446

CRYSTALLINITY DETERMINATION

Vol. 9

CRYSTALLINITY
DETERMINATION

Introduction

A variety of natural and synthetic polymers are well known to crystallize on
cooling from the melt or from solution. In fact, roughly 1/2 to 2/3 of useful poly-
mers are crystalline or crystallizable. Unlike many low molecular weight organic
compounds and inorganic crystals, polymeric materials are semicrystalline and
the extent of crystallinity formation is critically important in determining ulti-
mate physical properties, and hence suitability for particular applications (see
S

EMICRYSTALLINE

P

OLYMERS

). Degrees of crystallinity vary over a wide range be-

tween crystallizable polymers, as well as within a given polymer system. The
crystallinity developed is dependent on average molecular weight and distribu-
tion (which influence crystallization kinetics), crystallization/processing condi-
tions (and its relationship to the glass-transition temperature, T

g

, of the poly-

mer), and the chemical structure of the chains [flexible like polyethylene (PE)
and poly(ethylene oxide) (PEO), or more rigid like poly(ethylene terephthalate)
(PET) and poly(ether ether ketone)]. Structural regularity of the chain is also of
paramount importance: ie, whether the chains contain comonomer units, stereoir-
regularity, etc.

This article provides an overview of common methods for determining de-

grees of crystallinity in polymer systems, focusing on isotropic materials.

X-ray Diffraction

Traditional Approaches.

For many purposes, relative degrees of crys-

tallinity (sometimes referred to as the crystallinity index) are sufficient and a
number of approximate wide-angle X-ray diffraction (WAXD) methods have been
reported and utilized. Before the advent of modern computational tools, WAXD
patterns were simply resolved into contributions from crystalline and amorphous
reflections, and a “polynomial” background (1). The degree of crystallinity was
then calculated as I

c

/(I

c

+ I

a

), where I

c

is the diffracted intensity from all resolved

crystalline reflections, I

a

the diffraction intensity under the amorphous halo, and

(I

c

+ I

a

) the total intensity. This was an improvement over older work where peak

separation was carried out arbitrarily (eg, by simply constructing a line between
intensity minima; this results in lower determined crystallinity).

Similar approaches are used today to obtain relative degrees of crystallinity

from WAXD experiments, but modern curve resolving routines are frequently
employed. Of course, the “raw” WAXD data must be of sufficiently high quality,
having low signal-to-noise ratio (ie, the data must be obtained at sufficiently slow
angular scan speeds or for sufficiently long count times), to permit reliable curve
fitting. If one is interested in absolute values of the crystallinty, it is important to
multiply (ie, correct) the observed diffraction peaks by Lorentz and polarization
factors to obtain a more accurate estimate of the total intensity diffracted by the
crystalline phase (2).

Encyclopedia of Polymer Science and Technology. Copyright John Wiley & Sons, Inc. All rights reserved.

Vol. 9

CRYSTALLINITY DETERMINATION

447

Fig. 1.

Resolution of the WAXD pattern of a multiphase POM/PS/PPO blend. The exper-

imental WAXD pattern of the ternary blend is represented by the full bold line and that of
the amorphous component by the dashed line. The result after subtraction is given by the
dotted line and this can be considered as the WAXD pattern of the POM phase alone. The
full fine line represents normalization to 100% POM. From Ref. 3

As an example, determination of the crystallinity index of polyoxymethylene

(POM) in a ternary blend with polystyrene (PS) and poly(phenylene oxide) (PPO)
is presented (3). Although the latter polymer in this mixture is crystallizable,
it does not do so under the processing conditions used in Reference 3. Degrees
of crystallinity were estimated as the ratio of the integrated areas under the
POM (100) and (105) diffraction peaks (A

110

and A

105

, respectively) to the total

integrated diffracted intensity, ie, X

c

,WAXD

= (A

110

+ A

105

)/(A

110

+ A

105

+ A

a

), where

A

a

is the area under the amorphous halo (see Fig. 1). In this case, the amorphous

halo was fit using a Lorentzian function and the crystalline reflections with a
Pearson function, using commercial peak fitting software.

As another example, see the WAXD pattern for a monoclinic isotactic

polypropylene (PP) in Figure 2. In this case, the amorphous halo of the semicrys-
talline material was obtained by scaling the diffraction pattern of noncrystalline
atactic PP to obtain the best fit to the experimental spectrum. The authors of Ref-
erence 2 demonstrated that the WAXD pattern of isotactic PP at a temperature
(T) greater than the melting point (T

m

, 180

◦

C in this case) was the same as that

from atactic PP at the same temperature. The entire diffracted intensity from the
crystalline reflections (I

c

) was determined, and the crystallinity determined from

I

c

/(I

c

+ I

a

).

