On the relationship between radial structure heterogeneities and efficiency of chromatographic columns

G Model

ARTICLE IN PRESS

CHROMA-359086; No. of Pages 15

Journal of Chromatography A, xxx (2017) xxx–xxx

Contents lists available at ScienceDirect

Journal of Chromatography A journal homepage: www.elsevier.com/locate/chroma

On the relationship between radial structure heterogeneities and efficiency of chromatographic columns Fabrice Gritti Waters Corporation, Instrument/Core Research/Fundamental, Milford, MA 01757, USA

a r t i c l e

i n f o

Article history: Received 3 August 2017 Received in revised form 11 September 2017 Accepted 23 September 2017 Available online xxx Keywords: Column efficiency Transverse structural column heterogeneities Structure–performance relationship Aris general dispersion theory Asymptotic and pre-asymptotic dispersion regimes Sub-2 ␮m particles

a b s t r a c t The general dispersion theory of Aris is applied to predict the virtual asymptotic dispersion behavior of packed columns. The derived model is also used to estimate the actual pre-asymptotic dispersion behavior of modern 2.1 mm × 50 mm columns packed with sub-2 ␮m fully porous particles (FPPs) during the transient dispersion regime. The model accounts for the actual radial distribution of the flow velocity across the column diameter. From the wall to the center of the column, focused-ion-beam scanning electron microscopy (FIB-SEM) experiments were recently performed to reveal the existence of a thin (0.15dp wide, dp is the average particle diameter) hydrodynamic boundary layer (THBL), a thin (3dp wide) and loose orderly packed layer (TLOPL), a 60dp wide and dense randomly packed layer (WDRPL), and a large (460dp ) randomly packed bulk central region [1]. The theoretical calculations of the actual pre-asymptotic reduced van Deemter curves (2.1 mm × 50 mm column, sub-2 ␮m BEH-C18 FPPs, n-hexanophenone analyte, acetonitrile/water eluent, 80/20, v/v, flow rate from 0.05 to 0.35 mL/min) confirm that the impact of the sole THBL on column dispersion can be neglected. In contrast, the contribution of the TLOPL to the reduced plate height (RPH) is about 0.2 h unit at optimum reduced velocity. Most remarkably, the negative impact of the TLOPL on column performance may be fully compensated by the presence of the adjacent WDRPL if the depth of the velocity well were to be 5% of the bulk velocity. In actual 2.1 mm × 50 mm columns packed with sub-2 ␮m FPPs, this velocity depth is as large as 25% of the bulk velocity causing a significant RPH deviation of 0.7 h unit from the RPH of the bulk packing free from wall effects. Maximum column performance is expected for a reduction of WDRPL density. This suggests optimizing the packing process by finding the proper balance between the stress gradient across the WDRPL (responsible for the deep velocity well) and the friction forces between the packed particles (responsible for the rearrangement of the particles during bed consolidation). Past and recently reported RPH data support the theoretical insights: the stress gradient/particle friction balance in the WDRPL is better realized when packing superficially porous particles (SPPs) rather than FPPs in 2.1–4.6 mm i.d. columns (the RPH deviation is reduced to 0.4 h unit) or sub-2 ␮m particles in 100 cm × 75 ␮m i.d. capillaries combining high slurry concentrations and sonication (the RPH deviation is reduced to only 0.15 h unit). © 2017 Elsevier B.V. All rights reserved.

1. Introduction The performance of modern narrow-bore (2.1 mm i.d.) ultrahigh-pressure liquid chromatography (UHPLC) columns packed with either sub-2 ␮m FPPs or sub-2 ␮m SPPs is limited by the presence of the column hardware embedding the packed particles [2–4]. While the expected minimum RPH related to axial dispersion along random sphere packings is expected to be in the range from 0.5 for non-porous particles (NPPs) to 0.9 for FPPs in the

E-mail address: Fabrice [email protected]

absence of geometrical confinement [5,6], the observed minimum RPHs are typically 1.4 and 2.0 for commercial UHPLC columns optimally packed with SPPs and FPPs, respectively [7–9]. On one hand, the presence of the frits, the inlet flow distributor and of the outlet flow distributor may affect the performance of short columns (< 5 cm long) when poorly retained compounds are injected [10]. On the other hand, during the packing process, the presence of the stationary wall of the column (stainless steel tube for particulate columns) causes a higher stress in the peripheral region than that in the central bulk region [11,12]. This stress gradient induces density gradients in the bed and non-uniform flow profiles across the column diameter. Additionally, the flat surface of the column wall forces a two-dimensional geometrical arrangement of the spher-

https://doi.org/10.1016/j.chroma.2017.12.030 0021-9673/© 2017 Elsevier B.V. All rights reserved.

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030

G Model CHROMA-359086; No. of Pages 15 2

ARTICLE IN PRESS F. Gritti / J. Chromatogr. A xxx (2017) xxx–xxx

ical particles versus a three-dimensional packing arrangement in the bulk region. This structural heterogeneity applies to any column irrespective of their length and inner diameter unless the ratio of the inner column diameter to the particle diameter falls below 7 [13–15]. For such small column to particle diameter ratio, the geometrical wall constraint extends to the entire column diameter and the bed structure is found radially uniform [15]. For column aspect ratios larger than 7, the bed structure is no longer uniform across the entire column diameter: serious velocity disturbances are expected in the wall region relative to those across the uniform bed structure in the central region. These trans-column velocity biases may seriously affect the performance and resolution power of UHPLC columns [16–18]. Indirect experimental evidences of trans-column velocity biases and structural heterogeneity of the packed bed have been reported in the past literature. This was achieved for wide i.d. columns with microvoltammetric electrodes [19–21], optical fibers in a fluorescence-detection scheme [22], flow reversal [23] or with X-ray computed tomography [24], among others. Very recently, these observations were refined for narrow-bore UHPLC columns (2.1 mm × 50 mm) packed with sub-2 ␮m particles using FIB-SEM followed by image processing and fluid-flow simulation [1]. The observed radial distribution of the flow velocity revealed four distinct concentric zones of the packed bed from the wall to the center of the column: (1) an extremely thin (0.15dp wide, dp is the particle diameter) hydrodynamic boundary layer (THBL); (2) a thin (3dp wide) loose and orderly packed layer (TLOPL); (3) a wide (60dp ) dense and randomly packed layer (WDRPL); and (4) a large (460dp ) central bulk region. The local velocity increases from 0 to 2.2ubulk (ubulk is the average velocity in the bulk region) across the THBL, oscillates and decreases from 2.2ubulk to 0.75ubulk in the TLOPL, increases from 0.75ubulk to ubulk across the WDRPL and remains uniform at the bulk velocity across the entire central region. Active flow technology (AFT) has helped solving this problem of transcolumn flow heterogeneity. Shalliker and co-workers have amply proven the benefit of AFT on column performance [25–29] while others independently confirmed this advantage [30–32]. Regrettably, this approach remains impractical because it requires using proper outlet restrictors to optimize the flow rate ratio between the wall and the center region of the column. Additionally, the optimum flow rate ratio is somewhat dependent on column length, column diameter, retention mode, and on the nature of the small analyte. Therefore, progress in column performance is better linked to the development of more uniformly packed columns, ordered structures [33], or of 3D printing technologies [34] than it is to AFT. The unprecedented work and achievement made by Reising and Tallarek regarding the complete structural heterogeneities of the packed bed across a narrow-bore UHPLC column have opened new fundamental and practical roads towards the complete understanding of the complex relationship between bed structure and column performance. Ultimately, this work should guide the practitioners towards the proper development and optimization of packing procedures. In this work, on the basis of the accurate radial distribution of the fluid-flow velocity observed across 2.1 mm i.d. UHPLC columns [1], the impacts of the radial structure heterogeneities of packed beds (including the THBL, the TLOPL, the WDRPL, and the central bulk region) on the performance of narrow-bore UHPLC columns is investigated in depth. First, the exact expression of the asymptotic (long-time) dispersion coefficient along short 5 cm long UHPLC columns is derived by extending the applications of the Aris general dispersion theory [35] from open coaxial cylinders to filled packed columns. It is important to keep in mind that the present theoretical development aims at studying the impact of the structural heterogeneities of columns on column efficiency. By no means is it about investigating the fundamental impact of thermal effects

and radial temperature gradients on column performance when operating UHPLC columns at very high pressures (>10 000 psi) and high flow rates (> 1 mL/min). This problem was tackled in depth during the 2000s when UHPLC technology was emerging [36–41]. Secondly, the relationship between the long-time asymptotic RPHs (Aris) and the observed pre-asymptotic RPHs is empirically determined from the study of the transient dispersion behavior of a small molecule (n-hexanophenone) on a 2.1 mm × 50 mm column packed with sub-2 ␮m BEH-C18 particles. Thirdly, the application of Aris dispersion theory enables to evaluate the impact of the combined existence of four different fluid-flow velocity regions on column performance for various and arbitrarily designed radial velocity profiles. The results are reported and discussed in terms of the structure–performance relationship for a chromatographic column. Finally, solutions are proposed to minimize the performance losses of narrow-bore columns packed with sub-2 ␮m particles by acting properly on the selection of the packing process conditions and on the morphology of the particles to be packed.

2. Theory The first part of this section summarizes the main results derived by Aris in 1959 for the prediction of the asymptotic axial dispersion coefficient of analytes by convection, diffusion, and exchange between two cylindrical and coaxial phases [35]. These results are valid for infinitely long columns or at infinitely long times, e.g., when the concentration distributions tend towards a symmetrical Gaussian distribution [42]. In the second part, the formalism of Aris is directly applied to the determination of the asymptotic dispersion coefficients of analytes along UHPLC narrow-bore columns packed with sub-2 ␮m spherical particles. The fundamental equations enable the calculation of the asymptotic trans-column eddy dispersion RPH from the actual distribution of the fluid-flow velocity recently obtained by flow simulations in a FIB-SEM based 3D reconstruction from a 2.1 mm × 50 mm column packed with sub2 ␮m BEH-C18 particles [1].

