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Transition-Site Model for the Permeation of Gases andVapors through Compact Films of Polymers

PHILIP MOLYNEUX

Macrophile Associates, 53 Crestway, Roehampton, London SW15 5DB, United Kingdom

Received 27 October 1999; accepted 2 March 2000

ABSTRACT: The theory of the permeation of effectively spherical molecules of gases andvapors (permeants) through films of compact amorphous solids such as polymers isdeveloped using the activated-jump model (AJM). The conventional view is that per-meation has to be analyzed as the resultant of sorption (solution) and diffusion effects.By contrast, in the present article, it is proposed that permeation may be viewed as asimple or fundamental process. This is suggested by a number of experimental obser-vations: (a) the permeation correlations of Stannett and Szwarc (19551956); (b) thePermachor concept of Salame (19611973); (c) the ideal permeation behavior ofwater vapor through moderately polar polymers; (d) the absence of any effect ofoxidation on the water-vapor permeability of polyethylene (PE); and (e) the isokineticcorrelations between the Arrhenius parameters for permeation. The conventional AJMfor diffusion is analyzed using the principle of microscopic reversibility, which shows

that the average jump is characterized by a transition site L at its midpoint, analo-gous to the transition state in chemical reactions. For amorphous solids, these transi-tion sites would be structural features, distributed at random and with their axespointing at random. This leads to the present transition-site model (TSM) of perme-ation, where, at the steady state, a certain fraction of these sites will be transientlyoccupied by molecules of the permeant in equilibrium with the free molecules at thatlevel. The concentration of these free molecules corresponds to the thermodynamicactivity at that point, that is, for gases and vapors, the partial pressure. The rate-determining step of the permeation process is then taken to be the release of thepermeant molecule from the transition site according to classical transition-state the-ory. Using an idealized cubic-lattice model for the distribution of the transition sites,this is shown to lead to the observed proportionalities of the permeation rate to the areaof the film, the pressure difference across it, and the reciprocal of the film thickness. Italso accords with the observed Arrhenius-type dependence of the permeability coeffi-

cient on temperature, where the Arrhenius parameters relate to the thermodynamicparameters for the transfer of the permeant molecule from the gas phase and itsinsertion in the transition site. The Arrhenius parameters from the literature (Polymer

Handbook) for 16 homopolymersNR, PA 11, PC, PDMB, PDMS, three PEs (HDPE,LDPE, and hydrogenated polybutadiene), PETFE, PEMA, PET, PP, PTFE, PVAC,PVBZ, and PVCwith 16 simple permeantsH2, He, CH4, Ne, N2, CO, O2, HCl, Ar,CO2, SO2, Cl2, Kr, SiF4, Xe, and SF6 as well as H2O vaporare used as the dataset.These Arrhenius parameters are first discussed in relation to isokinetic behavior. Theyare then correlated according to the TSM theory with the van der Waals moleculardiameter of the permeant G, and its absolute entropy S

0. With certain exceptions,linear correlations are obtained with the 10 smaller-molecule permeants (He to CH4)

Correspondence to: P. Molyneux.Journal of Applied Polymer Science, Vol. 79, 9811024 (2001)

2000 John Wiley & Sons, Inc.

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that show that they use the same set of transition sites, below and above the glasstransition temperature, with each polymer; the permeant molecules evidently behavehere as hard spheres, regardless of their other chemical characteristics. This enablesestimates to be made of the four characteristic parameters for the polymer: the intersitespacing (equivalent to the lattice parameter of the idealized model and to the jumplength of the AJM); the size of the transition-site aperture, L; the force constant associated with expansion of the aperture by the permeant molecule; and the entropyincrement also associated with this expansion. For most of the systems, the site-spacing is of the order of 10 nm, and the aperture L is about 200 pm. The theoryprovides a molecular basis for the interpretation and design of the permeation charac-teristics of polymers. 2000 John Wiley & Sons, Inc. J Appl Polym Sci 79: 9811024, 2001

Key words: activated process; jump length; permeability coefficient; thermodynam-ics of permeation; transition-site model

INTRODUCTION

General Features of Permeation Through Polymers

The ability of solids such as polymers to transmitsmall-molecule substances has many practicalapplications and consequences, as well as being ofgreat theoretical interest. This transmission in-

volves the three interlinked processes of sorption(solution), diffusion, and permeation. In the caseof polymers, permeation is important in a wholehost of technical processes: dialysis, osmosis, re-

verse osmosis, pervaporation, ultrafiltration, and

other membrane-separation processes; the dyeingof fibers and textiles; the behavior of barrier ma-terials (balloons, geomembranes, gloves, packag-ing, tires, and so forth); the design of controlleddrug-release systems; and the migration of plas-ticizers and other additives in polymers.18

In the present context, permeation refers to theprocess in which the permeant gas (or vapor) Gpasses through a layer (membrane, film, sheet,etc.) of the polymer (the permeate). This transmis-sion from the bulk of the medium containing thepermeant on one side of the layer to the bulk ofthe medium on the other side of the layer takesplace under the influence of a difference in ther-modynamic activity; this would be a pressure dif-ference in the case of a gas or vapor or a concen-tration difference in the case of a solute in solu-tion. Likewise, for volatile solids (camphor,disperse dyes, iodine, solid plasticizers, etc.), thedriving force would be a difference in the vaporpressure of the solid.

With solids that are porous, the permeationinvolves the mass transport of the permeant.However, in the present article, we are concernedspecifically with the behavior of solids that are

compact, that is, that lack any well-defined pores,whether macroscopic or microscopic. The perme-

ation must then involve the transport of individ-ual molecules through the matrix of the solid. Ofcourse, certain compact materials can acquire aporous structure by swelling when brought intocontact with a solution of the permeantthis ap-plies particularly to regenerated cellulose and hy-drogels when brought into contact with the aque-ous system involved.

Many polymers and related materials aresemicrystalline, in that they contain both amor-phous and polycrystalline regions, with the per-meation taking place primarily through the amor-

phous region because of its liquidlike nature.Although in the present article we are concernedprimarily with amorphous solids, the picture maybe extended to the permeation behavior of com-posite materials, although there will be uncer-tainties of interpretation arising from ambigu-ities in differentiating between the amorphousand the crystalline regions in these materials.

The present treatment will also focus on per-meants whose molecules are either inherentlyspherical (as with the monatomic noble gases) orwhich become effectively so by rotation (as withdiatomic molecules and the simpler polyatomic

molecules). For chain-molecule polyatomic per-meants, reptation effects would come into play;the present treatment may cast some light onsuch reptation processes by providing a model forthe behavior of the simpler spherical molecules.

Analysis of the Permeation Process

The process of permeation through a layer of acompact amorphous solid comprises a sequence ofthree stages:

(i) Sorption (dissolution) of the permeant atthe surface layer of the solid in contact

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with the phase of higher permeant activ-ity;

(ii) Diffusion of the permeant moleculesthrough the bulk of the solid to the otherside of the layer under the influence of theconcentration gradient across it;

(iii) Desorption (evaporation) of the permeantinto the phase of lower permeant activity.

Some DefinitionsIdeal and Nonideal Behavior

We may define ideal permeation as the simplestform observed experimentally, where the rate ofpermeation Q (the amount passing through thelayer of sample in unit time) is given by

Q APp/b (1)

where A is the exposed area; p, the pressuredifference; b, the thickness of the film; and P, the

permeability coefficient. The negative sign in eq.(1) takes account of the flow being from high tolow permeant activity, that is, in the oppositedirection to the difference p. The proportionali-ties between Q and the area A, and the reciprocalthickness 1/b, are almost invariably observed. De-

viations are more often seen with the proportion-ality to p; such cases may be referred to asnonideal permeation and may be associated withthe plasticization of the polymer by the sorbedpermeant. For such systems, the value ofP in thelimit of very low pressures may be termed theideal permeability coefficient, denoted by PI:

PI lim Pp 30 (2)

The units used for the permeability coefficient Pare discussed in Appendix A.

Likewise, in ideal diffusion, the rate of migra-tion within the material is given by

Q ADc/b (3)

where c is the concentration difference of thepermeant across the bulk of the solid, and D, thediffusion coefficient. This is, of course, a state-ment of Ficks First Law of Diffusion. In this case,nonideal diffusion relates to systems where D isdependent on the value of c. For such systems,the value ofD in the limit of very low concentra-

tion may be termed the ideal diffusion coefficient,denoted by DI:

DI lim Dc 30 (4)

Likewise, ideal sorption may be defined by

c Bp (5)

where B is the sorption coefficient. This is also astatement of Henrys Law, so that B is one form ofthe Henrys Law coefficient for the system. Non-ideal sorption is then characterized by the valueof B being dependent on c (and p). The limiting

value of B for low values of c (and p) may betermed the ideal sorption coefficient and is de-noted by BI:

BI lim Bc 30 (6)

Combining eqs. (1), (3), and (5) leads, for a triplyideal system, to the three coefficients being re-lated by

P BD (7)

Likewise, for all systems, in the limit of low p andc, the relationship is that between the three idealcoefficients as specified earlier, that is:

PI BIDI (8)

Restriction to Ideal Behavior

In the discussion that follows, unless otherwisespecified, the permeation, diffusion, and sorptioncoefficients refer to the ideal values. The subscriptI has accordingly been omitted from the symbols.

Temperature Dependence of the Coefficients

In considering the temperature dependence of

these three coefficients, in the simplest cases,they all follow the same exponential-reciprocalform. For permeation and diffusion, they followthe Arrhenius forms

P PAexpEP/RT (9)

and

D DAexpED/RT (10)

where R is the gas constant; T, the absolute (ther-

modynamic) temperature; and EP and ED, theactivation energies for the respective processes.

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Likewise, the sorption coefficient follows the anal-ogous vant Hoff form for equilibria:

B BVexpHS/RT (11)

where HS is the standard enthalpy change forthe sorption process.

