Czichos H., Saito T., Smith L.E. (Eds.) Handbook of Metrology and Testing

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428 Part C Materials Properties Measurement

to C [7.272]

eff

) =



+ p







, (7.128)

which requires the pressure dependence of the per-

meability and solubility to calculate D

eff

. Methods to

measure pressure dependence of permeability and solu-

bility are discussed in this chapter.

Gas sorption isotherms in polymers are typically

described using simple models, such as Henry’s law,

the Flory–Huggins theory or the dual-mode sorption

model [7.264]. These models have been extensively in-

vestigated in the literature [7.273] and, therefore, will

only be brieﬂy introduced. Once the gas sorption in

a polymer has been measured, values of dp/dC

can

be easily calculated for use in (7.128).

Henry’s law corresponds to a linear relationship be-

tween C and p [7.274]

C = k

p , (7.129)

where k

is the solubility coefﬁcient, or Henry’s law

constant. This equation is typically valid for light gas

sorption in rubbery polymers at low concentrations,

such as 1 cm

(STP)/cm

polymer. For example, N

and O

sorption in poly(dimethylsiloxane) (PDMS)at

◦

C obey Henry’s law over a rather wide pressure

range [7.272].

For highly sorbing penetrants in rubbery polymers,

such as organic vapors, penetrant concentration in the

polymer is often described using the Flory–Huggins

model [7.274]

ln a = ln φ

+(1 −φ

)+χ(1 −φ

)

, (7.130)

where a is the penetrant activity, χ is the Flory–Huggins

interaction parameter, and φ

is the volume fraction of

penetrant dissolved in the polymer matrix. For ideal

gases, the activity is equal to p/p

, where p

is the

penetrant vapor pressure at the experiment temperature.

The volume fraction φ

is given by [7.274].



C/22 414





C/22 414



, (7.131)

where V

is the partial molar volume (cm

/mol) of

the penetrant in the polymer, and C has units of

(STP)/(cm

polymer). For low concentrations of

sorbed gas, φ

 1, (7.130)and(7.131) reduce to

Henry’s law [7.275]. For high concentrations of sorbed

gas, this model predicts an increasing value of solubil-

ity (i. e., C/ p) as pressure increases. Equation (7.130)

successfully describes the sorption of strongly sorb-

ing penetrants in rubbery polymers, such as CO

poly(ethylene oxide) at 35

◦

C[7.275].

The dual-mode sorption model was originally used

to describe gas sorption in glassy polymers, which

contain both equilibrium volume (i. e., Henry’s law

sorption sites) and nonequilibrium excess volume (i. e.,

microvoids or Langmuir sorption sites). The model is

expressed as follows [7.276]

C =C

p +



1 +bp

, (7.132)

where C

and C

are the gas concentrations in the

Henry’s law sorption sites and Langmuir sites, re-

spectively, b is the Langmuir afﬁnity constant, which

characterizes the afﬁnity between penetrants and mi-

crovoids, and C



is the Langmuir capacity constant

[cm

(STP)/cm

polymer], which characterizes the

maximum sorption capacity of the nonequilibrium ex-

cess volume. This model has also been applied to

describe the gas sorption in hybrid systems such as

blends of rubbery polymers and additives [7.277, 278].

These additives could be ﬁllers that physically adsorb

the gas or dissolved species that chemically interact

with the gas.

The dependence of permeability on pressure is of-

ten described using empirical equations. For example,

the following equation is often used to describe the

pressure dependence of gas permeability in rubbery

polymers [7.272,275]

= P

(1 +mΔp) , (7.133)

where P

and m are constants, and Δp = p

− p

.For

glassy polymers at p

=0 (or, equivalently, Δ p = p

the following equation, based on the dual-mode model,

has been used [7.279]



1 +

1+bp



, (7.134)

where K =C



b/k

,andD and F are constants ob-

tained by ﬁtting the model to experimental sorption and

permeation data. For example, Koros and coworkers ob-

tained k

, b and C



by ﬁtting gas sorption isotherms

to (7.132) and then ﬁtting the permeability data at var-

ious upstream pressures to (7.134) to obtain D and

F [7.279]. Equation (7.133)or(7.134) can be used to

calculate dP

/dp, which can be substituted into (7.128)

along with dp/dC

to calculate D

eff

The ideal selectivity of a membrane for gas A over

gas B (i. e., α

A/B

) is the ratio of their pure-gas perme-

Part C 7.6

Mechanical Properties 7.6 Permeation and Diffusion 429

abilities [7.262]

A/B









, (7.135)

where D

is the diffusivity selectivity, which is the

ratio of the diffusion coefﬁcients of gases A and B. The

ratio of the solubility coefﬁcients of gases A and B,

, is the solubility selectivity. For a binary gas

mixture, the selectivity of gas A to gas B (α

∗

A/B

)is

deﬁned as

∗

A/B

≡

=α

A/B

Δp

, (7.136)

where y

and x

are the mole fractions of component i in

the upstream and downstream gas phases, respectively,

and Δp

is the partial pressure difference of compo-

nent i across the membrane. When the downstream

pressure is much less than the upstream pressure, α

∗

A/B

approaches the ideal separation factor α

A/B

7.6.2 Kinetic Measurement

The kinetics of gas transport through a thin uniform ﬁlm

have been modeled, and such measurements can yield

solubility, diffusivity and permeability in a single ex-

periment. The penetrant concentration as a function of

time and position in the ﬁlm is given by Fick’s second

law [7.270]

∂C

∂t

∂

∂x



eff

∂C

∂x



. (7.137)

The analytical solution of (7.137) depends on initial

conditions, boundary conditions and the concentra-

tion dependence of the diffusion coefﬁcient, if any.

