Molyneux JMS 42

8/14/2019 Molyneux JMS 42

1/15

Journal of Membrane Science 320 (2008) 4256

Contents lists available at ScienceDirect

Journal of Membrane Science

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / m e m s c i

Permeation of gases through microporous silica hollow-fiber membranes:

Application of the transition-site model

Philip Molyneux

Macrophile Associates, 33 Shaftesbury Avenue, Radcliffe-on-Trent, Nottingham NG12 2NH, UK

a r t i c l e i n f o

Article history:

Received 29 September 2007

Received in revised form 1 February 2008

Accepted 2 March 2008

Available online 18 March 2008

Keywords:

Activation energy

Hollow-fiber membranes

Microporous silica

Molecular sieving

Permeability theory

Transition-site doorways

a b s t r a c t

In a previous paper [P. Molyneux, Transition-site model for thepermeation of gasesand vapors throughcompact films of polymers, J. Appl. Polym. Sci. 79 (2001) 9811024] a transition-site model (TSM) for the

activated permeation of gases through compact amorphous solids was developed and applied to organic

polymers; the present paper examines the applicability of the TSM to permeation through microporous

silica. The basis of the TSM theory for amorphous solids in general is outlined; the present extension to

inorganic glasses has revealed that the transition sites (TS) of this theory, which are the three-dimensional

saddle-points critical in the molecular sieving action, equate to the doorways long recognized in per-

meation through amorphous silica and other inorganic glasses. The TSM, which views permeation as a

primary process, is contrasted with the conventional sorptiondiffusion model (SDM) for permeation. It is

pointed out that in the SDM, the widely accepted analysis into two apparently distinct factors sorption

(equilibrium) and diffusion (kinetic) has the fundamental flaw that these factors are not independent,

since both involve the sorbed state. By contrast, the TSM focuses on the permeant molecule in only

two states: as the free gas, and as inserted in a doorway D; hence the characteristics of these doorways

(unperturbed) diameter D, spacing , and the thermodynamic parameters (force constant) and

(entropy increment) for the insertion process can be evaluated. The theory is applied to literature data

[J.D. Way, D.L. Roberts, Hollowfiber inorganic membranesfor gas separations, Sep. Sci. Technol. 27 (1992)

2941; J.D. Way, A mechanistic study of molecular sieving inorganic membranes for gas separations, FinalReport submitted to U.S. Department of Energy under contract DE-FG06-92-ER14290, Colorado School of

Mines, Golden, CO, 1993, www.osti.gov/bridge/servlets/purl/10118702-ZAx4Au/native/1011872.pdf; M.H.

Hassan, J.D. Way, P.M. Thoen, A.C. Dillon, Single component and mixed gas t ransport in silica hollow fiber

membrane, J. Membr. Sci. 104 (1995) 2742] on the permeation through microporous silica hollow-fiber

membranes (developed by PPG Industries Inc.) of the nine gases: Ar, He, H2, N2, O2, CO, CO2, CH4 and

C2H4, over the temperature range 25200C. The derived Arrhenius parameters for the permeation of

these gases (excepting He) lead to estimates of the four doorway-parameters: D , 125 pm; , ca. 30nm;, 0.43 nN; , 1.7pNK1; these values lie within the ranges of those obtained with the glassy organicpolymers. Some secondary effects, shown particularly by CO and CO 2, are interpreted as hostguest

interactions at the doorway. The behavior of He is anomalous, the permeation rising linearly with tem-

perature. This study confirms that the TSM may be applied to gas permeation by activated molecular

sieving for this type of inorganic membrane.

2008 Elsevier B.V. All rights reserved.

1. Introduction

The migration of gases in and through solids has many prac-

tical and theoretical implications, particularly in the use of films

as barriers, and in separation processes [1,2]. In these processes,

we can distinguish permeation as the transfer of the gas through a

layer of the solid; solubility or sorption as the equilibrium process

Tel.: +44 115 933 4813; fax: +44 115 933 4813.

E-mail address: [email protected] .

between the gas and that held within by the solid; and diffu-

sion as the movement of the sorbed gas molecules within the

solid.

In current thought, theprimacyin migrationis given todiffusion

[36]. With solubility as the equilibrium-based process, perme-

ation is then viewed as the composite of solubility and diffusion,

leading to the sorptiondiffusion model (SDM). By contrast, in a

previous paper [7] dealing with migration of gases and vapors

in organic polymers, a model is presented which gives primacy

to permeation; this was termed the transition-site model (TSM).

In the present paper this model is applied to simple gases per-

0376-7388/$ see front matter 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.memsci.2008.03.001
http://www.sciencedirect.com/science/journal/03767388http://www.osti.gov/bridge/servlets/purl/10118702-ZAx4Au/native/1011872.pdfmailto:[email protected]://dx.doi.org/10.1016/j.memsci.2008.03.001http://dx.doi.org/10.1016/j.memsci.2008.03.001mailto:[email protected]://www.osti.gov/bridge/servlets/purl/10118702-ZAx4Au/native/1011872.pdfhttp://www.sciencedirect.com/science/journal/03767388


2/15


3/15

44 P. Molyneux / Journal of Membrane Science 320 (2008) 4256

In parallel with the energy profile shown in Fig. 1, there must

also be an entropy profile; this is seldom considered, but evidently

must be taken into account to give the true, free energy descrip-

tion of the processes involved. In this case, the profile would have

troughs where molecules are restricted; a further difference is that

in contrastto theenergysituation where theenergy level is zero for

all dilute gases, in the case of entropy the starting levels would be

different following the absolute entropies of the gases considered.

This has to be taken into account in interpreting the vant Hoff and

Arrhenius prefactors as discussed below. In previous cases where

entropy effects have been considered [9,10,14], theyhave dealt only

with the diffusive jump process.

2.2. General relations

On the basis of ideal conditions that is, low pressure p of

the permeant gas and low concentration c of dissolved perme-

ant, the three determining parametersthe Henrys Law solubility

coefficient, S,1 the diffusion coefficient D, and the permeability

coefficient Pare related [2] by

P= DS. (1)

It is this equation that is used as the basis for the conventional

SDM, which considers that permeation must be a composite quan-

tity, depending on both the sorptive and the diffusive properties of

the system.

In considering the molecular bases of these processes, the

temperature dependence of the three parameters is of particular

importance. These generally take the similar vant Hoff (sorption)

and Arrhenius (permeation and diffusion) forms. In the light of Eq.

(1), this gives for the corresponding prefactors

P0 = S0D0 (2)

and for the energy quantities

EP=HS

+ED

(3)

where EP and ED the activation energies for permeation and diffu-

sion, and HS is the isosteric enthalpy change of sorption.In Eq. (3), if the sorption process is exothermic (so that

HS is negative), then the value of the permeation activation

energy EP becomes the difference between the numerical values

ofHS and ED; for example, in the case of H2O with poly(ethylmethacrylate) (PEMA), the studies of Stannett and Williams [15]

gave HS =34kJmol1 (exothermic) and ED =36kJmol

1 so that

EP =2kJmol1 (compare Fig. 1B).

As a general point, inasmuch as each permeant has spe-

cific Arrhenius and vant Hoff parameters, the plots for different

permeants may cross. This is particularly important with the

increasing emphasis on the use of membranes over wide tempera-

ture ranges. For example, the recent work of Wilks and Rezac [16]showed, for the competitive permeation of CO2 and H2 through

poly(dimethylsiloxane) (PDMS) membranes, that the selectivity

factor is 4.2 at 35 C (CO2 faster), but that this has fallen to 0.8 (H2faster) at 200 C. Thus the comparison of the permeation behav-

ior at a single temperature is not sufficient to elucidate migration

behavior.

Furthermore, any molecular interpretation should seek to

explain and correlate not only the activation energy, but also the

prefactor. In particular, using the TSM [7], valuable information is

1 This symbol for solubility coefficient, used here in conformity with customary

practice, should not be confused with that for entropy ( S0, etc.) used later in the

paper.

obtained from P0 on the entropy effects in the migration process,

and on the spacing between transition-sites (Fig. 1).A more general point in connection with these energy levels is

that they allow the permeation activation energy EP to be zero or

negative ifED becomes equal to or less than HS, so long as ED hasa positive value as required for the jump process.

2.3. Critique of the sorptiondiffusion model (SDM) for analyzing

permeation

The conventional and widely accepted belief is that permeation

must be considered as composite, that is, the resultant of sorption

(equilibrium) and diffusion (kinetic) effects. This seems to be sup-

ported, for example, by the form of Eq. (1), and by the simplicity of

the units for diffusion coefficient (m2 s1) compared with those for

permeability coefficient (mol m m2 s1 Pa1).

However, as detailed in the previous paper [7], there are a num-

ber of instances where permeation is a simpler process than would

be expected if it were the composite of independent effects. In this

section, this is taken a step further by showing that, despite the

almost universal view that diffusion must be considered as the pri-

maryprocess in solid-state migration, thereare a number of aspects

that raise doubts as to its validity for this purpose. These may be

considered under three headings.

