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

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418 Part C Materials Properties Measurement

the temperature of fracture test, z is distance of the notch

opening gage location above the surface of the speci-

men and V

is the plastic component of critical notch

opening displacement, as deﬁned by Fig. 7.89.

CTOD Testing of Welds

Fracture toughness evaluation of welds is more compli-

cated than that of base metal because the welds have

heterogeneities in terms of microstructure and tough-

ness. Therefore, toughness values of welds inherently

show large scatter. The existence of local brittle zones

(LBZ) in the heat-affected zone (HAZ) sometimes gives

extremely low toughness values. From this viewpoint,

special attention should be paid to CTOD testing for

welds.

Figure 7.90 shows the variation of critical CTOD

values of high-strength-steel weld HAZ as a fraction of

the coarse-grained (CG) HAZ along the fatigue precrack

front in the thickness direction [7.245]. This result indi-

cates that at least 15% of the CG HAZ should be sampled

by the fatigue precrack to assess the lower-bound CTOD

value; see below for the deﬁnition of %CG HAZ.Most

of the CTOD testing standards of weld HAZ specify

the %CG HAZ for this reason. Typical standards of for

CTOD testing method for welds are speciﬁed in [7.246,

247]. Among these, API RP 2Z [7.246] was the ﬁrst to

specify a CTOD testing method for weld HAZ.Other

standards follow the same concept. The API RP 2Z stan-

dard is intended for preparing and certifying fabrication

welding procedures and assuring that the steel to be sup-

plied is inherently suitable for welding.

Critical CTOD (mm)

0 5 10 15 20 25 30 35 40 45 50

0.1

0.01

% CG regions

Fig. 7.90 Critical CTOD values versus % CG regions for steels

showing some LBZ behavior (after [7.245])

API RP 2Z refers to BSI 7448 part 1 [7.238]and

ASTM E 1290-02 [7.243]fortheCTOD testing method.

Additional requirements are speciﬁed regarding the

notch positioning in the weld and post-test examinations

for sampling the CG HAZ regions, which is thought

to have lower-bound toughness. Figure 7.91 shows the

HAZ regions in a multipass weld. Because of the multi-

thermal cycles induced by multipass welding, the HAZ

has an extremely complex microstructure distribution.

It is intended that the fatigue precrack certainly sam-

ples the CG HAZ. For this purpose, the fatigue precrack

should be carefully introduced. Welding should be made

carefully so that the fusion line of the weld is straight

enough to sample as much CG HAZ as possible. It

should be noted that proper welding consumables and

welding procedures should be adopted to maintain a cer-

tain level of critical CTOD value for the weld metal. In

addition, welding residual stress may be relieved prior to

fatigue precracking to promote crack-front straightness.

This is done by lateral compression (Fig. 7.92)[7.248].

API RP 2Z requires three levels of weld heat inputs.

At least ﬁve or eight, depending on the heat input, CTOD

tests should be conducted for each welding condition. At

least three specimens aim to sample the CG HAZ regions

close to the fusion line and at least two specimens aim to

sample the boundary region between the HAZ and base

metal.

Post-test metallographic examinations are speciﬁed.

Figure 7.93 shows the sectioning method for this pur-

pose. The broken half of the specimen is sectioned at the

fatigue precrack front, etched and its microstructure is

revealed. The CG HAZ regions sampled by the fatigue

precrack tip is measured (Fig. 7.94), and the %CG is cal-

culated as

%CG =

100



, (7.115)

where L

is length of the i-th CG region sampled by the

crack front and B is the plate thickness. The CG regions

need not be continuous. The value of %CG should be

at least 15% for the three specimens. For the other two

specimens, the crack front should sample at least 50%

of the boundary region between the HAZ and the base

metal.

The CTOD toughness values are inﬂuenced by many

other factors than the local brittle zones. One is the de-

gree of strength matching, i. e. the ratio of yield strength

of the weld metal to that of base metal. An overmatch

may cause excessive constraint in the HAZ,whichmay

cause unexpectedly low critical CTOD values.