2.1. Aris general theory of analyte dispersion by convection, diffusion, and exchange between phases Aris rigorously derived the increase rate of the spatial peak variance, d2,z /dt, of the concentration zone migrating along an open tube filled with two coaxial annular phases in contact. Phase 1 is located in between the radial coordinates r0 and r1 , phase 2 in between radial coordinates r1 and r2 and the volume embedded in between the radial coordinates r = 0 and r0 is impermeable to the analyte. At equilibrium, the concentrations, c1 and c2 , of the analyte in phases 1 and 2, respectively, are linked by the equilibrium constant ˛ (c2 = ˛c1 ). When the two phases are not in thermodynamic equilibrium, the mass transfer between them is assumed to be given by the linear driving force model [43] in phase 2: the mass flux is then k(c2 − kc1 ), where k is the rate constant (see Fig. 1). The mathematical formalism of Aris provides the time-independent expression of d2,z /dt under asymptotic conditions. This expression is given by [35]:  1 d2,z = ˇ 2 dt

 D2 + 2

where s =

 D1 + 1

U22 [r22 − r12 ]



D2

U12 [r12 − r02 ] D1



 +

 + (1 − ˇ)

2

sˇ[1 − ˇ][U1 − U2 ] 2k˛r1

(1)

(r12 − r02 ) + ˛(r22 − r12 ) and ˇ = r12 − r02 /s2 .

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030

G Model

ARTICLE IN PRESS

CHROMA-359086; No. of Pages 15

F. Gritti / J. Chromatogr. A xxx (2017) xxx–xxx

3

Fig. 1. Description of the separation system treated by Aris in his general dispersion theory along two coaxial phases (1 and 2) by diffusion (D1 and D2 ), convection (U1 and U2 ), and exchange (rate constant k) between them.

In Eq. (1), Di (i = 1,2) is the mean diffusion coefficients and Ui is the mean linear velocity in phase i. 1 and 2 are the integrals given by [35]:



r1



1 = r0



r2

2 =



2

R r 2 − r02 1 (r) −  r2 − r2 1 0 2 (r) −

r1

dr 2r1 (r)

2 2

R r22 − r  − 1 r2 − r2 2 1

dr 2r2 (r)

(2)

i (r) =



r 



2r i (r )dr

2 ri2 − ri−1



i (r) =

1 2 ri2 − ri−1

2r 

i (r



)dr



r1 = rc

(7)

(inner radius of the packed column)

(5)

ri−1

where i (r)Di is the local diffusion coefficient in phase i at radial coordinate r. 2.2. Application of Aris theory to the asymptotic dispersion along UHPLC columns In this section, the analogy between Aris description of the dispersion problem along a tube containing two coaxial annular phases and the dispersion along particulate columns is first presented. Then, the asymptotic dispersion coefficient of the analytes in actual UHPLC columns is rigorously derived from Aris formalism. 2.2.1. Analogy The analogy between the axial dispersion phenomena taking place along Aris column and UHPLC packed columns is shown schematically in Fig. 2. The uniform phase 1 in Aris description is assumed to be represented by the packed bed immersed in the eluent. There is no impermeable cylindrical core in the center of the UHPLC column, which has an inner radius rc . The uniform phase 2

(8)

The mean velocity U2 , the concentration c2 , and the equilibrium constant ˛ are all equal to zero because the analyte does not adsorb (partition) onto (into) the stainless steel tube. Accordingly, the Aris parameters , s, ˇ, and R take a simpler form:

(4)

where i (r)Ui is the local linear velocity in phase i at radial coordinate r. and r

(6)

U1 = 1 U1 − U2

 =

ri−1



r0 = 0 (no impermeable cylindrical core in packed columns)

r 2 is irrelevant for packed columns (3)

where  = U1 /U1 − U2 and R = ˇ +  − 1. In Eqs. (2) and (3), the functions i (r) and i (r) are given by [35]: 1

in Aris column is represented by the impermeable stainless steel column tube embedding the packed bed. Accordingly, the radii r0 , r1 , and r2 in Aris mathematical formalism become:

(9)



(r12 − r02 ) + ˛(r22 − r12 ) = rc

s =

r12 − r02

ˇ =

s2

= 1

(10) (11)

R = ˇ+ −1 = 1

(12)

One fundamental difference between Aris separation system and randomly packed beds is that each mean diffusion coefficients (D1 or D2 ) is no longer the same across (transverse dispersion coefficient) and along (axial dispersion coefficient) a chromatographic column. In packed beds, the mean transverse dispersion coefficient (Dt,1 ) differs from the mean axial dispersion coefficient (Da,1 ) because the net linear velocity of the eluent in any directions perpendicular to the flow direction is zero [44,45]. Therefore, the general Aris Eq. (1) to be applied to the study of dispersion along UHPLC chromatographic columns is reduced to: d2,z dt

= 2Da,1 + 1

2U12 rc2 Dt,1

(13)

where the new expression for 1 becomes:



rc

1 =





1 (r) − 0

1



1 (x) − x2

= 0

2

r2 rc2

2

dr 2r1 (r)

dx 2x1 (x)

(14)

(15)

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030

G Model

ARTICLE IN PRESS

CHROMA-359086; No. of Pages 15

F. Gritti / J. Chromatogr. A xxx (2017) xxx–xxx

4

Fig. 2. Description of the separation system for conventional packed columns. Note the correspondence between Aris description in Fig. 1 and the present one: for packed columns, there is no impermeable cylindrical core, phase 1 is the packed bed immersed in the eluent, phase 2 is the impermeable stainless steel tube, and there is no longer any mass transfer between phase 2 and phase 1. The properties in phase 1 (concentration, diffusion, and velocity) are the effective or apparent properties of the binary mixture made of packed particles and inter-particle eluent.



1

= 0

1 (x)2 dx − 2 2x1 (x)



1

0

= 11 − 212 + 13

x1 (x) dx + 21 (x)

 0

1

x3 dx 21 (x)

(16) (17)

where x = r/rc is the dimensionless radial coordinate and 11 , 12 , and 13 are the three integrals given in the right-hand-side of Eq. (16) and previously derived by Martin and Guiochon in turbulentflow liquid and gas chromatography [46] and in open-tubular capillary liquid chromatography with electroosmotic flow [47,48]. Similarly, the functions 1 (x) in Eq. (4) and 1 (x) in Eq. (5) are rewritten:



x

2x 1 (x )dx

1 (x) =



(18)

0

where 1 (x )U1 is the local linear velocity of the analyte at the dimensionless radial coordinate x . and



x

2x

i (x) =

 1 (x )dx



(19)

0

where 1 (x )Dt,1 is the local transverse dispersion coefficient of the analyte at the dimensionless radial coordinate x . By construction, the functions 1 (x) and 1 (x) obey the following normalization relationships:



1

c1 = e cm + (1 − e )cs = e (1 + k1 )cm

(22)

where k1 is the zone retention factor defined by [16]: 1 − e cs e cm

k1 =

(23)

The mass balance equation is written in the absence of axial dispersion for an elementary slice of bed volume (in between axial positions x and x + dx) during the time period dt. Accordingly [43],

e

∂cm ∂cs ∂cm + (1 − e ) + e u = 0 ∂t ∂t ∂x

(24)

where u is the interstitial linear velocity of the mobile phase. It is given by: u =

Fv

(25)

e rc2

where Fv is the applied flow rate. After simplification, Eq. (24) is rewritten as a function of the apparent concentration c1 :

2x1 (x)dx = 1

(20)

∂c1 u ∂c1 + = 0 1 + k1 ∂x ∂t

= 1

(21)

The velocity U1 is equal to the propagation velocity of any fixed concentration c1 . This defines a characteristic line of the mass balance equation. The slope of this characteristic line is given by [43]:

0



between the external inter-particle eluent and the packed particles. The overall concentration, c1 , of the analyte in the bed is the average concentration taken over the inter-particle volume (bulk eluent, volume fraction e , concentration cm ) and the particle volume (porous particles, volume fraction 1 − e , concentration cs ). Accordingly, c1 is given by:

1

2x

1 (x)dx

0

In the next section, the expressions of the system parameters U1 , Da,1 , and Dt,1 in Eq. (13) are derived for a packed column. Because a packed bed immersed in the eluent is not a uniform phase, the effective or apparent properties of this complex binary material are needed in order to be directly applied in Aris general formalism. 2.2.2. Effective properties: velocity, axial dispersion, and transverse dispersion 2.2.2.1. Sample zone velocity U1 . The apparent velocity, U1 , of the chromatographic zone depends on the distribution of the analyte

dx  dt

c1

= U1 =

(26)

u 1 + k1

(27)

2.2.2.2. Apparent axial dispersion Da,1 . In presence of axial sample dispersion in the eluent (Da ) along the column, the mass balance equation is completed by adding a dispersion term in the righthand-side of Eq. (26) [43]: 2

∂c1 ∂ cm ∂c1 + U1 = e Da ∂t ∂x ∂x2

(28)

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030

G Model CHROMA-359086; No. of Pages 15

ARTICLE IN PRESS F. Gritti / J. Chromatogr. A xxx (2017) xxx–xxx

2

=

is then written:

Da ∂ c1 (1 + k1 ) ∂x2

(29)