2.3.1. The sorption and diffusion processes are not independent

The primary point of weakness of the SDM arises in the uni-

versally assumed analysis of permeation into the two assumedly

distinct aspects, sorption (equilibrium factor) and diffusion (kinetic

factor). Any analysis of a physical or chemical process requires that

the proposed component effects are independent; for example, in

analyzing thefactors controllingthe volume of a fixed amount of an

idealgas, thesetwo factors would be the pressureand the (absolute)

temperature,each of whichmay be varied independently. However,

in the case of the SDM, the two coefficients S and D are clearly not

independent, since they both involve the sorption state. For the

sorption coefficient S relates to the equilibrium between the gasstate and the sorption state; whilethe diffusion coefficient D relates

to the ease of a jump from the sorption state across the intervening

barrier (doorway), or (on transition-state theory) to the equilibrium

between the sorption state and the transition state (Fig. 1B).

Of course, this is not to argue against the classical picture of

permeation being a sequence of processes that we call sorption,

diffusion and desorption; nor is it to argue with the mathemat-

ical validity of Eq. (1). But neither of these leads to, or necessarily

requires, the conventional SDM analysis.

2.3.2. The sorption sites are heterogeneous

Considering the sorption of a single simple-molecule permeant

in an amorphous solid, such as an organic polymer, or an inorganic

glass, because of the disordered molecular structure of the hostmatrix, the permeant molecules will be held in a variety of sites

and locations. For example, the variation of solubility and partial

molar volume with uptake indicates a Gaussian distribution for the

sorption sites both with glassy organic polymers [17a] and with

vitreous silica [17b]. It is therefore evident that the starting point

for the diffusive jump (Fig. 1) is not well defined.

2.3.3. It is the pressure gradient and not the concentration

gradient that is the true driving force for transport

The deficiencies in taking the concentration gradient as the

driving force, are seen from the molecular viewpoint, when we

compare the behavior of two permeants O2 and H2O, which have

similar molecular diameters (293 pm and 289pm from critical

constantssee Appendix A and Ref. [7]), in a polar polymer such as


4/15

P. Molyneux / Journal of Membrane Science 320 (2008) 4256 45

PEMA [15]. For a given partial pressure of the penetrant, the uptake

oftheH2O is much greater than that of the O2 because the H2O will

hydrogen bond to the ester groups in the polymer chains. However,

for the same concentration gradient, the actual amount of H2O free

for diffusion will be much smaller than that for the O 2, because of

this hydrogen bonding. Thus the use of the concentration gradient

clearly does not represent the true driving force for the migration

process.

In this example, where the structures and interactions for the

solid and the migrant molecule are well established, the molecular

basis is self-evident, but it must apply more generally where these

forces may not be well understood.

These objections do not apply when we consider the pressure

gradient as the driving force, since this is a true measure of the

gradient of chemical activity (Fig. 1A). The present TSM uses this

picture,and shows howthis gradientmay be linked to the migration

process.

The validity of the pressure gradient as the driving force

is revealed in numerous cases where, when the conditions are

changed, the values ofD and S vary markedly but that ofPremains

essentially constant. Since the sorbed state enters in opposite ways

in the sorption and diffusion processes, its effect cancels out in

permeation.

For example, considering a case with the same permeant in the

same solidat different pressure and concentration levels, Rundgren

etal. have shown recently [18] withH2/vitreous silica at 550C, that

when the upstream pressure was varied up to 0.12 MPa, the value

ofPremained essentially constant (10%) although the value of S

decreased by a factor of approximately three while that of the dif-

fusion coefficient D correspondingly increased by the same factor.

Much the same effect can apply to the same permeant in differ-

ent solids, as with the case of H2O in moderately polar polymers

previously discussed [7]. A striking example of this type of effect

is shown in the work of McCall et al. [19] on the permeation of

H2O through poly(ethylene) (PE) films that had been air-oxidized

in processing to varying extents, leading to the incorporation of

hydrophilic groups on the polymer chains; such oxidation up to3% combined oxygen gave a 50-fold increase in the sorption coeffi-

cient S compared with the starting PE,but an essentially equivalent

decrease in the diffusion coefficient D, so that the permeability

coefficient Pwas essentially unchanged (20%) by the oxidation.

The above arguments apply not just to organic polymers or vit-

reous silica from which the examples have been chosen, but to any

system where the SDM analysis is currently used.

2.4. Analysis of the jump process

In the field of migration through organic polymers, little atten-

tion has been paid to the nature of the peak on the path of the

jump, with greater attention being paid to the free volume which

relates to the environment of the sorption sites [2]; however, thepreviously developed TSM [7] focuses particularly on the state of

the permeant molecule at this peak. It turns out, that in the field

of migration in inorganic glasses, these transition sites have been

recognized for over half a century, and were termed doorways [20],

a name that has been retained since [2125]. It is therefore conve-

nient to use term doorway (D) interchangeably with transition-site

(TS) in the rest of the paper.

A simple analysis of the jump process further highlights the

significance of the TS doorway. Firstly, if we apply the Principle

of Microscopic Reversibility to a jump, it follows that the reverse

process must also be acceptable as a jump; thus on the average

the jump will be symmetrical about the midpoint. Secondly, the

requirement for activation of the molecules to enable the jump to

take place indicatesthe presence of an energybarrier, which would

be located on the average at this midpoint. Finally, because of the

Boltzmann factor exp(E/RT) that controls the fraction of perme-

ant molecules with the required excess energy E, in making the

jump the penetrant molecule will expend the minimum amount of

energy necessary for a successful jump, that is, barely passing over

the central point. This location is the transition-site doorway D,

which has the nature of a three-dimensional saddle-point (Fig. 1B

and C).

2.5. Derivation of equation for permeability coefficient on the

TSM

The present TSM is then a direct application of the well-

established standard theory of the transition state (also termed the

activated complex) for chemical reactions [26]. For an elementary

chemical reaction, there is postulated to be equilibrium between

the starting reactants and those in the transition state, this being

the position of maximum potential energy on the lowest energy

path from the reactant to the products; this transition state then

converts into the products of the reaction (or may revert to the

reactants). For example, for the transfer of the atom B between the

species A and BC, the two-step sequence involved is:

A+ BCReactants

[A B C]Transitionstate

[A B+C]

Products

. (4)

Although this picture is well accepted for reactions in the gas

phase, there is some hesitation in applying it to processes in con-

densed phases. However, it is well established as the explanation of

the primary salt effectfor reactions in aqueous solution, that is, the

effect on the rates of reactions between ions in solution of added

neutral salt (an electrolyte with non-common ions); here it is the

net charge on the transition state that is important, and this state is

evidently sufficiently long-lived for the DebyeHuckel ionic atmo-

spherefromthe added neutral salt tobe established [26]. Moreover,

the theory has been applied successfully in considering perme-

ation in physiological systems [27], while the concept of transition

state analogs in now widely used in enzyme chemistry [28]. Morespecifically, this type of picture was used some time ago to inter-

pret quantum effects in the permeation of light gases (He, H2,

D2) through organic polymers [10], and more recently to similar

systems using molecular dynamics simulations [29].

With permeation through an amorphous solid,the pressure dif-

ferencep acrossthe layer of solidalso corresponds toa gradient ofpartial pressure of the gas within the solid, representing a gradient

of chemicalactivityacrossthe layer. Forthe idealcase underpresent

discussion (dilute gases, and dilute solutions of the penetrant in

the solid, so that there is no plasticization or other modification

of the host solid by sorbed permeant), this gradient will be linear

(Fig. 1A). Concretely, the partial pressure at any level would be that

within a microscopic cavity at that level; alternatively, it would be

thepressure realized if thesolidweresliced through in theyz-planeat that level, the two parts slightly separated, and the permeation

flux resumed. This does not mean that there are actual free per-

meant molecules in the solid, but simply that any depth x can be

assigned its value of the partial pressure px.

Applying these concepts to the cubic lattice picture having lat-

tice parameter (Fig. 1), the occupancy of doorways at depth x iscontrolled by the insertion equilibrium:

G(x)+ D(x) [GD(x)] (5)

where D(x) is an unoccupied doorway at the depth x, GD(x) is an

occupied doorway at that depth (labeled with the double cross

to show its analogy with the transition state in Eq. (4)), and G(x)

relates to the partial pressure at that level as already discussed.

The rate of loss of G from the occupied doorways (downstream or


5/15


upstreamFig. 1B) is then taken to be controlled by the standard

transition-state rate coefficient kkBT/h where kB is the Boltz-

mann constant and h is the Planck constant [26]. On the TSM, it

is this gradient of occupied sites [GD(x)] that gives rise to the

observed flux in the downstream direction.

This then leads [7] to an expression for the flux that correctly

incorporates its experimentally observed proportionality to the

area, the pressure difference across the film, and the reciprocal of

the film thickness. The permeability coefficient Pthen becomes

P=

kBT

ehNA

exp(S) exp

U

RT

(6)

where e is the exponential number, NA the Avogadro constant, S

the entropy change for the insertion of a mole of the gas G into a

mole of the doorways D according to Eq. (5), andU is the internalenergy change for this same process. The Arrhenius prefactor and

the activation energy then become

P0 =

kBT

ehNA

exp

S

R

(7)

EP =U

. (8)Considering Eq. (7), evidently the prefactor P0 is not expected

to be truly temperature independent since the expression has the

factor Tin front, but over narrow temperature ranges the variation

is much less than that of the exponential factor, so that it is per-

missible to use the average temperature in applying this equation.