Part C 7.5

Mechanical Properties 7.5 Fracture Mechanics 419

Etched HAZ

in weld metal

(CG, FG and

ICHAZ)

SCHAZ

in weld

metal

Columnar weld metal

Etched HAZ

Unaltered FGHAZ

Unaltered CGHAZ

Unaltered ICHAZ

Unaltered SCHAZ

Etched HAZ SRCG

IRCG

Fig. 7.91 Schematic of microstruc-

tures in HAZ of a multi-pass welding

(after [7.246])

7.5.3 Fatigue Crack Propagation Rate

Fatigue crack propagation results from the cyclic plastic

strain cumulation at the crack tip, especially in sta-

ble crack propagation. It is expected that the crack

propagation rate da/dN can be related to the stress-

intensity factor range ΔK (= K

max

−K

min

). Since Paris

et al. [7.249, 250] ﬁrst demonstrated the characteriza-

tion of crack growth by fatigue in the early 1960s,

a considerable number of works have covered the de-

termination of crack growth behaviors using fracture

mechanics. Figure 7.95 shows typical fatigue crack

growth behavior in metallic materials, which is ex-

pressed in a schematic log–log relationship between

da/dN and ΔK; ΔK is usually deﬁned as the posi-

tive component of the total stress-intensity factor range,

which is

ΔK = K

max

−K

min

=(1 −R)K

max

for R ≥ 0

ΔK = K

max

for R ≤ 0 (7.116)

where R is the stress ratio (=σ

min

/σ

max

= K

min

max

The sigmoidal curve in Fig. 7.95 can be divided into

three regions. Region I is a lower bound against crack

growth, which is characterized by a threshold value

ΔK

.BelowΔK

, there is no observable fatigue crack

growth. In region II, for intermediate ΔK values, the re-

B dia.

≤1% B

(or 0.5% B on each side)

P=1× 4 B

Section A–A

Fig. 7.92a,b Lateral compression prior to fatigue precracking, for

relieving welding residual stress (after [7.248])

Part C 7.5

420 Part C Materials Properties Measurement

Fracture

face

Section A–A

Machined notches

Fatigue crack

Fig. 7.93 Sectioning the weld half of

a CTOD specimen for post-test metal-

lographic examination (after [7.246])

lationship in Fig. 7.95 can be ﬁtted with a linear line,

which is formulated as

da/dN =C(ΔK)

, (7.117)

where C and m are both materials constants, which

should be determined experimentally. The fatigue crack

growth rate in region II is mainly controlled by ΔK,

and is insensitive to the R ratio. This power-law rela-

tionship for fatigue crack growth in region II was ﬁrst

proposed by Paris and Erdogan [7.250]and(7.117)

is often referred to as Paris’ law. In region III, crack

growth is accelerated. Even if the applied stress am-

plitude Δσ is constant, ΔK shows a rapid increase

due to fast crack extension and the increased ΔK en-

hances the acceleration in crack growth. When the

max

[=ΔK/(1 −R)] reaches the fracture toughness

of the material K

, the crack growth transforms into

instability, which corresponds to an inﬁnite value in

da/dN.

The ASTM has standardized the measurement

methodology for the fatigue crack propagation rate

based on fracture mechanics since 1978 [7.251]. The

following is a summary of the ASTM standard E 647.

The compact tension C(T) specimen with a side-edge

crack and the middle tension specimen M(T) with

a center crack are prescribed. The conﬁgurations of

the C(T) and M(T) specimens are shown in Figs. 7.96

and 7.97, respectively. The C(T) specimen is recom-

mended only for tension–tension loading while the

M(T) specimen can be loaded in either tension–tension

Part C 7.5

Mechanical Properties 7.5 Fracture Mechanics 421

Section A–A

Fig. 7.94 Calculation of %CG HAZ sampledbythefa-

tigue precrack tip (after [7.246])

or tension–compression. The conﬁgurations of both

specimens are given with proportional dimensions of

the specimen width W. The C(T) specimen prescribed

here has the same conﬁguration as the C(T) spec-

imen in the toughness measurement except for the

specimen thickness B. It is not necessary to adopt

a large thickness in the fatigue crack growth, be-

cause the plane–strain concept is not required in the

da/dN measurement. Small specimen thicknesses are

rather better because a thumbnail crack front possi-

bly enhances the uncertainty in the measurement of

the crack length. From these points of view, the spec-

imen thickness requirement in a fatigue crack growth

measurement can be ﬂexibly chosen within the recom-

mended range.

