American Journal of Analytical Chemistry, 2011, 2, 619-625
doi:10.4236/ajac.2011.25070 Published Online September 2011 (
Copyright © 2011 SciRes. AJAC
On a New Mechanism for Separating two Components in a
Stationary Flow through Mesopores
Harald Morgner
Wilhelm Ostwald Institute for Physical and Th eoretical Chemistry, University Leipzig, Leipzig, Germany
Received February 24, 2011; revised April 6, 2011; accepted July 11, 2011
When simulating the behavior of fluids in a stationary flow through mesopores we have observed a phe-
nomenon that may prove useful in some cases as basis for separating fluid components. The scheme works at
constant temperature which makes it energy efficient as are other schemes like (molecular) sieves or chro-
matography. Sieves rely on differences in molecular size and chromatography on different affinity of com-
ponents to the solid material of the ‘packing’. The scheme presented here may sometimes complement the
established techniques in that it is based on a different mechanism. The fluids to be separated can have the
same molecular size and the same affinity to solid material they are in contact with. The only requirement for
the scheme to work is that the miscibility behavior varies somewhat with pressure or density. From literature
it is known that virtually any mixture reacts on strong variations of pressure. Even a mixture that behaves
almost ideally at ambient pressure will show slight deviations from ideal miscibility when exposed to ex-
treme pressure. The strong differences in pressure are not created by external means but by exploiting the
spontaneous behavior of fluids in mesopores. If the experiment is designed correctly, strong pressure gradi-
ents show up in mesopores that are far beyond any gradient that could be established by technical means.
Our simulations are carried out for situations where pressure inside the pores varies between a few hundred
bar positive pressure and a few hundred bar negative pressure while the pressure in the gas phase outside the
pores amounts to ca.170 mbar.
Keywords: Fluids in Mesopores, Negative Pressure States, Pressure Dependence of Miscibility
1. Introduction
It is common knowledge that the vapor of a fluid may
condense in mesopores even if the pressure of the gas
reservoir is below the saturation pressure at the given
temperature. It further is known that the density of the
liquid formed in the pores is somewhat smaller than of
the liquid under normal conditions at equilibrium. Thus,
the situation of capillary condensation leads to so called
‘expanded’ liquids [1]. The related pressure in the liquid
is not only small, but usually takes on even negative
values. The occurrence of negative pressure in the pore
turns out to be of importance for the phenomenon that
we propose to employ for separation purposes. Negative
pressure states in macroscopic systems have been ob-
served long before by Torricelli and by Huygens, but are
still today subject to investigations [2,3].
Recently, we have studied a fluid in a mesopore with
two open ends, cf. upper panel of Figure 1. The bound-
ary conditions were chosen so as to bring the system into
a state pertaining to the curve of states COS(), i.e. to
create a liquid in the pore. Usually, such a system is
studied experimentally and by simulation with equal
pressure in both gas reservoirs. We found that the liquid
structure with negative pressure persists even, if a slight
difference in gas pressure is applied between both gas
reservoirs [4]. The fluid flows through the system, leav-
ing the structure of the liquid phase in the pore unaf-
fected. This stationary state remains stable over a range
of pressure differences between both gas reservoirs [5],
cf. Figure 2.
So far we have not found an upper limit for the pres-
sure difference. The flow does not affect noticeably the
pressure profile in the pore compared to the equilibrium
state. It is shown in Figure 3.
A further observation has been made when a second
curves of s tates
straight pore with two open ends
de sorption switch
adsorption switch
-6700 -6600 -6500 -6400 -6300 -6200 -6100
chem. pot. [J/mol]
load [ mol/cm
Figure 1. Isotherm of a model system. The cylindrical pore
is open at both ends with diameter 5 nm and length 50 nm.
The parameters are chosen to represent argon in silica
pores (SBA-15) at 77.35 K. The isotherm displays two
separate curves of states, COS(α) and COS(β). The fact that
the switching points for adsorption and desorption differ,
leads to the adsorption hysteresis. The pore is filled with
liquid when in a state pertaining to the COS(β). The con-
cept of COS(= curves of states) has been introduced in a
previous publication [5].
stationary state of negative pressure
0. 00 E+00
00.0050.01 0.0150.020.025 0.03 0.035 0.04
pressure di ff er ence [bar]
flow [mol /s]
Figure 2. Flow through pore from Figure 1 with negative
pressure state. Different gas pressures at both pore ends.
pressure profile along pore axis
-30 -20 -100102030
position [nm]
pres sure [bar]
pressure profile along pore diameter
-3-2-10 12 3
position [nm]
Figure 3. (a) pressure profile in the cylindrical pore in Fig-
ure 1 with two open ends in states pertaining to COS(
This panel: along symmetry axis; (b) pressure profile in the
cylindrical pore in Figure 1 with two open ends in states
pertaining to COS(
). This panel: along diameter.
component B is added to the gas reservoir upstream of
the pore. After some time a new stationary state is estab-
lished. Both components flow through the pore. Again,
the existence of the vapor/liquid interfaces is not affected.