In any case, there may also be a further complicating effect from

changes in S with temperature.The entropy of activation S can be expressed in terms of the

entropiesof the reactants and products for the insertion process

(5) as

S = S(GD) S(D) S0(G) (9)

where S(GD) is theentropy of 1 molof occupied doorways, S(D)that

of the occupied sites, and S0(G) is the absolute molar entropy of the

gas. Substituting in Eq. (7) and rearranging gives

P0 exp[(S0(G)/R)] =

kBT

ehNA

exp

[(S(GD) S(D)]

R

. (10)

It is convenient to define a quantity Y, the entropic coefficient,

relating the original permeability coefficient P and the absolute

entropy by

Y P exp[(S0(G)/R)]. (11)

Correspondingly, for the limiting case of 1/T0,this definesthe

entropic prefactor Y0 as

Y0 P0 exp[(S0(G))/R]. (12)

This last expression is used in evaluating Y0 from permeation

experiments, as discussed below. In these expressions for Yand Y0,

theunits of pressureimplicitin the absolute entropy and explicit in

the permeability coefficient have to be brought into concordance,

as discussed in Appendix A.

Substituting from Eq. (12) in Eq. (10) and taking logs gives

logY0 = log

kBT

ehNA

log + (log e)

[S(GD) S(D)]

R

. (13)

As discussed below, this relation is useful in making estimates

of the doorway spacing .Reverting to Eq. (8), the permeation activation EP is revealed as

the internal energy changeU toinsert the molecule of G intothe

doorway D (Eq. (5) and Fig. 1C). Likewise, the entropy difference

[(S(GD) S(D)] in Eqs. (9), (10) and (13) is the equivalent entropy

change for this same process.

It is useful to analyze the value of EP into three presumably

independent contributions to the insertion process of Eq. (5):

EP = EMM + Eex EMG (14)

where EMM is the energy to overcome any noncovalent interactions

between the hostmatrix (M) across the unperturbed doorway aper-ture, Eex is the energy to expand the matrix by the insertion of

the (presumably) larger gas molecule G (taken to be a incompress-

ible sphere), and EMG is the energy released from any interactions

(presumably, noncovalent) between the molecule of G and the sur-

rounding host matrix in the doorway.2 The first contribution, EMM,

is neglected here, and this is supported by the fact that the door-

ways are later deduced to have a diameter greater than 100 pm

(Section 3.5).

In previous analyses of this process as applied to silica and

derived inorganicglasses,attention hasbeen focused on thesecond

contribution Eex. In its earliest development the doorway was con-

sidered as a spherical cavity, but this was later replaced [20] by the

somewhat more realistic picture of a cylindrical cavity; this would

then lead, using the mechanics of elasticity for the expansion of a

thick-walled pipe, to the relation

Eex =GD(G D)

2

4(15)

where G is the shear modulus of the bulk solid, D the diameter ofthe unperturbed doorway, and G is the diameter of the permeant

molecule (see Fig. 1C). However, this commonly applied formula-

tion [18,2025] suffers fromtwo defects: firstly, the doorwaywould

be saddle-shaped (Fig. 1C), rather having parallel sides as Eq. (15)

requires; and secondly, there is no reason to believe that the elastic

behavior of the solid at the molecular level (relating to the rotation

and flexion of individual bonds) follows that for the bulk solid as

defined by the shear modulus G.

In many previous formulations [10,18,2025,29], Eex has been

equated to the activation energy for diffusion, which ignores thepart played by the sorption enthalpy in the diffusive jump process

(Fig. 1B).

Summarizing, in contrast to the diffusive jump picture, in the

TSM the permeation activation energy is referred to a well defined

initial state, that of the free gas, with reference energy of zero for

anygas(Fig. 1B); however, the prefactor P0 hasto be adjusted forthe

different absolute entropies of different gases to give the entropic

prefactor Y0 according to Eq. (12).

2.6. Application of the TSM to experimental data for organic

polymers

In thepreviousapplicationof the TSMto organic polymers [7], it

was tested on the collected literature data [30] for the permeation

of a variety of gases (ranging from He to SF6) with 16 amorphous orsemi-crystalline polymers using three test plots, involving the two

parameters EP andlog Y0 togetherwith the permeant gas molecular

diameter G:

Plot A: EP versus GPlot B: log Y0 versus GPlot C: log Y0 versus EP.

Previously [7] there had been four test plots, with the first of

these (Case 1) a direct plot of the original Arrhenius parame-

2 Kirchheim has shown the importance of this site expansion energy term Eex for

the sorption of gases both by organic polymers [17a] and by vitreous silica [17b].


6/15


ters, that is, log P0 versus EP, but in view of the better theoretical

basis for the entropy-adjusted form logY0, this firstplot is not now

considered to be useful.

With certain exceptions [7], these three test plots were found

to be essentially linear, and may be fitted to the respective conven-

tional forms

PlotA : EP = m1G + c1 (16)

PlotB : logY0 = m2G + c2 (17)

Plot C : log Y0 = m3EP + c3. (18)

In the context of the present paper, and to anticipate the

discussion on microporous silica in Section 3 below, these

plots are exemplified by Figs. 35, respectively, below (Section

3.4).

Before continuingwith the interpretation of theseEqs. (16)(18),

it shouldbe emphasized that their linearitydoes notfollow directly

from the theory, neither is it in any sense an integral part of it. It

should also be evident that these are not three independent equa-

tions, since they involve only three independent variables; this

leads to the parameters for the third equation as

m3 =m2m1

(19)

c3 =(c2m1 c1m2)

m1. (20)

As noted above, with the organic polymers not all the sys-

tems from the literature conformed to these linear relations of

Eqs. (16)(18). In the case of the permeants G, the commonest

anomalies were seen [7] with the two permeants CO2 and H2O.

The anomalies were in most cases only seen in the Plot A and

Plot C, and not in the Plot B (see Eqs. (16)(18)), indicating that

this is related to an anomaly specifically in the permeation activa-

tion energy EP, with this value being lower than expected from

the behavior of other gases; this seems to be the cause of CO 2

being a relatively fast gas in permeation [1,7]. At the time ofthe previous publication [7] no clarification could be given of

this effect with CO2. However, re-evaluation of this data [31]

indicates that there an essentially constant energy anomaly (low

EP) of 18(5)kJmol1 for the organic polymers studied with this

permeant, with little if any correlation of the individual values

with such parameters as the cohesive energy density of the poly-

mer, or the enthalpy change of sorption HS of CO2 into thebulk polymer. It seems that this effect results from to an ener-

getically favorable interaction between the quadrupole moment

of CO2 molecule at the transition site doorway and the polar

groups in the surrounding host matrix. This is thus an example

of the matrix-permeant energy contribution EMG towards EP in

Eq. (21). This interpretation is supported by the fact that CO2 is

well known to interact in a specific way with the bulk matrix ofmany polymers [32,33]. This effect is also seen with silica (Sec-

tion 3 below) and is discussed further there. A similar energy

anomaly occurs with H2O [7], where parallel interactions could

occur due to polar interactions, or possibly hydrogen bonding, at

the transition site doorway; this would also explain why H2O is

also characterized as a fast gas (vapor) in membrane separation

processes [1].

2.7. Interpretation of the linear plots

The simplest explanation of these linear relations in Eqs.

(16)(18) seems to be as follows.

In the first place, the linear relation of Eq. (16) (Plot A) indi-

cates that with increase in molecular diameterG there is a parallel

increase in the internal energy change U, shown to be equal tothe activation energy EP (Eq. (8)), to insert the molecule of G into

the doorway D (Fig. 1). Focusing on the expansion contribution Eexin Eq. (14), if we follow this linear plot back down to its intersec-

tion point with the x-axis (EP = 0), this point then represents the

condition where there is no energy required, that is, the molecule

exactly fits into the doorway (see Fig. 3 below Section 3.4). This

indicates that for the set of permeant molecules considered, thereare a fixed number of pre-existing doorways of diameter D thatcontrol the migration of the permeant molecules through the host

medium. The energy dependence of Eq. (16) may then be put more

specifically in the form

EP = (G D) (21)

where the coefficient is theforce constantfor the expansion of thedoorways for this host solid.

Likewise, the second linear relation Eq. (10) (Plot B) shows that

there is correspondingly a linear increase in the entropy of the

matrix at the doorway when the molecule is inserted, due to the

expansion by the inserted molecule (see Fig. 4 below Section 3.4);

this follows because logY0 is directly related to the entropy change

[(S(GD) S(D)] byEq. (13), and the firsttwotermson the right handside of this equation are independent ofG. This may be put in themore specific form:

log Y0 = log Yz +(log e)(G D)

R(22)

where the coefficient is the corresponding entropy incrementforthe doorway expansion process for that solid, and Yz is the limiting

value ofY0 for G =D.These two parameters and therefore define the thermody-

namics of the expansion process. They are unusual in that, in the

case of the energy change, because this increases linearly with the

expansion,this representa constant force being exertedby the door-

way, rather than an increasing force as might be expected for a

bulk solid with constant elastic modulus. In particular, it is in con-trast with the form of Eq. (15) that is derived from the mechanics

of elasticity for a cylindrical cavity in the bulk solid; however, the

defects of this approach have already been noted in the discus-

sion of this equation. Speculatively, the linear behavior may arise

from the transition-site doorways being asymmetric (transversely

elliptical or slot-shaped) rather than being symmetric (circular) as

depicted by Fig. 1C.