The formula for the stress-intensity factor K for each

type of specimen is given in the standard, being a func-

tion of the specimen dimensions and applied load P.It

is the ΔK that is calculated from the positive component

da/dN (m/cycle)

–4

–5

–6

–7

–8

–9

–10

Shedding ΔK

ΔK

start

log ΔK

Quick dropped ΔK

Region IIIRegion IIRegion I

Paris low

da/dN = C (ΔK)

max

= K

Fig. 7.95 Schematic relation between stress intensity

range ΔK and crack growth rate da/dN

0.25 W Dia.

0.6 W

0.275 W

0.2 W

1.25 W

Fig. 7.96 Compact tension C(T) specimen

of an applied load amplitude ΔP (referring to (7.116))

and an extending crack length a (=a

+Δa).

A sharpened starter crack must be introduced by

fatigue from a machined notch. Sufﬁcient size and

straightness of a fatigue crack must be kept to elim-

inate the effect of the machined starter notch on the

K calibration and the effect of the crack-front shape

on the subsequent crack growth rate data. The ma-

chined notch outline must lie within the 30

◦

angled

lines whose apex is at the end of the fatigue precrack

showninFig.7.98. The length of the precrack should

Part C 7.5

422 Part C Materials Properties Measurement

W/3 Dia.

W W

W/2

min.

1.5 W min.

0.2 W

Fig. 7.97 Middle tension M(T) specimen

not be less than 0.1B, h or 1 mm, whichever is greater.

The K

max

value during the fatigue loading for precrack-

ing must be suppressed below the initial K

max

value

for which the da/dN data are of interest. Precrack-

ing growth rates less than 10

−8

m/cycle are suggested,

especially when near-threshold growth rates are to be

obtained. The precrack sizes measured on the front and

back surfaces must not differ by more than 0.25B.

In da/dN measurements, the crack extension

should be measured as a function of elapsed cycles

under a controlled load amplitude. The crack can be

visually measured with a traveling microscope. The

compliance variation that is the crack opening displace-

ment response for a speciﬁed load and the electric

potential difference can be also used to estimate the

crack growth, similarly to the detection of the J

test methodology prescribed in ASTM E 1820 [7.237].

Bisual measurement is recommended, together with

either the compliance technique or the electric potential-

difference technique. In all methods, a minimum

resolution below 0.1mm or 0.002W (whichever is

greater) is required. Crack size extension is measured

at a Δa interval larger than 0.25 mm. However, the

Δa measurement in each crack extension of 0.25 mm

is sometimes difﬁcult in near-threshold (ΔK

) testing,

where it is required that there are at least ﬁve da/dN

data points despite the fact that crack extension has al-

most stopped. In this case, the measuring interval Δa

can be reduced by ten times the crack size measurement

precision.

A loading procedure with either ΔK increasing or

ΔK decreasing must be selected, depending on the

da/dN value that is to be measured. The ΔK-increasing

procedure is well suited to fatigue crack growth rates

above 10

−8

m/cycle. However, it becomes gradually

W/16

Chevron

Straight through

Hole and slot for M(T) sp.

30°

Fig. 7.98 Examples of fatigue precrack from machined

notch

more difﬁcult for small initial da/dN values because

the applied ΔK should be smaller than that in the pre-

cracking.

The ΔK-decreasing test procedure can be con-

ducted for rates below 10

−8

m/cycle. In the ΔK-

increasing test procedure, an applied load range (ΔP)

and other loading variables (stress ratio and frequency)

are ﬁxed constant. The schematic variation of the ΔK

value and the crack growth Δa as a function of elapsed

cycles are shown in Fig. 7.99a.