The diffusion coefﬁcient could be a function of con-

centration, which makes the solution to (7.137)more

complex [7.261]. However, Crank and Park have shown

that solutions to (7.137) assuming constant D

eff

val-

ues are also applicable when the diffusion coefﬁcient

depends on concentration [7.268]. In general, this

approximation is sufﬁciently accurate to model experi-

mental data [7.268], except that the diffusion coefﬁcient

obtained from such an analysis should be interpreted

as an average effective diffusion coefﬁcient as deﬁned

in (7.126).

Constant Diffusion Coefﬁcient

In the simplest scenario, the gas diffusion coefﬁcient

is constant over the concentration range of interest.

This assumption is usually valid for low-sorbing pene-

trants in rubbery polymers. However, care must be taken

since gas diffusion coefﬁcients can exhibit concentra-

tion dependence at concentrations where gas sorption

still obeys Henry’s law, i. e., when gas solubility is in-

dependent of pressure [7.280]. Nevertheless, in a typical

experiment, initially (i. e., at t = 0) the membrane is at

a uniform concentration C

.Att > 0, one face (x =0)

of the membrane is exposed to a constant concentra-

tion C

by changing the gas pressure in contact with

the membrane, and the other face at x =l is exposed

to C

[7.261]. The analytical solution to (7.137) under

these conditions is [7.261]

C = C



−C



∞



n=1

cos nπ −C

×sin



nπ x



exp



−Dn

t/l



∞



m=0

2m +1

sin





2m +1



πx



×exp



−D



2m +1



t/l



. (7.138)

The direct measurement of penetrant concentration in

the ﬁlm as a function of position and time is very difﬁcult

and is usually not done. Instead, a much more straight-

forward experiment, measurement of the gas ﬂux out of

the ﬁlm is usually performed. Two methods are typically

used to monitor this kinetic process: (1) transient perme-

ation, and (2) sorption (or desorption).

In a transient permeation study, C

and C

are

zero. The gas molecules diffusing out of the mem-

brane at x =l (i. e., the downstream side) are collected

in a closed volume of known size. The gas ﬂux [i. e.,

−D(∂C/∂x)

x=l

] can be easily calculated from the rate of

gas accumulation in the closed volume. The total amount

of gas transported across the membrane at time t, Q

,is

given by



−D



∂C

∂x



x=l

DtC

−

2lC

∞



n=1



−1



exp



−Dn

t/l



, (7.139)

which reduces to the following equation when the gas

ﬂux reaches a steady state or, equivalently, as t →∞



t −



(7.140)

Part C 7.6

430 Part C Materials Properties Measurement

Verification

curve

Time

Fig. 7.106 A typical curve for the amount of accumulated

permeate Q

as a function of time in a transient permeation

experiment

A typical curve for the amount of accumulated gas as

a function of time appears in Fig. 7.106. The intercept

of the steady-state region of Q

with the time axis (i. e.,

/6D) is deﬁned as the time lag θ. The diffusion coefﬁ-

cient is given by

D =l

/6θ. (7.141)

The above theoretical analysis was developed by Daynes

in 1920 [7.281]. In 1939, Barrer and coworkers reported

a system to measure the time lag and steady-state ﬂow

in polymers. Using this instrument, permeability, dif-

fusivity and solubility can be estimated from a single

experiment [7.282].

In a kinetic sorption or desorption study, the migra-

tion of gas into or out of a ﬁlm is monitored by measuring

the weight change of the polymer ﬁlm containing the

sorbed gas as a function of time. Typically, a polymer

ﬁlm with an initial gas concentration C

is suddenly ex-

posed to an environment with a constant gas pressure

,i.e.,att > 0, C = C

at x = 0andl. C

is the con-

centration of gas in the polymer in equilibrium with the

applied gas pressure p

.WhenC

> C

, sorption occurs,

and penetrant diffuses into the ﬁlm. Desorption happens

when C

< C

. The kinetics of these processes are given

by [7.261]

∞

=1−

∞



n=0



2n +1



exp



−D



2n +1



t/l



(7.142)

where M

and M

∞

denote the total amount of penetrant

sorbed or desorbed in the ﬁlm at time t and at equilib-

rium, respectively. For a ﬁlm of cross-sectional area A

and thickness l, M

is given by







x, t



dx −C



(7.143)

and M

∞

is given by

∞

−C

Al . (7.144)

Therefore, by determining the weight change of the

sample with time, diffusion coefﬁcients can be ob-

tained by ﬁtting the experimental data to (7.142). When

∞

> 0.7, all terms with n > 0in(7.142) can be

neglected, and (7.142) becomes

∞

=1 −

exp



−Dπ

t/l



, (7.145)

which can be rearranged to [7.261]



1−M

∞



= ln

−

Dπ

t . (7.146)

In this way, the diffusion coefﬁcient can be easily ob-

tained by ﬁtting the experimental data to (7.146).