The relationship between the apparent (relative to the total sample concentration c1 ) axial dispersion coefficient, Da,1 , and the actual dispersion coefficient Da (relative to the bulk sample concentration cm ) is then: Da,1 =

Da 1 + k1

H h u Dm Da = bulk = bulk 2 2

(31)

where  = udp /Dm is the reduced interstitial linear velocity, hbulk = Hbulk /dp is the reduced plate height, Dm is the diffusion coefficient in the eluent, and dp is the average particle diameter. The accurate estimation of the reduced plate height, hbulk , in the bulk region of random packings involves longitudinal diffusion [49], eddy dispersion in the mobile phase including trans-channel and short-range inter-channel velocity biases [5,50,51] (long-range velocity biases are not considered as they represent a property of the entire column), and solid–liquid mass transfer resistance between the mobile and stationary phases (assuming spherical FPPs). Accordingly, the local reduced plate height in the bulk region of a randomly packed bed is best described by the following expression [4,49]: Deff 1 + Longitudinal diffusion Dm 

(32) (33)

ω2  + Short-range inter-channel dispersion 1 + (ω2 /2 2 )

(34)

 k 2 1 1 1 + k1



Solid − liquid mass transfer resistance(35)

where  = Dp /Dm is the ratio of the particle diffusivity to the bulk diffusion coefficient measured from peak parking experiments [52,53]. In this work,  = 0.46 and k1 = 2.8 (BEH-C18 FPPs, n-hexanophenone, acetonitrile/water, 80/20, v/v, T = 297 K). ω1 = 0.0041, 1 = 0.45, ω2 = 0.13, and 2 = 0.23 are the best parameters obtained by fluid-flow simulation through the S×2 computer-generated random sphere packing and by fixing the external porosity at 0.38 [50,5]. Finally, in Eq. (32), Deff is the effective diffusion coefficient of the analyte along the packed bed given by Torquato’s equation [54–57] which was validated both experimentally and numerically [58,59]: Deff Dm

1 = e (1 + k1 )



1 + 2(1 − e )ˇ − 2e 2 ˇ2 1 − (1 − e )ˇ − 2e 2 ˇ2



(36)

where the parameter  2 is set at 0.627 for randomly packed spherical particles and ˇ is defined by [56,57]: ˇ =

−1 +2

+

ω1 2 ω2 2 + 2 + (ω1 / 1 ) 2 + (ω2 / 2 )



k12 1 e 1 2  60 1 − e (1 + k1 )3 

(38)

2.2.2.3. Apparent transverse dispersion Dt,1 . Similarly to the determination of the axial dispersion coefficient, the transverse dispersion coefficient [44] of the analytes has been derived from flow simulation across randomly packed beds. For NPPs, the corresponding RPH ht , is best written [45]: ht =

2e + 2at ˇt −1 

(39)

where  e is the external obstruction factor of the packed bed ( e  0.6), at = 0.146, and ˇt = 0.758 for wide particle size distribution packings [45]. ht is then written as the sum of a diffusion and convection RPH terms. For FPPs, Eq. (39) is simply rewritten by assuming that the effective diffusion (in the whole column volume including the particles) and convection (in the external bulk volume only) terms remain independent and additive. Therefore, at a first approximation: ht = 2(1 + k1 )

Deff 1 + 2at ˇt −1 Dm 

(37)

Finally, by combining Eqs. (31)–(32), the final expression of the apparent (relative to the total sample concentration c1 ) dimensionless axial dispersion coefficient in the bulk region of a packed bed

(40)

By definition [9], Dt =

ht Dm 2

(41)

In presence of transverse sample dispersion in the eluent (Dt ) along the column, the mass balance equation is completed by adding a second dispersion term in the right-hand-side of Eq. (28) [43]. Accordingly,



2

ω1  + Trans-channel dispersion 1 + (ω1 /2 1 )

1 e 30 1 − e



Deff Da,1 1 = + Dm Dm 1 + k1

(30)

The actual axial dispersion coefficient in the mobile phase is lumping all dispersion phenomena taking place in a local bulk region of the chromatographic column. This is the basis for the derivation of the equilibrium-dispersive model of chromatography in Eq. (28) [43]. By definition, Da is given from the local plate height, Hbulk , in the bulk region of the packed bed and the interstitial linear velocity u [9]:

hbulk = 2(1 + k1 )

5

2

∂c1 ∂c1 ∂ c1 ∂ cm 1 ∂cm + e Dt + + U1 = Da,1 r ∂r ∂t ∂x ∂x2 ∂r 2 2

= Da,1

Dt ∂ c1 + (1 + k1 ) ∂x2



2

∂ c1 1 ∂c1 + r ∂r ∂r 2



(42)

(43)

The effective transverse dispersion coefficient is then: Dt,1 =

Dt 1 + k1

(44)

Therefore, the apparent (relative to the total sample concentration c1 ) transverse dispersion coefficient Dt,1 is written: Deff Dt,1 at ˇt = + Dm Dm 1 + k1

(45)

2.2.3. Experimental radial velocity profile: the 1 (x) function The calculation of the integrals 11 , 12 , and 13 in Eq. (16) necessitates the exact knowledge of the experimental radial distribution of the local migration linear velocity 1 (r)U1 across the whole column diameter. The data for the local intertitial linear velocities u(r) have been obtained by flow simulations in a FIB-SEM based 3D reconstruction from a standard 2.1 mm × 50 mm column packed with sub-2 ␮m particles (1.99 ␮m BEH-C18 particles) [1]. The results (full blue circles data points) are shown in Fig. 3 as the plot of the experimental ratio u(r)/ubulk (ubulk is the reference interstitial linear velocity in the center bulk region of the column) versus the normalized radial position r/dp (or the number of particle diameters) from the wall (r = 0) towards the center of the column. The function (its mathematical expression is purely empirical, so, it is irrelevant and is not provided here) that best fits these data

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030

G Model CHROMA-359086; No. of Pages 15

ARTICLE IN PRESS F. Gritti / J. Chromatogr. A xxx (2017) xxx–xxx

6

Fig. 3. Radial interstitial velocity profile obtained by flow simulations in a FIB-SEM based 3D reconstruction (full blue circles data) from a 2.1 mm × 50 mm column packed with sub-2 ␮m BEH-C18 particles [1]. The solid red line is the best mathematical fit to these data. Note that the three visible experimental oscillations observed near the wall (within 3 particle diameters dp from the column wall) were represented by a smooth function with a single maximum located at the extremity of the hydrodynamic boundary layer at a distance of 0.15 dp from the wall surface. No slip-flow conditions were assumed, so, the velocity is strictly zero at the column wall. The velocity far from the wall towards the center of the column is the bulk velocity ubulk . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

is represented by the thick solid red line. The data reveal a slow and continuous diminution of the eluent velocity over a distance from about 60 (u = ubulk ) down to 3 (u = 0.75ubulk ) particle diameters from the column wall. In this wall region (about 120 ␮m wide or one tenth of the column inner radius), the bed density is higher than that in the central bulk region as a result of the slurry packing process. Within 3 particle diameters (6 ␮m) from the column wall, particles are more orderly distributed and the local velocity passes through a succession of maxima (at exactly 0.2, 1.1, and 2.1 particle diameters from the wall) and minima (at exactly 0.7, 1.7, and 2.7 particle diameters from the wall) to reach 2.2ubulk at a distance of 0.15dp from the wall. The eluent velocity drops across the hydrodynamic boundary layer (0.3 ␮m thick) to reach zero at the very wall surface where no slip-flow conditions apply. The relationship between the local migration linear velocity U1 (r) and the local interstitial linear velocity u(r) is: u(r) 1 + (1 − e (r)/e (r))K

U1 (r) =

2

e (x)

2

u(x) =

dp2 P Kc L

= K0

(47)

where Kc is the Kozeny–Carman constant, K0 is a constant, P is the constant pressure drop along the column, and  the constant eluent viscosity. The exact solution of Eq. (47) for e (x) is:

e (x) =

−1 +



K0 /u(x)

(K0 /u(x)) − 1



1

e =

2xe (x)dx

(49)

0

and

1 k1 =

0

2x [1 − e (x)] cs dx

1 0

2xe (x)cm dx

= K

1 − e

e

(50)

Accordingly, for e = 0.38 and k1 = 2.8, K0 = 2.6594 and K = 1.7161. The corresponding function 1 (x) is finally represented in Fig. 4. The mean migration linear velocity is somewhat smaller than the bulk migration velocity by about 2% at U1 = 0.9812Ubulk,1 .

(46)

where K is the unique equilibrium constant (independent of r) and e (r) is the local external porosity at the radial position r. At a very first approximation, the mobile phase is considered as incompressible, its viscosity is uniform everywhere in the column, and the pressure profile is radially uniform (capillary bundle approximation). Accordingly, the radial distribution of the external porosity, e (r), may be directly estimated from the Kozeny–Carman relationship for any radial position x: [1 − e (x)]

The unknown constants K0 and K are unambiguously obtained from the two constraints related to the observed average bed porosity and average zone retention factor:

(48)

2.2.4. Experimental radial transverse dispersion profile: the 1 (x) function The calculation of the integrals 11 , 12 , and 13 also requires the knowledge of the experimental radial distribution of the local transverse dispersion coefficient 1 (r)Dt,1 across the whole column diameter. Dt,1 is the sum of a diffusion term (Torquato model of effective diffusion in packed beds) and convection term (Tallarek simulations). In this work, experimental data were recorded for a weakly retained compound (n-hexanophenone, k1 = 2.80) in order to estimate the diffusion term. Accordingly,  = 0.46 so the diffusion term Deff /Dm in Eq. (45) is equal to 0.44. The convection term in Eq. (45) depends on the reduced velocity . For the sake of illustration, the maximum column efficiency was observed for  = 5 (Fig. 6). Eventually, Fig. 5 represents the plot of the corresponding function 1 (x) at  = 5, which shows that the transverse dispersion coefficient is more uniformly distributed than the migration velocity across the column diameter. Eventually, the average transverse dispersion coefficient is within less than 0.3% of the bulk dispersion coefficient at Dt,1 = 0.9978Dt,bulk,1 .