This same Plot B also provides a method to obtain an estimate

of the doorway-spacing parameter, which equivalent to the jumplengthof thediffusion model(Fig.1). Considering Eq. (13) for logY0,

the limit EP = 0 corresponds toG =D, with the gas molecule G just

fitting into the doorway, there will be no perturbation of the door-

way by the penetrant; so that not only will there be zero energy

change but there will also be zero entropy change, that is theentropy term on the right of Eq. (13) will be zero; this corresponds

to the value of log Yz from Eq. (22). Thus Eq. (13) gives

log Yz = log

kBT

ehNA

log (23)

from which in principle the value of may be estimated since allthe other quantities in this equation are known.

Considering the third linear relation, Eq. (18) (Plot C), as already

noted its linear form follows directly from Eqs. (16) and (17) (see

Fig. 5 below Section 3.4).

The goodness of fit of these linear Eqs. (16) and (17) depends

necessarily on well-defined values of the molecular diameter G,both with these organic polymers and with the inorganic amor-

phoussolid, silica, considered in Section 3 of this paper. Thesources


7/15


and suitability of these values are therefore discussed critically in

Appendix B.

2.8. Effects of finite uptake

As noted initially in Section 2.2, the present treatment applies

specifically to the ideal case of low pressures and low concen-

trations. In real systems there are often deviations at the higherpressures and sorbed concentrations, so that the sorption coeffi-

cient S depends upon p, and D upon c.In the TSM, these effects may be viewed in the first instance as

the result of the sorbed G on the characteristics of the doorways;

taking the spacing to be fixed for the material, this then involvesthe remaining three parameters D, , and . To disentangle theseeffects, it would be necessary to measure EP and Y0 as before over

a range of uptakes c, with care being taken that the uptake is the

same for each range of temperaturefor example, that there is the

same value ofc at 25 C as there is at 200 C.

In some cases the sorbed molecules may be so strongly bound

as to have little if any effect on the transition-site doorways, so

that the value of P remains essentially constant. This seems to be

the situation in the case of H2O with medium-polarity polymers[7], and likewise of H2O/oxidised PE [19] considered in Section 2.3.

This may also apply to the similar behavior for H2 with vitreous

SiO2 [18] also considered in that section.

At higher pressures there may be the purely hydrostatic effect

of the gas G upon the insertion process of Eq. (11), which may be

formalised as a volume change of activation V; this would leadto a contributionpV to thefree energychangeG andthence to

the activation energy EP, which would thus show up in the Arrhe-

nius plots. At such higher pressures, it may be necessary to use the

gas fugacity as the activity-corrected form in place of the pressure.

At higher uptakes still there may be effects from blocking of the

sorptioncavities tothe release of the moleculefrom the TS doorway,

or plasticization of the whole solid matrix, butin this case the solid

must then be considered to be different in nature from the purematerial.

3. Application of the TSM to permeation through

microporous silica hollow-fiber membranes

3.1. Inorganic membranes for permeation

Although organic polymerscontinue to playan importantrole as

membranesfor separationprocesses, there is an increasing interest

in inorganic membranes, because of favorable properties such as

greater heat stability [11,34]. It is therefore useful to see to what

extent the TSM model applies to this latter type of membrane. In

thepresent case, this is appliedspecificallyto the inorganicglasses,

notably silica as the paradigm simplest case; some other types ofinorganic membranes are discussed briefly in Section 3.9.

Thereis a greatbulk ofpreviousliteraturestudieson permeation

through and diffusion in glasses [3,5,6]. Unfortunately, it is difficult

to link these studies into a coherent picture, not least because there

are so many varieties of silica; forexample, Doremus [35] has listed

four distinct commercial types of compact (nonporous) vitreous

silica. Furthermore,numerousnoveltypes ofmicroporoussilica have

been developed recently, made for example, by the acid leaching

of borosilicate glass in hollow-fiber form [3642], or as a solgel

form by coating a polymeric silica sol onto a more porous substrate

such as such as alumina [34,4346]. In this context, microporous

implies a substance with pore diameter less than about 2 nm. Such

porosity is intended to confer a higher permeability than that for

the compact form, while at the same time retaining the activated

characterof themigration soas togiveuseful separationof different

gases.

3.2. Permeation data of Way et al. for microporous silica

hollow-fiber membranes

In applying the TSM to microporous silica, what is required is a

set of data for a set of permeant gases having a range of moleculardiameters, studied over a sufficiently wide range of temperature

that good Arrhenius data may be obtained and the test plots of Eqs.

(16)(18) may be applied as already discussed in Section 2.6. This

mirrors the requirements for the data sets for organic polymers

usedin the previous development of the TSM [7]. This requirement

is largely fulfilled by the literature data on the microporous sil-

ica hollow-fiber membranes that were produced and patented by

PPG Industries [36,37], and that were usedin the three permeation

studies published by Way and colleagues [3840], where they were

referred to as developmental products. As detailed in the refer-

ences, the hollow-fiber membranes had been produced by melt

extrusion of a borosilicate glass, followed by exhaustive acid leach-

ingto remove Naand B, producing a network of poreswith diameter

claimed to be less than 20

A (2nm) [36,37]. For the present pur-poses, it is convenient to refer to this form of microporous silica as

SiO2, to distinguish it from (for example) normal compact vitre-ous silica, vSiO2. However, this is only one example from a diverse

range of these materials.

Before considering these data further, it should be noted that

there are parallel data, on another sample of the PPG product, pub-

lishedby Shelekhinet al.[41,42]. Since thereare marked differences

in the results from these two groups, the latter are considered sep-

arately below in Section 3.8.

Thefibers studied intwo main papersfromWay etal. [39,40] had

inside and outside diameters of the fibers of 35and 45m,giving anominal wall thickness of 5m, while thematerialitself was foundto be amorphous, with about 20% porosity and with polydisperse

poresaveraging about 10 nm. Therewas alsophysisorbedmolecular

water (not quantitified) that was removed on heating above 450 K,

with exposure to the atmosphere at normaltemperature leading to

its resorption.

The present paper focuses on the main studies in the two

later publications [39,40], which used nine permeant gases: Ar,

H2, He, N2, O2, CO, CO2, CH4 and C2H4, whose permeation rate

was measuredover 25200 C. A pressure differentialp = 21.4 atm

(2.17 MPa) was used for the main permeation studies.

The studies also included measurements with gas mixtures;

such studies are naturally important in relation to gas separation

[1,11,41,45,47]. However, only the single-gas measurements have

been considered here.

For convenience of the future discussion, some important phys-

ical properties of these nine permeant gases are listed in Table 1:

(electrical) polarity factors (molar polarizability, dipole moment,and quadrupole moment) [48,49], the absolute entropy S0 [50] (see

Appendix A), the critical constants [51], and various sets of values

of the molecular diameter G [5256] (see Appendix B).3

The data from first of the three Way publications [38] show

certain inconsistencies both internally (nonlinear Arrhenius plots)

and with the later data [39,40]; they must therefore be viewed

as only preliminary, and (except for He) they are not considered

further here.

3 Datain thispaper aregiven in the internationally standardconciseform: mean

value (limits of error in the last decimal place of the mean value), so that for

example, 1.23(4) represents 1.230.04 (see: http://physics.nist.gov/cuu/ Con-

stants/index.htm).
http://physics.nist.gov/cuu/%20Constants/index.htmhttp://physics.nist.gov/cuu/%20Constants/index.htm


8/15


Table 1

Physical and molecular properties for the gases G that were studied as permeants with the microporous silica hollow-fiber membranes [3840]

G Polarity factors S0d Critical constantse Molecular diameters (pm)f

e a peb Qec pc Tc Vc Breck LJ CC

Ar 1.00 0 nd 251 4.87 151 75 340 343 (5) 295

H2 0.49 0 1.7 227 1.29 33 64 289 292 (4) 260

He 0.13 0 nd 222 0.23 5 57 260 258 (3) 210

N2 1.03 0 4.7 288 3.39 126 90 364 371 (6) 313

O2 0.94 0 nd 256g 5.04 155 73 346 348 (8) 294

CO 1.18 0.4 9.5 294 3.50 133 93 376 367 (7) 315

CO2 1.51 0 14.3 310 7.38 304 94 330 398 (8) 324

CH4 1.47 0 nd 282 4.60 190 99 380 380 (6) 324

C2H4 2.52 0 13.1 315 5.04 282 130 390 416 (7) 359

a Molar electric polarisability (cm3 mol1) [48,49].b Electric dipole moment (Cm)1030 [49].c Electric quadrupole moment (C m2)1040 [49].d Absolute entropy at 298K (J K1 mol1) [50], adjusted to 1 Pa as standard pressuresee Appendix A.e Critical constants: pressure pc (MPa), temperature Tc (K), molar volume Vc (cm

3 mol1) [51].f Molecular diameter values (pm) with sources: Breck [52], LennardJones (LJ) [5356], and critical constants (CC) [51]see Appendix B.

g Oxygen special value ofS0 see Appendix A.

3.3. Permeability coefficients and the Arrhenius plots

Considering the two main papers, the data in earlier paper [39]werereported as permeability coefficients P, whereas those in later

paper [40] were only quoted as permeances Q, that is, the ratio P/L

where L is the membrane thickness. Correlation of the two sets

of data indicated values of L between 4.4 and 4.9m; this is inaccord with the direct measurements for the internal and exter-

nal diameters [40] that gave L = 5m. The permeance values [40]were accordingly converted into values of P using L calculated for

that sample. The Arrhenius plots for the combined data [39,40] are

shown in Fig. 2. The linear plots shown fit the experimental values

ofPwith an average deviation of 6%.The anomalous behavior of He

is considered below in Section 3.7 (see also Fig. 7).