Another way to analyze kinetic sorption data is based

on an alternative analytical solution to (7.138), which

is [7.261]

∞





1/2



−1/2

∞



n=1



−1



i erfc

√



. (7.147)

The above solution can be simpliﬁed to the following

equation at short times (i. e., M

∞

< 0.6)

∞



16D

πl



1/2

. (7.148)

Equation (7.148) predicts that the amount of penetrant

sorbed or desorbed exhibits a linear dependence on t

1/2

and the diffusion coefﬁcient can be calculated from the

slope of the fractional uptake (or desorption) kinetics

versus t

1/2

In general, penetrant solubility can be derived from

∞

and the initial concentration C



∞



, (7.149)

where p

is the gas-phase pressure at equilibrium. The

+sign is for sorption, and the −sign is for desorption.

Variable Diffusion Coefﬁcient

In many situations, penetrant diffusion coefﬁcients

vary with concentration [7.261]. For example, strongly

Part C 7.6

Mechanical Properties 7.6 Permeation and Diffusion 431

sorbing penetrants in rubbery or glassy polymers

often swell the polymer at sufﬁciently high activ-

ity, leading to changes in polymer properties (e.g.,

glass-transition temperature, fractional free volume,

etc.) and penetrant diffusion coefﬁcients [7.264].

Frisch developed an explicit expression for the time

lag when the diffusion coefﬁcient is concentration-

dependent [7.283]. Crank treated this subject in detail

using both analytical and numerical methods [7.261].

Strictly speaking, most of these treatments require

knowledge of the functional form of the relation-

ship between diffusion coefﬁcient and concentra-

tion [7.261]. In contrast, the steady-state measure-

ments discussed in Sect. 7.6.1 can provide a reliable

way to evaluate diffusion, solubility and permeability

coefﬁcients without making a priori assumptions re-

garding the concentration dependence of the diffusion

coefﬁcient.

Various experimental approaches have been re-

ported to measure gas transport through polymeric

ﬁlms. In principle, most of the techniques directly mea-

sure gas ﬂux through the ﬁlm or gas uptake by the

ﬁlm (i. e., solubility). Diffusion coefﬁcients are typi-

cally derived indirectly from permeability and solubility

measurements based on the theory introduced earlier. In

the following sections, general strategies are described

for measuring gas transport properties in polymers.

7.6.3 Experimental Measurement

of Permeability

Film Preparation

The measurement of gas ﬂux requires the success-

ful preparation of a uniform, nonporous thin ﬁlm. In

general, polymer ﬁlms for permeation testing have

thicknesses of less than 250 μm, unless special situa-

tions require a thicker ﬁlm. For example, thicker ﬁlms

may make it easier to prepare nonporous specimens

for study, or they may be used to increase the time

lag when studying penetrants with high diffusion co-

efﬁcients (e.g., He or H

). Uniformity of thickness is

important for reducing the uncertainty in permeability,

as illustrated in (7.124). A nonporous ﬁlm is required

if one wishes to determine the inherent properties of

the material under study, because gas ﬂow through

a pinhole obeying Knudsen diffusion can be orders of

magnitude faster than that through a dense ﬁlm obeying

the solution–diffusion mechanism [7.258]. Thus, any

pinhole defects in a sample can compromise the accu-

rate measurement of gas ﬂux through a polymer ﬁlm.

The ﬁrst challenge in successfully measuring gas per-

meation properties typically lies in the ability to prepare

uniform, pinhole-free ﬁlms.

Polymer ﬁlms are typically prepared in the labora-

tory by melt-pressing and solvent-casting [7.284, 285].

Melt-pressing is used less often than solvent-casting

to prepare ﬁlms for study. In melt-pressing, one ap-

plies heat to melt a polymer powder, and then a ﬁlm is

made by pressing the molten polymer under high pres-

sure. Solvent-casting is a widely used method to prepare

ﬁlms. In this method, solid polymer is dissolved in a sol-

vent, the resulting solution is cast on a leveled support,

and the solvent is allowed to evaporate slowly, leaving

a solid ﬁlm behind.

Gas permeation properties of ﬁlms are often inﬂu-

enced by many processing factors. One critical factor

is the solvent used in preparing the ﬁlm, which can

strongly affect polymer morphology and, in turn, gas

permeation properties. For example, poly(4-methyl-1-

pentene) ﬁlms were prepared by solvent-casting using

chloroform and carbon tetrachloride [7.285]. The N

permeability at 35

◦

C and 2 atm is 1.1 Barrers for

a ﬁlm prepared from chloroform solution, but it is

6.0 Barrers for a similar ﬁlm prepared from carbon

tetrachloride solution, even though the two ﬁlms ex-

hibit almost the same crystallinity values (i. e., 57%

and 56%, respectively) [7.285]. Other possible factors

inﬂuencing gas transport properties include polymer

concentration, evaporation temperature, annealing con-

ditions, etc. Pinholes in a solid ﬁlm can be introduced

during the solvent-casting process by bubbles present

in the solution before casting, the presence of insoluble

impurities in the solution (such as dust), and very rapid

solvent evaporation. Typically, before casting, the solu-

tion is ﬁltered to remove any impurities, and the solvent

evaporation rate is controlled during ﬁlm drying. For

highly crystalline polymers such as poly(ethylene ox-

ide), thermal annealing above the melting temperature

could be critical to form defect-free ﬁlms, although the

reason for this is not clear [7.275].

