editorial


HUNGARIAN JOURNAL
OF INDUSTRY AND CHEMISTRY

VESZPRÉM
Vol. 42(1) pp. 25-29 (2014)

IMPACT OF SIZE HETEROGENEITY OF CORE-SHELL PACKING MATERIALS
ON CHROMATOGRAPHIC SEPARATION OF LARGE BIOMOLECULES

DIÁNA LUKÁCS AND KRISZTIÁN HORVÁTHB

Department of Analytical Chemistry, University of Pannonia, 8200 Veszprém, Egyetem St. 10., HUNGARY
BE-mail: raksi@almos.uni-pannon.hu

The effect of particle size heterogeneity of core-shell stationary phases on the efficiency of chromatographic separation of
large biomolecules was studied. It was shown that the column efficiency was affected significantly by the breadth of particle
size distribution. The chromatographic efficiency decreased as the heterogeneity of particle sizes increased. Although the
absolute decrease of separation efficiency was affected by the linear velocity, u, of the eluent, the relative change of HETP
was independent of u in the practical range of eluent velocities. The results showed that the affect of particle size distribution
was the highest in case of fully porous phases, and it decreased as the diameter of the inner core decreased. It was shown
that, in the usual range of particle size heterogeneity of core-shell phase, the peak capacities did not change significantly
even at high eluent velocities.

Keywords: high performance liquid chromatography, particle size heterogeneity, separation efficiency, core-shell phases

Introduction

Modern analytical applications of liquid chromatography
require efficient stationary phases. Before the
introduction of core-shell particles – particles with
a porous layer surrounding a solid core (Fig. 1) – the
use of monolithic silica rods [1, 2], or of sub-2 µm
particles offered today the most satisfactory results.
Due to their excessive radial heterogeneity, however,
the efficiency of monolithic columns currently is lower
than that of sub-2 µm particles. Columns packed with
fine particles, however, have a low permeability [5–7].
Thus, efficiency of columns packed with core-shell
particles similar to the columns packed with sub-2 µm
particles, but can be operated with the same instruments
as those used for conventional columns. The use of
core-shell phases is particularly advantageous in the
separation of large biomolecules [8]. Recently, core-
shell phases were optimized for the separation of large
biomolecules [9, 10]. Besides their unique structure, the
high efficiency of core-shell phases is in part due to the
very narrow particle size distribution, with a relative
standard deviation around 5–10% vs. 20–40% for most
totally porous particles [7].

In literature, fairly contradictory information can be
found about the effect of particle size distribution, PSD,
on the separation efficiency. Some results suggest that
the large particle size variance has no influence on the
column efficiency, while according to other data the
wide PSD decreases the efficiency of chromatographic
separation. Halász and Naefe [11] were the first ones
who examined the effect of particle size distribution

Porous shell

Non-porous core

r
core

r p

�
shell

Figure 1: Structure of core-shell particles

on the column efficiency in case of particles that were
larger than 50 µm. In their work, no experimental result
was found for the change of chromatographic efficiency
even in case of wide PSDs (⇠ 40%). Two years after
Halász’s work, significantly smaller, 1-10 µm, particles
were tested by Endele et al. [12]. No change in the
permeability and in the chromatographic efficiency were
observed as long as the average particle size remained
the same. Dewaele and Verzele prepared columns packed
with mixtures of different compositions of reversed phase
particles of 3 and 8 µm and investigated the effect of the
PSD on the column efficiency [13] with similar results.

Billen et al. found that the breadth of the distribution
of the stationary phase particles has no effect on
the chromatographic efficiency as long as no fines
are present. In the presence of fines, the efficiency
decreases significantly [14]. They emphasized that



26

different conclusions can be drawn about the goodness
of a chromatographic column depending on the definition
used for the particle size distribution.

The effect of the PSD on the chromatographic
efficiency of sub-2 µm particles was studied by Cabooter
et al. by analyzing kinetic plots of these phases [15].
In their work, they showed that not just the PSD is
important but also the separation efficiency one wants to
achieve. It was concluded that the greater the achievable
number of theoretical plates, the more the PSD affects
the separation efficiency. In their experiments, the authors
obtained better results by using columns with a narrower
PSD than by using columns of the same dimensions but
packed with particles with broader PSD.