3.4. Arrhenius data and derived parameters for the main gases

Table 2 lists the Arrhenius parameters for the eight gases (thatis, excepting He) derived from the linear plots in Fig. 2, together

Fig. 2. Arrhenius plots for the permeation of gases as labeled through the micro-

porous silica hollow-fiber membranes. He: quadratic fit (but see also Fig. 6); other

gases: best linear fits. Derived from the data of Way et al. [39,40].

withthevaluesoflog Y0 (entropic prefactorEq. (12)) derived using

the values of the absolute molar entropy S0 (at 298 K, adjusted to

the standard pressure 1 Pa) as given in Table 1. These Arrheniusparameters may be taken to refer specifically to 298 K, since the

Arrhenius plots are linear at least down to this temperature (Fig. 2)

and the entropy values S0 do refer specifically to this value. Table 2

also show, for comparison, the corresponding literature values of

the Arrhenius parameters for permeation through normal vitreous

silica, vSiO2; only with four gases Ar, H2, N2 and O2 are com-

parative data available for both types of silica [57], while there are

only from single literature sources for Ar [58] and for O2 [59]. The

comparison shows that the activation energies EP are much larger

with normal vitreous silica than they are with the present microp-

orous type. The EP-ratio for H2 is about 1.2, whereas those for the

other three gases the ratios are very similar, averaging 2.14(5); this

latter fact suggests that the structure of the present SiO2 differs

in a consistent way from that of the normal compact form.Reverting to the SiO2 data, they are plotted along with themolecular diametersG to give the three plots as already used with

organic polymers [7] as discussed above (Section 2.4):

Fig. 3, Plot A: EP versus G;Fig. 4, Plot B: log Y0 versus G;

Fig. 5, Plot C: log Y0 versus EP.

Table 2

Arrhenius parametersfor the permeation of the gases G through microporous silica

(SiO2 ) hollow-fiber membranes (Fig. 3), derivedin this paper fromthe data ofWay

et al. [39,40], compared with literature data for compact vitreous silica (vSiO2 ) [57]

G SiO2 vSiO2

9+log P0a

EPb

log Y0c

15+log P0a

EPb

Ar 0.87(4) 48.6(4) 4.98(5) 0.42d 106d

H2 0.38(5) 31.2(4) 3.18(5) 1.9(1) 37(1)

He nde nde nde 2.1(1) 22(1)

N2 0.12(13) 46.3(10) 6.11(13) 4.0(4) 100(8)

O2 0.64(6) 44.4(4) 5.01(6) 0.61f 93f

CO 0.70(5) 49.2(4) 7.06(5) nd nd

CO2 0.26(12) 38.9(10) 7.45(12) nd nd

CH4 0.89(10) 54.8(8) 6.62(10) nd nd

C2H4 1.00(11) 57.8(8) 8.51(11) nd nd

a P0 is the permeation prefactor (molm m2 s1 Pa1); note the different additive

factors for the SiO2 and the vSiO2 results.b Permeation activation energy (kJ mol1).c Y0 is the entropic prefactor (molm m2 s1)see Eq. (12).d Single value [58].e Nonlinear Arrhenius plotsee Figs. 2 and 7, and Section 3.7.f

Single value [59].


9/15


Fig. 3. Plot Apermeation activation energy EP versus molecular diameter G for

gases throughthe PPGmicroporoussilicahollow-fiber membranes.Key: sixgases

as labeled the straight line is least-squares fit for this set; CO2 ; CO2 inter-polated value, with the broken line indicating the energy anomaly EP; CO;

Heextrapolated EP value. The extrapolation of the linear fit to EP = 0 gives the esti-

mate of the (unperturbed) doorway diameter D as labeled. Derived from the data

of Way et al. [39,40].

The molecular diameters values G used here are the criti-cal constant values CC listed in the last column of Table 1 (see

Appendix B), as previously used with the organic polymers [7]. In

each case the three plots are essentially linear, as previously seen

with the organic polymers [7]. The original papers [39,40] gave

similar plots to the present Plot A, using the Breck values [52] for

molecular diameter (see Table 1 and Appendix B), and also noted

the essential linearity.

Again,as with theorganicpolymers [7], deviations are seen withCO2, as discussed in Section 3.6 below; however, closer examina-

tion of thedata indicatedthat thebehaviorof CO is also anomalous,

Fig.4. PlotBlog Y0 (entropicprefactor)versus moleculardiameterG forpermeant

gases with the PPG microporous silica hollow-fiber membranes. Key: ; six gases

as labeled the straight line is least-squares fit for this set; CO2; CO; : He

extrapolated log Y0. The extrapolation of the linear fit to EP =0 is labeled with the

estimate of the (unperturbed) doorway diameter D from Fig. 3. Derived from the

data of Way et al. [39,40].

Fig. 5. Plot Clog Y0 (entropic prefactor) versus permeation activation energy EP ,

for permeant gases with the PPG microporous silica hollow-fiber membranes. Key: sixgases aslabeled thestraight line is least-squares fit forthisset;CO2; CO;

: He extrapolated (see Figs. 3 and 4). Derived from the data of Way et al. [39,40].

inthat inthecaseof PlotB (Fig.4) omitting this permeant as well as

CO2 gives a very good linear fit,with any deviations for the remain-

ing six permeants (H2, Ar, O2, N2, CH4 and C2H4) similar to those

suggested by the Arrhenius plots (Table 2). This indicates that, at

least with this set, the log Y0 values are well defined; this indicates

in turn that the EP values are also well defined, and that any devia-

tions inthePlot A (Fig. 3) and PlotC (Fig. 5) from linear correlations

represent systematic effects. The good linear fit in Plot B (Fig. 4)

also indicates that the G values for these six permeants are welldefined, supporting the self-consistency of the critical constant

values CC used here and derived as discussed in Appendix B.4.It is therefore convenient to divide the interpretation of thedata into the primary effects, relating to the (average) straight lines

in the test plots, and the secondary effects, relating to systematic

deviations from these linear correlations.

3.5. Primary effects: H2, Ar, O2, N2, CH4, C2H4

Considering the primary effects, the behavior of this group of

six gases relates mainly to the expansion of the SiO2 matrix, corre-

sponding to the energy contribution Eex of Eq. (14), the data from

the three linear Plots AC results indicate, as discussed for the

organic polymers in Section 2.7, that the permeation of this set

of six gases involves a single fixed set of doorways with defined

diameter and defined spacing. The linear increase ofEP and log Y0results from the energy and entropy effects of inserting increas-

ingly large permeant molecules to expand the silica matrix at the

TS doorway (Fig. 1).4 Using Eqs. (21) and (22) gives the values of the

four characteristicparameters for theTS doorways listedin Table 3.

These values are similar to those seen with the organic poly-

mers [7], although in the latter casethe thermodynamic parameters

in particular show a wide range of values because of the diver-

sity of polymers considered. As with the organic polymers, these

parameters must be considered average values.

It will be noted that these expansion effects are seen for perme-

ants ranging in size and molecular complexity from H2 up to C2H4,

4

See footnote 2.


10/15


Table 3

Parameters for the transition-site doorways in the permeation of gases through the

PPG samples ofSiO2 studied by Way et al. [39,40]a

Name Symbol Value Units Figure Equation

Unperturbed diameter D 125(10) pm 3 (21)

Force constant 0.43(7) nN 3 (21)

Entropy increment 1.7(2) pN K1 4 (22)

Doorway spacing 30(4) nm 4 (22)

a Primaryeffectdata fromthe linear fitsin theFigures specifiedfor thesix gases:

Ar, H2, N2, O2, CH4, C2H4see Section 3.5.

and including the intrinsically spherical (Ar) as well as the intrin-

sically planar (C2H4). This also shows the value of studying a wide

number and range of permeants, since it allows for a few to turn

out to be markedly anomalous (here, He, CO and CO 2) while still

allowing sufficient remaining to define the main effects.

3.6. Secondary effects

It is presumed in discussing these effects that the behavior all

the gases, including CO and CO2 as the most deviant cases, refers to

thesame values of theaveragedoorway spacing and unperturbeddoorway diameter D that were estimated in the previous section(Table 3). It is then convenient to quantify these secondary effects

in terms of the deviations in the two parameters EP and logY0 from

the linear trends in Figs. 3 and 4, with the values of the respective

energy anomaly EP and entropy anomaly (log Y0) given by:

EP = EP(cal) EP(exp) (24)

(log Y0) = (log Y0)(cal) (log Y0)(exp) (25)

where cal indicates the value calculated from the trend Eqs.