The use of traditional WAXD in the determination of degrees of crystallinity

for some other semicrystalline polymers can be found in References 4–8.

As a result of the high X-ray flux available at synchrotron X-ray sources such

as that at the National Synchrotron Light Source at Brookhaven National Labs,
WAXD patterns (and hence degrees of crystallinity) can be determined during
the course of crystallization at a specific temperature, or at programmed heat-
ing or cooling rates (see S

YNCHROTRON

R

ADIATION

). Data is typically acquired over

448

CRYSTALLINITY DETERMINATION

Vol. 9

Fig. 2.

X-ray diffraction patterns taken at room temperature: (a) isotactic polypropylene

crystallized at 145

◦

C and (b) a sample of atactic polypropylene. From Ref. 2.

a limited angular range in such experiments, and a crystallinity index or rela-
tively crystallinity is generally the best one can do. In addition, such capabili-
ties are often linked with small-angle X-ray scattering (SAXS) instrumentation
allowing one to follow the development of the lamellar microstructure in real
time, along with crystallinity development. References 9–13, and the references
therein, provide details of such experiments on a variety of crystallizing polymer
systems.

As an example, Figure 3 shows the sequence of WAXD patterns acquired

(one every 30 s) during isothermal crystallization of high molecular weight PEO
using a specially designed sample holder to allow for a rapid jump between the
melt temperature and selected crystallization temperatures (T

c

) (14). Degrees of

crystallinity were determined using a curve-fitting program where the diffraction
profile was separated into three crystalline PEO reflections and an amorphous
halo (10). The apparent degree of crystallinity was defined as the ratio of the
area under the resolved crystalline peaks to the total unresolved area. The first
several patterns in Figure 3 originate from amorphous PEO. Crystallization be-
gins subsequently and is largely complete at a time

<900 s. No changes in the

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CRYSTALLINITY DETERMINATION

449

Fig. 3.

Development of WAXD diffraction profiles as a function of crystallization time for

PEO during crystallization at 45

◦

C. 2

θ is relative to λ = 0.154 nm. From Ref. 10.

positions of the WAXD crystalline reflections (indicative of changes in the crys-
talline unit cell or unit cell parameters) or the widths at half peak height (in-
dicative of “crystallite size”) were noted during the crystallization of PEO at this
temperature.

Absolute Degrees of Crystallinity.

Determination of the absolute de-

gree of crystallinity from WAXD requires diffraction data over a wide angular
range and correction for air scattering, incoherent scattering, and diffuse scatter-
ing arising from thermal fluctuations and paracrystalline disorder. Only a brief
summary of this approach is presented here. The interested reader is directed to
the original Ruland and Vonk References 15–17 as well as several more recent
publications (18–20).

This approach (often referred to as the Ruland method) is based on the con-

servation of the total scattered intensity by a set of atoms, independent of their
state of structural order. The experimental X-ray diffraction pattern is first ac-
quired over a broad angular range [or equivantly, a broad s range: the reciprocal
space parameter s is used here for consistency with References 16–21, where
s

= (2sinθ)/λ and λ is the X-ray wavelength] and corrected for air and incoherent

scattering. It was recognized that even completely crystalline solids exhibit dif-
fuse scattering with corresponding loss of intensity in the crystalline reflections,
as a result of thermal vibrations and imperfections in the crystalline lattice (ie,
paracrystalline disorder). The absolute degree of crystallinity (X

abs

,WAXD

) is then

determined as

X

abs

,WAXD

=

s

2

#

s

1

s

2

I

c

(s) ds

s

2

#

s

1

s

2

I(s) ds

· K

(1)

450

CRYSTALLINITY DETERMINATION

Vol. 9

where s

1

are s

2

are indicative of the angular limits of the experiment, and

K

=

s

2

#

s

1

s

2

¯

f

2

ds

s

2

#

s

1

s

2

¯

f

2

D ds

(2)

where ¯

f

2

is the mean-square atomic scattering factor for the polymer. D(s) is the

generalized Debye–Waller factor that takes into account thermal atomic vibra-
tions as well as paracrystalline disorder, both of which result in crystalline peak
intensities decreasing as

θ (or s) increases. As for other methods discussed above,

a critical feature of this procedure is the accurate separation of crystalline reflec-
tions from the overall background scattering.

Small-Angle X-ray Scattering.