Uniform thickness ﬁlms can be obtained by spread-

ing the solution onto a support using tools such as

a Gardner knife or a doctor blade, which controls the

liquid ﬁlm thickness [7.284]. Alternatively, the solution

may be poured into a glass ring resting on a ﬂat solid

support. The glass ring is typically stuck to the sup-

port using silicon caulk to achieve a leak-free boundary.

The ﬁlm thickness is controlled by the amount of liquid

solution added and its concentration. The solid support

needs to be leveled to obtain ﬁlms of uniform thickness.

The choice of the support material can be important. If

the solution cannot wet the support, the liquid polymer

Part C 7.6

432 Part C Materials Properties Measurement

solution will bead up, and it will not form a continu-

ous ﬁlm. If the solid ﬁlm adheres too strongly to the

support, it can be difﬁcult or impossible to remove the

ﬁlm intact from the support. Common supports are glass

plates, Teﬂon plates, metal plates, plastic plates (which

do not interact with the solvent) and even liquid surfaces

such as water and mercury [7.284].

Permeation Cell

Gas permeability is the steady-state ﬂux through a ﬁlm

normalized by the pressure difference and ﬁlm thick-

ness, as indicated in (7.124). Pressure is measured using

commercial sensors or gauges with high precision in

different pressure ranges. For example, Dresser Instru-

ments (Shelton, USA) provides a digital indicator for

pressure measurement with an accuracy of 0.025% of

full scale over the pressure range 0–20 atm. Film thick-

ness can be measured using a digital micrometer (Mi-

tutoyo Corp., Japan), which can measure thickness to

the nearest micrometer. To measure the thickness of rub-

bery polymers, the ﬁlm should be coveredwith a uniform

thin plate (quartz, glass or other ﬂat, rigid materials) to

prevent direct contact of the measuring tip of the digital

micrometer with the soft ﬁlm. The measurement tip can

press into ﬂexible ﬁlms or even penetrate them; in either

case, the thickness value provided by the micrometer is

an underestimate of the true ﬁlm thickness.

The gas ﬂux through the ﬁlm, which is the key

parameter for permeability measurements, is typically

measured from the permeate side (x =l) using a perme-

ation cell. Figure 7.107a–c presents schematics of typ-

ical permeation cell designs for constant-pressure var-

iable-volume, constant-volume variable-pressure and

mixed-gas systems, respectively. These cells are com-

O-ring

Feed

Membrane

Permeation

cell

Feed

Residue

Feed

Residue

Sweep in

Sweep out

Permeate

Purge

Permeation

cell

Permeate

Metal mesh

support

Flange

a) c)b)

Fig. 7.107a–c Schematic of a typical permeation cell (a) for pure-gas measurements using the continuous ﬂow method;

(b) for pure-gas measurements using the constant-volume variable-pressure method and (c) for mixed-gas measurements

mercially available from Millipore Corporation (Biller-

ica, USA) under the tradename of high-pressure ﬁlter

holder. A ﬂat polymer ﬁlm with the same size as the inner

diameter of the cell is placed on a sintered-metal sup-

port inside the cell, dividing the system into upstream

(i. e., above the ﬁlm) and downstream (i. e., below the

ﬁlm) compartments. A silicon or Viton O-ring is often

used to prevent gas leakage at the ﬁlm edge from the

upstream (i. e., high-pressure) to the downstream (i. e.,

low-pressure) side of the cell and to avoid gas exchange

between the upstream portion of the cell and the exterior

atmosphere. In this way, only the feed gas permeating

through the ﬁlm is collected in the downstream section

of the cell. For a pure-gas study, the upstream portion of

the cell can be designed as a dead end, and gas permeat-

ing into the downstream side of the ﬁlm will be collected

as illustrated in Fig. 7.107a and b. For mixed-gas stud-

ies, the gas ﬂow pattern requires a special cell design to

minimize gas composition changes in the upstream and

downstream chambers of the permeation cell. The up-

stream gas enters the cell at the center of the ﬁlm, ﬂows

radially to the ﬁlm edge and then ﬂows out of the cell,

which eliminates any trapped gas or dead zones in the

cell [7.286]. Typically, trapped gas does not mix well

with the entering feed gas, which results in gas concen-

tration variations across the ﬁlm and, therefore, errors in

estimating the partial pressure that the ﬁlm experiences

during the measurement. The design of the permeate side

follows similar principles. Additionally, to reduce the

change of feed gas composition during ﬂow from the

ﬁlm center to the edge, due to the preferential permeation

of a gas component, the gas ﬂow rate in the upstream

is much higher than the gas ﬂow rate due to permeation

through the ﬁlm. Typically, the ratio of the permeate gas

Part C 7.6

Mechanical Properties 7.6 Permeation and Diffusion 433

ﬂow rate to feed gas ﬂow rate (the so-called stage cut)

wouldbe1%orless.

Films are typically supported on ﬁlter paper and then

on a sintered-metal support. In general, the mass transfer

resistance of the paper and the metal support should be

negligible relative to that of the polymer ﬁlm so that the

measured gas ﬂux reﬂects only the inherent properties of

the ﬁlm. The practical ﬁlm area available for gas diffu-

sion is not the same as the area of the entire ﬁlm, because

the contact area between the O-ring and the ﬁlm area is

not accessible to gas diffusion. Typically, after the cell

is clamped in place for measurements, the O-ring would

leave an imprint on the ﬁlm. The inside diameter of the

imprint is often used to calculate the active test area. In

many cases (for example, ﬁlms that are too soft or vis-

cous or when the area of the ﬁlms is smaller than the cell

size), ﬁlms need to be partially masked using imperme-

able aluminum tape so that there is no direct contact of

the O-ring with the ﬁlms [7.275,287, 288].