It is more customary, in these eqs. (9)(11), touse a subscript zero with the preexponential fac-tors, but in the present case, the subscript A(Arrhenius) or V (vant Hoff) is used to differen-tiate these from the ideal values in eqs. (2), (4),and (6), which are commonly likewise character-ized by a subscript zero. This also reminds us that

these preexponential factors are the values in thehypothetical limit of very high (infinite) temper-ature, rather than of very low (zero) absolute tem-perature as a subscript zero would suggest.

It should be noted that eq. (11) has a thermo-dynamic basis, in that, in principle, the value ofthe enthalpy change could be determined directlyby calorimetry and should agree with that fromthe vant Hoff eq. (11). By contrast, the activationenergies EP and ED are derived solely from eqs.(9) and (10).

Comparing the temperature-independent andtemperature-dependent parts of eqs. (9)(11) inthe light of eq. (8) gives

PA DABV (12)

EP ED HS (13)

Effects from the Glass Transition andMelting Points

One complicating effect here is that frequentlythe Arrhenius plot shows a break at the glasstransition temperature, T

g

, indicating that theglass state (T Tg) and the rubber state (T Tg) have different values of the Arrhenius pa-rameters and, hence, presumably show differentbehavior at the molecular level. This is discussedlater in the article. This situation means that,strictly speaking, the parameters should be la-beled with the state that is involved.

Likewise, breaks may be expected if the melt-ing temperature is within the experimental tem-perature range, although this is less commonlyencountered in practice.

Even for a purely amorphous polymer at tem-

peratures away from the Tg, deviations may beexpected from the simple equations above if the

temperature range is wide enougheven in theabsence of glass transition or melting effects, a

polymer at 100C is a different material from thatat 0C; however, such effects are not generallyencountered because the temperature rangesused are much narrower than this.

PERMEATION VIEWED AS A PRIMARYPROCESS

The foregoing analysis of permeation into thethree successive steps (sorption, diffusion, anddesorption) is long-established, while the deriva-tion of eq. (7) and the establishment of the Arrhe-nius behavior is only somewhat less mature.917

The entrenched character of this overall picturehas led to the firmly held assumption that, ac-cording to eq. (7), permeation is a secondary pro-cessto be interpreted as a composite of sorptionand diffusion as two primary processes. Evi-dently, even from a purely practical viewpoint,this is a hindrance in (say) the design of a barriermaterial, since it means that the sorption anddiffusion characteristics must be separately de-signed in, but, at the same time, balanced in such

a way that the correct permeation characteristicsare obtained.

However, there are a number of instances inthe published literature where the permeationbehavior is simpler than might be expected withthe above picture and in such a way as to suggestthat permeation may be a primary or fundamen-tal process. We cite five examples here:

StannettSzwarc Correlation

More than 40 years ago, Stannett, Szwarc, andtheir coworkers pointed out that, although thepermeation coefficient for a particular gas in arange of polymers may vary over many orders ofmagnitude, the ratio of the permeability coeffi-cients for a pair of gases can be almost con-stant.18,19 They used the permeant gases N2, O2,CO2, and H2S (at 30C), with a dataset of 22polymers ranging from low permeability onessuch as Mylar (PET) to highly permeable onessuch as the elastomers natural rubber (NR) andpolyisobutene (PIB) (Table I). Although the P val-ues ranged over three orders of magnitude, nev-

ertheless, the mean value of the ratio P(O2)/P(N2)was 3.8(8), while for P(CO2)/P(N2), it was 24(7),

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and for P(H2S)/P(N2), it was 22(7) (Fig. 1).* Thisenabled them to assign multiplicative numericalfactors to the individual gases and polymers, us-ing an equation of the form

P FMFGFGM (14)

where FM

is the factor for the polymer; FG

, thatfor the permeant gas; and FGM, that accountingfor any specific interaction effects between thetwo (in the original articles, these three factorswere designated F, G, and ). The derived valuesofFM, using FG 1 for N2 and assuming that FGM 1, are listed in Table I.

This correlation was noted in a chemical engi-neering text4 and in a polymer chemistry text.8

However, the lack of a theoretically justifiableexplanation, coupled with the belief that perme-

ation must be composite, has evidently inhibitedits use even as an empirical correlation.

The Permachor Concept of Salame et al.

A parallel approach to correlations for the perme-ability coefficient was that of Salame et al. whointroduced the concept of Permachor S as an

additive property related to the molecular struc-ture of the polymer and/or the permeant.2023 Inthe early publications,20,21 attention was focusedon the permeation of liquids through a single typeof polyethylene (PE). The logarithm of the perme-ability coefficient was related to the sum of thePermachor values of individual groups (or atoms)in the permeant molecule

log P KS 0.22SG (15)

with

SG Sgroups (16)

* See the Nomenclature section at the end of this article forthe method of representing mean and standard deviation val-ues.

Table I List of Polymers for the StannettSzwarc Correlation

Polymer

Code Designation Details17

log FM

A Buna S Poly(Bu-co-S) 4.678B Butyl rubber Polyisobutene 3.369C Cellulose acetate (plasticized) 3.574D Cellulose acetate P-912 3.322E Ethyl cellulose (plasticized) 4.799F Hycar OR 15 Poly(An-co-Bu): 39:61 3.246G Hycar OR 25 Poly(An-co-Bu): 32:68 3.656H Kel F (Trithene) Polychlorotrifluoroethylene 2.989I Methyl rubber Polydimethylbutadiene 3.556

J Mylar Poly(ethylene terephthalate) 1.574K Natural rubber cis-Poly(isoprene) (NR) 4.782L Neoprene Polychloroprene 3.947M Nylon 6 Polyhexanoamide 1.875N Perbunan Poly(An-co-Bu): 27:73 3.900O Pliofilm FM NR hydrochloride 3.021P Pliofilm NO NR hydrochloride 1.778Q Pliofilm P4 NR hydrochloride 3.667R Polybutadiene 4.685S Polyethylene 4.154T Poly(vinyl butyral) 3.273U Saran Poly(vinylidene chloride) 0.848V Vulcaprene Polyurethane 3.565

Code: letter codes for the polymers in Figure 1. Designation: names used in the original articles.18,19 Details: from the literature(An acrylonitrile, Bu butadiene, S styrene).6 FM: factor for the polymer in the StannettSzwarc correlation at 30C (303 K),equal by definition to the permeability coefficient for N2.


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where the Salame constant KS is a parameterthat depended only on the temperature. This wasextended to other types of PE, and to polypro-pylene (PP), with different values for the param-eter KS.

From the present viewpoint, the permeationcoefficients for these liquids would need to becorrected appropriately if they are to be comparedwith the data for gases and vapors, by introducingthe vapor pressure of the liquid. At the same time,these might not be expected to be the ideal values

PI because of possible plasticization of the poly-mers (to different extents) by the different liquids.

In later work on permeation of gases (O2, N2,CO2) through a wide range of polymers, the Per-machor concept was applied to the polymers, us-ing contributions from the structural groups ofthe polymer chain.22 It is interesting that theseparticular data were found to follow the Stan-nettSzwarc-type behavior (see above), in that forthe 84 polymers listed and a temperature of 25Cthe value of the ratio P(O2)/P(N2) was 4.0(23) andthat ofP(CO2)/P(O2) was 3.6(20). This appears to

be have been noted independently of the Stan-nettSzwarc work.18,19

The same concept was subsequently applied tothe permeation of gases and water vapor though

the acrylonitrile copolymer barrier resins.

23

These correlations by Salame were noted in achemical engineering text,4 but not apparently inthe main polymer literature; presumably, theyhave been considered as purely empirical corre-lations, lacking any fundamental significance be-cause of the presumed composite nature of thepermeability coefficient value.

Permeation of Water Vapor Through Polymers

In general, a polymer/permeant system followsone of two forms of behavior, that is: (a) It is triply

ideal in that all three eqs. (1), (3), and (5) areobeyed or (b) it is triply nonideal, in that all threecoefficients show dependence on the pressure andconcentration levels of the permeant.

In the case of water as the permeant, behavior(a) applies to nonpolar polymers such as (pure)PE, and behavior (b), to highly polar polymerssuch as poly(vinyl alcohol), nylon, and regener-ated cellulose.24,25

However, as the present author noted,25 aninteresting intermediate situation applies in thecase of water vapor with moderately polar poly-

mers, in that the permeation behavior is idealalthough both sorption and diffusion are non-ideal. This is applies to silicone rubbers,26,27 poly-(alkyl methacrylate)s,28,29 natural and syntheticrubbers,30 (oxidized) PE,31,32 poly(vinyl bu-tyral),33 poly(vinylidene chloride-co-acryloni-trile),33 natural rubber hydrochloride,34,35 cellu-lose acetate,33 and ethyl cellulose.3537 However,the behavior with polyoxymethylene and poly(vi-nyl chloride) is less clear-cut.38

In this context, water might be thought not tobe a suitable example, since its behavior is wellknown to be anomalous. However, this appliesprimarily to water in the liquid state, whereas inpermeation through compact polymers, we areconcerned with isolated water molecules in thepolymer or, at most, clusters of small size.24

Again, this simple permeation behavior hasevidently not been considered to be significant,but, rather, presumed to be the result of a coinci-dental cancellation of the deviations from idealityof the sorption and diffusion processes.

Permeation of Water Vapor Through Oxidized PE

With the above-cited early studies by Rouse onPE, the water sorption was high in comparison

Figure 1 StannettSzwarc correlation: loglog plotof the permeability coefficient P (in standard unitsAppendix A) for the permeant G versus that for N2 at30C (303 K); see Table I for the polymer codes. Per-meant symbols (with value for the ordinate scalingfactor N): (E) N2 (N 17reference line); () O2 (N 17); () CO2 (N 17); () H2S (N 18). Data fromrefs. 18 and 19.