In the ΔK-decreasing test procedure for da/dN

measurements below 10

−8

m/cycle, ΔK and K

max

must start from a level equal to or greater than the ter-

minal precracking values. Subsequently, the load ΔP

is shed as the crack grows, until the lowest ΔK or

crack growth rate of interest is reached. Load shed-

ding is usually performed as decreasing load steps at

speciﬁed crack size intervals, as shown in Fig. 7.99b.

The shedding of load should be gradual enough to

preclude anomalous data resulting from sudden reduc-

tions in the stress-intensity range. The load reduction

in each shedding step should not exceed 10% of the

maximum load P

max

in the previous load step. Af-

ter a step reduction, a minimum crack extension of

0.5 mm is recommended. The da/dN measured us-

ing these shedding rules sometimes shows a trace

different from the true material properties, as shown

in Fig. 7.95. This mainly results from strain harden-

ing and compressive residual stress in the crack–tip

Part C 7.5

Mechanical Properties 7.5 Fracture Mechanics 423

Force range ΔP

Applied load range ΔP

Stress intensity factor range Δ K

Crack growth behavior

Applied load range ΔP

ΔK(ΔP, a)

Force range Δ P

Crack length a

Cycles N

a) b)

ΔK-increasing test ΔK-decreasing test

Fig. 7.99a,b Correspondences of ap-

plied load with crack growth and

stress-intensity factor range. (a) ΔK-

increasing test, (b) ΔK-decreasing

test

plastic zone formed in the previous load step. This

mis-measured da/dN–ΔK relation usually gives an

overestimation in the fatigue crack growth resistance of

the materials.

The crack growth rate is determined from the crack

size versus elapsed loading cycles data (a–N curve).

The secant and incremental polynomial methods are

recommended. Both methods are suitable for the ΔK-

increasing test. For the ΔK-decreasing tests, where load

is shed in decremented steps, the secant method is rec-

ommended. A crack growth rate determination should

not be made over any increment of crack extensions that

include a load step. The secant method simply involves

a calculation of the slope of the straight line joining two

adjacent data points on the a–N curve. It is formally

expressed as follows

(da/dN)

=(a

i+1

−a

)/(N

i+1

−N

) . (7.118)

Since the da/dN value calculated in this way is an

average rate over the a

i+1

−a

increment, the average

crack size

a =(a

i+1

−a

)/2 is used to calculate ΔK.

On the other hand, the incremental polynomial method

involves ﬁtting a second-order polynomial to a set of

(2n +1) successive data points, where n is usually taken

to be 1–4. The form of the equation for the local ﬁt is

as follows

−C

)/C



−C

)/C



(7.119)

where −1 ≤ (N

−C

)/C

≤+1andb

, b

,andb

are the regression parameters that are determined

by the least-square approximations over the range

i−n

≤a ≤a

i+n

. The value

is the ﬁtted value of the

crack size at N

. The parameters C

= (N

i−n

i+n

)

and C

=(N

i−n

−N

i+n

) are used to scale the input data.

The crack growth rate da/dN at N

is calculated from

the derivative of the parabola above, as follows

(da/dN)

+2b

−C

)/C

. (7.120)

The ΔK value associated with this da/dN value is

calculated using the ﬁtted crack size

corresponding

to N

Both recommended methods can give the same av-

erage da/dN response. However, the secant method

often results in increased scatter in da/dN relative to

the incremental polynomial method, since the latter nu-

merically smoothes the data.

The K

max

value of all data must be examined within

the linear elastic fracture mechanics. When the data

Part C 7.5

424 Part C Materials Properties Measurement

Fig. 7.100a–c Three of the most common fracture mechanisms:

(a) microvoid coalescence fracture; (b) cleavage fracture; (c) inter-

granular fracture

Fig. 7.101a–d SEM micrographs for most common fracture ap-

pearances: (a) microvoid coalescence fracture/dimple pattern;

(b) microvoid coalescence fracture/dimple pattern; (c) cleavage

fracture/river pattern; (d) intergranular fracture

satisfy the specimen size requirements below, the spec-

imen can be predominantly identiﬁed as elastic.