SAXS is an X-ray scattering method that

is used frequently to investigate the lamellar microstructure of semicrystalline
polymer systems; that is, to characterize structural periodicities on the order of
5–50 nm. For melt-crystallized systems, one typically observes a maximum in the
scattered intensity at low angles associated with the so-called long period (L),
which is associated with the mean center-to-center between lamellae. Figure 4 il-
lustrates the room temperature SAXS profiles of a poly(

L

-lactide/

D

-lactide) random

copolymer crystallized isothermally at selected temperatures. The parameter q in
Figure 5 is the scattering vector [q

=

4

π

λ

sin(

θ/2) and θ is the scattering angle]. The

Fig. 4.

Room temperature small-angle X-ray scattering patterns of a poly(

L

-lactide/

D

-

lactide) random copolymer containing 1.5%

D

-lactide crystallized at the T

c

s shown in the

figure. The scattered intensities have been Lorentz-corrected (by q

2

). From Ref. 22.

Vol. 9

CRYSTALLINITY DETERMINATION

451

Fig. 5.

Schematic of components in absolute crystallinity determination of an unoriented

sample of PET. The dashed line represents the incoherent scattering and the dotted line
represents the sum of the incoherent scattering and the scattering due to the amorphous
component. The solid line (not bold) represents the global background scattering. The solid
line at the bottom of the figure represents the residuals. The absolute crystallinity of this
sample was determined to be 51%. From Ref. 18.

long periods (L

= 2π/q

max

) increase progressively with T

c

, as is normally observed

for crystalline polymers. Weak second-order reflections are also observed for sev-
eral of these samples, particularly at higher T

c

, indicating relatively well-ordered

lamellar stacks.

It is sometimes assumed that the bulk degree of crystallinity is same as the

SAXS “linear crystallinity.” The linear crystallinity (w

c

) is defined on the basis of a

one-dimensional view of a lamellar stack morphology. For a stack of lamellae, the
linear crystallinity is defined simply as l

c

/L, where l

c

is the crystalline thickness

(L

= l

c

+ l

a

, where l

a

is the amorphous layer thickness between lamellae). In the

case of volume-filling lamellar stacks (the so-called “infinite stack” model), as
might be anticipated for polymers exhibiting relatively high degrees of crys-
tallinity like linear PE and PEO, the bulk crystallinity measured by another
method such as WAXD or differential scanning calorimetry (DSC) would be
expected to be near w

c

. This has been shown to be the case in a number of instances

(21). Therefore, to a first approximation, one can estimate the mean lamellar
thickness for a given polymer that exhibits an infinite stack morphology from l

c

=

L

· φ

c

, where

φ

c

is the volume fraction crystallinity determined from DSC, WAXD,

or another method.

However, this simple model is not generally valid, particularly for lower

crystallinity materials. In the latter, the (linear) crystallinity of lamellar stacks
can be relatively high, but the bulk crystallinity low. In this case

φ

c

= V

s

× w

c

(where V

s

is the volume fraction of lamellar stacks) and V

s

< 1 and w

c

> φ

c

.

452

CRYSTALLINITY DETERMINATION

Vol. 9

Thermal Analysis

Probably the most widely used technique for determining degrees of crystallinity
is differential scanning calorimetry (see T

HERMAL

A

NALYSIS

). This is due to the

ready availability of such instrumentation, rapid turn around time, and apparent
simplicity. The heat of fusion is measured experimentally and the weight-fraction
degree of crystallinity defined as:

X

c

,DSC

= H

f



H

0

f

(3)

where

H

f

is the measured heat of fusion (strictly, referred to temperature T)

and

H

f

0

is the heat of fusion of the 100% crystalline polymer (again, strictly,

determined at T). Since

H

f

is measure at the melting point and

H

f

0

is estimated

at comparable temperatures, it is important to recognize that X

c

,DSC

is therefore

defined near T

m

, not at ambient temperature where values from other analytical

methods are usually determined.

The main experimental problem encountered in measuring heats of fusion

is the construction of an appropriate baseline delineating the melting endotherm
from the underlying heat capacity contribution. The general equation relating
measured heat capacity to thermal measurement is (23)

H

= X

c

· H

c

+ (1 − X

c

)H

a

+ (dX

c

/dT) H

f

(4)

where H is the measured heat capacity and the subscripts c and a refer to the
crystalline and amorphous states, respectively. In the melting region, the last
term in equation 4 dominates and in the vast majority of the cases in the litera-
ture, a straight line is used to connect the onset to the last trace of melting. For
relatively sharp-melting materials, the error involved in using a flat baseline is
typically small. Some polymers, on the other hand, may melt over a wide temper-
ature range, and it can be difficult to determine where melting actually begins. In
addition, appreciable baseline curvature can arise from instrumental factors or
changes in heat capacity with temperature. As a result, the simple linear baseline
construction is not necessarily correct.