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with that from other studies, evidently arisingfrom the oxidation of the polymer in processing.32

Subsequently, McCall and coworkers31 studiedthe permeation of water vapor (from the pureliquid) through three specimens of oxidized PE.These had been oxidized to controlled extents (bymilling in the air in the absence of any antioxi-dant) to give combined oxygen contents up toabout 3%. They found that, although this natu-rally enhanced the sorption coefficient (up to 50-fold) compared with the starting polymer, the dif-fusion coefficients suffered compensating reduc-tions, so that the permeability coefficient wasessentially unchanged by this treatment (Fig. 2).Here, again, this simple permeation behaviorhas been considered as accidental, that is, theby-product of coincidental canceling of sorptionand diffusion effects.31

Correlations Between the Arrhenius Parametersfor Permeation

When the Arrhenius parameters for permeationfor ranges of polymers and permeants are exam-

ined, it is frequently found that there is an essen-tially linear correlation between log PA and EP for

various permeants with the same polymer; this isdiscussed with specific examples later in the ar-ticle. This type of isokinetic relationship againsuggests that permeation may be viewed as aprimary process, although, similarly, it may beascribed as the result of such relationships apply-ing to both the component processes of sorptionand diffusion.

THE ACTIVATED-JUMP MODEL OFDIFFUSION

Before developing the present transition-sitemodel (TSM), which is the main theme of thisarticle, it is necessary to outline the activated-

jump model (AJM) which has been used widelyfor modeling diffusion in solids including poly-mers.3948 This outline serves both to point outthe features of the model widely accepted atpresent and to show how a closer examination ofthese features leads to the concept of the transi-tion site.

In the AJM, the sorbed molecules of the per-

meant are assumed (implicitly or explicitly) tooccupy definite equilibrium positions in the ma-trix of the solid. The molecule is then assumed tomigrate between two such adjacent positions by athermally activated jump; a lattice picture for thisis shown in the upper part of Figure 3. The mainevidence for this jumping is the Arrhenius form ofeq. (10) for the temperature dependence of thediffusion coefficient. The observed diffusion theninvolves a sequence of such jumps in a biasedrandom flight, with the net movement takingplace down the concentration gradient; this arisessimply because, for two such adjacent sorptionlocations, the greater number of sorbed permeantmolecules at the upstream location leads to agreater number of jumps in the downstreamdirection.

The activation energy ED in eq. (10) is thenidentified with the extra energy (above its normalthermal energy) that the molecule must acquirefor it to be able to perform its jump. With thesimplest picture, this energy is accumulated byrandom fluctuations in the thermal energy at thelocation of the sorbed molecule, with a probabilitygiven by the Boltzmann factor: exp(ED/RT). A

more complex picture is the zone theory, wherethe energy is taken to be accumulated from many

Figure 2 Reciprocal diffusion coefficient,D1, versusliquid solubility S at 298 K (25C) for H2O with threesamples (Specimens B, C, and D) of PEs after oxidation.Key: (E), Specimen B; (), specimen C; (), specimenD; (and running numbers), combined oxygen contentin percent. The straight line through the origin corre-sponds to permeability coefficient P 5.4(11) 1012

standard units (Appendix A) for H2O/PE. Data ofMcCall et al.31


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degrees of freedom of the thermal energy in thesurrounding matrix.13,14

To analyze this further, we may apply the Prin-ciple of Microscopic Reversibility (PMR); in thechemical context, this is referred to as the Prin-ciple of Detailed Balancing. The PMR requiresthat, for any molecular process, the forward andreverse forms of that process have the same prob-ability.49 In the present context, this requiresthat if we reverse the sequence of events in the

jump process this gives an equally probable formof jump, so that the (average) jump must be sym-metrical about its midpoint. In particular, thebehavior in the second part of the jump mustmirror that in the first part; the permeant mole-cule must arrive at its final location with thesame energy with which it started out, and thisenergy must be dissipated into the matrix of the

polymer in a fashion mirroring that by which itwas acquired at the start of the jump.

If we analyze the activation energy for diffu-sion, ED, then this is composed, in part, of theenergy to free the molecule from its sorption site,and if the molecule is then taken to be completelyfree at the start of the jump, then the energyrequired will be the enthalpy change for desorp-tion, HS. Thus, by eq. (13), the energy left willbe the permeation activation energy, EP. Thus,one feature of permeation makes its appearanceeven at this early stage of the analysis.

The fact that the molecule has to be activatedin this way shows that it needs to overcome aresistance to its motion in its jump, which slows itdown over the early part of its jump, while by thePMR it will be speeded up progressively over thelater part. To make the process symmetrical asrequired by the PMR, the early part will be thefirst half and the later part will be the second

half of the jump, with the velocity being a mini-mum at the midpoint of the jump. Now because of

Figure 3 Diagram for permeation through a polymer film of thickness b. Thepermeation takes place in the direction of the x-axis. The sample has a permeantpressure p0 on the front face (x 0) and a pressure pb on the back face (x b). Thecenter of the diagram represents a cubic lattice, with lattice parameter ( intersite-spacing diffusive jump length) . he closed volume is shown as heavily shadedareas, and the open volume, as the lighter-shaded remaining area. Open squares ()represent the preferred (lowest free energy) sites for sorption; filled circles (F) representmolecules of the permeant G. The upper part of the diagram illustrates the AJM fordiffusion, showing the process of the activated jump from one sorption site to the nextone, through the permeation barrier (transition site). The lower part of the diagramillustrates the present TSM for permeation, showing the release of a molecule of G fromthe transition site L.

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the exponential dependence of the Boltzmann fac-tor on the energy term, this energy must be the

absolute minimum that is required if the jump isto be successful, and the midpoint velocity mustlikewise be the absolute minimum.

The PMR does not seem to have been usedpreviously, at least explicitly, as a criterion inchoosing acceptable mechanisms for diffusion andpermeation. It should be noted, in particular, thatany (average) jump sequence that is proposedmust be symmetrical; any unsymmetrical pro-cessfor example, a permeant molecule jumpinginto a hole formed in its neighborhoodis notacceptable.

TSM FOR PERMEATION

Basis of the Model

In this model, we focus our attention on the be-havior of the midpoint of the activated jump al-ready discussed. In the transition-state theory ofchemical reactions, the reactants are required topass through a transition state before they canform the products.50,51 By analogy, this midpoint

is referred to here as the transition site, whoselocations are symbolized by L. Following theframework of the AJM, where the sorption sitesmust be taken to be fixed, the transition sitesmust likewise be taken to be at fixed positions inthe polymer matrix. The transition state in chem-ical reactions is the point of maximum (free) en-ergy on the path along the reaction coordinatefrom the reactants to the products, while in thisstate the (free) energy is locally a minimum in alldirections at right angles to the reaction coordi-nate. Likewise, in the transition site L, the per-meant molecule has maximum (free) energy inthe direction along the axis of the site (which isthe direction of the jump) and a minimum in alldirections at right angles to this axis.

Lattice Model

With an isotropic amorphous material such as apolymer, the transition sites L will be distributedat random in the matrix, with their axes likewisedirected at random. Such a picture is difficult todeal with mathematically. For tractability, there-fore, we will work with an idealized lattice model,

whose consequences should still apply to the realsituation.

We therefore use a simple cubic-lattice model,as shown in Figure 3. The lattice cube is viewed

as having a central cavity for the sorption of thepermeant, communicating with the adjacent cav-ities by way of the transition sites L on the facesof the cubes that represent restrictions or bottle-necks.25 The closed volume (that not accessibleto G) is that represented in Figure 3 by the dark-shaded circles, while the open volume (that ac-cessible to G) is the lighter shaded parts of thediagram. The transition sites that connect theselatter parts would be radially symmetrical abouttheir axes.

The jump length is then the lattice spacing,symbolized here by . In the later application of

the model, when we have to revert to a nonlatticerandom picture, the parameter becomes sim-ply the average intersite spacing, albeit stillequivalent to the jump length of the AJM.

For further simplicity, the lattice is taken to bealigned with the surface of the sheet (of area A) inthe yz-plane so that the permeation takes placealong the x-axis of the system; here, the x-valuewill run from zero at the front face to the thick-ness b at the back of the sheet. The transitionsites then divide into three types by the directionof the axis, that is, Lx, Ly, and Lz. By this means,

only the transition sites of type Lx will contributeto the permeation process that is taking placealong the x-axis.

The permeation will be taken to be ideal, sothat the value of the permeability coefficient isthe same throughout the layer. The permeantmolecules will be taken to be either spherical (i.e.,the noble gases) or to be effectively so at thesetemperatures by reason of their rotation. Thisshould therefore apply to diatomic molecules suchas N2, H2, and O2 and even to simple polyatomicmolecules such as CH4, CO2, and H2O.

Permeant Insertion Equilibrium at theTransition Sites

Considering Figure 3, for the permeant G at (par-tial) pressure p0 on the front of the sheet (x 0)and pb on the back (x b) the pressure differenceacross the sheet is

p pb p0 (17)

With a steady-state flux of permeation, there willbe a linear gradient of permeant pressure within

the sheet. This may be visualized (and could berealized) as the pressure that would be set up in a


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small (microscopic) cavity at that point; alterna-tively, it would be the pressure set up if the sheet

were sliced across in the yz-plane at this level, thetwo parts slightly separated, and the steady stateresumed. This gradient was constructed mathe-matically for the system (oxidized) PE/water byRouse.32 In more general terms, it is the gradientof thermodynamic activity of the permeant,whose significance in these processes was dis-cussed by Kuppers and Reid52 and by Reiss.53

Thus, at the level at general distance x fromthe starting plane, the partial pressure px is givenby

px p0 xp/b (18)

For a particular transition site L, the insertionprocess of its occupation by a molecule of per-meant G will be controlled by the equilibrium

Gx LxGLx (19)

where the depth is specified because the partialpressure of G will depend on this.

The equilibrium of eq. (19) will be governed byan equilibrium constant K (using the superscript

double dagger , which is the conventional sym-bol for the transition state) whose value will becharacteristic of the system. It will relate theequilibrium concentrations of the three species ateach particular level according to

K GLx/Gx Lx (20)

where [G(x)] is the partial pressure of the per-meant; [L(x)], the concentration of unoccupiedtransition sites L; and [GL(x)], the concentrationof occupied transition sites, all at the depth x.