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030

G Model CHROMA-359086; No. of Pages 15

ARTICLE IN PRESS F. Gritti / J. Chromatogr. A xxx (2017) xxx–xxx

7

Fig. 4. (Left graph) Representation of the Aris function 1 (x), which is the ratio of the local migration velocity at the radial position x (U1 (x)) to the average migration velocity U1 of the analyte. The function was calculated for an actual 2.1 mm × 50 mm column packed with sub-2 ␮m BEH-C18 particles and is represented from the center (x = 0) to the column wall (x = 1). (Right graph) Zoom in the wall region from x = 0.95 to x = 1.

Fig. 5. (Left graph) Representation of the Aris function 1 (x), which is the ratio of the local transverse dispersion coefficient at the radial position x (Dt,1 (x)) to the average transverse dispersion coefficient Dt,1 of the analyte. The function was calculated for an actual 2.1 mm × 50 mm column packed with sub-2 ␮m BEH-C18 particles and is represented from the center (x = 0) to the column wall (x = 1). (Right graph) Zoom in the wall region from x = 0.95 to x = 1.

Fig. 6. Plots of the different reduced plate heights versus the reduced velocity of a 2.1 mm × 50 mm column packed with sub-2 ␮m BEH-C18 particles. In blue, for the sake of reference, is shown the RPH plot associated with the dispersion along the bulk structure of the bed free from the wall effects. This is the performance of the infinite diameter column. The RPH plot in purple is related to the column performance if operated in the asymptotic dispersion regime. The open green circles are experimental data recorded for n-hexanophenone at 297 K using a mixture of acetonitrile and water (80/20, v/v) as the eluent. The solid green line is the best Knox plot enabling extrapolation of the pre-asymptotic dispersion data up to  = 25. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030

G Model

ARTICLE IN PRESS

CHROMA-359086; No. of Pages 15

F. Gritti / J. Chromatogr. A xxx (2017) xxx–xxx

8

columns (2.1 mm × 50 mm) from the difference between the temporal moments recorded by the two detection cells.

2.2.5. Asymptotic plate height of a UHPLC column According to the formalism of Aris and Eq. (13), d2,z

d2,z dz dt dz dt u = hcolumn dp 1 + k1 =

= 2Da,1 + 1

2U12 rc2

(51)

3.3. Columns

(52)

The commercial 2.1 mm × 50 mm column packed with 1.7 ␮m (brand size) fully porous BEH-C18 particles was obtained on-site (Waters, Milford, MA) were used. The actual average particle size was corrected to 1.99 ␮m according to SEM measurements [1]. The external porosity was measured on-site at 38.0% by inverse sizeexclusion chromatography. The RPHs of n-hexanophenone were recorded at seven different flow rates of 0.04, 0.09, 0.13, 0.19, 0.24, 0.30, and 0.35 mL/min. No higher flow rates could be applied due to the pressure limitation of the pre-column detection cell 1. The experimental RPHs were corrected for the dispersion taking place from one to the second detection cell in absence of the chromatographic column. Therefore, the experimental RPH is calculated from the expression:

(53)

Dt,1

After identification and simplification, the final expression of the reduced plate height, hcolumn , is then given by: hcolumn

(Da /Dm ) = 2 + 21 



= hbulk + hwall

rc dp

2

 (Dt /Dm )

(54) (55)

In conclusion, the specific contribution of the trans-column interstitial linear velocity biases (or that of the wall effects which affect the intensity of the parameter 1 ) to the total asymptotic reduced plate height of a chromatographic column of inner radius rc and packed with particles of average diameter dp is directly given by the second term in the right-hand-side of Eq. (54). 3. Experimental 3.1. Chemicals The mobile phase used was a mixture of acetonitrile and water (80/20, v/v). Both solvents were HPLC grade from Fisher Scientific (Fair Lawn, NJ, USA). The selected sample compound was n-hexanophenone and it was purchased from Sigma Aldrich (Milwaukee, WI, USA) with a purity exceeding 99%.



hexp





    L 2 (2) − 2 (1) − 2,ex (2) − 2,ex (1)  = dp [1 (2) − 1 (1)] − 1,ex (2) − 1,ex (1)

(56)

where 2 (i) and 1 (i) are the second central moment and the first moment, respectively, of the temporal concentration distribution recorded by the low-dispersion detection cell i (i = 1 for the precolumn cell and i = 2 for the post-column cell). The subscript ”ex” refers to the same recorded moments but in the absence of the chromatographic column. 4. Results and discussion 4.1. Comparison between bulk, pre-asymptotic, and asymptotic dispersion in a 2.1 mm × 50 mm column packed with sub-2 m particles

3.2. Instrument and materials The i-class Acquity UPLC system (Waters, Milford, USA) was used to record the concentration profiles of n-hexanophenone. The system is operated by the Empower 3 Chromatography Data Software (Waters, Milford, MA, USA). An updated version of this software was used in order to measure the first and second central temporal moments of the recorded concentration profiles at various flow rates. The standard configuration of this instrument includes a binary solvent delivery system, an auto-sampler equipped with a 15 ␮L flow-through-needle (FTN) loop connected to the injection valve by a 30 cm × 100 ␮m tube, which generates a pre-valve sample dispersion volume variance of 0.5 ␮L2 [60], an air-oven compartment (not used in this work), an active column preheater (not used in this work), an outlet 26.7 cm × 100 ␮m PEEK tubing (not used in this work) and a 500 nL flow-cell TUV-detector (not used in this work). The flow rate may range from 0 to 2 mL/min. The system pressure limitation is 15 000 psi up to 1 mL/min. The average lab temperature was 297 K with a maximum daily variation of ± 1 K. This standard configuration of the i-class Acquity UPLC system was modified by building two identical low-volume detection cells 1 and 2 (100 ␮m × 3 mm, 25 nL volume each) connected immediately before (cell 1) and after (cell 2) the chromatographic column through a 100 mm × 75 ␮m Zenfit perfect connection tubing. The total cell-to-cell volume was measured at 1.0 ␮L in absence of the chromatographic column. Each flow cell has a maximum pressure tolerance of only 3000 psi. The pre-column detection cell was connected to the injection valve of the Acquity UPLC system through a 250 mm × 75 ␮m Zenfit perfect connection tubing. This modified version of the i-class Acquity UPLC system enabled us to measure accurately the RPHs of low dead volume UHPLC

4.1.1. Asymptotic dispersion regime Fig. 6 compares four different plots of the RPHs of nhexanophenone versus the reduced velocity  for a 2.1 mm × 50 mm column packed with 1.99 ␮m BEH-C18 particles. In blue color, the reduced van Deemter plot is represented for the so-called “infinite diameter column”, e.g., for a virtual packed column free from the negative impact of the wall effects on the column performance. This corresponds to sample dispersion taking place in the homogeneous random bulk packing (h = hbulk as given by Eq. (32)). In purple, the reduced van Deemter plot is calculated from the application of the Aris general dispersion theory to packed columns (see details in the theory Section 2). It represents the expected column performance when the asymptotic dispersion regime has been reached, e.g., when the column is long enough to enable equilibration of the sample concentration across the whole column diameter (2.1 mm). The open green circles are the experimental RPHs measured from Eq. (56) at seven different flow rates from 0.04 to 0.09, 0.13, 0.19, 0.24, 0.30, and to 0.35 mL/min. The peak shape are slightly tailing due to the contribution of the extra-column system. The largest USP tailing factor was recorded at 1.078 (see supplementary material S1). The RPH data were corrected for the low cell-to-cell volume variance, which increases from 0.14 to 0.23, 0.27, 0.35, 0.43, 0.47, and to 0.48 ␮L2 at the same flow rates as those above-mentioned. For the sake of comparison, the total (detection cell 1 + column + detection cell 2) volume variances (the total elution volume is 252 ␮L, the zone retention factor of the analyte is k1 = 2.80) were recorded at the same flow rates at 13.2, 7.14, 5.30, 4.46, 3.99, 3.95, and to 4.02 ␮L2 , respectively. The parameter  (the ratio of the particle diffusivity to the bulk diffusion coefficient [52]) was measured at 0.46 assuming the Torquato model of effective diffusion along packed beds. Note that the observed minimum RPH is equal to 1.6, an unusually low

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030

G Model

ARTICLE IN PRESS

CHROMA-359086; No. of Pages 15

F. Gritti / J. Chromatogr. A xxx (2017) xxx–xxx

minimum RPH for a UHPLC column packed with sub-2 ␮m FPPs (the minimum RPH is typically around 2.0 for FPPs). The explanation is based on the fact that the RPH data were all calculated for the actual average particle diameter dp = 1.99 ␮m as accurately measured by SEM [1]. Based on the commercial particle diameter (1.7 ␮m), the minimum RPH would actually be 1.9. Finally, the solid green line is the best Knox RPH plot, which extrapolates the experimental data to higher reduced velocities up to 25. The general Knox equation is given by: hKnox = 2(1 + k1 )