The two components are modeled in a way that is de-
scribed below. Here it suffices to state that their behavior
as single component is identical and they have the same
interaction with the pore wall. The interaction between
the two components is treated to make the miscibility
between the components pressure (density) dependent. It
is important to note, however, that the components are
miscible in all proportions over the entire range of pres-
sures that may play a role. In spite of the similarity be-
tween the two components we observe an interesting
phenomenon: at the outlet of the pore the two compo-
nents are not homogeneously distributed any more.
Rather an enhancement of the second component in the
center of the flow is observed. In Figure 4 we present
the molar fraction across the streaming fluid a few nm
behind the outlet. While at the entrance the molar frac-
tion is constant over the entire cross section of the sys-
tem, a spatial separation occurs in the flow behind the
outlet. We observe that the minority component B is en-
riched in the center of the flow while it is depleted at
greater distance from the symmetry axis.
We further observe that the degree of local deviation
from the mean concentration fades away with increasing
distance from the outlet. There clearly is an optimum
distance where the separation is most pronounced. In
order to make use of the scheme for separation one
would have to introduce a structure in the vapor flow
behind the pore that collects either the central part or the
outer annual part. The fluid captured from the latter part
would constitute a fluid with reduced concentration of
the minority component. Thus, this part of the scheme
would represent an elementary step in the attempt to pu-
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profile of molar fraction perpendicular to axis
position [nm]
mol ar f ract ion B
before pore
Figure 4. Profile of molar fraction of component B perpen-
dicular to flow direction at different distances from the
pore outlet.
rify the component A from the impurity B. Of course one
would have to repeat the procedure several times as is
common practice in other schemes of separation, cf.
chromatography where the number of effective stages
(theoretical plates) may be of the order of 105. We be-
lieve that the efficiency of the present scheme is so high
that the required number of stages would be much
smaller than the number quoted.
The particularly interesting aspect of the strategy
found here is the notion that the two components can
well have very similar properties and even mix ideally at
ambient pressure. Even if both components have identi-
cal interaction with the pore wall the separation still
takes place. If the two components have different inter-
actions with the pore wall, the separation effect may be
further enhanced, but this is not prerequisite for the
separation as such. The only requirement for the scheme
to work is that the miscibility behavior varies somewhat
with pressure or density. From literature it is known that
virtually any mixture reacts on strong variations of pres-
sure [6,7]. Even a mixture that behaves almost ideally at
ambient pressure will show slight deviations from ideal
miscibility when exposed to extreme pressure.
So far we have presented results obtained before [5].
In the present communication we will built on the ob-
servation described above and explore the influence of
different parameters in order to achieve a more system-
atic picture. We will vary the molar fraction of the sec-
ond component B at the entrance and we will allow the
fluid/wall interaction to be different for both fluids. From
a number of situations being simulated we establish a
more general description. Further, we will make an at-
tempt to incorporate the effect of a mechanical skimmer
that allows separating the central fraction of the flow
with enhanced concentration of component B from the
outer part of the flow with depleted concentration of B.
We will see that introducing the skimmer is not detri-
mental to the separation power of the scheme.
2. Remarks on the Simulation Technique
Both fluids A and B employed for the present simulation
are modeled in the same way as the fluid in [8]. There
the adsorption of argon in porous silica at 77.35 K is
modeled by an approach very akin to density functional
theory. The fluid is described by a modified van der
Waals equation which reads
The parameters a and b retain their usual meaning in
the vdW equation, while the power s is an additional
parameter. This equation is somewhat more flexible than
the simple vdW equation. The parameters a, b, s are fit-
ted to experimental data. The interaction between any
two components X, Y is controlled by a convolution
function fXY(r) and a parameter XY. The convolution
function is normalized to
rr r
. Its shape
reflects the product of the pair distribution function gXY(r)
and the interaction potential VXY(r). Parameter XY mea-
sures the strength of the interaction while fXY (r) contains
information of the range of the interaction. The interact-
tion between both fluids as well as the interaction be-
tween the fluids with the material D of the pore wall is
modeled in this way.
In all calculations reported here the two fluids A, B as
single components are treated by the same parameters.
