Theoretical Analysis of Efficiency of Multi-Layer Core-Shell Stationary Phases in the High Performance Liquid Chromatography of Large Biomolecules

Modern analytical applications of liquid chromatography require columns with higher and higher efficiencies. In this work, the general rate model (GRM) of chromatography is used for the analysis of the efficiency of core-shell phases having two porous layers with different structures and/or surface chemistries. The solution of the GRM in the Laplace domain allows for the calculation of moments of elution curves (retention time and peak width), which are used for the analysis of the efficiency of bi-layer particles with and without a non-porous core. The results demonstrate that bi-layer structures can offer higher separation power than that of the two layers alone if the inner layer has smaller surface coverage (retentivity) and the pore size and pore diffusion of the outer layer is either equal to or higher than that of the inner layer. Even in the case of core-shell phases, there is an increase in resolution by applying the bi-layer structure; however, we can always find a mono-layer core-shell particle structure with a larger core size that provides better resolution. At the optimal core size, the resolution cannot be further improved by applying a bi-layer structure. However, in case of the most widely produced general-purpose core-shell particles, where the core is ∼70% of the particle diameter, a 15–20% gain of resolution can be obtained by using well-designed and optimized bi-layer core-shell phases.


Introduction
Higher separation efficiency and faster speed have always been of great interest in high performance liquid chromatography (HPLC) [1][2][3]. The diameter of HPLC particles has been shrinking through the years, so that sub-2 µm totally porous particles are now used widely for separating small molecules. Columns of superficially porous particles (SPP) [4,5] have shown even further efficiency advantages, such that some users prefer these over totally porous particles (TPP). Recent studies have reported both the advantages and disadvantages of columns packed with core-shell and totally porous sub-2 µm particles [6,7].
Using very fine particles (sub-2 and sub-1 µm), due to the narrow peaks, sensitivity and separation are improved at the cost of pressure. Knox and Saleem were the first to discuss the compromise between speed and efficiency [8]. A critical aspect is the effect of frictional heating at ultra high pressure, causing temperature gradients within the columns [9]. The radial temperature gradient, due to the heat dissipation at the column wall, can cause significant losses in plate count [10]. Gritti et al. concluded that both longitudinal and radial temperature gradients are more significant when the column length

Structure of Multi-Layer Core-Shell Particles
Multi-layer core-shell particles consist of three different parts (see Figure 1). 1. A non-porous core with a radius r c : where r p is the radius of particle ( Figure 1) and ρ is the factor of proportonality between the radius of the inner solid core and the radius of the particle. This region is impermeable to the compounds analyzed and to the molecules of eluent. Note that, if ρ is equal to 0, there is no solid core inside the particle, while, in the case of ρ = 1, the whole particle is non-porous, such as the Kovasil phases [22]. Accordingly, 0 ≤ ρ ≤ 1. 2. A porous inner layer with a thickness δ i " where r i is the radius of the outer surface of the inner porous layer ( Figure 1) and β is the factor of proportonality between r i and r p . This layer has a given porosity (ε i ) and surface chemistry. Note that, if β is equal to ρ or 1, the particle has only one porous layer. If ρ ≤ β ≤ 1, the particle has two porous layers. 3. A porous outer layer with a tickness δ o : Depending on the manufacturer, this layer may or may not have different porosity (ε o ) and surface chemistry than the inner porous layer.  The porosity of the shells (ε p ) depends on the thickness of the different layers, and the size of the non porous core: The total porosity of the column packed with multi-layer core-shell particles can be calculated as where ε e is the external porosity of the column.