Deff 1 1 e + A1/3 + Dm  30 1 − e

 k 2 1 1 1 + k1



 (57)

where the best A coefficient accounting for eddy dispersion in the column is A = 0.51. The large h differences between the Aris RPH plot (solid purple line) and the Knox RPH plot (solid green line) demonstrates that the dispersion regime of n-hexanophenone along the 2.1 mm × 50 mm column is clearly pre-asymptotic. This is not a surprising result given the column i.d. (2.1 mm), the weak intensity of the apparent transverse dispersion coefficient in random sphere packings (Eq. (44), 8 × 10−6 cm2 /s) and the rather short residence times inside the column (from 0.7 to 7 min). At most for the lowest applied flow rate (tR = 7 min residence time and Dt,1 = 6.4 × 10−6 cm2 /s), the largest standard deviation  of the characteristic transverse length in a circular geometry ( 4Dt,1 tR ) is just 1.04 mm. At best, n-hexanophenone may statistically sample only one inner column radius. At worst, for the highest applied flow rate (tR = 0.7 min residence time and Dt,1 = 8.0 × 10−6 cm2 /s) the characteristic transverse length is only 0.37 mm. Therefore, the sample concentration cannot be strictly uniform across the entire column diameter in presence of flow heterogeneities. The question is to know when the asymptotic dispersion regime is reached. Taking open tubes and a parabolic Poiseuille-flow profile, the asymptotic dispersion regime is reached within 1% when the following condition is met [42]: 4Dt tR > 2800rc2

(58)

where Dt is the transverse diffusion coefficient of the analyte and tR its residence time in the cylindrical tube. This amounts to saying deviation of the radial distance deviation should that the standard √ be at least 2800/2 25 times larger than the inner diameter of the tube in order to ensure a quasi-asymptotic dispersion regime in the presence of a parabolic flow profile. Extrapolating this result to packed columns in which the fluid-flow profile is not as severely deformed as a parabolic flow profile, the condition expressed by Eq. (58) would impose a minimum column length, Lmin , given by: Lmin =



700rc2

dp (1 + k1 )(Deff /Dm ) + at ˇt



(59)

For the sake of application, if  = 5 (close to the observed optimum experimental reduced velocity), rc = 1.05 mm (inner radius of D

= 0.44 (the measured effective longitudiour UHPLC column), Deff m nal coefficient of n-hexanophenone), k1 = 2.8 (the measured zone retention factor of n-hexanophenone), Dm = 1.37 × 10−5 cm2 /s (the measured diffusion coefficient of n-hexanophenone in the acetonitrile/water mixture, 80/20, v/v, at 297 K), and dp = 1.99 ␮m (the actual average size of the BEH-C18 particles), then, the asymptotic dispersion regime would be reached within 1% if the column length were at least 900 meters long. A more relevant estimation of the minimum column length required to reach the asymptotic dispersion regime was provided by Daneyko et al. regarding the actual transient dispersion regime along random sphere packings confined in different conduit geometries [45]. They demonstrated from fluid-flow and advection-diffusion simulations that the asymptotic time

9

scale beyond which axial dispersion becomes time-independent (2rc /dp = 20) is [2,61]: t>

2.25rc2 2Dt

(60)

or Lmin =

2.25rc2



dp (1 + k1 )Deff /Dm + at ˇt



(61)

Accordingly, the minimum column length required to reach the asymptotic dispersion regime in 2.1 mm i.d. columns packed with 2 ␮m particles is at least 3 m at  = 5. 4.1.2. Transient dispersion regime The results of the previous section made it indisputable that a pre-asymptotic dispersion regime prevails with actual 2.1 mm × 50 mm UHPLC columns packed with sub-2 ␮m particles when they are run at flow rates from 0.05 mL/min (Lmin =64 cm) to 0.35 mL/min (Lmin =3.5 m). For instance, at  = 5, the ratio of the asymptotic to the pre-asymptotic RPH is as large as 55. The calculation of the asymptotic RPH from Aris theory should then be considered with great caution regarding the anticipated performance of the short 2.1 mm × 50 mm UHPLC column. The relevant correcting factor between the calculated hAris and the observed hKnox data will be needed in the next section when considering arbitrary flow profile distributions across the 2.1 mm i.d. column. In essence, this correcting factor depends on the column length L (the longer the column is, the closer the observed dispersion regime is to the asymptotic dispersion regime), on the linear velocity u (the slower the speed is, the larger the residence time inside the column is, and the closer the observed dispersion regime is to the asymptotic dispersion regime), on the column inner radius rc (the wider the column is, the longer time it takes for the analyte to sample any radial position), and on the transverse dispersion coefficients (the larger Dt is, the faster the asymptotic dispersion regime is reached). More quantitatively, Tallarek and co-workers have shown that the temporal evolution of the pre-asymptotic axial dispersion coefficient, Da (t), towards the long-time dispersion coefficient, Da (∞), follows an exponential law. The characteristic time of this exponential law is the transverse dispersive time tD = Lc2 /2Dt [3,61] where Lc is the characteristic transverse length. The transient dispersion regime is then quantitatively represented by: Da (t) = Da (∞)



1 − exp −

t tD



(62)

where the constant  = 7 for confined random sphere packings in a cylindrical conduit geometry of aspect ratio rc /dp = 20 [2,62] and 10 [63]. Since t = L/u = Ldp /Dm and Lc = rc for circular conduits, Eq. (62) can be rewritten in a dimensionless form:



Da , L/rc , rc /dp , Dt /Dm Da (∞)





= 1 − exp

2 Dt L dp −  Dm rc rc



(63)

For the sake of illustration, in this work, L/rc =48, rc /dp = 525, and Dt /Dm = 2.2 at  = 5. Accordingly: Da (t) = 0.43 Da (∞)

(64)

In theory, one would then expect that the pre-asymptotic axial dispersion coefficient is as much as 43% of the axial asymptotic dispersion along the 5 cm long UHPLC column. Experimentally, hKnox = 1.66 so hAris is expected to be 3.86. However, the calculations predict that hAris = 91.4. The duration of the transient dispersion regime is probably longer for packings confined in wide cylindrical conduit (rc /dp 500) than in narrow ones (rc /dp 10) [63]. For instance, at  = 10, the duration of the transient regime increases

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030

G Model

ARTICLE IN PRESS

CHROMA-359086; No. of Pages 15

F. Gritti / J. Chromatogr. A xxx (2017) xxx–xxx

10

Fig. 7. Representation of the different flow velocity profiles used in this work: in red, the velocity distribution is ideal and uniform. In green, the sole presence of the thin hydrodynamic boundary layer (THBL) is considered. In purple, the velocity profile combines the contributions of the THBL and that of the thin loose orderly packed layer (TLOPL). All the other velocity profiles account for the contributions of the THBL, the TLOPL, and of the wide dense randomly packed layer (WDRPL): they differ by the depth of the velocity well, which locate the minimum of the local velocities in the column. These minima were varied from 0.75 (actual velocity profile in the 2.1 mm × 50 mm column packed with sub-2 ␮m BEH-C18 particles), 0.80, 0.85, 0.90, 0.95 × the bulk velocity ubulk . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

from circular (characteristic transverse dispersion length Lc = 10dp ) to square (+18%, Lc = 12.5dp ), rectangular (+50%, Lc = 14dp ), and to semicircular (+155%, Lc = 20dp ) conduit geometries [63] despite the same intensities for the bulk transverse dispersion coefficient Dt and for the cross-section area. Eventually, the observed preasymptotic axial dispersion coefficient is only 1.8% of the axial asymptotic dispersion coefficient in 2.1 mm i.d. UHPLC columns. Accordingly, the coefficient  in Eq. (62) appears dependent on the characteristic transverse length Lc and it was empirically adjusted to 0.23 for Lc = 525 dp . The correcting factor sought for the estimation of the pre-asymptotic experimental RPHs from the asymptotic RPHs calculated for variable and arbitrary radial flow profile distributions across the UHPLC column is then given by:



hKnox 0.0414 Dt = 1 − exp −  Dm hAris

 (65)

In the next section, it is investigated how the positive deviations of the pre-asymptotic and asymptotic RPHs from hbulk are impacted by the combined presence of (1) an extremely thin (0.15dp wide) viscous hydrodynamic boundary layer (THBL) at the inside wall surface of the stainless steel tube, (2) a thin (3dp wide) and loose orderly packed layer (TLOPL) of particles in contact with the column wall, and of (3) a wide (60dp ) and dense randomly packed layer (WDRPL) of particles in the peripheral region of the column and how they can be minimized for maximum performance of UHPLC columns.

4.2. Relationship between radial structural heterogeneities and performance of UHPLC columns In the next sections, different flow velocity profiles are considered across the packed column. We start from the presence of the sole THBL and progress in a continuous way towards the full observed flow velocity profile combining the presence of the THBL, TLOPL, WRDPL, and of the bulk central region.