The interaction with the solid material D of the pore wall
is modeled by the same convolution function, thus fAD(r)
= fBD(r) which leads to the same range of the interaction
force between the fluids and the pore wall. Different be-
havior of both fluids with respect to the pore wall is de-
scribed in this communication by choosing different
values for AD and BD.
The interaction between the components A and B is
expressed in the same way as derived in the supporting
information to [8]. The chemical potential of both com-
ponents is augmented by a term that contains the ener-
getic influence of the other component
 
and 2
The quantities
r are the convo-
luted densities that incorporate the shape of the interact-
tion potential between both components. The parameter
controls the strength of the interaction, as explained
above. The entropy of mixing between both fluids is as-
sumed to follow the rule for regular solutions.
As we plan to simulate situations with a flow, i.e. non
equilibrium situations, we have to handle building up of
Copyright © 2011 SciRes. AJAC
a velocity field in the fluid. The way to treat simultaneously
the transport processes of diffusion and convection has
been described before [5,8]. The formulae are given
below for reference.
Both transport processes are based on the gradient of
the chemical potential as driving force and can be de-
scribed by the generalized diffusion equation plus the
slightly reformulated Navier Stokes equation
 
 
stands for the viscosity and M for the molecular
weight of the component treated while denotes the
vector of velocity. Here, transport is accounted for by the
first equation while the second equation controls the
changes of the velocity field. We have now a formulation
that describes non-equilibrium and equilibrium correctly
even in the presence of interfaces or other inhomogene-
ous regions while it converges to the familiar equations
in the limit of weakly inhomogeneous or homogeneous
systems. Onsager’s diffusion coefficient L is usually not
found in the literature. Therefore, it is extracted from
Fick’s diffusion coefficient D via L =
D/R, R being the
gas constant.
If more than one component is present, the same
strategy can be applied. For a binary mixture of two
components A, B we get
ax gradx grad
 
 
3. Results
3.1. Extending the Range of Parameters
Compared to Previous Results
In the following we measure the separation power of the
system by evaluating the ratio
xin .
stands for the constant molar fraction of
component B in the gas phase before entering the pore
center outflow refers to the molar fraction of
B in the center of the flow about 2.5 nm downstream of
the pore outlet (cf. Figure 4). This ratio is referred to as
factor of enrichment in some of the figure
This ratio has been studied for a few different values
of two parameters. Firstly, the value of
in has been
varied over one order of magnitude. Secondly, the pa-
rameter BD has been varied against the parameter AD.
A ratio of unity indicates that the interaction of both
components with the pore wall is identical. If the ratio
BD/AD drops below unity the component B is less
strongly attracted to the pore wall than the component A.
The results from our simulation calculations are com-
prised in Table 1.
We have found that all data can be reproduced by a
simple analytical form, i.e. by an incomplete Taylor ex-
pansion around the point
1, 2
It reads
 
12 1
11 2
aa bx
bx incx in
 
 
 
 
A comparison of the simulated data with the fit by the
above expression is shown in Figure 5. As the fit repre-
sents all data with good accuracy, we can rely on its va-
lidity and evaluate a contour plot, Figure 6. This plot
shows how the factor of enrichment depends on the pa-
in . In particular, we ob-
serve that separation is feasible even for 1
i.e. if both fluids exert the same interaction with the pore
wall. This situation is displayed again in Figure 7. Here
the only asymmetry between both fluids A and B is the
different concentration at the entrance. If this asymmetry
is removed by setting
xin, the factor of enri-
chment reduces to unity as has to be expected.
Table 1. Results from the simulation calculation. The mean-
ing of the quantities is explained in the text.
center outflow ratio
0.8 0.0385 0.0821 2.13
0.8 0.385 0.543 1.41
0.95 0.385 0.461 1.19
0.95 0.1925 0.277 1.43
0.99 0.385 0.431 1.12
0.99 0.0385 0.0647 1.68
1 0.0385 0.0617 1.60
1 0.385 0.424 1.10
Copyright © 2011 SciRes. AJAC
fa ct or of e nr i c hm ent , com par is on t o fit
number of si m ulati on
facto r
Figure 5. Factor of enrichment from simulation in com-
parison to a simple fit (see text). The quoted number of
simulation corresponds to the entries in Table 1.
f actor of enrichment in pore center
0.80.820.84 0.860.880.90.920.940.960.981
xB infl ow
Figure 6. Ratio of molar fraction of xB on symmetry axis
downstream of the pore over molar fraction xB before the
entrance of the pore.
f acto r of enri ch m ent for
00.1 0.2 0.3 0.4 0.5
at inflow
rati o
Figure 7. Ratio of molar fraction of xB[center outflow] on
symmetry axis downstream of the pore over molar fraction
xB[in] before the entrance of the pore. Here the results are
displayed for the special case with interaction of both com-
ponents with pore wall being ide ntical. Separation ceases to
be feasible for xB[in] = 0.5, but works for smaller values of
xB[in]. The performance improves with decreasing xB[in].