Temperature Control

Thermal uniformity is an important factor since gas per-

meability can depend strongly on temperature [7.265].

Two typical media are used as temperature-control ﬂu-

ids: air and liquids. An air bath can conveniently provide

operational temperatures above room temperature. Such

a system includes a fan to circulate air past a heating

element and the permeation instrumentation, a heater

element (which can be as simple as a light bulb for

temperatures near ambient or commercially available

heating elements from, for example, Omega Engineering

Inc. (Stamford, USA)) and a temperature control system.

Commercially available temperature controllers, such as

a CN76000 microprocessor-based temperature/process

controller made by Omega Engineering, work well

for this purpose. A liquid bath, such as a mixture of

methanol and water (50/50 by weight) can provide tem-

peratures as low as −30

◦

C. For low-temperature opera-

tions, cooling is typically accomplished using a chiller.

Generally, good circulation in the bath is required to

achieve a uniform temperature around the cell and other

components of the measurement system.

7.6.4 Gas Flux Measurement

Gas ﬂux is commonly measured using one of the fol-

lowing three methods: (1) constant-pressure variable-

volume, (2) constant-volume variable-pressure and

(3) a method using a special sensor for mixed-gas per-

meation measurements. These techniques are discussed

below.

Constant-Pressure Variable-Volume Method

A schematic of a constant-pressure variable-volume

(or continuous ﬂow) permeation system is shown in

Fig. 7.108. A schematic of a typical cell for such a sys-

tem is given in Fig. 7.107b. The feed gas enters the cell

and leaves in the residue stream, and the downstream

is typically purged using the feed gas before beginning

measurements to remove any impurities from the down-

stream. This apparatus operates by applying a target gas

at a constant pressure to the upstream face of the ﬁlm

and measuring the resulting steady-state gas ﬂux on the

downstream side of the ﬁlm or membrane being tested

using a ﬂow meter. Electronic ﬂow meters are commer-

cially available, such as the Agilent model ADM1000

(Wilmington, USA). However, a simple and convenient

device is a soap-bubble ﬂow meter (Alltech Associates,

Inc. Deerﬁeld, USA). By timing the movement of the

soap ﬁlm in a graduated capillary tube, the steady-state

permeability (cm

(STP)cm/cm

scmHg)isgivenby

− p

273p

atm

TA76





, (7.150)

where the downstream pressure p

is atmospheric pres-

sure (cmHg) in this case, p

atm

is the atmospheric

pressure (cmHg), T is the absolute temperature of the

gas in the bubble ﬂow meter (K), and dV/dt is the

steady-state volumetric displacement rate of the soap

ﬁlm (cm

/s). The accuracy of this method depends

on the sensitivity of the bubble ﬂow meter. Typically,

the accuracy of the bubble ﬂow meter becomes higher

as the capillary tube diameter decreases. This method

is typically used to study polymeric ﬁlms with high

gas ﬂuxes, such as poly(dimethylsiloxane) (PDMS),

poly(1-trimethlsilyl-1-propyne) (PTMSP) and compos-

Gas cylinder

Vent

Cell

Valve

Bubble flow

meter

Fig. 7.108 Schematic of a constant-pressure variable vol-

ume apparatus for gas permeability measurements (R: reg-

ulator; P: pressure transducer). The parts within the dashed

box are in a temperature-controlled chamber

Part C 7.6

434 Part C Materials Properties Measurement

ite membranes with very thin polymer coatings on

porous membrane supports [7.289]. The authors have

good experience with such materials and bubble ﬂow

meters with ranges such as 0–1.0ml, 0–10ml and

0–100 ml, which are made by Alltech Associates, Inc.

(Deerﬁeld, USA).

Constant-Volume Variable-Pressure Method

As illustrated in Fig. 7.109, a constant-volume variable-

pressure system measures permeate ﬂux by monitoring

the pressure increase of collected permeate gas in

a closed volume using a pressure transducer. A typi-

cal cell is presented in Fig. 7.107a. Unlike the cell used

for the continuous-ﬂow method, the cell used here does

not require a residue or purge stream since any volatile

impurities or air gases can be removed from the sam-

ple by exposing the whole system to vacuum prior to

beginning a measurement. The apparatus operates by

initially evacuating the upstream and downstream vol-

umes to degas the ﬁlm. Then, the valve connecting the

permeation cell to the vacuum pump is closed, and

a slow pressure rise in the downstream volume (i. e.,

the leak rate of the system) is observed. This rate of

pressure rise should be at least ten times less than the

estimated steady-state rate of pressure rise due to per-

meation to obtain accurate permeability estimates. The

feed gas is then introduced to the upstream side of the

membrane, and the pressure rise in the downstream vol-

ume is recorded as a function of time. Gas permeability

(cm

(STP)cm/(cm

s cmHg)) is calculated from the

Gas cylinder

Cell

Va lve

Vacuum

pump

Computer

Fig. 7.109 Schematic of a constant-volume variable-pressure appa-

ratus for gas permeability measurements (R: regulator; P: pressure

transducer; V

, V

and V

: tubing volume, volumes 1 and 2,

respectively). The parts within the dashed box are in a temperature-

controlled chamber

following expression [7.275, 290]