(21) and (22) for that value ofG, and exp indicates the exper-imental value (Table 2). These anomaly values are plotted against

one another for display in Fig. 6. These effects may be discussed

on the basis of the three contributions to EP shown in Eq. (14):

EMM, Eex and EMG, but with the value of the first contribution

(matrixmatrix noncovalent interactions across the unoccupied

doorway) again neglected because of the width of the doorway

(125 pm). The discussion of the primary effects focused specif-

Fig. 6. Entropy anomaly (log Y0) (Eq. (25)) versus energy anomaly EP, (Eq. (24))

for permeant gases with the PPG microporous silica hollow-fiber membranes. Key:

: six gases as labeled; CO2; CO. Derived from the data of Way et al. [39,40].

ically on the expansion contribution, Eex, as analyzed by Eqs. (21)

and (22). Thesecondaryeffects may be ascribed tothe gasmolecule

in the transition-site doorway (Fig. 1C) interacting with the sur-

rounding silica matrix, presumably by polar interactions with the

electrical distribution on the Si and O atoms of the matrix; this cor-

responds to the contribution EMG of Eq. (14). Although no attempt

is made here to quantify these interactions, it is useful to note

the degree of polarity for the various species involved. The com-

mon factor here is the SiO2 matrix. There seems to have been little

mention on its charge distribution in discussions of its doorway

structure [2025]; however, both Pauling electronegativity values

[60] and electron density measurements [61] give essentially +1.0

unit charge on each Si atom and 0.5 unit charge on each O atom.

Carbon dioxide: The deviations with CO2 parallel those previ-

ously seen with organic polymers [7]; they correspond to the value

ofEP being lower than that expected byEP =12kJmol1, but with

in this case also a marked entropy effect as shown by the apprecia-

ble (log Y0) value (Fig. 6). The effect seen so markedly with thisgas seems to be the result of the interaction of its large quadrupole

moment with the polar silica matrix as discussed; this quadrupole

momenthas been modeled bya chargedistribution with +0.66 unit

charge on the C atom and 0.33 unit charge on each O atom [62].

Carbon monoxide: This gas is characterized by a marked

(log Y0) value (Fig. 6), essentially the same as that for CO2but a small if not zero value of EP. This anomaly also showsitself in the Arrhenius plots (Fig. 2), where the plot for CO will

clearly cross that for N2 at lower temperatures. The difference in

behavior of these two particular molecules is remarkable, since

in their electronic structure and their critical constants they are

very similar and differ only in the presence of a small dipole

moment with CO (Table 1), which with the internuclear dis-

tance of 113 pm [49] corresponds to partial charges of only 0.02

units on the respective atoms. Evidently the polarity of the SiO 2matrix again leads to a stronger effect than might have been

expected.

Six gases: H2,Ar,O2, N2, CH4, C2H4.Itwillbeseenfrom Fig.6 that

these gases show appreciable and apparently significant values forthe energy anomaly EP and with smaller and less significant val-ues of the entropy anomaly (log Y0). The values however seem toshow no clear correlations with their molecular features (Table 1).

For example, C2H4 has a similar quadrupole moment (albeit of the

opposite sign) to that of CO2, but shows much lesser secondary

effects than the latter. Similarly, it is difficult to see why such three

differentmolecules as H2, N2 andC2H4 should be clusteredtogether

(Fig. 6).

Summarizing, thesesecondaryeffects evidentlyarise because of

the much higher polarity of SiO2 than that of the organic polymers;

indeed, these effects may serve to probe the fine structure of the

TS doorways. However, although the deviations seen with CO 2 are

understandable and parallel those seen with the organic polymers,

thepicture forthe other gases is much less clear. If the option existsto extending these studies to other gases, the most useful would

be the other noble gases (Ne, Kr, Xe), since the simplicity of their

structure would greatly simplify the interpretation of the effects

observed.

3.7. Helium

As Fig. 2 shows, the Arrhenius test plot for He is markedly non-

linear (the curve drawn in the figure is an arbitrary quadratic best

fit). However, Fig. 7, which now includes the He data from the ear-

liest paper [38], shows that a simple plot ofP(molmm2 s1 Pa1)

versus T(K) gives an essentially linear dependence

10

13

P=

0.030(T

295). (26)


11/15


Fig. 7. Permeabilitycoefficient Pversus temperaturefor He andH 2 throughthe PPG

microporous silica hollow-fiber membranesdata of Way et al. [3840]. He, Ref.

[38];He,Ref. [39]; +H2 [39,40]. Thestraight line is best fitto theHe data;the curve

is the Arrhenius fit to the H2 datacompare with Fig. 2.

The sameplot also shows the datafor H2 as the next lightest gas,

with the best fit of the Arrhenius equation (Table 2).

This simple behavior of the temperature dependence of per-

meation for He does not seem to have been observed previously

in the literature, where for compact vSiO2 the dependence is

essentially Arrhenius even down to these low temperatures [57].

The parallel work of Shelekhin et al. [41] (see Section 3.8) also

gave an essentially linear Arrhenius plot for He with their sam-

ple of the PPG SiO2, while their EP values correspond to an EPvalue of about 30kJ mol1 for the present Way sample (compare

with Figs. 3 and 5). If the He were escaping by Knudsen flow

through microscopic pores, the temperature dependence would be

as (1/T)1/2.

At the same time, it is suspicious that the permeation flux

begins at a temperature (295 K) that is only a little below the

starting experimental temperature (298K = 25 C), as if heating the

equipment were opening some apertures in the membrane sys-

tem through thermal expansion. However, the effect seems to

occur both with the data in first paper [38] and those in the third

paper [40], where different samples of the hollow-fiber membrane

were used (Fig. 6). It is also curious that with rise in temper-

ature the permeation rate for H2 eventually overtakes that of

He (Figs. 2 and 6); any anomalous permeation mechanism forHe would be expected to be additional to the normal Arrhe-

nius rate, which should be should be markedly higher than that

of H2.

This simple but curious behavior of He therefore remains unex-

plained.

3.8. Permeation data of Shelekhin et al. on the PPG microporous

silica hollow-fiber membranes

Ithasalreadybeennoted,inSection3.2, thatparallelpermeation

studies on the PPG microporous silica hollow-fiber membranes

have also been carried out by Shelekhin et al. [41,42]. Although

these data, and those of Way et al. [3840], already considered

in some detail, were obtained on what was nominally the same

type of type of PPG product [36,37], it is evident that there were

marked differences between the two materials used. The Shelekhin

fiber sample, for example, had a smaller outside diameter (32m)andinside diameter (22m),although with thesame nominal wallthickness (L = 5m). Five gases were studied with this sample: He,H2, O2, N2, CH4, CO2; the permeation of these was studied over

the temperature range 30250 C, except for H2 that was studied

at only 30 C. Both sets of Arrhenius parameters were lower than

those for the Way samples (Table 2); the Shelekhin sample values

of log P0 were 3.9 units lower for O2 and N2, and 4.6 units lower

for CH4 and CO2; for EP there were a fairly consistent lowering of

33(2)kJ mol1 for all four of these gases. The lower values of EPseem to be an extension of the reductions seen in going from the

compact vitreous silica vSiO2 to the microporous silica studied by

Way et al. (Table 2).

Their data were interpreted by the authors [42] on the basis of

the SDM. However, using the present TSM, the EP values gave a

rather scattered linear Type A plot (not shown) for the four gases:

He, O2, N2, and CH4, where the downwards shift in EP corresponds

to a higher value of the doorway diameter D = 240(20) pm (com-

pare with Fig. 4); the EP value for CO2 was again low, in this case by

about 11 kJ mol1.

It seems that thelarger doorway size with the Shelekhin sample

compared theWaysample is theresult of differences in theproduc-

tion histories of these samples. Comparative results of this kind on

suitably diverse sets of permeant molecules with membrane sam-

ples preparedunder different controlledconditions promise to give

some clues as to the factors influencing the four TSM parameters

(D, , , ) that determine permeation behavior.

3.9. Other microporous inorganic media

The microporous silica discussed in Section 4 is only one type

in a wide range of such media whose gas permeability behav-

ior is important in practice [1,11,14,34,52]. Two other important

types are the zeolites, with their open-network crystalline struc-ture, and where the crystallographically defined windows are the

equivalent of the present doorways [1,9,11,12,14,52], andthe micro-

porous carbon membranes, also referred to as carbon molecular

sieve membranes (CMSM) [11,13,34,47].

With these materials, insofar as the pore size and the perme-

ant molecular diameter lead to activated migration and molecular

sieving, the TSM should again be applicable. As with the organic

polymers [7] and the present microporous silica(Table 1), this again

requires experimental permeation data over a wide temperature

rangewitha setof permeantshaving a spectrumof moleculardiam-

eters to probe the doorwaysand define theircharacteristics; the use

of nine permeantsin thepresent case seems exemplary. Once again

the TSM obviates the need, inherent with the SDM in experimental

and in modelling studies, to deal separately with the sorption and

the diffusion processes.