To avoid potential inconsistencies and uncertainties noted above, three simi-

lar methods have been proposed in literature. A brief summary of these approaches
is provided below, and the interested reader is encouraged to consult the original
publications. Although by all accounts these are preferred methods, they are not
routinely applied in the literature.

The first of these is now often referred to as the total enthalpy method (24,25).

In one version, the crystallinty is determined as follows (25). At a temperature T

1

below where melting begins, the total heat capacity can be written simply as

H

1

= X

c

,1

· H

c

,1

+ (1 − X

c

,1

)H

a

,1

(5)

where X

c

,1

is the crystallinity at T

1

and H

c

,1

and H

a

,1

are the heat-capacity con-

tributions from the crystalline and amorphous components, respectively, at T

1

.

At a temperature T

2

where the sample is completely molten: C

p

,2

= C

p

,a,2

. By

Vol. 9

CRYSTALLINITY DETERMINATION

453

difference,

H

2

,1

= H

2

− H

1

= H

a(2

,1)

+ H

0

f

,1

· X

c

,1

(6)

where H

a(2

,1)

= H

a

,2

− H

a

,1

and

H

f

,1

0

is the enthalpy required to convert 1 g of

completely crystalline material into 1 g of completely amorphous material at T

1

.

Solving for X

c

,1

leads to

X

c

,1

= m · H

2

,1

− b

(7)

where m

= ( H

f

,1

0

)

− 1

and b

= H

a(2

,1)

/

H

f

,1

0

. Equation 7 demonstrates that the

degree of crystallinity at T

1

is related to the total enthalpy absorbed between T

1

and T

2

(measured from the DSC trace). The advantage of this method is that it

provides the correct crystallinity at T

1

regardless of the path the polymer takes

to T

2

. However, the disadvantage is that

H

a(2

,1)

and

H

f

,1

0

must be known to

calculate X

c

,1

.

A similar approach for determining X

c

,1

has been proposed by Mathot and

Pijpers (26,27). Finally, the so-called “First Law method” has been proposed by Hay
and co-workers (28,29). This involves two measurements: (1) a DSC experiment
that determines the enthalpy change on heating from T

1

to T

2

(

>T

m

) and (2) a

virtual experiment determining the enthalpy change on cooling from T

2

to T

1

without crystallization taking place. The difference between steps 1 and 2 leads
to a residual enthalpy; ie, the heat of fusion of the sample at T

1

· H

f

0

reported

in the literature is normally determined near the equilibrium melting point (T

m

0

)

but to determine X

c

,1

,

H

f

,1

0

is needed. It has been shown that the latter can be

obtained from the experiments as

H

0

f

,1

= H

0

f

−



Tm

0

T

1

(H

a

− H

c

) dT

(8)

where H

a

and H

c

are the heat capacities of the completely amorphous and crys-

talline states, respectively. Once

H

f

,1

0

is determined from equation 8, X

c

,1

can

be calculated readily.

References 23 and 30 remain the best sources of a collection of

H

f

0

values,

but there has been considerable work on a variety of polymers since their pub-
lication and it is recommended to verify particular values by reference to more
recent literature. Since perfectly crystalline materials are rarely available,

H

f

0

values must be estimated, typically by an “extrapolation” method. For example,
a series of samples of a particular polymer can be crystallized to varying degrees,
and the experimental

H

f

measured along with the sample densities. The mea-

sured

H

f

values are then plotted vs density (or equivalently, specific volume) and

the data linearly extrapolated to the density of the perfectly crystalline version
of the polymer in question (19) [a value which is generally well established (see a
following section)]. This extrapolation can be rather lengthy, especially for poly-
mers of low crystallinity. In a similar fashion, measured

H

f

values are plotted

vs l

c

− 1

and

H

f

0

is obtained by extrapolating to l

c

− 1

→ 0 (31). Crystalline lamel-

lar thicknesses are typically determined from SAXS experiments, although other

454

CRYSTALLINITY DETERMINATION

Vol. 9

methods like the Raman longitudinal acoustic mode or microscopic techniques
(ie, transmission electron or atomic force) are sometimes used. For polymers for
which a series of analogs of various low molecular weights are available (eg n-
alkane analogs for linear PE), measured heats of melting of the analogs can be
plotted vs n

− 1

(where n

= number of C atoms in an n-alkane) and H

f

0

for the

polymer obtained by extrapolation of the measured heats to n

− 1

→ 0 (32). Finally,

H

f

0

has been estimated from the melting point (dissolution temperature) depres-

sion observed in the presence of a miscible diluent (33). Particularly care must be
exercised when using this latter approach to use equilibrium dissolution temper-
atures (ie, extrapolated values representing the dissolution of a perfect crystal
of the polymer at a particular diluent concentration), not simply experimentally
measured quantities.