4.2.1. Hydrodynamic boundary layer Reising et al. [1] observed that the interstitial linear velocity rapidly decreases from about 2.2ubulk down to zero at the inner wall of the column tube. This is expected under no-slip velocity boundary condition at the solid–liquid interface of the column inner wall and can be seen in Fig. 3 across a layer of thickness equal to only 0.15 times the particle diameter. This defines the THBL of eluent at the inner surface of the stainless steel column tube. Its thickness is then of the order of 300 nm for 2 ␮m particles. The corresponding flow velocity profile in the absence of TLOPL and WDRPL is represented by the solid green curve in Fig. 7. It is strictly flat at u = ubulk from x = 0 up to x = 0.9997 and quadratically decreases down to u = 0 at x = 1. For the sake of comparison, the red horizontal line represents the ideal flat velocity profile over the entire column diameter from x =0 to x =1. In this reference case, 1 and hwall are strictly zero, so, hAris = hbulk . This serves as the reference bulk column performance. The calculated plot of the asymptotic RPH hAris versus  in the presence of the sole THBL is shown in Fig. 8 (solid green line). It reveals that the impact of the THBL on the column performance under asymptotic conditions can be neglected as hAris deviates by less than 2.4% from hbulk . The anticipated pre-asymptotic RPHs hKnox would then differ by less than 0.01% from hbulk (see Fig. 9). To summarize, in packed 2.1 mm i.d. columns, the inevitable thin stagnant layer of eluent at the inner surface of the column tube is not a concern whatsoever regarding column performance. 4.2.2. Hydrodynamic boundary layer combined with a thin loose orderly packed layer In addition to the presence of the THBL, the TLOPL was also taken into account in the description of the velocity profile across the 2.1 mm i.d. UHPLC column. It is represented by the solid purple line in Fig. 7. A clear positive deviation of the flow velocity relative to ubulk is now obvious. The maximum velocity excess (+1.2ubulk ) was kept the same as that observed in the BEH-C18 column. The excess velocity decreases and persists over a layer thickness of about three particle diameters towards the center of the column. Note that the actual oscillations of the velocity profile were ignored since the

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030

G Model CHROMA-359086; No. of Pages 15

ARTICLE IN PRESS F. Gritti / J. Chromatogr. A xxx (2017) xxx–xxx

11

Fig. 8. Same figure as in Fig. 7, except the corresponding asymptotic RPH plots.

Fig. 9. Same figure as in Fig. 8, except the corresponding projected pre-asymptotic RPH plots.

average velocity was considered in this work (see in Fig. 3 the best fit of the mathematical velocity profile to the experimental data points in the vicinity of the wall). The corresponding plot of hAris versus  calculated under asymptotic conditions is shown in Fig. 8 (solid purple line). The results reveal that the TLOPL is responsible for a non-negligible fraction of the RPH deviation from hbulk . At the optimum reduced velocity of about 10, hAris is already as large as 35 (solid blue line) in comparison to hbulk = 1.0 but still much lower than hAris = 159 (solid blue line) for the actual complete velocity profile. Fig. 9 compares the different pre-asymptotic hKnox versus  plots for all the different radial velocity profiles shown in Fig. 8. At  = 10, the positive deviation of hKnox relative to hbulk is only 0.15 h unit for a minimum RPH of 1.10 versus 0.95 for hbulk . In conclusion, the impact of the inevitable TLOPL in a 2.1 mm i.d. column packed with 2.0 ␮m particles on the observed RPH plot is almost negligible.

4.2.3. Hydrodynamic boundary layer combined with a thin loose orderly packed layer and a wide dense randomly packed layer In addition to the THBL and TLOPL, the WDRPL was considered in the construction of the complete radial velocity profile. This dense region of the packed bed located in the peripheral region within about 120 ␮m from the column wall is causing the local velocities to be significantly smaller than the bulk velocity ubulk . Typically, it was observed as low as 75% of the bulk velocity. This velocity well is represented in Fig. 7 for different minimum velocities equal to 0.75 (observed, solid blue line), 0.80 (solid pink line), 0.85 (solid green line), 0.90 (solid red line), and 0.95 (solid orange line) times the bulk velocity. The deeper the velocity well is, the wider the WDRPL as it expands further towards the center of the cylindrical bed. The corresponding asymptotic plots of hAris versus  are shown in Fig. 8. Remarkably, the data reveal that the coexistence of the TLOPL and WDRPL (orange, red, green, and pink lines) could

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030

G Model CHROMA-359086; No. of Pages 15

ARTICLE IN PRESS F. Gritti / J. Chromatogr. A xxx (2017) xxx–xxx

12

Fig. 10. Variation of the deviation of the pre-asymptotic RPH from hbulk normalized to the maximum deviation as a function of the integral velocity deviation (IVD) defined in Eq. (66). The IVD was calculated for the velocity profiles shown in Fig. 7 involving the contributions of the THBL, the TLOPL, and of the WDRPL. hmax is calculated for the maximum IVD of 15.3. Note that the increase of this normalized RPH is independent of the reduced velocity  and grows fast as the power 2.35 of the IVD.

be advantageous with respect to the sole presence of the THBL and TLOPL (purple line). Indeed, the high local velocities in the TLOPC are somewhat compensated by the low local velocities in the WDRPL. Because these two layers are adjacent to each other, transverse dispersion averages rapidly this velocity bias and enables the analyte molecules to travel in the 130 ␮m wide wall region at a velocity which is very close to the bulk velocity experienced by the molecules in the central region of the column. For instance, for a depth of the velocity well equal at 0.05ubulk (orange color) and for  = 5, the resulting deviation of the asymptotic RPH hAris from hbulk is only 1.5 h unit. This means that the anticipated preasymptotic RPH hKnox would deviate from hbulk by no more than 1.3%. Additionally, the relative increase of hAris with increasing the depth of the velocity well from 0.10 (red color) to 0.25ubulk (blue color) becomes increasingly significant and may dramatically affect the column performance under the observed pre-asymptotic dispersion regime. Experimentally, the relative deviation of hKnox from hbulk was measured at 81%. In this case, the higher than average local velocities in the TLOPL are more than compensated by the lower than average local velocities in the WDRPL. The column eventually suffers from serious efficiency losses as the average analyte velocity in the wall region becomes smaller than the bulk velocity in the central region of the column.

4.2.4. Integrated velocity deviation and column performance Based on the results of the previous section, an attempt is made to establish a quantitative relationship between the structural heterogeneity of the packed bed across the column diameter (revealed by the radial distribution of the velocity profile) and the column performance (measured from the deviation of the observed preasymptotic RPHs with respect to hbulk ). Tallarek and co-workers have introduced the integral porosity deviation (IPD) and the integral velocity deviation (IVD) in order to quantify flow heterogeneity in a packed column [64]. In this work, we consider the IVD defined for a cylindrical column as:

 IVD = 2

1

[u(x) − ubulk ] xdx 0

(66)

The reference IVD is zero for the ideal column which is characterized by a perfectly flat velocity profile from the wall to the center of the column (u(x) = ubulk for all x). In the combined presence of the THBL, TLOPL, and WDRPL, the IVD (×1000) was calculated at −0.47, −4.41, −8.14, −11.8, and −15.3 for depths of the velocity well in the WDRPL equal to 0.05, 0.10, 0.15, 0.20, and 0.75ubulk (actual column), respectively. Note that the IVD parameter would be exactly zero if the depth of the velocity well was exactly equal to 0.045ubulk . The absolute value of the IVD divided by the maximum value (15.3) is taken as the dimensionless IVD variable. The column performance was measured from the deviation of the predicted RPHs hKnox with respect to hbulk divided by the maximum deviation predicted for the maximum IVD of 15.3. Accordingly, the column performance is measured by the parameter CP: CP =

hKnox − hbulk hKnox,max − hbulk

(67)

Fig. 10 plots the variation of CP as a function of the dimensionless IVD variable for three different reduced velocities  = 0.1, 1, and 10. Remarkably, irrespective of , CP increases as the power 2.35 of the dimensionless IVD. Accordingly, the absolute IVD (× 1000) should not exceed about 5 in order to maintain an observed RPH within 0.1 h unit from hbulk at  = 10. Based on flow simulations in a FIB-SEM based 3D reconstruction, the absolute IVD (× 1000) of actual 2.1 mm × 50 mm UHPLC columns is as large as 15 causing a plate height deviation of 0.7 h unit. There is definitely some room left for further improvement of column performance: while the presence of both the THBL and TLOPL are inevitable, new packing strategies have to be developed [65] in order to either reduce the stress differential between the bulk and the wall region of the bed (in order to reduce the depth of the velocity well in the WDRPL) or to increase the friction forces experienced by the packed particles in the WDRPL (in order to reduce the rearrangement of the particles during bed consolidation). These packing conditions are somewhat achieved for the packing of sub-3 ␮m superficially porous particles (SPPs) since the observed minimum RPH is within 0.4 h unit from hbulk [7]. The stress gradient over the WDRPL region is not reduced (since higher than usual packing pressures are actually applied to pack SPPs effi-

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030

G Model CHROMA-359086; No. of Pages 15

ARTICLE IN PRESS F. Gritti / J. Chromatogr. A xxx (2017) xxx–xxx

ciently) but the friction forces between the particles is significantly increased which limits their spatial rearrangement in the WDRPL during bed consolidation. As a result, the average porosity in the wall region is very close to the bulk porosity and the efficiency loss is minimum. The higher than usual friction forces between SPPs may result from the specific preparation method used to synthesize SPPs (layer-by-layer deposition or coacervation approaches) compared to the traditional sol-gel process used to prepare FPPs. Therefore, the reason why 2.1 mm i.d. columns packed with sub3 ␮m SPPs are more efficient than those packed with sub-3 ␮m FPPs is most likely due to the higher surface roughness of SPPs relative to that of FPPs. At least, this interpretation is fully consistent with the trend reported in this work. It should also be emphasized that 1 m × 75 ␮m capillary columns packed recently with the same 2.0 ␮m BEH-C18 FPPs deliver a minimum RPH of only 1.05 [66]. The deviation from hbulk is then minimized at only 0.15 h unit. In this case, high slurry concentrations (200 g/L) and sonication during packing were combined to achieve consistently this exceptional kinetic performance. While sonication is likely needed to lower the number of large packing voids, high slurry concentrations are needed to suppress the rearrangement of the particles during the consolidation step. Again, the exceptional performance of these capillary columns is consistent with the ideal combination of the TLOPL and WRDPL for which the IVD is close to zero. This has yet to be confirmed from the structure reconstruction of the packed bed using confocal laser scanning microscopy and image processing as was done in the past for various capillary columns [67–70].

