classic time-dependent solution approaches used in computational fluid dynamics [40], [41], [42], [43], [44], [45], [46], [47], [48], [49]. It has been extensively validated on a broad range of test geometries. In a previous study [26], we already studied the BO-case and observed a strong anomaly among the order of the plate height curves for a different zone retention factor k’’.

Even without being covered by a retentive layer, the presence of the top and bottom wall can be expected to significantly increase the band broadening compared to the hypothetical 2D-case (no top and bottom wall). This due to the so-called wall effect, first pointed out by Golay [50], [51], [52] or the case of open-tubular channels with a flat rectangular cross-section. He showed that, compared to a hypothetical flow between two infinite parallel plates (no side-walls), the plate height in a non-retentive system can be expected to increase with a factor of 7.95 times as soon as the zone between the parallel plates is delimited by adding side-walls, no matter how far these side-walls would be spaced apart (in fact, the factor 7.95 becomes smaller when the side walls are brought closer together, eventually reaching a value of 1.81 in case of a square cross-section). A detailed physical explanation for this effect in an open-tubular non-retentive channel is given in [53]. In brief, the reason for the excess band broadening emanating from the side-walls is to be found in the fact that the velocity is locally lower near the side-walls than in the rest of the channel because of their flow arresting effect. This creates a lateral velocity non-equilibrium. As the lateral distance in flat-rectangular channel is much larger than the distance between the two long walls, the corresponding mass transfer resistance is much larger as well. In the micro-pillar array columns, a similar effect occurs [54] as the top and bottom-wall take on the role of the side-walls, while the narrow zone between the pillars has a flat-rectangular cross-section.

To better understand the results for the three micro-pillar array geometries (cf. Fig. 1a), we compared them with the band broadening in true parallel flat-rectangular channels (= array of parallel plates, Fig. 1b) with similar configurations ie. a “bottom only” (BO-PP), a “no coating on top or bottom” (NTB-PP) and a ‘‘coating on both top and bottom” (TB-PP). By including this parallel plate geometry, the present study also provides an extension of the classic Poppe paper [36] wherein he derived the analytical plate height expressions for open-tubular channels with a flat rectangular cross-section. In [36], the retention is supposed to occur exclusively in a monolayer coated on solid and impermeable channel walls. As a consequence, the plate height values given in [36] are for the mobile zone mass transfer only (so called HCm-term). Since the present study considers a geometry where the retention occurs in a mesoporous layer with finite thickness and internal diffusion coefficient, the plate height data obtained for the geometries shown in Fig. 1b also include the stationary zone mass transfer (HCs-term). Another difference is that we assume the diffusion in the retentive layer is isotropic, while the computational method adopted by Poppe intrinsically assumes the surface diffusion in the retained state is zero.