Note that this assumes that the system can becharacterized by a unique value of K, whereasbecause of the amorphous character of the mate-rial it is likely that there is a distribution of

values. However, the distribution cannot be toobroad; otherwise, we would not get the observed

Arrhenius form for the temperature dependenceof P.

Evaluation of the Rate of Permeation

To calculate the rate of permeation for this model,as shown in Figure 3, we chose as the plane of

permeation that at depth d that passes throughthe center of the cubes. This has a plane of tran-

sition sites L (partially occupied by the permeant)at distance /2 on either side. This choice of the

plane of permeation ensures that the moleculesreleased from the transition sites on either sidereach it at the same time. At the correspondinglevels of these transition sites L, the partial pres-sure of the permeant will be given, following eq.(18), by

G p0 d /2p/b (21)

where the positive and negative signs, respec-tively, denote the downstream (d /2) andupstream (d /2) sides of the central plane.

Since with ideal permeation the degree of oc-cupancy of the transition sites L is (vanishingly)small, by putting these in terms of the number ofsites exposed over the area considered (A), theratio of the number of sites occupied, n, to thetotal number of sites, ntot, will be given by

n/ntot GL/L (22)

The total number exposed over this area will be

ntot A/2 (23)

since each transition site is at the center of asquare of edge (Fig. 3), so that combining eqs.(20)(23) gives

n A/2Kp0 d /2 p/b (24)

We can now apply the standard transition-statetheory50,51 to obtain the rates at which the per-meant molecules at these transition sites are re-leased toward the plane of permeation. Theserates will be obtained by multiplying the numberof transition-site-attached molecules by the ratecoefficient k of the transition-state theory. Thus,the rate of release from the upstream side to-ward the central plane, q

, will be positive (since

it is in the direction of increasing x) and given by

q kn (25)

while from that from the downstream side to-ward the central plane, q

, will be negative (since

it is in the direction of decreasing x) and given by

q kn (26)

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so that the net rate of molecules flowing acrossthe plane of permeation, q, will be the sum of

these two rates:q q q k

n n (27)

Combining eqs. (24)(27) gives the net molecularrate of permeation as

q AkKp/b (28)

Note that the central-plane depth d has disap-peared, confirming that the rate is independent ofthis depth.

Dividing eq. (28) throughout by the Avogadroconstant, NA, gives the macroscopically observedrate, Q, that is, the amount (moles) of permeantpassing across the layer in unit time, as

Q AkKp/bNA (29)

Comparing this expression for Q obtained fromthe TSM with the standard expression for idealpermeation, eq. (1), shows that this TSM expres-sion correctly incorporates the observed propor-tionalities of Q with the area, A, the pressuredifference, p, and the reciprocal of the thickness,b. The permeability coefficient is thus given by

P kK/NA (30)

Permeability Coefficient and Thermodynamics ofthe Insertion Process

The rate coefficient k in the above expressions isgiven from transition-state theory as

k kBT/h (31)

where kB is the Boltzmann constant and h is thePlanck constant.50,51 The equilibrium constant K

can be expressed in thermodynamic terms by

K expG/RT (32)

where G is the standard free-energy change forthe insertion process , that is, transferring 1 molof permeant from the gas phase at unit pressureto 1 mol of the transition sites. Expanding the freeenergy into its enthalpy (H) and entropy (S)components and combining eqs. (30)(32) gives

P kBT/hNAexpH/RTexpS/R (33)

The enthalpy component may be further ex-panded as

H U pV (34)

where U and V are the internal energy andvolume changes for the transfer process, respec-tively. Since the volume change is that for thecondensation of the 1 mol of gas onto the sites,then neglecting deviations from ideal gas behav-ior, and any volume effects at the transition site,this becomes

H U RT (35)

Substituting in eq. (33) and rearranging slightly,this gives the TSM relation for the permeabilitycoefficient:

P kBT/ehNAexpS/RexpU/RT 36

where e is the exponential number (derived fromthe factor RT in eq. (35)).

This TSM expression is seen to conform withthe experimentally observed Arrhenius tempera-ture dependence of the permeability coefficient,

eq. (9). Equating the preexponential parts and theexponential parts of these two equations gives

PA kBT/ehNAexpS/R (37)

and

EP U (38)

Considering eq. (37), evidently, the preexponen-tial factor PA is not expected to be truly temper-ature-independent, since it does contain T, but

over the temperature ranges used, the variationis much less than that in the exponential factor,so that it is permissible to use the average abso-lute temperature in applying this equation. Theentropy of activation S can be expressed interms of the entropies of the reactants andproducts as

S SGL SL S0G (39)

where S(GL) is the entropy of 1 mol of occupiedsites; S(L), that of the vacant sites; and S(G), the

molar absolute entropy of the permeant gas G(see Appendix C). Substituting in eq. (37) and


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transferring the absolute entropy term over to theleft-hand side gives

PAexpS0G/R

kBT/ehNAexpSGL SL/R (40)

The quantity on the left-hand side is experimen-tally accessible, as are all the quantities on theleft-hand side except for the intersite spacing (lat-tice parameter) and the entropy difference in-side the exponential term. It is convenient to sym-bolize the left-hand side of eq. (40) by Y, that is:

YPAexpS0G/R (41)

where this quantity Y may be termed the entro-py-adjusted Arrhenius preexponential factor or,more simply, the entropy function. For laterapplication, substituting for Yin eq. (40) and tak-ing logarithms gives

log Y logkBT/ehNA log

log e SGL SL/R (42)

Considering eq. (38), the activation energy for

permeation is revealed as the internal energychange, U, required to expand the matrix of thepolymer at the transition site L to accommodatethe molecule of the permeant. Likewise, the dif-ference [S(GL)S(L)] in eqs. (40)(42) is the ac-companying entropy change; for a polymer, itwould be that associated with the altered confor-mations of the polymer chains around the transi-tion site L when the molecule of the permeant isintroduced.

Status of Permeation Versus Diffusion

Standing back for a moment from this mathemat-ical jungle, the TSM has therefore led to a decou-pling of the sorption process from the permeationprocess. Indeed, with this picture, it is now diffu-sion which is to be considered the secondary orderived quantity, with eq. (7) being inverted intothe form

D P/B (43)

This is, of course, not to deny that the moleculesof the permeant have to be absorbed by the poly-

mer matrix, and to move by diffusive jumps, toarrive at some point inside the polymer, such as a

transition site L. But it does indicate that fromthe molecular viewpoint, as well as in interpret-

ing and predicting such behavior, the permeabil-ity coefficient has a simpler basis. In particular,the AJM for diffusion involves the features of thepermeant molecule G at the sorption site andthose for G at the transition site Lboth of whichare situations that are not clearly defined. Bycontrast, the TSM involves the equilibrium be-tween the molecule in the gas or vapor (a gener-ally well-defined and well-understood state) andthat in the transition site L. Experimental dataon permeation therefore enable us to use the per-meant molecule to probe the properties of thetransition sites, as shown later in the article.

Molecular Basis of Ideal and Nonideal Permeation

It is also possible to see why certain systemsshould show ideal permeation and others non-ideal. If the amount of the permeant sorbed issmall, then this will not affect the properties ofthe transition sites L, so that the permeation willbe ideal. For high degrees of sorption, the sorbedpermeant molecules are, on average, closer totransition sites L, so that a plasticization effectwill change the thermodynamic character of thetransition-site behavior. The ideal permeationbehavior of water with moderately polar polymersalready discussed may be ascribed to the watermolecules being so strongly bound (hydrogen-bonded) to the sorption sites that they do notaffect the transition sites L; the deviation fromideal sorption (Henrys Law) has conventionallybeen ascribed to clustering, but, once again, thiswould tend to localize their radius of action that,therefore, does not extend to influencing the tran-sition site L.

EFFECTS OF THE VARYING THE NATUREOF THE POLYMER AND THE PERMEANT

In the above development of the TSM, it has beenassumed that we are dealing with a single poly-mer and a single permeant. In extending this to arange of polymers, and a range of permeants,there are three features that would be expected to

vary: the nature of the transition site L; the valueof the site-spacing ; and the nature of the inter-actions between the polymer matrix at the tran-sition site L and the permeant molecule. For ex-

ample, even with a single polymer and a range ofpermeants, if there is a fixed type of transition

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site L with a defined hole size, then the smallestpermeants (such as He) may pass through the

hole so easily that it may not represent a barrierto it, while the larger quasi-spherical permeants,such as sulfur hexafluoride (SF6), may be so largethat they would not be able to pass through thebarrier at all. Even for the more realistic pictureof a range of hole sizes (and barrier heights), thesmallest permeants would be using one end of thedistribution and the larger ones the other end, sothat the set of transition sites used by one groupwould be different from that used by the othergroup.

Furthermore, such considerations apply onlyfor a particular choice of temperature, for exam-

ple, one that is convenient experimentally, or thatis important in the application of the polymer.This applies to the single-temperature correlationof Stannett and Szwarc and the Permachor con-cept of Salame that were considered earlier.

From the more fundamental viewpoint, there-fore, we have to consider the temperature-inde-pendent quantities, that is, the Arrhenius param-eters. It should be possible to use these to probethe mechanical and thermodynamic characteris-tics of the polymer transition site L, which mustnot be expected to be the same as that of the bulk

of the polymer. To this end, it is necessary tochoose a suitable dataset of polymers and per-meants from the literature.

DATASET OF ARRHENIUS PARAMETERSFOR PERMEATION

Source of the Arrhenius Parameter Data

The TSM theory has been tested by applying it tothe literature permeation data compiled by Pauly

in the Polymer Handbook.6

These data are evi-dently accepted as authoritative and may, in mostcases, be confirmed by referring back to the orig-inal articles.

Polymers in the Dataset

Focusing on homopolymers, 16 such polymerswere found in the Polymer Handbook compila-tion6 for which the Arrhenius parameters for per-meation were determined for at least four sim-ple permeants (see below), so as to make themsuitable for further analysisin particular, to

look for linear correlations in the four test plotsused.