ART





−





leak



, (7.151)

where V

is the downstream volume (cm

), l is the ﬁlm

thickness (cm), p

is the upstream absolute pressure

(cmHg), A is the ﬁlm area available for gas trans-

port (cm

), the gas constant R is 0.278 cmHg cm

(cm

(STP)K), T is absolute temperature (K) and

(dp

/dt)

and (dp

/dt)

leak

are the steady-state rates

of pressure rise (cmHg/s) in the downstream vol-

ume at ﬁxed upstream pressure and under vacuum,

respectively. The downstream pressure must be kept

much lower than the upstream pressure to maintain an

effectively constant pressure difference across the mem-

brane. In our laboratory, the downstream pressure is

typically lower than 0.03 atm, compared with a typi-

cal upstream pressure of 4.4 atm or more. If steady-state

permeation cannot be achieved before the downstream

pressure increases to a signiﬁcant value, the down-

stream should be evacuated using the vacuum pump,

and the recording of the pressure rise in the downstream

volume is restarted. Additionally, the design of the

downstream in Fig. 7.109 allows one to choose a suit-

able downstream volume to achieve a reasonable rate of

pressure rise for ﬁlms of different ﬂuxes. If the ﬂux is

low, the smallest volume (i. e., the volume of the down-

stream tubing only) can be used to observe the pressure

rise. On the other hand, one or two additional volumes

can be used if the ﬂux is high.

In this method, the upstream and downstream pres-

sures can be accurately measured using commercially

available pressure gauges or transducers. The key pa-

rameter to be measured is the downstream volume, for

example V

(deﬁned as the tubing volume), which in-

cludes part of the permeation cell (i. e., beneath the

ﬁlm), tubing, tubing connections and valves, as illus-

trated in Fig. 7.109. The direct measurement of V

impossible. One way to circumvent this issue is to mea-

sure volume 1, V

, including a short section of tubing

and a vessel (as illustrated in Fig. 7.109) ﬁrst by liquid

ﬁlling followed by Burnett gas expansion [7.291]. Vol-

ume 1 is ﬁlled with a volatile liquid, such as methanol,

and the value of V

equals the volume of the liquid re-

quired to completely ﬁll volume 1. To obtain the value

of V

, an impermeable aluminum foil ﬁlm is installed in

the permeation cell to isolate the downstream from the

upstream. Initially, V

and V

are under vacuum (i. e.,

the pressure is 0 cmHg). Volume 1 is isolated from V

by turning off valve 1, and gas is introduced into the tub-

ing at a pressure of p

(which is typically much lower

Part C 7.6

Mechanical Properties 7.6 Permeation and Diffusion 435

than 1 atm). Light gases such as N

and He are typically

used since they exhibit ideal gas behavior at subatmo-

spheric pressure. After the value of p

stabilizes, the

valve is opened, and the gas expands into volume 1; the

pressure reaches p

in both volumes. Assuming ideal

gas behavior, the ratio of the two volumes is given by

= p

−1 , (7.152)

which permits one to obtain an accurate value of the

tubing volume. Following the same procedure, V

can

be estimated. V

can also be estimated using the liquid-

ﬁlling method.

To simplify the data acquisition process, automatic

recording of the downstream pressure can be employed.

Typically, transducers (such as those from MKS Instru-

ments, Wilmington, MA, USA) used in the downstream

sense the pressure and give an analog voltage output

signal. The pressure is typically linearly related to the

transducer output voltage, and this relationship can be

obtained by calibrating the transducer using a standard

pressure indicator. The voltage signal can be recorded

using a chart recorder. More conveniently, the signal

can be digitized and monitored using a computer via

data-acquisition programs such as Labtech Notebook

(Laboratory Technologies Corp., Andover, MA, USA)

or Labview (National Instruments Corp., Austin, TX,

USA), which show and record the pressure in real time

and allow one to analyze the data with widely used soft-

ware such as Microsoft Excel. The required hardware,

including computer boards, terminals, readouts etc., is

provided by commercial sources, such as Measurement

Computing Corp. (Middleboro, MA, USA).

Due to the ability to measure pressure accurately,

the constant-volume variable-pressure system is able to

measure a wide range of ﬂux values, including mater-

ials exhibiting ﬂux values too low to be measured by

a constant-pressure variable-volume system.

Mixed-Gas Permeability Measurement

The measurement of mixed-gas permeability coefﬁ-

cients follows principles similar to those used for

pure-gas permeability measurements, except that the

mixed-gas measurement requires a sensor to detect gas

concentration in the feed, residue and permeate streams.

A gas chromatograph (GC) is the most widely used con-

centration detection method [7.290, 292]. A schematic

of a mixed-gas permeation instrument is provided

in Fig. 7.110 [7.289]. In general, the permeate is swept

away from the membrane surface using a carrier gas

(e.g., He, H

or N

) at a pressure of about 1 atm. The

preferred carrier gas should have a thermal conductivity

that is very different from that of the permeate gas, since

thermal conductivity detectors are widely used for con-

centration detection in the GC. Consequently, helium or

hydrogen is often used as the carrier gas [7.293]. By

measuring the permeate gas concentration in the sweep

gas and the sweep gas ﬂow rate (i. e., S), permeability

can be calculated as follows [7.289]



− p



(7.153)

where x

and x

are the mole fractions of compo-

nent A and helium in the permeate stream, respectively;

is the mole fraction of component A in the feed gas.