5. Conclusion In this work, the impact of the radial structural heterogeneities of packed beds on their chromatographic behavior was investigated for modern narrow-bore (2.1 mm i.d.) columns packed with sub-2 ␮m particles. The method is based on (1) the actual radial distribution of the flow velocity across the entire column diameter as previously obtained by flow simulations in a FIB-SEM based 3D reconstruction by Reising and Tallarek [1] and (2) on the extension of the general Aris dispersion theory from open tubes to packed columns. The results of Aris theory provide the asymptotic (longtime) dispersion behavior which can be directly compared to the observed pre-asymptotic dispersion behavior. This enables the prediction of the actual column performance upon varying the nature of the radial distribution of the flow velocity across a packed column and, ultimately, to optimize the packing procedure of certain particles (size and nature) in particular columns (inner diameter). In 2.1 mm i.d. columns packed with 2 ␮m particles, this work demonstrates that the deviation between the observed RPH (1.6) and the bulk RPH (0.9) is primarily due to the existence of a wide and densely packed wall region. In theory, exceptionally well-packed columns (h  hbulk ) would be expected if the depth of the velocity well in this dense wall region would be controlled in a range from 5 to 10% of the bulk velocity. In general, the results show that the key to the production of highly efficient columns (narrow-bore or capillary columns) is to compensate the presence of the TLOPL by the presence of the adjacent WDRPL in terms of minimization of the IVD. For the sake of illustration, in the current state of the art of column packing, the stress gradient applied to 2 ␮m FPPs in the peripheral region of 2.1 mm × 50 mm i.d. columns is too large relative to the friction forces between these particles. This leads to a significant particle rearrangement and to an excessive densification of the packing in the 120 ␮m WDRPL. A severe decrease of the local flow velocity down to 75% of the bulk velocity is actually observed. Solutions to this problem are definitely possible accord-

13

ing to the prediction of column performance from the general Aris dispersion theory and the knowledge of the transverse structural heterogeneities of the bed: for 2.1 mm i.d. columns, the packing process (pressure, slurry solvent, slurry concentration, sonication, etc.) and the surface morphology of the particles should be selected so that the depth of the velocity well in the WRDPL does not exceed 10% of the bulk velocity. The minimum RPH would eventually drop to 0.9 instead of 1.7 as currently observed. In practice, this means either packing columns at smaller pressures (to reduced the stress differential over the WRDPL but with the risk of producing large packing voids) or increasing the friction forces between the particles (to limit the particle rearrangement). The second option was actually applied for the successful packing of sub-3 ␮m SPPs in 2.1–4.6 mm i.d. columns (minimum RPH = 1.3–1.4). Even better, the nearly perfect velocity compensation between the TLOPL and the WRDPL has been realized in the capillary format when packing sub2 ␮m particles in 75 ␮m i.d. 1 m long capillary columns using high slurry concentration [69] (to reduce particle rearrangement at the wall) and sonication [66] (to minimize the number of large packing voids) to provide consistently minimum RPHs as low as 1.05. Note that if the operator applies reduced velocities much larger than 100 (solid–liquid mass transfer resistance and/or viscous heating effects dominate over the impact of transcolumn eddy dispersion on column efficiency) and abandons the possibility of working close to the optimum performance of the column, then, any improvement towards a uniform flow velocity profile across the column would obviously be vain. The performance optimization would then switch towards the amelioration of the inner structure of the particles or towards the design of alternative stationary phases. In practice, from small molecules to peptides and for sub-2 ␮m particles, this is not the case because reduced velocities remain typically in a range from 2 to 25. Also, large biomolecules (proteins, monoclonal antibodies) are often eluted at low or moderate flow rates in order to approach the optimum velocity while keeping the analysis time reasonably short. The development of better packed and more uniform UHPLC columns is then still definitely needed for most applications in the pharmaceutical and biological industries.

Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.chroma.2017.12. 030.

References [1] A. Reising, S. Schlabach, V. Baranau, D. Stoeckel, U. Tallarek, Analysis of packing microstructure and wall effects in a narrow-bore ultrahigh pressure liquid chromatography column using focused ion beam scanning electron microscopy, J. Chromatogr. A 1513 (2017) 172–182. [2] A. Daneyko, S. Khirevich, A. Höltzel, A. Seidel-Morgenstern, U. Tallarek, From random sphere packings to regular pillar arrays: effect of the macroscopic confinement on hydrodynamic dispersion, J. Chromatogr. A 1218 (2011) 8231–8248. [3] S. Khirevich, A. Daneyko, A. Höltzel, A. Seidel-Morgenstern, U. Tallarek, Statistical analysis of packed beds, the origin of short-range disorder, and its impact on eddy dispersion, J. Chromatogr. A 1217 (2010) 4713–4722. [4] F. Gritti, G. Guiochon, Perspectives on the evolution of the column efficiency in liquid chromatography, Anal. Chem. 85 (2013) 3017–3035. [5] A. Daneyko, A. Höltzel, S. Khirevich, U. Tallarek, Influence of the particle size distribution on hydraulic permeability and eddy dispersion in bulk packings, Anal. Chem. 83 (2011) 3903–3910. [6] P. Magnico, M. Martin, Dispersion in the interstitial space of packed columns, J. Chromatogr. 517 (1990) 31–49. [7] F. Gritti, G. Guiochon, Comparison between the efficiencies of columns packed with fully and partially porous c18-bonded silica materials, J. Chromatogr. A 1157 (2007) 289–303. [8] F. Gritti, G. Guiochon, Facts and legends about columns packed with sub-3-␮m core–shell particles, LC-GC North Am. 30 (2012) 586–595.

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030

G Model CHROMA-359086; No. of Pages 15 14

ARTICLE IN PRESS F. Gritti / J. Chromatogr. A xxx (2017) xxx–xxx

[9] J. Delgado, A critical review of dispersion in packed beds, Heat Mass Transf. 42 (2006) 279–310. [10] F. Gritti, T. McDonald, M. Gilar, Impact of the column hardware volume on resolution in very high pressure liquid chromatography non-invasive investigations, J. Chromatogr. A 1420 (2015) 54–65. [11] B. Yew, E. Drumm, G. Guiochon, Mechanics of column beds: I. Acquisition of the relevant parameters, AIChE J. 49 (2003) 626–641. [12] B. Yew, J. Ureta, R. Shalliker, E. Drumm, G. Guiochon, Mechanics of column beds: II. Modeling of coupled stress–strain-flow behavior, AIChE J. 49 (2003) 642–664. [13] J. Knox, J. Parcher, Effect of column to particle diameter ratio on the dispersion of unsorbed solutes in chromatography, Anal. Chem. 76 (2004) 5777–5786. [14] S. Hsieh, J. Jorgenson, Preparation and evaluation of slurry-packed liquid chromatography microcolumns with inner diameters from 12 to 33 ␮m, Anal. Chem. 68 (1996) 1212–1217. [15] S. Hsieh, J. Jorgenson, In-depth characterization of slurry packed capillary columns with 1.0 ␮m nonporous particles using reversed-phase isocratic ultrahigh-pressure liquid chromatography, Anal. Chem. 76 (2004) 5777–5786. [16] J. Giddings, Dynamics of Chromatography, Marcel Dekker, New York, NY, 1965. [17] F. Gritti, G. Guiochon, Impact of retention on trans-column velocity biases in packed columns, AIChE J. 56 (2010) 1495–1509. [18] F. Gritti, G. Guiochon, New insights on mass transfer kinetics in chromatography, AIChE J. 57 (2011) 333–345. [19] J. Baur, E. Kristensen, R. Wightman, Radial dispersion from commercial high-performance liquid chromatography columns investigated with microvoltametric electrodes, Anal. Chem. 60 (1988) 2334–2338. [20] J. Abia, K. Mriziq, G. Guiochon, Radial heterogeneity of some analytical columns used in high-performance liquid chromatography, J. Chromatogr. A 1216 (2009) 3185–3191. [21] F. Gritti, G. Guiochon, Relationship between trans-column eddy diffusion and retention in liquid chromatography: theory and experimental evidence, J. Chromatogr. A 1217 (2010) 6350–6365. [22] T. Farkas, M. Sepaniak, G. Guiochon, Radial distribution of the flow velocity, efficiency, and concentration in a wide HPLC column, AIChE J. 43 (1997) 1964–1974. [23] M. Kaminski, Simple test for determination of the degree of distortion of the liquid-phase flow profile in columns for preparative liquid chromatography, J. Chromatogr. 589 (1992) 61–70. [24] D. Astrath, F. Lottes, D. Vu, W. Arlt, E. Stenby, Experimental investigation of liquid chromatography columns by means of computed tomography, Adsorption 13 (2007) 9–19. [25] R. Shalliker, H. Ritchie, Segmented flow and curtain flow chromatography: Overcoming the wall effect and heterogeneous bed structures, J. Chromatogr. A 1335 (2014) 122–135. [26] D. Foley, L. Pereira, M. Camenzuli, T. Edge, H. Ritchie, R. Shalliker, Curtain flow chromatography (the infinite diameter column) with automated injection and high sample through-put: the results of an inter-laboratory study, Microchem. J. 110 (2013) 127–132. [27] M. Camenzuli, H. Ritchie, J. Ladine, R. Shalliker, Enhanced separation performance using a new column technology: parallel segmented outlet flow, J. Chromatogr. A 1232 (2012) 47–51. [28] M. Camenzuli, H. Ritchie, R. Shalliker, Improving HPLC separation performance using parallel segmented flow chromatography, Microchem. J. 111 (2013) 3–7. [29] A. Soliven, D. Foley, L. Pereira, S. Hua, T. Edge, H. Ritchie, G. Dennis, R. Shalliker, Improving the performance of narrow-bore HPLC columns using active flow technology, Microchem. J. 116 (2014) 230–234. [30] F. Gritti, G. Guiochon, Effect of parallel segmented flow chromatography on the height equivalent to a theoretical plate. I – Performance of 4.6 mm × 30 mm columns packed with 3.0 ␮m hypurity-c18 fully porous particles, J. Chromatogr. A 1297 (2013) 64–76. [31] F. Gritti, G. Guiochon, Effect of parallel segmented flow chromatography on the height equivalent to a theoretical plate. II – Performances of 4.6 mm × 30 mm columns packed with 2.6 ␮m accucore-c18 superficially porous particles, J. Chromatogr. A 1314 (2013) 44–53. [32] F. Gritti, J. Pynt, A. Soliven, G. Dennis, R. Shalliker, G. Guiochon, Effect of parallel segmented flow chromatography on the height equivalent to a theoretical plate. III – Influence of the column length, particle diameter, and the molecular weight of the analyte on the efficiency gain, J. Chromatogr. A 1333 (2013) 32–44. [33] W.D. Malsche, H. Eghbali, D. Clicq, J. Vangelooven, H. Gardeniers, G. Desmet, Pressure-driven reverse-phase liquid chromatography separations in ordered nonporous pillar array columns, Anal. Chem. 79 (2007) 5915–5926. [34] C. Fee, S. Nawada, S. Dimartino, 3d printed porous media columns with fine control of column packing morphology, J. Chromatogr. A 1333 (2014) 18–24. [35] R. Aris, On the dispersion of a solute by diffusion, convection and exchange between phases, Proc. Roy. Soc. A 252 (1959) 538–550. [36] F. Gritti, M. Martin, G. Guiochon, Influence of viscous friction heating on the efficiency of columns operated under very high pressures, Anal. Chem. 81 (2009) 3365–3384. [37] F. Gritti, G. Guiochon, Ultra high pressure liquid chromatography column permeability and changes of the eluent properties, J. Chromatogr. A 1187 (2008) 165–179. [38] F. Gritti, G. Guiochon, Complete temperature profiles in ultra-high-pressure liquid chromatography columns, Anal. Chem. 80 (2008) 5009–5020.