W −a ≥(4/π)(K

max

/σ

)

for a C(T) specimen,

W −2a ≥1.25P

max

/(Bσ

) for an M(T) specimen.

(7.121)

This specimen size requirement here is rather more re-

laxed than that in the fracture toughness K

for unstable

fracture prescribed in ASTM E-399 [7.231].

The relation as shown in Fig. 7.95 appears on the

log–log graph plots between da/dN and valid ΔK.At

least ﬁve data points of approximately equal spacing at

growth rates 10

−9

–10

−10

m/cycle are needed to deter-

mine the threshold value of ΔK, denoted by ΔK

The best-ﬁt straight line from a linear relation of

log da/dN versus log ΔK using a minimum of ﬁve

data points of approximately equal spacing at growth

rates of 10

−9

–10

−10

m/cycle. The ΔK

value is deter-

mined from the ΔK that corresponds to a growth rate of

−10

m/cycle on the ﬁtted line.

The da/dN–ΔK relation and/or the ΔK

value,

which are measured with laboratory specimens, can be

applied to the crack growth assessment of the structures.

As ΔK is a function of both an applied stress range

and a crack length, an integration of the da/dN–ΔK

relation or (7.117) with an assumed initial crack size

predicts the crack growth for a elapsed loading cycles

in service. When the calculated ΔK for a stress range

in service and an initial crack size is smaller than the

ΔK

value, no crack growth is predicted.

However, it is necessary to examine the crack clo-

sure effect in testing. Compressive residual stress and/or

crack deﬂection often suppress crack–tip opening in

spite of the applied tensile load. The effective stress-

intensity factor range for a crack–tip opening of ΔK

eff

is smaller than the apparent value, which enhances the

overestimation in the ΔK

value of the material. The

da/dN–ΔK

eff

relation is sometimes evaluated using the

compliance variation procedure or the high-stress-ratio

The effect of small cracks in practical applications

should also be considered. When an initial crack in

a structure is smaller than a certain crack size, the crack

sometimes behaves differently from Fig. 7.95.Evenfor

smaller values of ΔK than ΔK

, small cracks can

grow. da/dN for a small crack is generally faster than

the value of da/dN estimated using the long-crack

method, even for the same ΔK. This small-crack effect

may result from both the effect of discontinuities in the

microstructure and limitations of fracture mechanics.

7.5.4 Fractography

Important information can be obtained from micro-

scopic examination of the fractured surface. This study

is usually called fractography. The scanning electron

microscope (SEM) is commonly employed in fractog-

Part C 7.5

Mechanical Properties 7.5 Fracture Mechanics 425

Cleavage fracture

Fatigue precrack

Machined notch

Fatigue

precrack

SZW

Microvoid

Cleavage

Cleavage trigger

Ductile crack

(microvoid)

a) b)

Fig. 7.102a,b Fracture appearance of toughness specimen in low carbon steel: (a) macroscopic observation, and (b) mi-

croscopic observation

raphy because of its large depth of focus. Fractography

is sometimes applied in order to ﬁnd the cause of acci-

dental failures.

Figure 7.100 shows schematically three of the most

common fracture mechanisms, which are caused by

monotonic tension in metallic engineering materials.

Ductile materials usually show microvoid coales-

cence type of fracture, shown in Fig. 7.100a. Microvoids

nucleate at second-phase particles or inclusions due to

either interface decohesion or particle cracking. Af-

ter growth of the voids, adjacent voids coalesce due

to ligament fracture. The appearance of the fracture is

characterized by the dimples showninFig.7.101a. In

material with many inclusions, the dimple size on the

fracture surface is small because they coalesce easily

(Fig. 7.101b).