4. Conclusions

In this paper, the previously presented transition-site model

(TSM) as applied to organic polymers has been re-presented

briefly with some changes in detail. The conventional and widely accepted sorption-diffusion model

(SDM) used to describe permeation through solids has now been

reviewed critically; one key feature of the present paper is the

observation that there is a fundamental point of weakness of the

SDM, in that the twoanalyzed factors, sorption and diffusion,

are not independent, since both contain the characteristics of the

sorptionstate. Other pointsnotedare that thesorptionsites must


12/15


be heterogeneous in character, and that the pressure differential

and not the concentration gradient is the true driving force for

the migration. By contrast, the strength of the TSM lies in its focus on the

transition-sites that it turns outare already accepted as doorways

for migration in nonporous inorganic glasses such as vitreous

silica. The TSM allows analysis of the linear test plots to estimate the

four characteristic quantities: the intrinsic diameter of the door-

way, D; the thermodynamic parameters for the expansion ofthe doorway, (force constant) and (entropy increment); andthe average spacing between neighboring doorways , which is

equivalent to the jump distance of the SDM. The TSM has the advantage over the SDM that if decouples the

migration process from the sorption equilibrium. Strictly, this

only applies in the limit of low pressures and low amounts of

sorption; the actual amount of sorption only comes in as a per-

turbationeffect, that is,of thesorbed moleculeson theproperties

and behavior of the doorways. The present work has shown that the TSM applies to one particu-

lar type of microporous silica hollow-fiber membrane produced

by PPG Industries, and that it gives values for the four character-

istic parameters that are similar to those obtained with organic

polymers. There are also some secondary effects that are inter-

preted as host-guest interactions at the doorway. These results suggest that the TSM should be applicable more

widely to inorganic membranes showing activated permeation

by molecular sieving, including othertypes of microporous solids

as well as zeolites and other crystalline solids.

Appendix A. Absolute molar entropy values,S0

The values of the absolutemolar entropy, S0,forthegasesplayan

important part in the TSM, in correcting for their different entropy

levels in the gas state on either side of the membrane, and putting

these on the same basis as the energy values which are all on thesame (zero) level in the gas state. There are two points to be noted

in connection with these values listed in Table 1.

The first point relates to the matter of the pressure units that

are used as the reference level in defining S0, since thismust bethe

same as the pressure units used for the permeability coefficient;

this reference level relates to the entropy change that takes place

when the solid (crystalline) form of the substance (with entropy

essentially independent of pressure) is convertedto thegas (vapor)

state. In a previous publication [63] this reference level has been

considered as a form of hidden unit; an alternative methodwould

be to simply use permeability coefficients that are referred to the

samestandard pressureunit of the absoluteentropy value, although

this is not such a transparentwayof working.In eithercase,the con-

version from the literature values of S0 [50] with the bar (=105 Pa)as the standard state pressure, to the pressure unit Pa used in the

permeability coefficient, requires the addition ofR ln(105), that is,

9 6 J K1 mol1; thevalues so adjusted are listedin Table 1, and used

in the calculation of the entropic prefactor Y0 using Eq. (12).

The secondpoint relates specifically to thevalueofS0 tobe used

for O2. In the case of the organic polymers [7], it was found that

using the value of 301 J K1 mol1 obtained by applying the above

pressure-unitadjustment, withall the polymerswhere O2 hadbeen

studied alongside a number of other permeant gases, in each case

it gave the same anomaly with the Plot B and the Plot C, but no

anomaly with the Plot A; this indicates that the anomaly relates

to a high value of the parameter log Y0 (entropic prefactor). Since

this occurred in a consistent way with all of the polymers involved,

andunless theanomalyrelatesto a differentvalue of theratecoeffi-

cientor lattice parameterforthis gas Eqs. (16)(18) then theonly

common factor is the gas itself. This was therefore dealt with prag-

matically by using the lower value of 256J K1 mol1 for this gas,

which removed the anomaly for this gas in all the cases involved.

This has also been done in the present case (Table 1), and leads to

a good fit in the Plot B (Fig. 4) and Plot C (Fig. 5), as well as in Plot A

(Fig. 3) where this factor is not involved.

Appendix B. Molecular diameters of permeant gas

molecules, G

B.1. Importance of molecular diameter values

In the evaluation and interpretation of the migration of guest

molecules in solids from the molecular viewpoint, it is evidently

necessary to have a self-consistent set of molecular diameters for

the gases used; however, there seems to have been little specific

or critical examination of these diameter values used in this area

[2,3,7]. This requirement is particularly important for the accu-

rate estimation of the diameter of the unperturbed doorway D,

of the thermodynamic parameters and for the expansion of thetransition-site doorway, and of the inter-site spacing . For each ofthese depends on the good linear definition of the three test plots

(see Figs. 35).

In quoting these values, it is found clearer to use picometers

(1pm=0.01 A = 0.001 nm) since these shows up better the effect of

small changes on the permeation behavior.

There are three main sets of values that have been applied in

the area of permeation through membranes, as considered below

in Sections B.3 and B.4. However, it is necessary firstly to consider

the effect of rotation on the effective molecular diameter.

B.2. Rotation of molecules in relation to molecular diameters

In considering the specific dimensions of molecules, and in

particular how they pack together in the solid-state at low tem-

peratures, it is evidently necessary to consider the detailed shapes.

Commonly, this can be summarized as spherical, rod-shaped, pla-

nar, etc. Frequently, in considering the migration properties of

molecules in solids, these shapes are referred to in order to clar-

ify or interpret the behavior; for example, a rod-shaped molecule

is commonly presumed to present its narrowest dimension to the

barrier to ease its passage. However, this ignores the significance of

rotation in the behavior of the molecule. At a high enough tem-

perature, a molecule will have an amount kBT/2 in each axis of

rotation; in macroscopic terms, this energy amounts at 25 C to

about 1 kJ mol1. Although rotation is suppressed for all molecules

at sufficiently low temperatures, as the temperature is raised quan-

tumeffects lead to theonsetof rotation. This is shown by thevalues

of the heat capacity at constant pressure Cp, which in the general

case rises with temperature as firstly translation, then rotation andthen vibration modes come into play [64]; the literature data for Cpshow that this rotational freedom is essentially complete at 298 K

for all the molecules considered here [65]. The angular velocity

will be given by the standard mechanics formula relating it to the

moment of inertia Iof the molecule about that axis

kBT

2=

I2

2. (B.1)

For molecules of the type used here as permeants

(Tables 1 and 2), this gives angular velocities in the rangeof 1012 to 1013 s1 [64]. Thus if a specific orientation is envisaged

for the permeant molecule in the doorway, this very rapid rotation

must be stopped. At the same time, the orientation involves an

entropy penalty, related to the degree of restriction envisaged,


13/15


since there must still some finite degree of motion (oscillation)

in the captured state. For example, an allowed oscillation of10

aboutthe long axis in alldirections reduces thedirectional freedom

by a factor of about 100, that is, only about 1% of the molecules

would be oriented in this way ready for insertion; from the

thermodynamic viewpoint, this would correspond to an entropy

penaltyS ofabout40J K mol1, leading at298K toan free energypenalty (T

S) of about 12kJ mol1. More stringent limitations on

the freedom on the molecule lead to a correspondingly smaller

fraction of molecules with the correct orientation, and paralleled

by correspondingly greater thermodynamic penalties; the latter

would also be greater still at higher temperatures. These consider-

ations strongly suggest that, in the first instance, molecules of all

kind should be considered as rotating essentially freely, that is, to

be spherically symmetrical, both in the gaseous state and when

inserted in the transition-site doorway, and that it is the diameter

of this sphere of rotation that determines the effective molecular

diameter.

B.3. Molecular diameter values of Breck

One widely used approach in permeation work is to use the set

of values listed by Breck [52] in connection with zeolite molecular

sieves, whichhas been also adopted forusein thefieldof membrane

processes in general [1], and more specifically in the publications

drawn upon in this paper [3840]; indeed, these values seem to be

viewed as definitive. The original values [52] were drawn from a

diversity of sources, including LennardJones parameters (Section

B.4) in a rather inconsistent fashion; being put in Angstrom (A)

units, it is not clear what precision is being claimed: 3.9 A may

imply either 1 or 5 on the derived value of 390pm.

In addition, in the cases of the isoelectronic pair, CO 2 and N2O,

the values have been taken to be those of the smallest diame-

ter, leading to quoted values of 0.33 A, that is 330 pm, giving the

anomaly that they have a lowervaluethan that for patently smaller

molecules such as CO, N2 and O2; this anomalous value has appar-

ently been accepted by the membrane community because it fitsin with this being a fast gas in permeation, although the present

treatment has shown that this is an energy effect (low EP) rather

than a molecular diameter effect. Further evidence on this point

is given by the electric polarisability e, which is another mea-sure of molecular volume; the value for CO2 is some 50% greater

than those for these same diatomic molecules (Table 1). Indeed, as

already noted (Section B.2), all these molecules must be taken to

be rotating freely in the gas state, and would tend to maintain this

rotationeven when held in theguestmatrix,so that thediameter to

be assigned to these triatomic molecules is the largest dimension.

Furthermore, it is not clear whether or not this adjustment for

CO2 and N2O has also been applied to other asymmetric molecules

(C2H4, C6H6, etc.).

A similar anomaly is evident for the pair: H2 (289pm) versusH2O (265pm) [52]. This anomaly is again highlighted by the fact

that the volume-related molar polarizability of H2O is nearlytwice that of H2 [48], whereas the converse situation should occur

if the quoted Breck values apply.

It is also not clear whether this set of parameters, which has

been selected specifically to apply to the zeolites, is necessarily

going to apply to such diverse other media as (for example) organic

polymers, silica, and microporous carbon.