Readers interested in additional background on Thermal Analysis (qv) tech-

niques applied to polymeric materials, may see, for example, References 34–36.

Vibrational Spectroscopy

Infrared (IR) and Raman spectroscopies have been used for decades to routinely
characterize polymeric and other materials. Vibrational Spectroscopy (qv), partic-
ularly Fourier transform IR (FTIR), has been used extensively to probe crystalline
and amorphous conformations in a wide variety of polymers, as well as to deter-
mine a measure of the crystallinity of such materials. In the FTIR spectra of crys-
talline polymers, one or more absorption bands are often observed that disappear
when crystallization is inhibited. Provided these bands can be genuinely assigned
to 3-D crystalline order, and if the absorbance of this band in the specimen under
examination is in the range for which the Beer–Lambert Law is applicable, then

A

= log(I

0

/I) = a

c

· X

c

,ir

· ρ · t

(9)

where A is the absorbance or optical density, I

0

and I the incident and transmitted

intensities, respectively, t the specimen thickness,

ρ the overall density, and a

c

the absorption coefficient of the 100% crystalline material. Since fully crystalline
specimens are rarely available, a

c

must be estimated, as will be discussed shortly.

If the spectrum contains bands indicative of noncrystalline material, an expression
similar to equation 9 can be written for the amorphous component.

Infrared bands should only be designated as crystalline bands if they disap-

pear on melting, are predicted from group theory, and X-ray or other data proves
the polymer to be crystalline. Only in a few rare cases have true crystalline bands
been observed; for example, the 720–730 cm

− 1

CH

2

rocking doublet of crystalline

PE is a reflection of a genuine crystallinity effect, as the splitting results from
intermolecular interaction of segments in the orthorhombic unit cell (37). More
frequently, reported “crystalline” bands are associated with a preferred crystalline
conformation that may also be present (in lower concentration) in the noncrys-
talline phase.

As an example of crystallinity determination by FTIR, the procedure used

in Reference 38 for polyethylene is summarized here. The infrared spectrum of a
solution-crystallized sample of polyethylene in the region of its CH

2

rocking (and

Vol. 9

CRYSTALLINITY DETERMINATION

455

Fig. 6.

FTIR spectrum of a solution-crystallized polyethylene (NIST standard SMR 1482,

M

w

= 1.36 × 10

4

) in the CH

2

rocking and bending (scissoring) regions. Pictured are the

crystalline and amorphous components in each of these regions, as resolved using a band
fitting procedure. From Ref. 38.

bending) modes is shown in Figure 6. The crystallinity was calculated from the
ratio of the integrated intensities (absorbances) of the crystalline component in
the CH

2

rocking region (I

722

+ I

730

) to the total absorbance in this region. The

latter is determined by I

722

+ I

730

+ γ I

723

, where I

723

is the contribution from

the amorphous component, and

γ is included to account for the fact that intrinsic

intensities of the crystalline and amorphous bands are not equal (

γ = a

c

/a

a

, where

a

a

is the absorption coefficient of the amorphous component). The value of

γ was

determined to be

∼1.2 in this case. Degrees of crystallinity were determined for

a variety of polyethylene samples using this method and found to be the same
within experimental uncertainty as those derived from DSC experiments on the
same samples. This result led these authors to conclude that this FTIR method
measures the “core crystallinity,” as does DSC (38).

A similar approach has been used to determine the crystallinity of poly(

ε-

caprolactone) by FTIR, in which

γ was measured by monitoring changes in the

absorbances of the amorphous and crystalline components of the carbonyl band
during isothermal crystallization (39). In many cases in the literature a “crys-
tallinity index” is effectively determined, either because true crystalline or amor-
phous IR bands are rarely available and/or

γ is assumed to be equal to 1. However,

such an index may prove to be suitable for a variety of purposes.

456

CRYSTALLINITY DETERMINATION

Vol. 9

Fig. 7.

Grazing incidence reflection FTIR spectra (p-polarized) of PEO (pyrene end-

labeled) ultrathin films on oxidized silicon with different thicknesses (crystallized isother-
mally at 40

◦

C). The negative bands are typical of grazing incidence reflection spectra on

nonmetallic surfaces. From Ref. 42.