Recently, Gritti et al. found that the chromatographic
efficiency of the columns can be optimized if a small
amount of greater particles is added to the sub-3 µm
particles which leads to better bead homogeneity [16].
Manufacturers practice matches these, because on their
confession they add a small amount of larger particles
to the bathes to optimize the pressure drop along the
column.

Experimental study of the effect of PSD on
chromatographic efficiency is rather complex, because
the column packing procedure is hardly reproducible.
It is particularly true when we work with stationary
phases which have different PSDs. The quality of column
packing also affects the column efficiency significantly.
Theoretical models can provide more reliable data
on this field because it does not affected by the
variability of column packing. The goal of this work
is to study the effect of particle size heterogeneity
on the chromatographic efficiency of separation of
large biomolecules on core-shell stationary phases on a
theoretical basis.

Theory

In HPLC, one of the measure of separation efficiency
is the height equivalent to a theoretical plate, HETP
[17]. The higher the plate height, the less efficient the
separation is. The general rate model of chromatography
[18] permits the calculation of HETP of columns packed
by core-shell phases [8]:

H(d
p

) =

2D
L

u
+

k2
1

1 + k2
1

 
⌦ u d2

p

30FD
p

+

u d
p

3Fk
f

!
, (1)

where d
p

is the diameter of particles of packing
material, u the linear velocity of the eluent in the
interstitial volume, k

1

the interstitial retention factor,
F the phase ratio, D

L

and D
p

the axial and pores
diffusion coefficients, respectively, and k

f

the external
mass transfer coefficient. ⌦ is given by the relationship

⌦ = (1 � ⇢)
1 + 3⇢ + 6⇢2 + 5⇢3

(1 + ⇢ + ⇢2)
2

, (2)

where ⇢ represents the ratio of the radius of the inner solid
core, r

core

, to the radius of the particle, r
p

(see Fig. 1):

⇢ =
r
core

r
p

. (3)

Accordingly, ⇢ is zero for fully porous particle and one
for non-porous particle.

For a core-shell particle, the interstitial retention
factor is

k
1

=

(1 � "
e

) (1 � ⇢3)
"
e

[K
a

(1 � "
p

) + "
p

] , (4)

where "
e

is the external bed porosity, "
p

is the porosity
of the porous shell, and K

a

is the adsorption equilibrium
constant (Henry constant).

The particle size distribution is usually described
by log-normal distribution [19]. The PSD of a packing
material with mean µ and variance �2 is

fdp =
1

d
p

p
2⇡✓

exp

0

B
@�

⇣
ln

dp
µ

+

1

2

✓
⌘
2

2✓

1

C
A , (5)

where

✓ = ln

✓
�2

µ2
+ 1

◆
. (6)

On the basis of the equations above, the probability
density function of local HETPs, fH , can be derived by
applying the change-of-variables rule [20]:

fH =

����
d

dH
d
p

(H)

����frp (dp(H)) , (7)

where d
p

(H) is the inverse function of H(d
p

) (see
Eq. (1)).

The observable HETP of the column is the first
moment of Eq. (7):

H
col

=

1Z

0

H fH dH. (8)

Methods of Calculations

For the calculation of H
col

(Eq. (8)) a software written
in C++ language, using the adaptive quadrature routine
provided by GNU Scientific Library (GSL, v. 1.15) [21].
The integration region was divided into subintervals,
and on each iteration the subinterval with the largest
estimated error was bisected. As a result, the overall
error reduced rapidly, as the subintervals became
concentrated around local difficulties in the integrand.
These subintervals were managed by the GSL library,
which handled the memory for the subinterval ranges,
results and error estimates too. The relative error of
integration was set to 10�10. The source code of
the program was compiled by g++ shipped by GNU



27

0

50

100

150

200

H
E
T
P
[
m
]

0 10 20 30 40 50

Eluent velocity [cm/min]

= 0.8 m
= 0.6 m
= 0.4 m
= 0.2 m
= 0.0 m

Figure 2: Van Deemter plots of large biomolecules in
case of different standard deviations of particle size
distributions of 2.6 µm core-shell stationary phases

Compiler Collection ver. 4.7.2 using O1 optimization.
The calculations were performed on a Pentium IV
computer (2.80 GHz) running GNU Linux operating
system (Debian Wheezy).