Further explanations in the text.
3.2. Introducing an Additional Structure for
Separating Fractions with Low and High xB
As we have seen, the flow through the pore filled with
liquid and held at negative pressure leads to an inhomo-
geneous distribution of both fluids in the liquid phase as
well as in the adjacent vapor phase. If one wishes to em-
ploy this phenomenon for a separation in preparation or
in analytical chemistry, one has to separate the fluid frac-
tions with different xB irreversibly to avoid mixing as
demonstrated in Figure 4. Here we simulate a possible
scheme for this purpose by introducing an additional
structure downstream of the pore outlet. It is a cylinder
that is meant to capture the fraction of the fluid in the
center of the flow. A cross section of the system treated
in the simulation calculation is shown in Figure 8. The
dark parts stand for solid material that forms the pore
wall. The additional cylinder adjacent to the outlet is
made from the same material. In the simulation the cyl-
inder is fragile due to its thin walls. When preparing an
experiment one would have to modify the structure in
order to give it more mechanical strength. Most likely a
wedge shaped wall would be a solution. The purpose of
the present simulation is only to explore whether an ad-
ditional structure would leave the inhomogeneous distri-
bution of xB intact or disturb it. We find that the task to
separate the fraction with higher xB in the center from the
outer fraction with lower xB is feasible in the presence of
the additional structure. The parameters of the simulation
are the same as correspond to the simulation described in
the second to last entry in Tab le 1. The factor of enrich-
ment quoted in Table 1 corresponds to the center of the
flow on the right side of the pore. As seen in Figure 4,
the separation effect is largest in the center. The equiva-
lent quantity cannot be evaluated for the situation shown
in Figure 8, because as soon as the fluid enters the cy-
lindrical structure the molar fraction xB gets more and
more homogeneous inside the cylindrical structure.
Therefore we average the molar fraction of component B
over the cross section of the cylinder. We get a value of
(to be compared with
At the first glance this looks disappointingly small com-
pared to the entry in Table 1 which yields
xcenter outflow
for the same boundary condition. However, one has to
realize that this value refers only to the center of the flow.
If we average in the simulation without the cylindrical
structure over the same cross section as before, we obtain
a value of 0.044
. Thus, the presence of the cylin-
drical structure does by no means deteriorate the separa-
tion performance; it rather seems to improve it some-
Copyright © 2011 SciRes. AJAC
Figure 8. Graph of the system treated in the simulation
calculation. Shown is a cross section of the cylindrical sys-
tem. The length of the pore is 22.6 nm and the diameter
amounts to 5 nm. The dark parts stand for solid material
that forms the pore wall. Adjacent to the outlet an addi-
tional cylindrical structure is added in order to separate the
fraction with higher xB in the center from the outer fraction
with lower xB. The material of this skimming structure is
treated as being identical to that of the pore. In practice it
might be preferable to choose a different material to avoid
with certainty recondensation of the fluid at the skimmer.
Light blue stands for vapor phase while darker blue indi-
cates the increase of fluid density up to the density of the
liquid in the pore. The vapor pressure on the left side is
about 177 mbar and, thus, 7 mbar larger than on the right
side. Accordingly, the flow direction is from left to right.
The velocity of the liquid in the pore is of the order of 0.01
cm/s. The flux through the pore is 6.9 × 10–17mol/s.
4. Summary and Outlook
On the basis of computer simulations of a mixture of two
fluids flowing through a mesopore we describe a –to our
knowledge– hitherto unknown phenomenon: if a situa-
tion of expanded liquid or negative pressure is main-
tained in the pore, a homogeneous mixture entering the
pore leaves the downstream end of the pore with a non-
homogeneous distribution of both components. It is shown
that the phenomenon exists even if both fluids behave
very similar, e.g. if the physical properties of both com-
ponents are identical and if the interaction of both fluids
with the material of the pore wall is the same. The inho-
mogeneous distribution downstream of the pore is caused
by the pressure dependence of the miscibility
by the strong density and pressure gradient in the
pore perpendicular to the flow direction
The effect of a single pore element is not sufficient to
allow perfect separation. In order to develop an efficient
method for practical application one would have to cou-
ple several stages in sequential order. We discuss the
situation assuming that we wish to tackle the task of pu-
rifying component A from the impurity B. In the context
of our results one would be interested in the fraction with
reduced xB, i.e. in the outer part of the flow. The molar
fraction xB is depleted in this fraction compared to the
inflow to 95% or less. Thus, any stage would reduce the
impurity fraction by a factor of 0.95. As the separation
scheme seems to gain efficiency with decreasing xB (cf.