The BCC and hexagonal close-packed (HCP) crys-

talline systems have cleavage planes along speciﬁc

crystallographic planes. Once cleavage fracture is trig-

gered, a crack along a cleavage plane propagates

unstably. The propagating crack changes its direc-

tion each time it crosses a grain boundary, as shown

in Fig. 7.100b. Figure 7.101c shows the appearance of

the fracture, observed by an SEM. The fracture surface

consists of several ﬂat facets. The facet edges converge

to a single line like tributaries of a river. When a prop-

agating cleavage crack encounters a grain boundary,

the nearest cleavage-plane angle in the adjacent grain

is often oriented at a twist angle to the current cleav-

age plane. Cracking in a number of parallel cleavage

planes just beyond a grain boundary accommodates the

twisted-angle mismatch. This river pattern is caused by

the convergence of multiple cracks into a single crack

during its propagation within a grain.

Under special circumstances, cracks can propa-

gate along grain boundaries, as shown in Fig. 7.100c.

This intergranular fracture sometimes appears when the

strength of the grain boundary is weakened by the

following mechanisms: segregation of embrittling ele-

ments or precipitation of a brittle phase on the grain

boundary, grain-boundary cavitation at elevated temper-

ature, or attack of the grain boundary by corrosion or

hydrogen. The grain boundary, which looks like rock

candy, is shown under SEM observation in Fig. 7.101d.

The macroscopic appearance of fractures such as the

ductile and brittle types described do not necessarily

correspond to the microscopic fracture. In low-carbon

steels, however, ductile fracture is generally caused by

microvoid coalescence, and cleavage fracture leads to

brittle fracture with less ductility.

In fracture toughness tests for low-carbon steels,

cleavage fracture is often triggered after a certain crack

extension due to microvoid coalescence. Figure 7.102a

shows the macroscopic appearance of a fracture tough-

ness specimen obtained at the ductile–brittle transition

temperature. The dark thumbnail area following the

fatigue precrack shows stable crack extension due to

microvoid coalescence. A triggered cleavage crack at

the ductile crack tip propagates unstably. Figure 7.102b

shows the fracture appearance transition observed by

SEM. The stretched zone (SZ) can be observed between

the fatigue precrack and the ductile crack. The SZ is

not caused by fracture, but by a blunting deformation of

the fatigue precrack. The stretched zone (SZ) conceptu-

ally corresponds to the crack–tip opening displacement

CTOD. The fracture toughness of material can be esti-

mated from the depth and/or the width of the stretched

zone on the fracture surface.

Part C 7.5

426 Part C Materials Properties Measurement

Fig. 7.103a–d Plastic blunting

process for striation formation (af-

ter [7.252])

Propagation

3 μm

a) b)

Fig. 7.104a,b Fatigue striations in: (a) aluminum alloy, and (b) steel

Fatigue crack propagation often produces striations

on the fractured surface. Figure 7.103 shows one of the

proposed mechanisms for striation formation [7.252].

When the tensile stress is applied, the crack tip

is blunted due to the concentrated slip deformation

(Fig. 7.103b). The fatigue crack incrementally extends

as a result of the formation of this stretched zone. When

the stress changes to compression, the slip direction

at the crack tip is reversed (Fig. 7.103c). This process

causes the buckling to form a resharpened crack tip. The

ripples left by these process above can be observed as

striations.

Figure 7.104 shows typical striations for aluminum

alloy and steel. Striations are formed in a similar

manner in materials with plasticity. According to the

mechanism in Fig. 7.103, the striation spacing is equal

to the crack growth rate da/dN.

7.6 Permeation and Diffusion

The diffusion of small molecules in materials has at-

tracted signiﬁcant interest, primarily because it is of

great signiﬁcance to many applications in the chem-

ical, petrochemical and medical industries [7.253,254].

For example, a polymer matrix has the ability to ex-

hibit different permeation rates for various chemical

species, thus providing the basis for separation [7.253].

Current applications include water desalination, gas

and vapor puriﬁcation, and medical applications such

as controlled-release devices and the artiﬁcial kid-

ney [7.253, 254]. Separations using membranes have

inherent advantages over conventional separation tech-

nologies, including low energy cost (since no phase

transition is involved in the separation), high reliabil-

ity (no moving parts) and small footprint [7.255]. Of

particular interest are gas separations, where polymeric

membranes behave as molecular ﬁlters. For example,

an O

selectivity of 5.9 can be achieved in poly-

meric membranes such as polysulfone, even though the

kinetic diameter difference between O

and N

is only

0.018 nm [7.256]. Additionally, if a polymeric material

is not very permeable to penetrants such as O

and H

it acts as a barrier, which can be useful for packaging

applications [7.257].