Mixed-gas measurements require a specially de-

signed permeation cell, as illustrated in Fig. 7.107c, to

ensure good mixing in the gas phase above and below

the membrane. Low stage-cut values (the stage cut is the

ratio of the permeate gas ﬂow rate to the feed gas ﬂow

rate) are used to improve mixing efﬁciency on the feed

side of the membrane. Typically, the stage cut is set to

be less than 1%; that is, the permeate ﬂow rate is less

than 1% of the feed ﬂow rate. Additionally, this method

can be used for pure-gas studies. Compared with the

continuous ﬂow system described earlier (i. e., constant

pressure/variable volume system), this apparatus is able

to measure low permeate ﬂux, mainly because of the

high sensitivity of GC for gas composition detection

and the ease of measurement of the high sweep gas ﬂux

using a bubble ﬂow meter.

An alternative technique can also be used for mixed-

gas permeability measurement [7.294]. Instead of using

a carrier gas on the permeate side of the membrane,

a closed downstream volume is used. When the gas ﬂux

Gas cylinder

R P

MFC

Cell

Carrier gas

Vent/GC

Fig. 7.110 Schematic of a mixed-gas permeability apparatus for gas

permeability measurements using a gas chromatograph for con-

centration detection (R: regulator; P: pressure transducer; MFC:

mass-ﬂow controller; GC: gas chromatograph). The parts within the

dashed box are in a temperature-controlled chamber

Part C 7.6

436 Part C Materials Properties Measurement

reaches steady state, as detected by monitoring the rate

of downstream pressure rise, the downstream is evacu-

ated to remove the gas accumulated during nonsteady-

state permeation. Afterwards, the downstream volume

is isolated, and pressure in this volume increases. Then,

a valve is opened, and the gas expands into an evacuated

line connecting the gas chromatograph to the down-

stream volume. The gas permeability can be calculated

from the steady-state rate of pressure rise in the down-

stream volume and the corresponding composition of the

permeate gas determined using the GC.

Besides GC, other sensors have been employed for

gas permeation studies. For example, oxygen transmis-

sion across ﬁlms is an important property in food pack-

aging, since the exposure of food and beverages to oxy-

gen can signiﬁcantly inﬂuence their shelf life [7.257].

The American Society for Testing and Materials

(ASTM) standard F1307 describes a constant-volume

variable-pressure system which employs an oxygen sen-

sor to detect the oxygen concentration in the downstream

volume [7.295]. Coupled with the rate of pressure rise

in the downstream volume, O

permeability can be eas-

ily calculated using (7.151). This method is particularly

convenient for mixed gas feeds containing O

and other

species, where only the O

permeability is of interest.

Transient Transport Measurement

Based on the pioneering work by Daynes and Bar-

rer [7.281,282], transient transport measurements (time

lag) have come to be widely practised. The most

common system used to detect the time lag is the

constant-volume variable-pressure system [7.266], al-

though systems employing a gas chromatograph have

also been reported [7.292]. As illustrated in Fig. 7.106,

the time lag is deﬁned as the intercept of the ex-

trapolated linear pseudo-steady-state region of the

downstream pressure rise back to the time axis. A key

issue is to ensure that the experiment has reached steady

Gas

cylinder

Vacuum

Valve

Volume

Polymer

Computer

Fig. 7.111 Schematic of a barometric pressure decay apparatus for

gas sorption measurement (dual-volume dual-transducer system)

(P: pressure transducer). The parts within the dashed box are in

a temperature-controlled chamber

state. Koros and Zimmerman proposed a procedure il-

lustrated in Fig. 7.106 [7.266]. The experiment is run

for a period that is approximately ﬁve to six time lags,

then the downstream volume is evacuated, and the mea-

surement is restarted. If the slope of the downstream

pressure rise versus time is consistent with the rate of

downstream pressure rise measured before evacuation,

then the experiment is considered to be at steady state.

This technique requires ﬁlms having good mechan-

ical properties to sustain the pressure difference across

the sample and a leak-free downstream volume. To in-

crease the sensitivity of this technique, i. e., to increase

the time lag, thicker ﬁlms can be used. As illustrated

in (7.141), time lag increases with the square of the ﬁlm

thickness. This strategy is especially useful for studies

of small penetrants such as hydrogen and helium, which

typically have high diffusion coefﬁcients and, therefore,

exhibit time lag values that are often too short to be

detected. On the other hand, (7.141) suggests that ﬁlm

thickness uniformity is very important for reducing the

uncertainty of this measurement.

7.6.5 Experimental Measurement of Gas

and Vapor Sorption

The sorption of gas in a polymer ﬁlm is usually detected

by either a decrease of gas pressure (i. e. gas-phase con-

centration) in the environment surrounding a polymer

in a closed chamber or the weight increase of a polymer

ﬁlm due to gas sorption. An apparatus is typically de-

signed to measure changes in either the amount of gas in

the gas phase or the weight change of the polymer ﬁlm

due to the sorption or desorption of the penetrant. In the

following section, common methods of measuring gas

sorption are described. They include the dual-volume,

pressure-decay method, the gravimetric method, and in-

verse gas chromatography.