The characteristics of these 16 polymers arelisted in Table II, where they are given the code

letters A through P. They comprise five hydrocar-bon polymersnatural rubber (A), polydimethyl-butadiene (D), three PEs (F, G, and H), andpolypropylene (L); three vinyl polymersPVAC(N), PVBZ (O), and PVC (P); a methacrylate es-ter, PEMA (J); two fluorinated polymers, PETFE(I) and PTFE (M); two chain heteraoatomic poly-mers PA (nylon 11) (B) and PET (K); and a sili-cone polymer, PDMS (E). With one exception,they are all true homopolymers; even the appar-ent exception, PETFE (I), is seemingly an alter-nating copolymer and, hence, can be viewed as ahomopolymer of the combined repeat unit.60

Although the present theory is aimed at amor-phous polymers, many of the polymers in thedataset are semicrystalline; also, the samplePDMS (E) is reported to have a minor content(10%) of filler (Table II). This will affect theirpermeability coefficients. However, because of un-certainties in the actual amorphous contents, andbecause of uncertainties in the effects of crystal-linity and fillers on permeation, no attempt hasbeen made to correct the permeation parame-ters to those for fully amorphous polymers.1,69

Permeants in the DatasetThe simple permeants for which Arrhenius pa-rameter data are available comprise also (almostcoincidentally) 16 gases that are listed in TableIII in the order of the molecular weight, MG, withcode letters ap and with other properties andcharacteristics that are important here. Thesecomprise the five noble gases He (b), Ne (d), Ar(i), Kr (m), and Xe (o); the six diatomic gases H2(a), N2 (e), CO (f), O2 (g), HCl (h), and Cl2 (l); thetwo triatomic gases CO2 (j) and SO2 (k); and thethree polyatomic molecules CH4 (c), SiF4 (n), andSF

6

(p). These permeants should have effectivelyspherical molecules. This is evidently the casewith the noble gases, and it is ensured by freerotation at these temperatures with the diatomicmolecules, and with the last three polyatomicmolecules because of their high symmetry, al-though it becomes somewhat less sure with theelongated cases of the triatomic gases CO2 (j)(which has a linear molecule) and SO2 (k) (whichhas an angular molecule).70

In addition, data for water vapor [H2O (w)] areincluded where they are available for the samepolymer sample; this permeant is considered sep-

arately from the other permeants because its per-meation behavior is also clearly different.


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Glass Transition Behavior in the Dataset

As noted earlier, the behavior of the permeationwith regard to the glass transition temperature(Tg) is often complex. In particular, the Arrheniusplot may show a break at the Tg, giving different

values of the Arrhenius parameters for the glassstate (below Tg) and the rubber state (above Tg).

In the present case, seven of the polymers inthe datasetNR (A), PDMB (D), PDMS (E), PE1(F), PE2 (G), PE3 (H), and PP (L)had beenstudied only above the Tg, so that no comparisonsbetween these two states can be drawn.

Furthermore, with PTFE (M), there is consid-erable uncertainty of interpretation because

Table II Features of the 16 Polymer Samples

# M Monomer Unit a

Tg Glass: T Tg Rubber: T Tg

Ref Lit Tmin Tmax Tave Tmin Tmax Tave

A NRa OCH2OCHACMeOCH2O 1.00 ? 203 ? ? ? 293 323 308B PAb ONHOCOO(CH2)10O 0.80 ? 315 283 315 299 315 333 324C PCc OOOCOOOOPhOCMe2OPhO 1.00 398 418 273 398 336 398 448 423D PDMBd OCH2OCMeACMeOCH2O 1.00 ? 262 ? ? ? 298 323 310E PDMSe OSiMe2OOO 1.00 ? 146 ? ? ? 233 298 266F PE1f OCH2OCH2O 0.23 ? 148 ? ? ? 278 333 306G PE2g OCH2OCH2O 0.57 ? 148 ? ? ? 278 333 306H PE3h OCH2OCH2O 0.71 ? 148 ? ? ? 278 333 306I PETFEi OCH2OCH2OCF2OCF2O 0.50 328 ? 278 328 303 328 373 350J PEMAj OCH2OCMe(COOOOEt)O 1.00 338 338 298 338 318 338 353 346

K PETk

OOO(CH2)2OOOCOOPhOCOO 0.60 353 342 293 353 323 353 403 378L PPl OCH2OCHMeO 0.52 ? 260 ? ? ? 303 328 316M PTFEm OCF2OCF2O 0.50 ? ? 278 373 326 278 373 326N PVACn OCH2OCH(OOCOOMe)O 1.00 296 305 273 296 284 296 313 304O PVBZo OCH2OCH(OOCOOPh)O 1.00 333 344 293 333 313 333 358 346P PVCp OCH2OCHClO 1.00 348 354 298 348 323 348 363 356

#, polymer code; M, polymer acronymdetails in footnotes ap below; a, volume fraction amorphous content of the sample; Tg,Ref: glass transition temperature from the original reference (footnotes ap), Lit: glass transition temperature from the litera-ture.54 Tmin, Tmax, and Tave, minimum, maximum, and average experimental temperature for all the permeants studied with thesample. ?, no data available.

a NR: natural rubber, cis-polyisoprene; unvulcanized smoked sheet Hevea; data of Michaels and Bixler.55,56b PA: polyamide (Nylon) 11, poly(imino-1-oxoundecamethylene); data of Ash et al.57c PC: polycarbonate, poly(oxycarbonyloxy-1,4-phenyleneisopropylidene-1,4-phenylene); General Electric Lexan; data of Nor-

ton.58d

PDMB: poly(2,3-dimethylbutadiene), methyl rubber; data compiled by Pauly.6

e PDMS: polydimethylsiloxane, vulcanized, with 10% Santocel CS filler; data of Barrer and Chio59; Arrhenius parametersrecalculated, correcting an error in the PA value for Ar.

f PE1: high-density polyethylene (HDPE); Grace Grex, 0.9640 g cm3, 0.15% Me groups; data of Michaels and Bixler.55,56g PE2: low-density polyethylene (LDPE); DuPont Alathon 14, 0.9137 g cm3, 3% Me groups; data of Michaels and

Bixler.55,56h PE3: hydrogenated polybutadiene (HPBD); Phillips Hydropol, 0.8940 g cm3, 5% Me groups, 1.3% CAC; data of Michaels

and Bixler.55,56i PETFE: poly(ethylene-alt-tetrafluoroethylene); Hoechst Hostaflon ET; a 0.50 value assumed; Tg value taken from the

fact that different parameters for H2O are quoted below and above this temperature (no literature value found); data of Pauly6; the

alternating structure was presumed from another literature source.60j PEMA: poly(ethyl methacrylate); data of Stannett and Williams29; parameters for SiF4 estimated from their graph.k PET: poly(ethylene terephthalate); data of Michaels et al.61,62l PP: polypropylene; pooled closely similar data for three polymers with 0.893 0.006 g cm3, a 0.50 008; data of

Sezi and Springer.63m PTFE: polytetrafluoroethylene; Hoechst Hostaflon FPA; data of Pauly6; there are two values of Tg (260 and 400 K) quoted

in the literature,54,64 so that the polymer is taken to be in either state for the present purposes.n PVAC: poly(vinyl acetate); data of Meares.65,66o PVBZ: poly(vinyl benzoate); data of Hirose et al.67; value of PA for Xe above Tg correctedthe quoted value

67 is 100-fold toohigh.

p PVC: poly(vinyl chloride); unplasticized sample; data of Tikhomirov et al.68; the data are mainly below Tg, and the valuesabove Tg are therefore less certain; the Arrhenius parameters for Ar were recalculated for the temperature ranges above and belowTg.

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there are two reported glass transition tempera-tures (260 and 400 K), along with multiplicity ofcrystalline transitions for the polymer.54,64 Toavoid having to make a decision in this case, ithas been treated as having the same parametersbelow and above the Tg, as with case (a) below,but with the conclusions to be interpreted bearingin mind this assumption.

For the remaining eight polymers, three casescan be distinguished: (a) PA (B), and PVC (P):There was no break with any of the gases studied,except that in the case of PVC the data for Ar (i)and Kr (m) did show a break; (b) PETFE (I) andPEMA (J): There was no break with any of thegases, but a break with H2O (w) which may betaken to define the value of Tg; (c) PC (C), PET(K), PVAC (N), and PVBZ (O): There was a breakat Tg for all the gases studied.

THE FOUR TYPES OF TEST PLOTS USED

Four types of plotsdesignated Type 1, Type 2,

Type 3, and Type 4 plotswere used in examin-ing the literature data of the polymer/permeant

data set and for fitting linear regression relationswhere they seemed justified by the graphs ob-tained. These plots involved the Arrhenius pa-rameters PA (plotted as log PA) and EP, along withthe entropy function Y (plotted as log Y ) and thepermeant molecular diameter G (Table III and

Appendix B).

Type 1 Plot: Log PA

Against Activation Energy EP

Background

In studies of the diffusion in polymers, when aparticular polymer is used with a range of per-meants, a linear correlation is often found be-tween the Arrhenius quantities for diffusion, log

DA and ED, obtained from eq. (10).1,14,16,59,66 This

type of correlation is commonplace in kinetics andthermodynamics, going by such names as thecompensation rule and the isokinetic relation-ship, as discussed below. This Type 1 plot is,therefore, a test for the behavior of the Arrheniusparameters for permeation and does not at this

stage involve the TSM directly, except to act ascomparison with the later plots.