[39] F. Gritti, G. Guiochon, Heat exchanges in fast, high-performance liquid chromatography. A complete thermodynamic study, Anal. Chem. 80 (2008) 6488–6499. [40] F. Gritti, G. Guiochon, Optimization of the thermal environment of columns packed with very fine particles, J. Chromatogr. A 1216 (2009) 1353–1362. [41] F. Gritti, G. Guiochon, Performance of new prototype packed columns for very high pressure liquid chromatography, J. Chromatogr. A 1217 (2010) 1485–1495. [42] A. Alizadeh, C. Nieto de Castro, W. Wakeham, The theory of Taylor dispersion technique for liquid diffusivity measurements, Int. J. Therm. 1 (1980) 243–284. [43] G. Guiochon, A. Felinger, A. Katti, D. Shirazi, Fundamentals of Preparative and Nonlinear Chromatography, 2nd ed., Academic Press, Boston, MA, 2006. [44] J. Knox, G. Laird, P. Raven, Interaction of radial and axial dispersion in liquid chromatography in relation to the “infinite diameter effect”, J. Chromatogr. 122 (1976) 129–145. [45] A. Daneyko, D. Hlushkou, S. Khirevich, U. Tallarek, From random sphere packings to regular pillar arrays: analysis of transverse dispersion, J. Chromatogr. A 1257 (2012) 98–115. [46] M. Martin, G. Guiochon, Influence of retention on band broadening in turbulent-flow liquid and gas chromatography, Anal. Chem. 54 (1982) 1533–1540. [47] M. Martin, G. Guiochon, Axial dispersion in open-tubular capillary liquid chromatography with electroosmotic flow, Anal. Chem. 56 (1984) 614–620. [48] M. Martin, G. Guiochon, Y. Walbroehl, J. Jorgenson, Peak broadening in open-tubular liquid chromatography with electroosmotic flow, Anal. Chem. 57 (1985) 559–561. [49] F. Gritti, G. Guiochon, Mass transfer kinetics, band broadening and column efficiency, J. Chromatogr. A 1221 (2012) 2–40. [50] S. Khirevich, A. Daneyko, A. Höltzel, A. Seidel-Morgenstern, U. Tallarek, Statistical analysis of packed beds, the origin of short-range disorder, and its impact on eddy dispersion, J. Chromatogr. A 1217 (2010) 4713–4722. [51] A. Daneyko, D. Hlushkou, V. Baranau, S. Khirevich, A. Seidel-Morgenstern, U. Tallarek, Computational investigation of longitudinal diffusion, eddy dispersion, and trans-particle mass transfer in bulk, random packings of core-shell particles with varied shell thickness and shell diffusion coefficient, J. Chromatogr. A 1407 (2015) 139–156. [52] F. Gritti, G. Guiochon, General HETP equation for the study of mass-transfer mechanisms in RPLC, Anal. Chem. 78 (2006) 5329–5347. [53] F. Gritti, G. Guiochon, Effect of the surface coverage of c18-bonded silica particles on the obstructive factor and intraparticle diffusion mechanism, Chem. Eng. Sci. 61 (2006) 7636–7650. [54] S. Torquato, Random Heterogeneous Materials. Microstructure and Macroscopic Properties, Springer, New York, 2002. [55] S. Torquato, Effective electrical conductivity of two-phase disordered composite media, J. Appl. Phys. 58 (1985) 3790–3797. [56] F. Gritti, G. Guiochon, Theoretical investigation of diffusion along columns packed with fully and superficially porous particles, J. Chromatogr. A 1218 (2011) 3476–3488. [57] F. Gritti, G. Guiochon, Diffusion models in chromatographic columns packed with fully and superficially porous particles, Chem. Eng. Sci. 66 (2011) 3773–3781. [58] D. Hlushkou, H. Liasneuski, U. Tallarek, S. Torquato, Effective diffusion coefficients in random packings of polydisperse hard spheres from two-point and three-point correlation functions, J. Appl. Phys. 118 (2015) 124901-1–124901-10. [59] H. Liasneuski, D. Hlushkou, S. Khirevich, A. Höltzel, U. Tallarek, S. Torquato, Impact of microstructure on the effective diffusivity in random packings of hard spheres, J. Appl. Phys. 116 (2014), 0349041-1–034904-10. [60] F. Gritti, T. McDonald, M. Gilar, Accurate measurement of dispersion data through short and narrow tubes used in very high-pressure liquid chromatography, J. Chromatogr. A 1410 (2015) 118–128. [61] S. Khirevich, A. Höltzel, U. Tallarek, Transient and asymptotic dispersion in confined sphere packings with cylindrical and non-cylindrical conduit geometries, Phil. Trans. R. Soc. A 369 (2011) 2485–2493. [62] S. Khirevich, A. Höltzel, S. Khirevich, U. Tallarek, Time and length scales of eddy dispersion in chromatographic beds, Anal. Chem. 81 (2009) 7057–7066. [63] S. Khirevich, A. Höltzel, D. Hlushkou, U. Tallarek, Impact of conduit geometry and bed porosity on flow and dispersion in noncylindrical sphere packings, Anal. Chem. 79 (2007) 9340–9349. [64] S. Khirevich, A. Höltzel, A. Seidel-Morgenstern, U. Tallarek, Geometrical and topological measures for hydrodynamic dispersion in confined sphere packings at low column-to-particle diameter ratios, J. Chromatogr. A 1262 (2012) 77–91. [65] M. Wahab, C. Darshan, R. Wimalasinghe, D. Armstrong, Fundamental and practical insights on the packing of modern high-efficiency analytical and capillary columns, Anal. Chem. 89 (2017) 8177–8191. [66] J. Godinho, A. Reising, U. Tallarek, J. Jorgenson, Implementation of high slurry concentration and sonication to pack high-efficiency, meter-long capillary ultrahigh pressure liquid chromatography columns, J. Chromatogr. A 1462 (2016) 165–169. [67] S. Bruns, U. Tallarek, Physical reconstruction of packed beds and their morphological analysis: core–shell packings as an example, J. Chromatogr. A 1218 (2011) 1849–1860.

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030

G Model CHROMA-359086; No. of Pages 15

ARTICLE IN PRESS F. Gritti / J. Chromatogr. A xxx (2017) xxx–xxx

[68] S. Bruns, D. Stoeckel, B. Smarsly, U. Tallarek, Influence of particle properties on the wall region in packed capillaries, J. Chromatogr. A 1268 (2012) 53–63. [69] S. Bruns, E. Franklin, J. Grinias, J. Godinho, J. Jorgenson, U. Tallarek, Slurry concentration effects on the bed morphology and separation efficiency of capillaries packed with sub-2 ␮m particles, J. Chromatogr. A 1318 (2013) 189–197.

15

[70] S. Bruns, J. Grinias, L. Blue, J. Jorgenson, U. Tallarek, Morphology and separation efficiency of low-aspect-ratio capillary ultrahigh pressure liquid chromatography columns, Anal. Chem. 84 (2012) 4496–4503.

Please cite this article in press as: F. Gritti, On the relationship between radial structure heterogeneities and efficiency of chromatographic columns, J. Chromatogr. A (2017), https://doi.org/10.1016/j.chroma.2017.12.030