B.4. LennardJones diameters

An alternative approach is to use the parameters derived from

the LennardJones(LJ) 612 potential for the intermolecularpoten-

tial energy u(r) of a pair of identical molecules at internuclear

separation r

u(r) = 4 LJ

LJr

12

LJr

6(B.2)

where LJ is the depth of the potential well and LJ is the LJmolecular diameter [2]. More exactly, this equation only applies to

nonpolar molecules (especially the noble gases); with polar gases

there would be a third term for the polar (electrostatic) contribu-tions, leading to the so-called Stockmayer equation. Table 1 shows

the values of the LJ diameters from the literature [5356]. The

values originate from one of three sources of experimental data:

gas viscosity, thermal conductivity, or equation of state (second

virial coefficient). The problem with all these methods is that they

may only give a combination of the two parameters LJ and LJ.This shows itself, when considering LJ parameters from different

sources, in a high value ofLJ being associated with a low valueofLJ, and vice versa. The data given in Table 1 have been evalu-ated in the light of these considerations, eliminating some values

thatwere evidentlyanomalous. Evenwith this procedure, as shown

in Table 1 the uncertainty in these values is still about 6 pm. In

addition, any polarity of the molecules makes the application of

the simple form of Eq. (B.2) less certain. In the case of CO2 in par-ticular, there are indications that the LJ parameters are markedly

temperature dependent [53]. There is also the general limitation

that these parameters require the availability of experimental data

from one of the three rather troublesome experimental methods

already specified.

B.5. Molecular diameters from critical constants via the van der

Waals equation

A third approach is to use parameters derived from the co-

volume b ofthe van derWaals equation,as in theprevious paper [7].

The rationale here is that the parameter b is presumed to be equal

to four times the volume of the molecules, leading to the relation:

=

3b

2NA

1/3. (B.3)

With values of b available from the literature [66], this enables to be determined for these same molecules. However, appearances

are deceptive since it turns out that the listed values ofb [66] are

not from a van der Waals fit, but are derived from critical constants

using the relation

b =RTc8pc

(B.4)

where Tc is the critical temperature and pc is the critical pressure;

it is somewhat odd that it is not the critical molar volume Vc that

is used, but presumably the logic is that the value of Vc is in many

cases either lacking or not well defined in the literature [51]. How-ever, this has the advantage that the critical constants Tc andpc are

known for a very wide range of substances [51], while unlike the LJ

values (Section B.3) there are no ambiguities from any polarity of

the molecules. This leads finally to:

CC =

3RTc

16NApc

1/3(B.5)

wherethe subscriptonCC is usedto indicatespecifically the sourceof the values. Judging from the data for the present substances

(Table 1) supplemented by that for the other noble gases, theseCCvalues are fairly consistently 15% less the correspondingLJ values.

Some ambiguities arise here with the light gases, notably He

and H2 (and their isotopes) and to a much lesser extent Ne, where

quantal (quantum mechanical) effects come into play. In the case


14/15


of the noble gases (He, Ne, Ar, Kr, Xe), this is evidently cause of

the anomaly, that Ne has a smaller derived value ofCC than He,although the values thereafter increase smoothly for NeXe. How-

ever, these directly calculated values were found to be satisfactory

in the previous application of the TSM model to organic polymers

[7], while quantal effects are themselves present in the perme-

ation of gases through amorphous media,for example with organic

polymers [10]. In the present case theCC

values for He and H2given in Table 1 have been adjusted by using the linear relation

that is observed between CC and 1/3, where is the volume-

dimensioned molar polarizability as also listed in Table 1.

In the present work with microporous silica, it is these CC val-

ues, as listed in the last column of Table 1, that have been used as

the molecular diameter G in the test plots ofFigs. 3 and 4, and inthe calculation of the derived parameters for the doorways listed

in Table 3.

Nomenclature

b van der Waals co-volume (m3 mol1)

D diffusion coefficient (m2 s1)D location of a doorway (transition-site)

D0 diffusion prefactor (m2 s1)

Eex energy to expand the TS doorway by the inserted G

molecule (J mol1)

ED activation energy for diffusion (J mol1)

EP activation energy for permeation (J mol1)

EMG energy of noncovalent interaction between the

inserted G molecule and the TS doorway matrix

(J mol1)

EMM energy of noncovalent interactions across the unoc-

cupied TS doorway (J mol1)

EP energy anomaly Eq. (24) (kJmol1)

G bulk shear modulus (Pa)

G permeant gas moleculeh Planck constant (6.6261034Js)

HS isosteric sorption enthalpy change (kJ mol1)

kB Boltzmann constant (1.3811023J K1)

k transition-state rate constant (s1)

L membrane thickness (m)

nd no data/not determined

NA Avogadro constant (6.0221023 moleculesmol1)

p pressure (Pa)

pc critical pressure (Pa)

pe electric dipole moment (C m)

P permeability coefficient (molm m2 s1 Pa1)

P0 permeation prefactor (molm m2 s1 Pa1)

PE poly(ethylene)

PEMA poly(ethyl methacrylate)p pressure difference across the membrane (Pa)Q permeance (mol m2 s1 Pa1)

Qe electric quadrupole moment (C m2)

R gas constant (8.3145 J K1 mol1)

S sorption (solubility) coefficient (mol m3 Pa1)

SDM sorption-diffusion model for permeation

S(D) entropy of 1mol of unoccupied TS doorways

(J K1 mol1)

S(GD) entropy of 1 mol of TS doorwaysoccupied bythe gas

G (J K1 mol1)

S0 sorption prefactor (mol m3 Pa1)

S0(G) absolute molar entropy of gas G (reference pressure

1Pa) (JK1 mol1 )

S molar entropy change for insertion of G into a TS

doorway (J K1 mol1)

T absolute temperature (K)

Tc critical temperature (K)

TS transition-site (doorway)

TSM transition-site model for permeation

U molar internal energy change for insertion of G intoa TS doorway (J mol1)

Vc critical molar volume (m3 mol1)

x general membrane depth in the direction of perme-

ation (m)

Y entropic coefficient Eq. (18) (molmm2 s1)

Y0 entropic prefactor Eq. (19) (molmm2 s1)

(log Y0) entropy anomaly (Eq. (25)) times-or-divided-by

Greek letters

e molar electric polarisability (m3 mol1)LJ LennardJones energy parameter (J) force constant for expansion of a TS doorway (N)

(average) spacing between neighboring TS door-ways (m)

entropy-change coefficient for expansion of a TSdoorway (N K1)

molecular (kinetic, collision) diameter (m)D diameter of an unoccupied TS doorway (m)G molecular diameter of the permeant gas G (m)CC value of estimated from critical constants

(Appendix B) (m)

LJ LennardJones molecular diameter (Appendix B)(m)

References

[1] J.D. Henry, et al., Gas-separation membranes, in: R.H. Perry, D.W. Green (Eds.),Perrys Chemical Engineers Handbook, McGraw-Hill, New York, 1997, pp. 22-6122-67.

[2] Yu. Yampolski, B.D. Freeman, I. Pinnau (Eds.), Materials Science of Membranesfor Gas and Vapor Separation, Wiley, New York, 2006.

[3] R.H. Doremus, Diffusion of Reactive Molecules in Solids and Melts, Wiley, NewYork, 2002.

[4] C.P. Flynn, Point Defects and Diffusion, Clarendon Press, Oxford, 1972.[5] J.E. Shelby, Handbook of Gas Diffusion in Solids and Melts, ASM International,

Materials Park, OH, 1996.[6] P. Shewmon, Diffusionin Solids,The Minerals, Metals& MaterialsSociety(TMS),

Warrendale, PA, 1989.[7] P. Molyneux, Transition-site model for the permeation of gases and vapors

through compact films of polymers, J. Appl. Polym. Sci. 79 (2001) 9811024.[8] R.M. Barrer, Activated diffusion in membranes, Trans. Faraday Soc. 35 (1939)

644656.[9] R.M. Barrer, Porous crystal membranes, J. Chem. Soc., Faraday Trans. 86 (1990)

11231130.[10] H.L. Frisch, C.E. Rogers, Quantum isotope effect in permeation, J. Chem. Phys.

40 (1964) 22932298.[11] A. Singh, W.J. Koros, Significance of entropic selectivity for advanced gas sepa-

ration membranes, Ind. Eng. Chem. Res. 35 (1996) 12311234.[12] A.J. Burggraaf, Single gas permeation of thin zeolite (MFI) membranes: theory

and analysis of experimental observations, J. Membr. Sci. 155 (1999) 4565.[13] J. Gilron, A. Soffer, Knudsen diffusion in microporous carbon membranes with

molecular sieving character, J. Membr. Sci. 209 (2002) 339352.[14] P. Bahukudumbi, D.M. Ford, Molecular modeling study of the permeability-

selectivitytrade-off inpolymeric andmicroporousmembranes, Ind.Eng.Chem.Res. 45 (2006) 56405648.

[15] V. Stannett, J.L.Williams,The permeabilityof poly(ethyl methacrylate) to gasesand water vapor, J. Polym. Sci. C3 (1965) 4559.

[16] B. Wilks,M.E. Rezac, Properties of rubbery polymersfor the recoveryof hydro-gen sulfide from gasification gases, J. Appl. Polym. Sci. 85 (2002) 24362444.

[17] (a) R. Kirchheim, Partial molar volume of small molecules in glassy polymers,J. Polym. Sci. Part B: Polym. Phys. 31 (1993) 13731382;(b)R. Kirchheim, Solubility of gases in vitreoussilicadescribed bya distribution

of dissolution energies, J. Am. Ceram. Soc. 84 (2001) 26992701.


15/15

56 P. Molyneux / Journa

Documents

Molyneux JMS 42