Table III Characteristics of the Permeant Gases and Vapors, G, in Order of Their Relative

Molar Mass

# G MG Z b G S0 Class

a H2 2.0 2 26.51 276 226 Smallb He 4.0 2 23.80 266 222 Smallc CH4 16.0 10 43.01 324 282 Smalld Ne 20.2 10 26.51a 276a 242 Smalle N2 28.0 14 38.70 313 287 Smallf CO 28.0 14 39.48 315 294 Smallg O2 32.0 16 31.86 293 256

b Smallh HCl 36.5 18 40.61 318 283 Smalli Ar 40.0 18 32.01 294 251 Small

j CO2 44.0 22 42.86 324 310 Largek SO2 64.1 32 56.79 356 344 Large

l Cl2 70.9 34 54.22 350 319 Largem Kr 83.8 36 39.60 315 260 Smalln SiF4 104.1 50 72.36 386 379 Largeo Xe 131.3 54 51.56 344 266 Largep SF6 146.1 70 87.86 411 387 Largew H2O 18.0 10 30.49 289 285 Small

#, code letter; G, molecular formula; MG, relative molar mass; Z, summed atomic numbers of component atoms (total numberof electrons); b, van der Waals covolume (cm3 mol1)see Appendix B; G, van der Waals molecular diameter (pm)see AppendixB; S0, absolute entropy (J K1 mol1 Pa)see Appendix C; Class, size classification (small or large) for the permeationplotssee text.

a See Appendix B for the assignment of these values.b See Appendix C for the assignment of this value.


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Results

The plots are shown for the present dataset inFigure 4(AP), with common scales for ease ofcomparison. Where such a plot is linear, it may befitted by the standard regression equation

log PA m1EP c1 (44)

The derived values of the regression parametersare listed in Table IV.

Isokinetic Temperatures

The term isokinetic relationship implies that for a

specific polymer there is a particular tempera-ture, the isokinetic temperature, Ti, at which

Figure 4 (AP) Type 1 plots (Arrhenius parameters): log PA versus activation energyEP; polymers as Table II, permeants as Table III. () glass state (T Tg); () rubberstate (T Tg).

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opposing effects of the two factors, log PA and EP,cancel out and the permeability coefficient hasthe same value, Pi, for all the permeants. It maybe evaluated by comparing eq. (44) with the log-arithmic form of the Arrhenius relation, eq. (9), atthis temperature:

log Pi log PA EPlog e/RTi (45)

which also represents a linear relation betweenlog PA and EP. This, therefore, gives

Ti log e/m1R (46)

The value of Ti may or may not be within the

experimentally accessible range. However, ifTiis similar to the average temperature, Tave,

Figure 4 Type 1 plots (Arrhenius parameters) (Continued from the previous page)


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then this suggests that the correlation is spuri-ous, arising from random fluctuations in theestimated parametersif the intercept of theplot is overestimated, then the slope will beunderestimated in a correlated fashion and vice

versa.

Correlation Behavior

Considering first the behavior for polymers inthe rubber state (T Tg) (filled diamond sym-

bols) from the broken-line bands (correspondingto 0.25 units on the ordinate scale), the plotsin Figure 4 are seen to be reasonably linear. Thecorrelation coefficient r1 (Table IV) averages0.95, the highest values being 0.992-0.996 forPC (C), PVAC (N) and PVC (P); the poorestcorrelations are seen with NR (A), 889 andPETFE (I), 0.823. The low r values may be

related to the restricted span of the plottedvalues, coupled with the fixed spread about the


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regression line. Since these systems are be-lieved to have reliable permeation data, and wewish to have an objective view of the behaviorand of the acceptability of the present theory topolymers in general, we will not reject thesefrom further consideration, but accept the iso-kinetic parameters derived along with their cor-respondingly greater uncertainties.

For the polymers in the glass state (T Tg),where there is no break at the Tg, then the corre-

lation coefficients are as for the rubber state Ta-ble IV), with a low of 0.823 for PETFE (I). In thecases where there is a break, the correlations aremuch poorer, with values ranging from 0.863 forPVAC (N) down to 0.152 for PC (C). These low

values must be taken to indicate the absence ofany overall correlations for the permeants in gen-eral.

If we accept that there is no such overall cor-relation in the glass state, then the values of the



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correlation coefficient obtained are indicative ofthe spurious correlation due to the isokinetic tem-perature being close to the (mean) experimentaltemperature.

The values of the regression parameters areplotted as a scattergram in Figure 5. It is diffi-cult to see any clear correlation between the plot-ted values or any correlation with the structure ofthe repeat units of the polymers. Nevertheless,these parameters may be taken to be character-istic quantities for the particular polymer for anypermeant.

However, all this having been said, we have to

take note of the fact that the later Type 3 plots,where the preexponential factorPA has been mod-ified into the entropy function Y according to eq.(41), show better linear correlations, at least forthe smaller-size permeants. This casts somedoubts on any of the correlations apparently seenwith these Type 1 plots, as discussed above.

Type 2 Plots: Activation Energy EP versus Permeant

Molecular Diameter G

Background

One popular approach in interpreting diffusiondata is to seek to correlate the activation energies

for diffusion with the molecular diameters of thepermeant molecules, or some power of this; themolecular basis of this has, however, always beenrather tenuous.1,44,56,58,62,6568 In the presentcase with the transition-site model for perme-ation, evidently, the molecule must be expected tostretch the transition site L to accommodate it-self; this would be expected to require a greateramount of energy, the greater the diameter of themolecule.

Results

The Type 2 plots of EP

versus the permeant mo-lecular diameter G for the 16 polymers areshown in Figures 6(AP), with common scales forease of comparison. For all the polymers exceptNR (A), PDMS (E), and PETFE (I), the Type 2plots are linear with a positive slope, at least forthe small molecules, that is, up to and includingCH4 (c) with G 324 pm, but excluding CO2 (j)with nominally the same value of G (Table III).This applies both above and below the glass tran-sition temperature. There is a clear distinctionhere between the 10 small permeants (lyingclose to the regression lines) and the six large

permeants (lying well away from the lines). WithNR (A) and PETFE (I), no clear regression lines

Table IV Type 1 Plot: Parameters for the Linear Correlations Between log PA and the Permeation

Activation EnergyEP

# Polymer

Glass State: T Tg Rubber State: T Tg

r1 c1 m1 r1 c1 m1 Ti Ti/Tave

A NR ? ? ? 0.889 3.23(55) 0.077(18) 678 2.20B PA 0.948 1.88(52) 0.078(15) 0.948 1.88(52) 0.078(15) 670 2.17C PC 0.152 * * 0.996 1.67(25) 0.087(4) 600 1.42D PDMB ? ? ? 0.986 1.46(52) 0.130(11) 402 1.30E PDMS ? ? ? 0.904 2.43(19) 0.105(17) 562 2.11F PE1 ? ? ? 0.904 1.16(76) 0.101(20) 522 1.71G PE2 ? ? ? 0.964 3.10(48) 0.108(12) 574 1.88H PE3 ? ? ? 0.967 2.61(53) 0.111(12) 528 1.73I PETFE 0.823 1.46(71) 0.090(25) 0.823 1.46(71) 0.090(25) 528 1.51

J PEMA 0.973 1.74(34) 0.096(9) 0.973 1.74(34) 0.096(9) 544 1.57K PET 0.737 * * 0.971 1.02(66) 0.094(13) 556 1.47L PP ? ? ? 0.985 1.44(68) 0.122(15) 428 1.35M PTFE 0.941 1.83(31) 0.078(13) 0.941 1.83(31) 0.078(13) 670 2.06N PVAC 0.863 * * 0.996 0.95(31) 0.130(5) 402 1.32O PVBZ 0.628 * * 0.943 3.04(37) 0.091(12) 574 1.66P PVC 0.992 0.84(34) 0.117(6) 0.992 0.84(34) 0.117(6) 446 1.25

See Table II for the polymer codes and details of the samples used. See Figure 4(AP) for the respective plots. r1, correlation

coefficient; c1, ordinate intercepteq. (44); m1, gradienteq. (44); Ti, isokinetic temperature (K)eq. (46); Tave,average experimental temperature (K) for the rubber state (Table III). ?, no data availablesample not studied inthis state; *, no meaningful datapoor linear correlation.

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can be seen; with PDMS (E), the line has a neg-

ative slope.The standard regression equation here is

EPm2G c2 (47)

The derived values of the regression parametersare listed in Table V.

Even without entering into a molecular inter-pretation of this data, the close fit to the straightlines seen in Fig. 6 with the small molecules and11 of the polymersPA (B), PDMB (D), PE1 (F),PE2 (G), PE3 (H), PEMA (J), PET (K), PP (L),PTFE (M), PVAC (N), PVBZ (O)gives us confi-dence both in the values of the molecular diame-ter for these molecules (Appendix B) and in liter-ature values of the activation energy EP for thesesystems.

Molecular Interpretation of the Type 2 Plots

The simplest interpretation of these linear plots(Figure 6) is that, for any particular polymer, thepermeants on the line all use the same set oftransition sites. The value of EP represents theenergy to expand the aperture to accommodate

the molecule [eq.(38)]. The intercept on the mo-lecular diameter axis, corresponding to the aper-

ture size for zero activation energy, thus repre-sents the size of the unperturbed aperture, L; the

average value of L for the polymers [exceptingPDMS (E) and the three PEs (F, G, H)] is 210(48)pm (Table VI). The permeant molecules evidentlybehave here as hard spheres of the stated vander Waals diameter G. The slope of the line thenrepresents the energy to expand the aperture toaccommodate the increasingly large molecules,and its linearity shows that this involves a fixedforce constant . o evaluate this from the gra-dient m2 from eq. (47), which is in units of (kJmol1)/(pm/molecule), it is necessary to convert toa molecular basis throughout by introducing the

Avogadro constant NA (6.023 1023 molecules

mol1) as well as to convert to consistent SI units(kJ 3 J, pm 3m). This gives

1.66m2 nN (48)

where nN is nanonewtons. The derived valuesof are listed in Table VI. Excluding againPDMS (E) and the three PEs (F, G, H), the 10polymers remaining give a mean value for of1.00(43) nN; for the three PEs (F, G, H), thismean is 0.34(4) nN.

One curious feature of the parameter is thatit represents a constant force, rather than aHookes Law/Youngs modulus coefficient asmight have been expected. It is as if the permeantmolecule being inserted into the transition-siteaperture were lifting a fixed weight.