There is widespread interest in thin and ultrathin (thickness

<100 nm) poly-

mer films for use in packaging, electronics and other applications, and infrared
spectroscopy is a particularly useful tool for characterizing the crystallinity in such
systems. For example, attenuated total reflectance FTIR has been used to provide
qualitative information on crystallinity and molecular orientation as a function
of depth into drawn films of PET (40). Crystallinity measurements on ultrathin
films are clearly difficult by conventional methods, but can be accomplished us-
ing FTIR spectroscopy in reflection mode (41,42). For example, grazing incidence
reflection FTIR has been used to provide insight on molecular orientation as well
as a qualitative measure of conformational order in ultrathin films of PEO on an
oxidized silicon surface (42). Figure 7 shows grazing incidence reflection spectra of
PEO films with different thicknesses. The absorbance near 1120 cm

− 1

is not sen-

sitive to conformational order (ie, is an “internal thickness” band), and all spectra
are therefore normalized to A

1120

to permit evaluation of the relative crystallinity

(conformational order) as a function of film thickness. The crystallinity was found
to decrease significantly with decreasing film thickness, in keeping with the usual
observation of slowing of crystallization for films with thicknesses below ca 100 nm
(43,44). It has been postulated that a reduction in molecular mobility occurs in ul-
trathin films, resulting from the onset of diffusion control of crystallization (45,46).

Volumetric Methods

Measurement of sample densities also provides an effective way of determining
degrees of crystallinity, (47–50), although this approach is not used as commonly
as in the past because of the availability and ease of utilizing DSC, FTIR, and
WAXD. The weight fraction degree of crystallinty can be determined from the

Vol. 9

CRYSTALLINITY DETERMINATION

457

measured density by

X

c

,d

=



ρ

− 1

a

− ρ

− 1



ρ

− 1

a

− ρ

− 1

c



= ( ¯V

a

− ¯V)/( ¯V

a

− ¯V

c

)

(10)

where

ρ

a

and

ρ

c

are the densities of completely amorphous and crystalline ver-

sions of the polymer in question, and V are the comparable specific volumes. The
relationship between the weight fraction crystallinty and volume fraction crys-
tallinty (

φ

c

), regardless of method used, is simply:

φ

c

= X

c

(

ρ/ρ

c

). Values of

ρ and

ρ

c

are generally such that

φ

c

and X

c

are within several percent of each other.

Sample density can be measured using a variety of methods: density-gradient

column, dilatometry, pycnometry, and flotation or buoyancy. The density-gradient
column approach is probably used most frequently to determine sample densities
for crystallinity measurements. When thermostated and appropriately calibrated
with floats, this approach permits measurements to accuracies of 0.2 mg/cm

3

or

better. Dilatometers are well suited for measuring specific volumes and following
crystallization, as a function of temperature.

Although the above-mentioned techniques are routine, a number of problems

can lead to errors in measured densities. For example, the solvent used must
completely wet the specimen but not cause swelling, and the sample must be free
of closed porosity. Solvents that induce further crystallization should be avoided.

Even in the absence of reliable

ρ

a

and

ρ

c

values, density measurements can be

used for qualitative ranking of specimens of a given polymer on the basis of degree
of crystallinity. Values for

ρ

c

are generally known reasonably well since they can

be calculated from the dimensions of the appropriate unit cell for the polymer
in question. The lattice parameters are determined from WAXD experiments, as
discussed elsewhere in this edition. Estimation of

ρ

a

is generally more difficult.

Some polymers can be prevented from crystallizing by quenching from above T

m

to below T

g

(eg isotactic polystyrene and PET) and their amorphous density can

therefore be measured directly at ambient temperature. In other cases,

ρ

a

has

been estimated by extrapolating specific volume data (from dilatometry) in the
melt to ambient temperature. This extrapolation can be lengthy and introduces a
degree of uncertainty in

ρ

a

.

It has been found in a number of cases that X

c

from density and WAXD

measurements is equivalent within experimental error, but is higher than that
from DSC (19,47). These observations have led to the conclusion that DSC provides
a measure of the core crystallinity only, while the crystallinity from density and
WAXD is a measure of the core crystallinity along with the crystal–amorphous
interphase.

Solid-State Nuclear Magnetic Resonance

This section provides a broad overview of the application of NMR spectroscopy
to crystallinity determination in polymers. The approaches illustrated are cer-
tainly not exhaustive of those available for the powerful NMR technique and the
interested reader is encouraged to consult one or more of the reference books on
NMR spectroscopic investigations of polymeric materials (51–53) (see N

UCLEAR

M

AGNETIC

R

ESONANCE

).

