Orientation of nano-crystallites and anisotropy of uniaxiallydrawn a-polyamide 6 films: XRD, FTIR, and microwavemeasurements

Hussein Shanak • Astrid Naumann •

Jan Lion • W. Gotz • Rolf Pelster

Received: 11 June 2014 / Accepted: 28 July 2014 / Published online: 15 August 2014

� Springer Science+Business Media New York 2014

Abstract We have used various techniques such as X-ray

diffraction, Fourier transform infrared spectroscopy as well

as dielectric measurements with a 4 GHz Microwave res-

onator to investigate the structure of uniaxially drawn

a-polyamide 6 films. The influence of uniaxial drawing on

several parameters such as degree of crystallinity, orienta-

tion and size of crystallites, permittivity and anisotropy was

studied as a function of the drawing ratio. The main axis (b-

axis) of the polyamide 6 nano-crystals is aligned in the

direction of drawing, while the hydrogen bonds (a-axis) are

oriented transverse to the drawing direction. Increasing the

drawing ratio yields a higher crystallinity, a better orienta-

tion of the crystallites, and a stronger dielectric anisotropy

of the films. The good agreement of the results demonstrates

that all the three experimental techniques are well suited to

characterize the microstructure of polyamide films.

Introduction

Polyamide 6 is an interesting polymeric material that has

attracted much attention in several fields of technology

[1–3]. It is used in different applications ranging from

carpets, automotive parts, packaging industry to intimate

apparel [2–6]. Despite the numerous important industrial

applications and several experimental investigations on

polyamide 6, there still remain many questions about the

details of different structures and the conversion between

them [4]. Stretching is an essential and necessary process

for the industrial production of polyamide 6 films; this is

done to improve properties like the mechanical stability.

The induced changes during drawing of polyamide 6

involve the microstructure of the semi-crystalline polya-

mides, i.e., degree of crystallinity, orientation of crystal-

lites, and the resulting anisotropy. In order to control

quality and homogeneity of the produced films, it is

important to quantitatively relate the previous parameters

to the production history. The use of techniques such as

X-ray diffraction (XRD), infrared spectroscopy, and

dielectric measurements is of particular interest in under-

standing the above drawing-induced changes [1, 7, 8].

Especially, the average orientation of molecules or crys-

tallites is strongly related to the mechanical properties of

polyamide films [1]. In previous studies, it was shown that

polyamide 6 films contain nano-crystallites of monoclinic

structure (so-called a-form), where the molecules are ori-

ented anti-parallel to the b-axis (see Fig. 1a) [9, 10]. The

hydrogen bonds connect the anti-parallel chains of the

polyamide. The unit cell parameters of the a-form are

a = 9.56 A, b = 17.24 A, c = 8.04 A, and b = 67.5�[10–12].

In this study, we investigate the effect of uniaxial

drawing on the structure formation of polyamide 6 films

using three techniques: XRD, Fourier transform infrared

spectroscopy (FTIR), and dielectric measurements with a

microwave resonator. All these techniques give us infor-

mation on the microstructure, and we shall compare the

H. Shanak � A. Naumann � J. Lion � R. Pelster (&)

FR 7.2 Experimentalphysik, Universitat des Saarlandes,

66123 Saarbrucken, Germany

e-mail: rolf.pelster@mx.uni-saarland.de

H. Shanak

e-mail: h_shanak@yahoo.com

H. Shanak

Physics Department, Palestine Technical University - Kadoorie

(PTUK), Tulkarm, Palestine

W. Gotz

BASF SE Company, 67056 Ludwigshafen, Germany

123

J Mater Sci (2014) 49:8074–8083

DOI 10.1007/s10853-014-8515-6

results to achieve a better understanding of the structure

development. To our knowledge, such a comparative study

of the same parameters obtained from different experi-

mental techniques does not exist for polyamide 6 films

(there is a recent comparative study of syndiotactic poly-

styrene films by Rizzo and Albunia where XRD and FTIR

techniques were used [13]). We want to find out to which

extent stretching has an influence on the orientation

strength of crystallites in commercially produced films and

to which extent it is possible to investigate the same

parameters by different techniques.

Experimental

Using the commercially available polyamide (UltramidR

B33L), un-stretched polyamide 6 films were produced as a

pre-product by the BASF Company on a Barmag cast film

line at 20 �C chill roll temperature. Afterward, mono-axi-

ally oriented films were made using a laboratory stretching

unit of Bruckner GmbH, Siegsdorf/Germany. Film samples

of 150-lm thickness were preheated to 190 �C and then

stretched with drawing ratios of 1:2, 1:3, and 1:4 (see

Fig. 1b). At the end, the films were cooled to 25 �C. All

further investigations were performed as a function of the

drawing ratios.