Figure 7) we can estimate that an n-fold array of stages
will reduce the concentration of impurity B by a factor
0.95n or better. This means that a sequence of a few hun-
dred stages would reduce the impurity concentration to
extremely low values which might be welcome in prepa-
ration tasks. On the other hand, if B is not considered as
an impurity, but as a substance to be identified, one
could make use of the B enriched fraction of the flow
and raise the concentration for the purpose of analysis.
In comparison to other strategies of separation, e.g.
chromatography, the requirement for accurately manu-
facturing the single stages is much higher for the present
scheme. On the other hand, an increasing number of re-
searchers/laboratories is able to produce very regular
pore shapes from an ever increasing number of materi-
als [9-11]. Thus, we believe that the technical basis is
sufficiently developed to turn already in the near future
the phenomenon presented in this communication into
practical application for preparative as well as analytical
chemistry. Another aspect, that might be of interest and
justify enhanced manufacturing effort, consists in the
fact that the new mechanism of separation lends itself to
continuous operation while chromatography relies on
retention time and, thus, requires pulsed operation.
5. References
[1] P. A. Monson, “Mean Field Kinetic Theory for a Lattice
Gas Model of Fluids Confined in Porous Material,”
Journal of Chemical Physics, Vol. 128, No. 8, 2008, pp.
084701-084711. doi:org/10.1063/1.2837287
[2] A. R. Imre, “On the Existence of Negative Pressure
States,” Physica Status Solidi B, Vol. 244, 2007, pp.
[3] Imre and R. Attila, et al., “Indirect Methods to Study
Liquid-Liquid Miscibility in Binary Liquids under Nega-
tive Pressure,” NATO Science Series, II: Mathematics,
Physics and Chemistry, Vol. 242, 2007, pp. 389-398.
[4] H. Morgner “Progress in Understanding Fluids in Me-
sopores,” Invited Talk at 3rd International Advanced Ma-
terial Summit, Chengdu/China May 6-7, 2010.
[5] H. Morgner, “Fluids in mesopores. A new theory and
applications,” Journal of Chemical and Chemical Engi-
neering, Vol. 5, 2011, pp. 456- 472.
[6] V. V. Tarsov and I. V. Persianova, “Compressibility of
Ideal Solutions and Mixtures of Non-associated Liquids,”
Nauchnye Doklady Vysshei Shkoly, Khimiya I Khi-
micheskaya Tekhnologiya, 1959, pp. 8-12.
[7] A. Schedemann, E. C. Ihmels and J. Gmehling “Liquid
Densities of THF and Excess Volumes Fort he Mixture
with Water in a Wide Temperature and Pressure Range,”
Fluid Phase Equilibria, Vol. 295, No. 2, 2010, pp.
201-207. doi:org/10.1016/j.fluid.2010.05.004
Copyright © 2011 SciRes. AJAC
Copyright © 2011 SciRes. AJAC
[8] H. Morgner, “Computer Simulation on Static and Dy-
namic Properties during Transient Sorption of Fluids in
Mesoporous Materials,” The Journal of Physical Chem-
istry C, Vol. 114, 2010, pp. 8877-83.
[9] M. B. Yue, W. Q. Jiao, Y. M. Wanga and M.-Y. He,
“CTAB-Directed Synthesis of Mesoporous Calumina
Promoted by Hydroxy Polyacids,” Microporous and
Mesoporous Mate rials, Vol. 132, No. 2, 2010, pp. 226-
231. doi:org/10.1016/j.micromeso.2010.03.002
[10] J. X. Jiang, J. H. Yu and A. Corma, “Extra-Large-Pore
Zeolites: Bridging the Gap between Micro and Mesopor-
ous Structures,” Angewwandte Chemie International Edi-
tion, Vol. 49, No. 18, 2010, pp. 3120-3145.
[11] B. Lorenzo, M. Giampaolo, L. F. Liu, L. Woo, G. Ulrich
and C. Benoit, “Capillary Condensation and Evaporation
in Alumina Nanopores with Controlled Modulations,”
Langmuir, Vol. 26, 2010, pp. 11894-11898.