The function fH (Eq. (7)) for the numerical
integration was derived symbolically by Mathematica 8.0
(Wolfram Research, Inc.). H (Eq. (1)) was calculated as
it is described in Ref. [8]. The internal ("

p

) and external
column porosities ("

e

), were 0.05 and 0.4, respectively.
The distribution coefficient of the compounds were set
to 3. The ratio of the molecule size and the average
pore diameter were assumed to be 0.68. The molecular
diffusivity was 2.5⇥10�5 cm2/min. The value of “eddy”
diffusion term, the internal and external obstruction
factors were 1.3, 0.31 and 0.6, respectively. The linear
velocity of the eluent was varied between 0.1 and
50 cm/min. The average pore size was 80 Å. It was
assumed that the quality of the column packing remained
identical in all cases.

Results and Discussion

In Fig. 2 the calculated van-Deemter curves of large
biomolecules separated on 2.6 µm core-shell phases can
be seen. The ratio of the radius of the inner solid core to
the radius of the particle, ⇢, is 0.7 representing the typical
core size of commercial core-shell phases. The standard
deviation of particle size distribution is varied between
0.0–0.8 µm that corresponds to 0–30% relative standard
deviation. Close examination of Fig. 2 highlights that the
HETP values of large biomolecules are approximately
directly proportional to the eluent velocity.

Fig. 2 shows that the heterogeneity of particle sizes
has a significant effect on the separation efficiency
of large biomolecules. The separation efficiency is the
highest when the variance of particle size distribution is
zero (solid black line). Increasing breadth of particle size
distribution decreases the efficiency of separation. The
decrease in the efficiency is more significant at higher
eluent velocities. At 10 cm/min, the difference between

1.00

1.05

1.10

1.15

1.20

R
e
la
ti
v
e
H
E
T
P

0 10 20 30 40 50

Eluent velocity [cm/min]

= 0.8 m
= 0.6 m
= 0.4 m
= 0.2 m

Figure 3: Relative van Deemter plots of large
biomolecules in case of different standard deviations of
particle size distributions of 2.6 µm core-shell stationary

phases

the two extremes (� = 0 and � = 0.8) is less than 5 µm
that is approximately 15% of the total plate height at this
eluent velocity in ideal case (� = 0). In case of 30, 40,
and 50 cm/min the differences are significantly larger:
⇠15, ⇠20, and ⇠25 µm, respectively. Note, however, that
the relative increase is ⇠15% in each case suggesting
that the relative change of HETP is independent from the
eluent velocity.

In Fig. 3 the relative HETP, H
rel

, values of large
biomolecules can be seen as a function of eluent velocity.
H

rel

were calculated as

H
rel

=

H�
H

0

, (9)

where H� and H0 are HETP values at �2 and 0 variance
of PSD.

In Fig. 3, it can be seen that raising eluent velocity
increases the relative HETP. Above ⇠8 cm/min, however,
a plateau is reached and the change of relative HETP
becomes negligible. It suggests that, at low eluent
velocities, the interstitial dispersion is the governing
effect in band dispersion. As the velocity of eluent
increases, the pore diffusion becomes more and more
significant. Accordingly, in practice (u > 10 cm/min), it
is the dominant effect in band dispersion.

Fig. 4 shows the effect of core size on the
chromatographic efficiency at different breadth of core-
shell particle distribution. The linear velocity of the
eluent was 15 cm/min that is a typical value in
chromatographic applications. It can be seen that, as
expected from Eq. (1), the efficiency of stationary phase
increases as the thickness of porous layer decreases.
The effect of width of particle size distribution is less
significant at larger ⇢ values. While in case of fully
porous particles, the difference is almost 17 µm between
the HETP values of the two extremes (� = 0 and � =
0.8), it is slightly more than 4 µm at ⇢ = 0.8.