The Dual-Volume Pressure-Decay Method

The dual-volume pressure-decay method employs dual

transducers and dual cells, and is widely used to deter-

mine pure-gas solubility in polymeric samples [7.296,

297]. A schematic diagram of such a system is shown

in Fig. 7.111. The system consists of a sample cell con-

taining a polymer sample and a charge cell connected to

a gas cylinder. In the ﬁrst step, the sample cell is evac-

uated and then isolated from the charge cell, which is

subsequently charged with a target gas at a known pres-

sure. The valve between the charge and sample cells is

opened brieﬂy to introduce gas to the sample cell. The

valve between the sample and charge cells is closed, and

Part C 7.6

Mechanical Properties 7.6 Permeation and Diffusion 437

the pressure in the sample cell is monitored as a func-

tion of time. The difference between the initial and ﬁnal

pressure in the charge cell can be used to calculate the

number of moles of gas admitted into the sample cell.

As the polymer sample sorbs gas, the pressure in the

gas phase of the sample cell decreases. When the sam-

ple cell pressure reaches a stable value, the polymer has

sorbed all of the gas that it can at the ﬁnal pressure in

the sample cell (i. e., equilibrium is achieved). From the

ﬁnal pressure in the sample cell, the number of moles of

gases in the gas phase of the sample cell can be calcu-

lated. The difference between the moles of gas admitted

to the sample cell from the charge cell and the ﬁnal

number of moles of gas in the gas phase of the sample

cell is the number of moles of gas sorbed by the sample

at the ﬁnal pressure of the sample cell. The second step

is to add more gas to the charge cell and repeat the pro-

cedure outlined above to obtain a data point at a higher

pressure. At step m, the number of moles of gas sorbed

in the polymer n

p,m

is given by [7.296]

p,m

p,m−1



c,m−1

RTZ

c,m−1

s,m−1



−V



RTZ

s,m−1



−



c,m

RTZ

c,m

s,m



−V



RTZ

s,m



, (7.154)

where V

is the volume of the charge cell, V

is the vol-

ume of the sample cell, and V

is the volume of polymer

sample. Z

c,m

, Z

s,m

, Z

c,m−1

and Z

s,m−1

are the com-

pressibility factors of gas in the charge cell at step m,

sample cell at step m, charge cell at step (m −1) and

sample cell at step (m −1), respectively. The subscripts

m and m −1 represent the properties at step m and

step (m −1), respectively. For example, n

p,m−1

is the

number of moles of gas sorbed in the polymer at step

(m −1). In the ﬁrst step (i. e., m = 1), the sample cell is

under vacuum, and n

p,0

is zero. From these sequential

measurements, the gas sorption isotherm in the polymer

can be obtained, and the gas solubility as a function of

gas pressure can be calculated based on the deﬁnition of

solubility: S =C/ p.

Compressibility factors are related to pressure

by [7.298]

Z =1 +B

∗

p +C

∗

, (7.155)

where B

∗

and C

∗

are virial coefﬁcients which depend

on temperature. They have been compiled by Dymond

and Smith for a large number of gases at various tem-

peratures [7.298]. Virial coefﬁcients are also available

for some gas mixtures from the same source [7.298].

In general, the amount of gas in the gas phase is

much greater than that sorbed in the polymer, i. e., the

values of the two terms in brackets on the right-hand

side of (7.154) are much larger than n

p,m

or n

p,m−1

.In

this sense, the value of n

p,m

is the result of a subtrac-

tion between two much larger numbers. Therefore, the

accuracy of n

p,m

is sensitive to the calibration of the

pressure transducers and volumes. A typical transducer

used for such studies is the Super TJE model (Sensotec,

Columbus, OH, USA), which measures a pressure range

of 0–34 atm with an uncertainty as low as ±0.05% of

full scale (i. e., ±0.017 atm). It is one of the most ac-

curate transducers commercially available. Generally,

the transducers sense pressure and provide an analog

signal such as a voltage, which can be related to the

pressure measured by a standard pressure gauge (such

as the PM indicator made by Dresser Instruments, Shel-

ton, CT, USA) using a linear or polynomial equation.

The volume calibration can be performed by a two-step

Burnett expansion process [7.291]. In the ﬁrst step, gas

(typically helium) in the charge cell at a pressure p

expanded to the empty sample cell, which is kept under

vacuum before the expansion. The pressure in both cells

reaches a stable value of p

. Based on a mass balance,

the following equation can be derived

RTZ





RTZ

, (7.156)

where Z

and Z

are compressibility factors at pres-

sures of p

and p

, respectively. Substituting (7.155)

into (7.156) and letting the third virial coefﬁcient,

∗

, equal zero (since C

∗

is a very small number for

helium [7.298]andC

∗

is typically negligible for he-

lium), yields

∗

. (7.157)

By varying p

, one can generate data to prepare a plot

of p

as a function of p

. The resulting line inter-

sects the p

axis at (1 +V

). The second step

is to vary the volume of the sample cell by inserting

some nonsorbing solid of known volume V

,suchas

stainless-steel spheres, into the sample cell. Repeating

the Burnett expansion step outlined above, the follow-

ing equation is obtained

∗



−V



−V

. (7.158)

Again, one can perform gas expansions as a function of

pressure to ﬁt this model and determine (V

−V

)/V

Part C 7.6