General Rate Model for Multi-Layer Core-Shell Particles
Many processes take place in the column, during the migration of solute molecules, which cause band broadening. There is no single rate-controlling process. The general rate model of chromatography considers the axial dispersion as the sum of axial and eddy dispersion, the external film mass transfer resistance, the intraparticle diffusion (including the pore and surface diffusions), and the rate of adsorption-desorption. As the processes taking place inside and outside a particle are considered separately in the GR model, two mass balance equations for the solute have to be written, one for the interstitial flowing mobile phase, and one for the stagnant mobile phase inside the particles.
For the processes in the interstitial liquid phase, the following mass balance equation can be written: where c e is the concentration of the solute in the interstitial volume; u e is the interstitial velocity of the eluent; D L is the axial dispersion coefficient defined for the external mobile phase, which is the sum of the molecular and the eddy diffusion coefficients; F is the phase ratio, defined as 1 − ε e ε e ; and q is the value of the stationary phase concentration, q, averaged over the entire particle.
For a spherical particle, it can be calculated as The derivative in the third term of Equation (6) is the rate of adsorption averaged over the particle. It is calculated as where k f is the external mass transfer coefficient and c o [r p , t] the concentration of the solute within the pores at the surface of the particle. For the processes taking place inside of the multi-layer core-shell particle, two differential equations have to be written: One for the outer porous layer, and another for the inner porous layer. Accordingly, where the superscripts i and o represent the inner and outer porous layers, respectively; c o and c i are the concentrations of the solute in the stagnant mobile phase of pores in the two layers; D o and D i are the diffusion coefficients of the solute in particle pores; and where K o and K i are the Henry coefficients of the solutes in the two layers. In the equations above, an infinitely fast adsorption-desorption kinetics was assumed (q[r] = Kc[r]). Note that, even if the retention of solutes is described here with simple equilibrium parameters (one for each layer), the kinetics of adsorption of solutes can be significantly more complex than this simple approach [23,24]. It was shown, in [24], that several thousands of similar but slightly different kinetic processes are lumped together in the adsorption of a similar-sized peptide as that used here as a model. The following boundary condition was used at the outer surface of the particle: At the boundary between the two porous layers, an infinitely fast mass transfer was assumed. Accordingly: and There is no mass flux through the porous core. Accordingly, the boundary condition at the surface of the non porous core can be written as For the sake of simplicity, a simple dirac delta injection was assumed as initial condition.

Height Equivalent to a Theoretical Plate of Chromatographic Columns
The performance of different chromatographic columns is conveniently compared on the basis of the values of their height equivalent to a theoretical plate (HETP), which can be calculated from the statistical moments of the peak eluted: where µ 1 is the first absolute, µ 2 the second centralized moment of the peak, and L the column length. A closed-form analytical solution of the system of partial differential equations defined in the previous section [Equations (6)- (17)] is impossible to derive in the time domain. Although the solution in the Laplace domain can be derived, it cannot be transformed back into the time domain. However, the moments of the peaks can be calculated from the Laplace transform of the solution easily [25][26][27]: where C(s, L) denotes the Laplace transform of the elution profile at the outlet of the column (L), s is the Laplace variable, and n > 0.

Methods
The separation power of the multi-layer particles was investigated for a typical large peptide (e.g., insulin: Molecular weight, M, ∼6 kDa; molecular diffusion coefficient, D m , ∼ 6 × 10 −5 cm 2 min ) by Equations (20)-(31) which was derived from Equations (6)- (19) using Mathematica 10.0 (Wolfram Research Inc., Champaign, IL, USA). The numerical calculations were carried out using 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 parameters, the particle size, and so on, are listed in Table 1.

General Solution of the GR Model
The first moment of the general rate model [Equations (6)- (17)]-in other words, the retention time-is given by where and The second moment is given by where δ ax , δ f , and δ d are the contributions of the axial dispersion, external film mass transfer, and intra-particle diffusion to the variance of peak eluted, respectively.
From Equation (18), it follows that the HETP of a column packed with multi-layer core-shell particles is given by where h ax , h f , and h d are the contributions of the axial dispersion, external film mass transfer, and intra-particle diffusion to the HETP, respectively. The apparent retention factor, k, of a compound can be calculated from Equation (20), considering that the hold-up time of the column, t M , is equal to the first statistical moment of the band of a non-retained compound. Accordingly, Equations (20)- (28) can be used as a general basis for the calculations of moments or HETP of columns packed with particles having different structures. For example, if β = 1 or β = ρ, the particle behaves as a core-shell particle; k 1 and δ d become which is identical to the results obtained previously for superficially porous particles.