The relative HETPs calculated as in case of Fig. 3
do not change significantly with the diameter of non-
porous core until ⇢ = 0.8 (Fig. 5). Above that value,



28

0

20

40

60

80

100

120

140
H
E
T
P
[
m
]

0.0 0.2 0.4 0.6 0.8 1.0

= 0.8 m
= 0.6 m
= 0.4 m
= 0.2 m
= 0.0 m

Figure 4: HETP of columns packed with 2.6 µm
core-shell particles

1.00

1.05

1.10

1.15

1.20

re
la
ti
v
e
H
E
T
P

0.0 0.2 0.4 0.6 0.8 1.0

= 0.2 m
= 0.4 m
= 0.6 m
= 0.8 m

Figure 5: Relative HETPs of columns packed with 2.6
µm core-shell particles

H
rel

s decrease abruptly, at ⇢ = 1 there is no difference
between the cases. It can be explained by considering that
the separation efficiency of large biomolecules is affected
mainly by the pore diffusion (second term of Eq. (1)).
As the thickness of porous layer of core-shell particles
decrease, the diffusion paths reduce as well. As a result,
the impact of pore diffusion on band broadening is less
and less significant. The difference between the distinct
particles vanishes. Finally, at ⇢ = 1, only the interstitial
band spreading (first term of Eq. (1)) influences the
efficiency of separation. Since D

L

is not affected by the
particle size distribution, H

rel

s become unity.
Besides the plate heights, the change of peak

capacities were also studied. Peak capacity, n
c

, is a
practical measure of separation potential in HPLC. n

c

is
defined as the maximum number of components that can
be resolved completely between the peaks of the least and
most retained solutes [22]. Several expressions exist for
the calculation of peak capacity [23] depending on the
mode of chromatography. In case of isocratic mode of
separation, nc can be calculated as

nc =
1

4

p
N (t

max

� t
min

) ln

t
max

t
min

, (10)

where t
max

and t
min

are the retention times of the first
and last eluting peaks, and N the number of theoretical

0.90

0.92

0.94

0.96

0.98

1.00

R
e
la
ti
v
e
p
e
a
k
c
a
p
a
c
it
y

0 10 20 30 40 50

Eluent velocity [cm/min]

= 0.2 m
= 0.4 m
= 0.6 m
= 0.8 m

Figure 6: Relative peak capacities of columns packed by
2.6 µm core-shell particles; L = 10 cm, t

0

= 10/9,
t
max

= 100/9

plates (N = L/H).
In Fig. 6 the peak capacities of 2.6 µm core-shell

phases can be seen as a function of eluent velocity in
case of separation of large biomolecules. The calculated
peak capacities were normalized for value that belongs
to zero variance of PSD. The figure shows that the
achievable peak capacity of a column packed with
core-shell particles is affected by the heterogeneity of
particle sizes. The decrease of peak capacities, however,
is significantly smaller than the decrease of column
efficiencies (Fig. 2). That is because n

c

is proportional
to the reciprocal square root of H. Accordingly, 15%
increase in the plate height results in ⇠7% loss of peak
capacity that is not significant in practice.

Conclusions

Since their introduction, core-shell stationary phases
became very popular for the analysis of large
biomolecules. The morphology of these phases results
in less band broadening compared to fully porous
particles and thus delivers extremely high efficiencies.
In this work, the effect of particle size heterogeneity
of core-shell stationary phases on the efficiency of
chromatographic separation of large biomolecules was
studied. The results showed that slopes of van Deemter
curves were affected by the breadth of particle size
distribution. As a result of widening PSD the column
efficiencies decreased significantly. The analysis of peak
capacities showed that the maximum number of large
biomolecules that can be resolved by core-shell phases
was not affected significantly, even if the PSD was wide.
Considering that the relative standard deviations of PDSs
of core-shell phases are between 0.2–0.4 µm for 2.6
µm particles, it can be concluded that the efficiency
of these phases is affected significantly by the size
heterogeneity of the particles. Further efforts from
column manufacturers in order to improve the PSD do
not give any more advantages and are not profitable.



29

Acknowledgements

This research was supported by the European Union
and the State of Hungary, co-financed by the European
Social Fund in the framework of TÁMOP-4.2.4.A/2-
11/1-2012-0001 ‘National Excellence Program’. The
research infrastructure was supported by the Hungarian
Scientific Research Fund (OTKA PD104819) and the
Hungarian State and the European Union under the
TÁMOP-4.2.2.A-11/1/KONV-2012-0071 project.

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