On a New Mechanism for Separating two Components in a Stationary Flow through Mesopores

doi:10.4236/ajac.2011.25070

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American Journal of Analytical Chemistry, 2011, 2, 619-625

doi:10.4236/ajac.2011.25070 Published Online September 2011 (http://www.SciRP.org/journal/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

E-mail: [email protected]

Received February 24, 2011; revised April 6, 2011; accepted July 11, 2011

Abstract

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

620 H. MORGNER

curves of s tates

straight pore with two open ends

de sorption switch

point

adsorption switch

point

0.E+00

1.E-09

2.E-09

3.E-09

4.E-09

5.E-09

-6700 -6600 -6500 -6400 -6300 -6200 -6100

chem. pot. [J/mol]

load [ mol/cm

]

COS(alfa)

COS(beta)

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

5.00E-17

1.00E-16

1.50E-16

2.00E-16

2.50E-16

3.00E-16

3.50E-16

4.00E-16

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

-350

-300

-250

-200

-150

-100

-50

-30 -20 -100102030

position [nm]

pres sure [bar]

(a)

pressure profile along pore diameter

-200

-100

100

200

300

400

500

600

-3-2-10 12 3

position [nm]

pressure

(b)

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-

H. MORGNER

621

profile of molar fraction perpendicular to axis

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

-4-3-2-101234

position [nm]

mol ar f ract ion B

0.6nm

2.5nm

6.3nm

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





 (1)

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



14πd

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

AABBAB

BABAAB





(2)

The quantities





,BAB





,AAB



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

622 H. MORGNER

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

Lgrad

grad







 



 





a

(3)

where



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



AAA

BBB

AABB

Lgrad

ax gradx grad









 











v





(4)

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







centeroutflow

xin .

Here







stands for the constant molar fraction of

component B in the gas phase before entering the pore

and



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

BD B

xin











It reads



 

12 1

11 2

BD BDB

AD AD

aa bx

bx incx in









 











 





 

 









(5)

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-

rameters

DAD



and





in . In particular, we ob-

serve that separation is feasible even for 1

BD AD





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.

BD/AD









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

H. MORGNER

623

fa ct or of e nr i c hm ent , com par is on t o fit

0.5

1.5

2.5

12345678

number of si m ulati on

facto r

simulation

fit

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

1.25

1.5

1.75

2.00

1.1

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

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



1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

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

0.0464



(to be compared with





0.0385

xin).

At the first glance this looks disappointingly small com-

pared to the entry in Table 1 which yields





0.0617

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-

what.

624 H. MORGNER

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.

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