Regarding the effect of the glass transition,there are five polymers that show different behav-ior below and above Tg: PC (C), PET (K), PVAC(N), PVBZ (O), and PVC (P). Here, although thereare differences between the derived L values(Table VI), these are within the limits of error ofthe intercepts on the molecular diameter axis

(Fig. 6), indicating the same set of sites acts aboveand below Tg. This is shown more clearly by theresults from the later plots.

Type 3 Plot: Log Y Against Activation Energy EP

Background

These plots are similar to those in Type 1 plots forthe Arrhenius parameters, but in this case, the

values of the preexponential parameter PA wereadjusted by introducing the absolute entropy of

the permeant, S0(G) to give the entropy functionY as defined by eq. (41).

Figure 5 Linear regression parameters m1 and c1 forthe Type 1 plots. Polymer codes as in Table II. ()Mean point with mean standard deviation error bars;other symbols as Figure 4(AP).


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Results

The plots for the 16 polymers are shown in Figure7(AP), with common scales for ease of compari-son. The pattern of behavior is similar to thatseen with the Type 2 plots, with straight lines ofpositive slope except for NR (A), PDMS (E), andPETFE (I); with NR (A) and PETFE (I), there isagain apparently only a cluster of points, whilewith PDMS (E), the line has again a negativeslope. As before, there is a distinction between the

10 small permeants and the six large ones[CO2 (j) and larger]. It is particularly remarkable

that with four of the polymers, PET (K), PVAC(N), PVBZ (O), and PVC (P), where the parame-ters differ for the glass and rubber states, thepoints in these plots (Fig. 7) are brought onto anessentially common single line; this is not thecase, however, with the corresponding fifth poly-mer, PC (C), although the data are evidently notas precise here as with the other polymers.

The standard form of the regression equationhere is

log Ym3EP c3 (49)

Figure 6 (AP) Type 2 plots: permeation activation energy EP versus permeantmolecular diameter G; polymers, permeants, and symbols as in Figures 4(AP).

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firm that the same set of transition sites is involved in the two states (glass and rubber) forthese polymers.

This also enables us to obtain an estimate ofthe site-spacing , as follows: Comparing eqs. (42)and (49), the ordinate intercept c3 is log YZ, whereYZ is the value of the entropy function Y for zeroactivation energy. Under these circumstances,the permeant molecule would have the same di-ameter as that of the transition-site aperture

[this is, of course, a hypothetical situation, sinceHe (b), the smallest molecule possible, is always

larger than the transition-site apertures]. Thereis then no perturbation of the nearby polymermatrix, and the permeant molecule is held by thetransition site with its translational entropic free-dom removed, so that the factor [S(GL) S(L)]becomes zero, and, hence, eq. (42) becomes

log YZ logkBT/ehNA log (50)

Taking antilogarithms and rearranging this gives

kBT/ehNA YZ1 (51)

Figure 6 Type 2 plots (Continued from the previous page)

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Type 4 Plot: Log Y Against Molecular Diameter G

Background

This third type of test plot for the TSM theorylooks for the correlations between the two param-eters, log Y and the molecular diameter G.

Results

The graphs for the 16 polymers are shown inFigure 10(AP), with common scales for ease ofcomparison. In contrast to the Type 2 and Type 3plots, these Type 4 plots give straight lines for allof the polymers, that is, now including NR (A) and

PETFE (I), which gave scatters of points previ-ously, and including the six large permeantsCO2 (j), SO2 (k), Cl2 (l), SiF4 (n), Xe (o), and SF6(p)which showed consistent deviations in theprevious two types of plot. As with the Type 2plots, with the five polymers where the parame-ters are different for the glass and rubberstatesPC (C), PET (K), PVAC (N), PVBZ (O),and PVC (P)two straight lines of different slopeare obtained.

The standard form of the regression equationhere is

log Y m4G c4 (53)

The derived values of the linear regression pa-

rameters m4 and c4 are listed in Table VIII.As with the previous two cases, the close fit to

straight lines seen here even more generally givesus further confidence both in the values of the vander Waals molecular diameters for these mole-cules (Appendix B) and in literature data of thepreexponential factor PA (from which the value ofthe entropy function Y is obtained) for these sys-tems.

Molecular Interpretation of the Type 4 Plots

Once again, the straight-line plots in Fig. 10 areconsistent with a single type of transition site foreach polymer. However, this evidently now in-cludes the cases of NR (A) and PETFE (I) whichdid not show this behavior in the other plots, aswell as the large permeants as listed abovewhich generally showed deviant behavior withthe two previous plots (Figures 4 and 6).

The straight lines evidently relate to a con-stant incremental effect of the permeant moleculediameter, G, on the entropy of the surroundingmatrix, S(L), since the translational entropy con-tribution of the permeant molecule has been elim-

inated by its insertion and entrapment in theaperture of the transition site. We may therefore

Table V Type 2 PlotParameters for the Linear Correlations Between Permeation Activation

Energy EP and Permeant Molecular Diameter G

# Polymer


r2 c2 m2 r2 c2 m2

A NR ? ? ? 0.311 * *B PA 0.965 106(27) 0.505(96) 0.965 106(27) 0.50(10)C PC 0.969 47(10) 0.24(3) 0.969 180(35) 0.74(11)D PDMB ? ? ? 0.994 130(11) 0.60(4)E PDMS ? ? ? 0.921 52(7) 0.14(2)F PE1 ? ? ? 0.965 19(9) 0.18(3)G PE2 ? ? ? 0.921 24(14) 0.23(5)H PE3 ? ? ? 0.943 14(10) 0.19(3)I PETFE 0.084 * * 0.084 * *

J PEMA 0.977 60(10) 0.327(35) 0.977 60(10) 0.33(4)K PET 0.946 52(19) 0.275(66) 0.993 145(16) 0.64(6)L PP ? ? ? 0.994 29(5) 0.13(2)M PTFE 0.979 59(10) 0.28(3) 0.979 59(10) 0.28(3)N PVAC 0.857 104(38) 0.43(13) 0.995 206(11) 0.90(4)O PVBZ 0.956 29(6) 0.17(2) 0.959 68(13) 0.36(4)P PVC 0.889 129(35) 0.60(12) 0.943 265(50) 1.10(17)

See Table II for the polymer codes and details of the samples used. See Figure 6(AP) for the respective plots. r2, correlationcoefficient; c2, ordinate intercepteq. (47); m2, gradienteq. (47). ?, no data availablesample not studied in this state; *, nomeaningful datapoor linear correlation.

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define the entropy increment for this insertion

process [eq.(19)] by

SGL/ G (54)

Combining this with eq. (42) gives

R/log elog Y/G (55)

and applying this to the regression expression eq.(53) gives

R/log em4 (56)

Since the entropy values S are in molar units,while the molecular diameters G are in molecu-lar units, it is again necessary to bring them intoline by introducing the Avogadro number NA, sothat the conversion becomes first

8.314/0.4343m4

J K1 mol1/pm molecule1 (57)

which then gives

31.8m4 pJ K1 m1 (58)

where pJ is picojoules (1012 J). The values of the

entropy increment so-derived are listed in Table VI.They are also plotted against the values of the forceconstant in Figure 11. This shows a positive linearcorrelation between the plotted quantities for the 14polymers [i.e., excepting NR (A) and PETFE (I)],but not a proportionality as might be expected; thiscorrelation includes the three PEs (F, G, H) andPDMS (E) which are anomalous in their values ofthe transition-site aperture L and the site-spacing (Figures 8 and 9).

It should be noted, in the molecular interpre-tation of these Type 4 plots, that the zero level oflog Y has no fundamental significance because itcan be shifted up or down by a change in the unitsof length or time. This contrasts with the status ofthe EP values in the Type 2 plots (Fig. 6).

The curious feature of this Type 4 plot is thatthe linear plots (Fig. 10) now include the polymersNR (A) and PETFE (I) which previously gavescattered plots and, moreover, include thelarge permeants. It seems that although the twodifferent groups of permeants (small andlarge) may have to use two different sets oftransition sites with different apertures, the en-tropy increment for the matrix surrounding them

is still the same, giving the same slope for allpermeants with a particular polymer.

Table VI Transition-site Characteristics

# Polymer


L log L log

A NR ? ? ? ? * * * 1.81(17)B PA 210(12) 0.84(16) 1.43(27) 2.83(24) 210(12) 0.84(16) 1.43(27) 2.83(24)C PC 196(15) 0.40(5) 0.85(107) 1.93(14) 243(12) 1.23(18) 0.85(107) 3.95(39)D PDMB ? ? ? ? 218(5) 0.99(6) 1.89(35) 4.49(17)E PDMS ? ? ? ? 369(11) 0.23(4) 5.00(31) 0.33(14)F PE1 ? ? ? ? 101(29) 0.31(5) 9.09(160) 2.36(15)G PE2 ? ? ? ? 105(35) 0.38(8) 7.99(293) 2.50(19)H PE3 ? ? ? ? 76(35) 0.32(6) 13.00(210) 2.46(16)I PETFE * * * 1.92(30) * * * 1.92(30)J PEMA 184(11) 0.54(6) 1.39(166) 2.26(21) 184(11) 0.54(6) 1.84(166) 2.26(21)

K PET 188(24) 0.46(11) 1.01(46) 1.90(28) 225(6) 1.07(9) 1.01(46) 4.01(42)L PP ? ? ? ? 216(8) 0.96(10) 2.42(78) 4.16(57)M PTFE 212(11) 0.46(6) 1.10(103) 2.21(33) 212(11) 0.46(6) 1.10(103) 2.21(33)N PVAC 239(11) 0.72(22) 0.52(24) 2.25(62) 230(3) 1.49(6) 0.52(24) 5.41(29)O PVBZ 176(14) 0.28(4) 0.39(54) 1.33(25) 188(12) 0.60(7) 0.13(54) 2.32(28)P PVC 216(13) 0.99(19) 0.15(30) 4.99(56) 241(7) 1.83(29) 0.15(30) 4.99(56)

See Table II for the polymer codes and details of the samples used. L, transition-site aperture diameter (pm)Figure 3; , forceconstant (nN)eq. (48); , average transition-site spacing (nm)Figure 3; , entropy increment (pJ K1 m1)eq. (54). ?, no dataavailablesample not studied in this state. *, no meaningful datapoor linear correlation. See Figs. 8, 9, and 11.