Separation Efficiency of Bi-Layer Fully Porous Particles
For the illustration of the separation efficiency of a superficially porous bi-layer particle, the HETP curves were plotted against the beta coefficient. The parameters of a typical large peptide (insulin) were used as the model molecule for our calculations. In respect of the surface coverage of the two porous layers and the diffusion constant of the molecules in the pores, 12 HETP curves could be plotted. The parameters presented in Table 2 were used to plot these curves.  (11) and (12)) and pore diffusion parameters (D i and D o ) of the inner and outer layers used for the calculation of separation efficiencies of superficially porous bi-layer particles.

No.
A Retentions in the inner and outer layers depend on the values of A i and A o , which can be calculated by Equations (11) and (12), respectively. Their values depend on the porosity (ε) and surface coverage (Henry coefficient, K) of the layer. The latter refers to the number of C 18 groups bonded to the surface of the given layer. The A i and A o values used in the calculations were 1.2 and 3, equivalent to retention factors, k, of 0.75 and 2.45 in the case of traditional totally porous particles, respectively. Besides the molecular diffusion coefficient of the solute, the values of pore diffusion parameters, D o and D i , depend mainly on the ratio of particle diameter to the nominal pore diameter. Accordingly, 10 −5 and 10 −6 cm 2 min pore diffusion coefficients correspond to pore sizes of 300 and 100 Å, respectively, in the case of a typical large peptide. Figure 2 shows the calculated HETP of totally porous (ρ = 0) bi-layer particles as a function of factor of proportionality between the radius of the outer surface of the inner porous layer, r i , and the particle radius, r p (β = r i /r p , as shown in Figure 1). The thickness of the inner and outer layers can be calculated by Equations (2) and (3) with ρ = 0. In order to demonstrate the effect of the bi-layer structure on the chromatographic efficiency, the contributions of axial and film mass transfer processes (δ ax and δ f ) were neglected. Accordingly, only δ d was taken into account during the calculations (see Equations (25)-(28)). Note, however, that for large peptides, pore diffusion is the most significant contribution of mass transfer processes to the overall HETP [25]. The calculated results correlate well with measured reduced HETPs on conventional and wide-pore core-shell phases [28,29]. Figure 2 shows that the structure of a bi-layer stationary phase affects its efficiency significantly. For a mono-layer totally porous phase, the best efficiency is given with A = 3 and D p = 10 −5 cm 2 min as could be expected. In that case, the reduced HETP is ∼2.3. Depending on the surface retentivity and pore diffusion of the inner and outer layers, the efficiency may increase or decrease as the ratio of porous layers changes. In general, it can be concluded that the efficiency increases as the ratio of the larger pore size layer increases, as it could be expected. In the cases of scenarios 5, 7, and 11, the efficiency decreases monotonically, while in cases of scenarios 4, 6, and 8, it increases monotonically as the inner layer increases. The HETP lines of scenarios 1, 2, 3, 9, 10, and 12 have either local minima (1, 2, 3) or maxima (9,10,12). The former cases reveal that it is possible to combine two different porous layers resulting in a higher efficiency than that of the two layers alone. The common factors in scenarios 1, 2, and 3 are that (1) the inner layer has smaller surface coverage (A i = 1.2 versus A o = 3) and (2) the pore size and pore diffusion of the outer layer is either equal to or higher than that of the inner layer. The highest efficiency was obtained in case of scenario 1 (black line, minimum point marked with a circle), where both layers had faster pore diffusion and the inner layer had a smaller retention than the outer one.  Table 2, where β is the factor of proportionality between the radius of the outer surface of the inner porous layer and the particle radius (see Figure 1 and Equations (2) and (3)).