This section reviews experimental techniques used

to study gas and vapor sorption, diffusion and per-

meation in polymeric ﬁlms. The more detailed fun-

damental theory and industrial applications pertinent

to gas transport in polymers are available in several

books [7.258–261] and review articles [7.262–264].

Speciﬁc reviews of experimental methods to measure

gas transport properties are also available [7.265–268].

In particular, Felder and Huvard reviewed experimen-

tal methods from a historical perspective and critically

evaluated the strengths and weaknesses of the various

techniques in 1980 [7.265]. Without intending to re-

view the subjective exhaustively, the main objective

of this section is to provide readers with a self-

Part C 7.6

Mechanical Properties 7.6 Permeation and Diffusion 427

consistent set of manual-like instructions to choose and

design the most suitable and reliable experimental tech-

niques for speciﬁc systems based on our laboratory’s

experience.

7.6.1 Gas Transport:

Steady-State Permeation

Gas transport through a dense or nonporous polymeric

ﬁlm is often described by the solution–diffusion mech-

anism [7.269]. As illustrated in Fig. 7.105, feed gas at

a high (i. e., upstream) pressure p

dissolves into the

feed-side surface of the ﬁlm, diffuses through the ﬁlm

due to a concentration gradient, and ﬁnally desorbs from

the permeate-side surface at the downstream (i. e., low

pressure) face of the ﬁlm. In the absence of chemical

reaction between the gas and polymer, the diffusion of

dissolved penetrant is the rate-limiting step in this pro-

cess. The one-dimensional ﬂux of gas A through the

ﬁlm in the x-direction (i. e., N

) can be described by

Fick’s Law [7.264,270]

=−D

) , (7.122)

where D is the gas diffusion coefﬁcient in the ﬁlm, C

is the local concentration of dissolved gas and w

the weight fraction of gas A in the ﬁlm; N

is the ﬂux

of the membrane, which is typically taken to be zero.

Consequently, (7.122) reduces to [7.264]

=−

1 −w

(7.123)

Steady-State Permeation

The steady-state permeability of gas A, P

, through

a ﬁlm of thickness l is deﬁned as [7.262,264,271]

≡

− p

, (7.124)

where p

and p

are the upstream (i. e., high) and down-

stream (i. e., low) pressures, respectively. Permeability

coefﬁcients are commonly expressed in Barrers, where

1 Barrer =1×10

−10

(STP)cm/cm

scmHg.Com-

bining (7.123)and(7.124) and integrating from x = 0

(C = C

)tox =l (C = C

), one obtains

= D

−C

− p

, (7.125)

Component A Component B

Upstream

Downstream

Fig. 7.105 Transport of gases A and B across a membrane

where D

is the concentration-averaged effective diffu-

sion coefﬁcient in the range C

–C

−C



1 −w

−C



eff

dC , (7.126)

where D

eff

is the local effective diffusion coefﬁcient.

In general, it can be challenging to measure the av-

erage gas diffusivity directly. Instead, gas permeability

and gas solubility are often measured independently,

and gas diffusivity is inferred from these measure-

ments, as described below. For simplicity, experiments

are often designed so that p

 p

and, consequently,

C

. In this limit, (7.125) reduces to

= D

× S

, (7.127)

where S

is the apparent sorption coefﬁcient

or solubility of penetrant A in the polymer. There-

fore, by measuring the gas permeability at an upstream

pressure of p

and solubility at a pressure of p

, one

can calculate the average gas diffusivity D

. As indi-

cated in (7.126), this diffusion coefﬁcient is an average

over the concentration range 0–C

. The local effective

diffusion coefﬁcient D

eff

, characterizing the penetrant

diffusivity in the polymer at a penetrant concentration of

, can be evaluated using an equation obtained by tak-

ing the derivative of both sides of (7.125) with respect

Part C 7.6