458

CRYSTALLINITY DETERMINATION

Vol. 9

Fig. 8.

Broadline

1

H NMR spectra: (a) polybutadiene at 20

◦

C; (b) poly(ethylene tereph-

thalate) at 20

◦

C; (c) polyethylene at 20

◦

C; and (d) polyethylene at

−150

◦

C. From Ref. 54.

Clasic Broadline

1

H NMR.

This technique has been utilized for many

decades. Broadline

1

H NMR spectra, the first derivatives of the NMR absorption

spectra, of several polymers are shown in Figure 8.

At temperatures above their T

g

, the resonance spectrum of noncrystalline

polybutadiene (PB) (Fig. 8a) is clearly different from that of the semicrystalline
polyethylene (Fig. 8c). Amorphous PB exhibits a narrow Lorentzian line shape
with a width of

∼0.2 G. In contrast, the PE spectrum comprises two components,

ie, narrow and broad line shapes. When the spectra of semicrystalline polymers are
recorded in the glassy state (Figs. 8b and 8d), only a broad component is observed.
This indicates that the line shape corresponds to molecular mobility and the line
width reflects a correlation (or relaxation) time. Therefore, the broad and narrow
components of semicrystalline PE (Fig. 8c) are related to protons of methylene
groups in rigid and mobile (amorphous) environments, respectively. On the basis
of this, it was proposed that the degree of crystallinity could be determined by
resolving the area of the broad component (rigid phase) from the spectrum.

For many polymers, however, the degree of crystallinity estimated from

the two-component analysis was found to be appreciably larger than that ob-
tained from WAXD or density measurements on the same materials. In addition,
although the crystallinity of PE determined by density methods remains constant
below room temperature (

∼60%), the broadline NMR “crystallinity” increases con-

tinuously to 100% at temperatures approaching T

g

(55). Thus the area under the

broad portion of the derivative curve is proportional to the number of immobile
protons in both crystalline and rigid noncrystalline phases. As a result, methods
to decompose such spectra into three components were developed (56,57).

Broadline NMR spectra can be expressed as

y(H)

= w

b

y

b

(H

,b

b

)

+ w

m

y

m

(H

,b

m

)

+ w

n

y

n

(H

,b

n

)

(11)

where y

b

, y

m

, and y

n

are the elementary spectra of the broad, medium, and nar-

row components, respectively, and w

b

, w

m

and w

n

are the weight fractions of these

Vol. 9

CRYSTALLINITY DETERMINATION

459

components (w

b

+ w

m

+ w

n

= 1). The line widths of the broad, medium, and nar-

row components are b

b

, b

m

, and b

n

, respectively. The calculated spectra of these

components can be obtained as described in References 56 and 57. In the fitting
process, the line widths and weight fractions of all elementary spectra are mod-
ified so as to minimize the difference between the calculated and experimental
spectra. A detailed discussion and examples of the broadline H

1

NMR technique

can be found in Reference 54.

Fourier Transform NMR.

In Fourier transform NMR (FTNMR), a repet-

itive radio frequency (RF) pulse is applied in order to excite all of the nuclei of
the particular nuclear species being studied. The sum of the free induction decay
(FID) curves from each pulse is analyzed by a Fourier transform method in order
to generate the familiar frequency domain spectra. Fundamentally, parameters
such as the frequency, intensity, application time of the appropriate RF pulse,
and time intervals between these pulses are important variables when using this
technique. The principle of the pulsed Fourier transform technique can be found
in books covering the fundamental concepts of NMR spectroscopy (58,59).

The application of a properly timed rf pulse disturbs the net equilibrium

magnetization aligned parallel to the magnetic field H

0

. The return of nuclear

magnetization to equilibrium for solid polymers depends on the chain mobility
and is monitored by the FID. The crystalline phase content can be extracted by
deconvoluting the decay curve into portions corresponding to crystalline and non-
crystalline components. Crystallinity determination of various polyethylene sam-
ples by FID

1

H NMR based on two- and three-component approaches, as well as

Fig. 9.

FID resolution into three components for ultrahigh molecular weight PE reac-

tor powder. The inset figure shows residuals between the observed and fitted data in the
long decay region (100–400

µs).

Observed,

 fitted,

crystalline,

 intermediate,

× mobile. From Ref. 62.

460

CRYSTALLINITY DETERMINATION

Vol. 9

Fig. 10.