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ANOMALIES IN THE TYPES 2, 3, AND4 PLOTS

Normal Polymers and Permeants

Of the 16 polymers for which literature data were

available (Table II), 10 of them showed consistentbehavior in the three types of plots and gave

consistent values of the four parameters derived(Table VI, and figs. 8, 9, and 11). Likewise, of the16 permeants that had been studied with thesepolymers (Table III), the 10 small permeantsshowed consistent behavior. We may refer tothese polymer/permeant pairs as the core group

and as representing the normal behavior. Weconsider here the six polymers and the six per-

Figure 7 (AP) Type 3 plots: log of the entropy function Y versus activation energyEP; polymers, permeants, and symbols as Figures 4(AP).

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meants that showed anomalous behavior in oneor more of these three types of plots.

NR (A)

This polymer showed only an apparent clusteringin the Type 2 plot [Fig. 6(A)] and the Type 3 plot[Fig. 7(A)], but a linear correlation in the Type 4

plot [Fig. 10(A)]. This polymer differs from theothers by being of natural origin. It may be spec-

ulated that there is a certain copolymer charac-ter for the polymer chain, which as seen withPETFE (I) leads to the same type of behavior inthe plots. It also possible that the significantamounts of nonrubber contaminants (especially,proteins and fatty acids) in commercial samples ofNR may play a part in this behavior. This evi-

dently needs further examination in view of theimportance of this polymerspecifically, by com-



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paring with the behavior of synthetic polyiso-prenes and with that of NR that has been appro-priately purified.

PDMS (E)

This polymer is anomalous in that it shows anegative slope in the Type 2 plot [Fig. 6(E)] and inthe Type 3 plot [Fig. 7(E)], albeit with a positive(but rather small) slope in the Type 4 plot [Fig.

10(E)]. [In evaluating these three plots, it becameevident that the data points for N2 (e) are out of

line. In particular, the log Yvalue is apparently 2units too high, which may have arisen from theoriginal value ofPA having been recorded too highby a factor of 100. The data for this permeanthave therefore been omitted in processing thedata for this polymer.] The Type 2 plot gave to a

value of the transition-site aperture (369 pm) thatis comparatively large but not unreasonable, butan apparently negative value for force constant ;the Type 3 plot gave an unreasonably small value

for the intersite spacing , while the Type 4 plotgave a value of the entropy increment that is


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relatively small although still positive (Table VIand figs. 8 and 11). From the molecular viewpoint,the difference between PDMS and the other poly-mers listed lies in the presence of the silicon at-oms in the backbone. This leads to an unusuallylarge flexibility of the chain and much easier seg-mental rotation.59 This needs to be investigatedby looking at the permeation behavior of othersilicone-type polymers.

PE1 (F), PE2 (G), and PE3 (H)

These three polymers show essentially normalbehavior in the Type 2 plots [Fig. 6(FH)], the

Type 3 plots [Fig. 7(FH)], and the Type 4 plots[Fig. 10(FH)]; the only anomaly here is that thedata points for the large permeant SF6 (p) lieclose to the lines, whereas the other large per-meant CO2 (j) shows the same deviations seen inother cases. However, the derived values of thetransition-site aperture L are relatively small(average 96 pm), while the values of the site-spacing are impossibly large (Fig. 8). Neverthe-less, the values of the thermodynamic properties and are in line with those of the other poly-

mers (Fig. 11). It is difficult to see why this poly-mer, which is among the simplest structured of all

Figure 7 Type 3 plots (Continued from previous page)


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polymers, should show these anomalies and thatthey occur even with some significant alterationsto the polymer structure (in terms of the contentof branches). By contrast, the closely related poly-olefin polypropylene (L) falls within the normalcore group in all its properties, although it doeshave a somewhat large value (260 nm) for thesite-spacing (Table VI and fig. 9).

PETFE (I)

This polymer is somewhat similar in its behaviorto NR (see above), showing only an apparent clus-tering in the Type 2 plot [Fig. 6(I)] and the Type 3plot [Fig. 7(I)] but a linear correlation in the Type4 plot [Fig. 10(I)]. It is possible that this anoma-lous behavior arises from the fact that the litera-ture sample6 was not a pure alternating copoly-mer as assumed.60 For example, it may have anadmixture with the homopolymers PE and PTFE.This would give a distribution of transition-siteapertures and site-spacings, which would cloudthe Type 2 and Type 3 plots; if, on the other hand,the entropy characteristics of the sites were sim-ilar, this could still give a linear Type 4 plot. The

behavior of this copolymer evidently needs fur-ther study.

Large Permeants

These are the six permeants that have moleculardiameters of 324 pm or greater (Table III), compris-ing CO2 (j), SO2 (k), Cl2 (l), SiF4 (n) ,Xe(o), and SF6(p). In the Type 2 plots, these show anomalously low

EPvalues [Fig. 6(BD,FH,JM,O,P)], although thepoints for SF6 (p) lie sometimes on or near thecorrelation line [Fig. 6(C,F)] and likewise for Xe (o)[Fig. 6(O)]. Similar behavior is seen in the Type 3plots (Fig. 7); although this could be equally wellinterpreted as low log Y values, the fact that suchanomalies are not seen in the Type 4 plots (Fig. 10)

confirms that this corresponds to low values of theactivation energy EP.

From the molecular viewpoint, if we considerthe type of transition site used by the smallmolecule permeants, it is evident that as the per-meant molecular diameter is increased a certaincritical size is reached at which the amount ofenergy required to insert the molecule becomestoo great for insertion in the site to be feasiblewhile still giving permeation at a detectable rate.The permeant molecules apparently are able tofind a set of transition sites with a larger value for

the aperture, giving a lower activation energyand, hence, a detectable rate of permeation. This

Table VII Type 3 PlotParameters for Linear Correlations Between log Y (Entropy Function) and

Permeation Activation Energy EP

# Polymer


r3 c3 m3 r3 c3 m3

A NR ? ? ? 0.695 * *B PA 0.998 1.51(27) 0.186(8) 0.998 1.51(27) 0.186(8)C PC 0.981 1.05(67) 0.232(23) 0.993 0.24(98) 0.162(14)D PDMB ? ? ? 0.999 1.98(35) 0.235(7)E PDMS ? ? ? 0.744 4.92(31) 0.063(28)F PE1 ? ? ? 0.986 9.17(160) 0.435(42)G PE2 ? ? ? 0.959 8.08(293) 0.383(65)H PE3 ? ? ? 0.985 13.0(210) 0.523(46)I PETFE 0.126 * * 0.126 * *

J PEMA 0.946 1.93(166) 0.232(46) 0.946 1.93(166) 0.232(46)K PET 0.988 1.10(46) 0.176(11) 0.988 1.10(46) 0.176(11)L PP ? ? ? 0.997 2.51(78) 0.243(18)M PTFE 0.981 1.18(103) 0.281(39) 0.981 1.18(103) 0.281(39)N PVAC 0.995 0.61(24) 0.176(5) 0.995 0.61(24) 0.176(5)O PVBZ 0.947 0.21(54) 0.175(17) 0.947 0.21(54) 0.175(17)P PVC 0.992 0.23(30) 0.149(6) 0.992 0.23(30) 0.149(6)

See Table II for the polymer codes and details of the samples used. See Figure 7(AP) for the respective plots. r3, correlationcoefficient; c3, ordinate intercepteq. (49); m3, gradienteq. (49); ?, no data availablesample not studied in this state; *, nomeaningful datapoor linear correlation.

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would correspond to correlation lines in the Type2 plot (Fig. 6) shifted to higher G values and,

hence, to lower EP values. However, there are notenough data on the large permeants in thepresent dataset to test this picture properly.

One of the curious common features in thisrespect is the sharpness of this transition, in thatCH4 (c), which has nominally the same moleculardiameter G (324 pm) as that of CO2 (j), shouldapparently always behave as a small moleculeand show normal behavior. The exception hereis with PVC (P), where Figure 7(P), in particular,shows that not only CH4 (c) but even N2 (e) (G 313 pm) behave as large molecules with this

polymer.

Water Vapor [H2O (w)]

This is a very important permeant, which has,nevertheless, been treated apart from the oth-ers because its behavior is evidently distinct.The term water vapor is applied here to em-phasize that it is the behavior of isolated watermolecules that is involved, that is, the values of

P are the ideal values PI. We stay within thepresent dataset of polymer samples for compar-ison with the other permeants, since it is evi-

dent that (particularly with this permeant) thepermeation behavior is sample-sensitivethus,

Barrie and Machin28 obtained different valuesfor the Arrhenius parameters for PEMA/H2O

from those (used here) obtained by Stannettand Williams.29 This, unfortunately, reducesthe polymer dataset to just three members:PETFE (I), PEMA (J), and PVC (P). Consider-ing the effects of change of state (glass or rub-ber) with these polymers, with the first two,there are different values for the Arrhenius pa-rameters in the two states (while the other per-meants have the same values), whereas in thelatter case, the Arrhenius parameters for H2Oare the same (while there are differences for thelarge permeants with this polymer).

One problem that arises with this permeant is

uncertainty in the value of the molecular diame-ter, which the van der Waals data indicate to be289 pm (Table III and Appendix B). Evidently,further work needs to be done with this permeantalong with related permeants on the same poly-mer sample. For example, there is scope for theuse of heavy water (D2O) to see the effect of thedifferent hydrogen isotope. Likewise, systematicstudies on the same polymer sample with theisoelectronic series: Ne, HF, H2O, NH3, CH4, mayshow up effects from hydrogen bonding at thetransition site; in the absence of such effects,

Figure 8 Transition-site characteristics: display plotof the log of the transition-site spacing (nm) versusthe site aperture diameter L (pm); polymers as listedin Table II; symbols as Figure 4(AP); see Figure 9 forinset.

Figure 9 Transition-sit

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