Separation Efficiency of Bi-Layer Core-Shell Particles
In Figure 3 the reduced HETPs of bi-layer core-shell phases can be seen against β at different core sizes. The black covering line (β = ρ, see Figure 1 and Equations (2) and (3)) corresponds to mono-layer particles. The best combination of the parameters from the previous section (scenario 1) was used to study the separation efficiency of the core-shell bi-layer phases. Close examination of Figure 3 highlights that, at any given core size, the efficiency can be further increased by applying a bi-layer structure. At the same time, however, we can always find a mono-layer core-shell structure which offers higher efficiency than any bi-layer solid core particle. Note, however, that the decreasing shell thickness causes a decrease of the surface area of the stationary phase, resulting in a decrease of retention and eventually of the actual resolution between the compounds when the porous layer becomes too thin. In chromatographic practice, resolution is a more important parameter than HETP. If we wish to study the resolution capability, we have to define a new compound. The A o and A i parameters of the new compound was multiplied with 1.2. In other words, the selectivity of the two compounds was 1.2. In Figure 4, the calculated resolution curves are plotted against β at different values of ρ. Similarly to the HETP curves shown in Figure 3, we can see that there is an increase in resolution by applying bi-layer superficially porous particles; however, up to a certain core size, we can always find a mono-layer core-shell particle structure with a larger core size that provides a better resolution. Further increase of the core size above a certain optimal value, however, decreases the chromatographic resolution, due to the significant decrease of retention. At the optimal core size, the resolution cannot be further improved by applying a bi-layer structure, since the less retentive inner layer decreases the retention time difference of the two compounds. Accordingly, the best separation power can be obtained by optimizing the core size of a mono-layer core-shell particle. It is important to note, however, that the optimal core diameter depends significantly on the type of analyte, as was shown in [20]. Economically, it is not feasible to design and optimize too many specific stationary phases. The core sizes of core-shell phases optimized for analysis of large macromolecules are typically 90-95% of the particle diameter (note that Figure 4 also suggests this core size). Our results clearly demonstrate that there is not any advantage of using a bi-layer structure for those phases. In the case of smaller core sizes, however, the advantages of a bi-layer structure are more significant. The size of core in the most-widely produced general purpose core-shell particles is ∼70% of the particle diameter (ρ = 0.7). As Figure 4 suggests, a 15-20% gain of resolution can be obtained by using well-designed and optimized bi-layer core-shell phases.

Conclusions
As was shown in [20], the general rate model is a powerful tool for designing and optimizing high-performance stationary phase particles, as it considers all the processes that take place in the flowing interstitial mobile phase and in the stagnant liquid phase inside the particles. In this work, the general rate model was used for the analysis of efficiency of bi-layer stationary phases with and without non-porous cores. The solution of GRM in the Laplace domain allowed for analysis of the efficiency of these phases. The results suggest that, by careful design and optimization, bi-layer structures can offer higher separation power than mono-layer phases if the outer layer has larger retentivity and pore diffusion than the inner layer. The results also demonstrated that resolutions cannot be further improved by a bi-layer structure if the size of non-porous core is optimized for the given separation. If the goal, however, is to develop a stationary phase for general purposes, a 15-20% gain of column efficiency can be reached by using a bi-layer structure.
The derived equations can also be used for the design of more exotic stationary phases, such as hollow particles that contain holes (eluent) at the center of the particles, allowing for fast diffusion and fast mass transfer. Even if the production of these exotic stationary phases is not feasibly technically at present, it may be interesting to theoretically analyze the applicability of such phases. It is important to note, however, that the approach introduced in this work cannot predict the mechanical stability of particles, nor the wall effect during column packing that influences the applicability and the efficiency of the column significantly. In spite of these limitations, the application of the general rate model can accelerate the design and optimization of novel stationary phases.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations and symbols are used in this manuscript: MDPI Multidisciplinary Digital Publishing Institute DOAJ Directory of open access journals HPLC High pressure liquid chromatography HETP Height equivalent to a theoretical plate GRM General rate model SPP Superficially porous particle TPP Totally porous particle r p particle radius r c core radius r i radius of the outer surface of the inner porous layer ρ factor of proportonality between the r c and r p β factor of proportonality between r i and r p δ i thickness of the inner porous layer Laplace transform of the elution profile at the outlet of the column d c column diameter d p particle diameter δ ax contribution of axial dispersion to the variance of peak eluted δ f contribution of external film mass transfer to the variance of peak eluted δ d contribution of intra-particle diffusion to the variance of peak eluted h ax contribution of axial dispersion to the HETP h f contribution of external film mass transfer to the HETP h d contribution of intra-particle diffusion to the HETP k i zone retention coefficient of the inner layer k o zone retention coefficient of the outer layer k 1 zone retention coefficient, sum of k i and k o A i retention parameter of the outer layer A o retention parmeter of the outer layer k apparent retention factor t R retention time t M hold-up time of the column z spatial variable s Laplace variable r radial variable t time