Pulse sequences for DD/MAS

13

C NMR (without CP). From Ref. 65.

comparison of these crystallinities with those determined from other methods, is
discussed in References 60–62. Figure 9 presents a summary of the FID

1

H NMR

spectral analysis of PE, based on a three-component method (62). The crystallinity
is determined by the initial normalized magnetization (zero time) of the calculated
decay of the crystalline component.

The degree of crystallinity can be determined via FT

1

H NMR spectra as well

(60,61) Similar to broadline

1

H NMR line shapes for the various components are

fit to the experiment spectrum in order to extract that of the crystalline compo-
nent, and the ratio of the area under the resolved crystalline spectrum divided
by the total area represents the weight fraction of the crystalline phase. The use
of this approach to determine the crystallinity of polyethylene, ethylene–vinyl al-
cohol copolymers and polypropylene can be found in References 60, 63 and 64,
respectively.

Fig. 11.

Fully relaxed DD/MAS

13

C NMR spectrum of metallocene-catalyzed lineer low

density polyethylene. From Ref. 67.

Vol. 9

CRYSTALLINITY DETERMINATION

461

Solid-State High Resolution C

13

NMR.

C

13

NMR with dipolar decou-

pling (DD), magic angle spinning (MAS), and cross-polarization (CP) can dis-
tinguish phases in semicrystalline polymers by differentiating their chemical
shifts and relaxation times. Useful pulse sequences utilized to obtain NMR spec-
tra are presented in Figure 10. Spectra with partial or total contribution from
the respective components (phases) can be obtained by appropriate choice of the
waiting times,

τ

l

and

τ

t

. The spectrum acquired from pulse sequences I and II

depends on the material relaxation times, ie spin-lattice relaxation time (T

1C

)

and spin–spin lattice relaxation time (T

2C

). With pulse sequence I (Fig. 10), if

τ

l

is longer than 5 times the longest spin-lattice relaxation time, the spectrum

Fig. 12.

A: Cross-polarized,

1

H-decoupled, static 25.152 MKz

13

C NMR spectrum of

Phillips 6050 polyethylene. The spectral window was 5 KHz, no line broadening, 24150
transients. B: Calculated “best-fit” spectrum, the sum of the crystalline and the amorphous
tensor components. C: The crystalline tensor. D: The amorphous tensor. E: Residual spec-
trum (A-B). The chemical shifts are relative to the methyl resonance of hexamethylbenzene
(17.6 ppm from TMS). From Ref. 68.

462

CRYSTALLINITY DETERMINATION

Vol. 9

(see Fig. 11) contains contributions from all structural phases. In order to ex-
tract the crystalline content, the location and shape of the resonance lines of each
component and proper composition model are required.

For PE, it has been shown by many research groups (65–67) that the reso-

nance line at 33, 31.3, and 31 ppm arise from the orthorhombic crystalline, the
crystalline-amorphous interfacial, and the amorphous (rubbery) components, re-
spectively. Detailed studies using different pulse sequences (varying

τ

l

and

τ

t

to

extract the information of individual component from the partial relaxed NMR
spectrum can be found in References 65–67. On the basis of chemical shifts of
all components, the experimental spectrum is analyzed using three calculated
Lorentzian curves centered at 33, 31.3, and 31 ppm, by least-squares fitting, as
seen in Figure 11. In the NMR study of PE in Reference 66, the monoclinic crystal
form (with the resonance line centered at 34.3 ppm) was observed in addition to
the resonance of the usual orthorhombic form. Spectra were therefore analyzed
using four components, using the same principles as described above.

In addition to DD/MAS

13

C NMR, the crystallnity of PE can be determined

from the

1

H-decoupled/CP

13

C NMR. Spectra consist of contributions from the

different phases, which are differentiated by their shielding tensors (anisotropic
chemical shift), as seen in Figure 12 (68). The crystalline fraction is estimated
from the ratio of the area under the crystalline tensor to the total area.

In summary, the key feature of these approaches is the deconvolution of the

spectrum of interest, which is sensitive to the presence of the different phases,
into spectra corresponding to each component. Although the above discussion fo-
cused on semicrystalline polyethylene, detailed discussion of crystallinity mea-
surements by

13

C NMR for other selected polymers can be found in the following

references: poly(vinyl chloride) (69), polypropylene (70), and polylactides (71).

Finally, in addition to

1

H and

13

C NMR,

19

F NMR has proven to be very

useful for determining the degree of crystallinity of fluoropolymers such as poly-
tetrafluoroethylene (72,73).

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J

AMES

R

UNT

M

ANTANA

K

ANCHANASOPA

The Pennsylvania State University