XRD measurements were carried out with a Bragg–

Brentano geometry [14, 15]. The measurements were per-

formed in a reflection mode using Cu Ka radiation. The

angular resolution was 0.02 degree, the measuring time per

step 20 s. The pole figure data were obtained with an

automated X-ray diffractometer (Panalytical X‘Pert MRD

System), using the geometrical configuration of Decker,

Asp and Harker [14] with Nickel filtered Cu Ka radiation.

The Bragg angle 2h was set at a particular (hkl) diffraction

peak, and the sample was tilted by an angle U and rotated

by an angle W as shown in Fig. 2a [15]. Here the intensity

I(U,W) was measured as a function of W for the different

steps of U, while U was held constant during each W scan.

U was varied from 0� to 90� in steps of 5�, and W from 0�to 360� in steps of 5�. The quantitative analysis for the

degree of orientation of the crystallites was performed by

evaluating the orientation function hcos2/i, where /denotes the angle between a crystal axis (a, b, or c) and a

film axis (drawing direction D, transversal direction T, or

normal N). A detailed description of this method is pro-

vided in the appendix.

Infrared spectra in the range of wave numbers from 5000

to 370 cm-1 with a resolution of 4 cm-1 were performed

using a Fourier transform spectrometer (FTIR Perkin Elmer

System 2000). A polarization filter allows us to perform

orientation-dependent measurements. Figure 3 shows the

geometry to determine the orientation angle from FTIR

measurements, where the polarization angle 0� corresponds

to an orientation parallel to the machine direction, i.e., to the

drawing direction (D) of the films. The polarization angle

ranges from 0� to 180�. At a polarization angle of 0�, the light

is vertically polarized, and at 90�, it is horizontally polarized.

The FTIR spectroscopy scans were performed with 5� steps

of the polarization angle.

The dielectric measurements were performed using a

rectangular home-made 4 GHz Microwave resonator in

the fundamental H101-mode with a HP8310B Network

Analyzer from Hewlett Packard Company [16]. The films

were placed in the middle of the resonator where the

electric field strength exhibits a maximum, the field

vector E being parallel to the surface of the film. The

shift of the resonance frequency compared to that of the

unloaded resonator allows us to evaluate the permittivity

of the film [16]. In order to quantify the anisotropy, the

films were rotated from hor = 0� to hor = 180, in steps

Fig. 1 a A representation for

the a-form of polyamide 6

according to [12]. The dotted

lines indicate the direction of

hydrogen bonds; the arrows

indicate the parallel and the

anti-parallel molecular chains.

b A schematic representation of

the uniaxial drawing of

polyamide 6 films. D denotes

the drawing direction and T the

transverse direction

J Mater Sci (2014) 49:8074–8083 8075

123

of 10�, where a orientation angle of 0� means that the

drawing direction of the film is parallel to E (see

Fig. 4b).

Results and discussions

X-ray diffraction

At first, we have measured XRD patterns of the uniaxially

drawn polyamide 6 films. All peaks are typical for the

monoclinic a-form, the main features being the 100/020

reflection at 2h = 11.77�, the 200 reflection at 2h = 20.5�,

and the 004 reflection at 2h = 49.1�. The maximum

intensity of these peaks increases with the increasing

drawing ratio. Additional measurements that confirm the

crystalline structure are described in the previous paper

[17]. Figure 5a presents the evaluated degree of crystal-

linity for the different films (for the method we refer to

Refs. [18, 19] ). It increases from 28 % at a drawing ratio

of 1:2 to about 50 % at a drawing ratio of 1:4. The grain

sizes were calculated from the XRD patterns by the use of a

Williamson-Hall-Plot based on an evaluation of peak

widths [20]. The result is displayed in Fig. 5b. The grain

size is in the range of 8–10 nm. It does not depend

Fig. 2 a Geometry for the pole

figure measurements: position

of plane normal Phkl by

spherical coordinates U and W.

U is the angle between the

sample axis N (the film normal)

and the reciprocal lattice vector

Phkl. b A schematic

representation showing the

angles U and W in a pole figure

Fig. 3 A sketch showing the polarization angle in the FTIR

measurements

Fig. 4 a A schematic

representation showing the

direction dependence of the

permittivity measurements in

the microwave resonator. H is

the angle between direction of

the electric field E and the

drawing direction, D. b Sketch

illustrating the orientation angle

hor, which is the angle between

the electric field vector E and

the direction of the maximum

permittivity

8076 J Mater Sci (2014) 49:8074–8083

123

considerably on the drawing ratio, since the size of a single

crystal is determined by equilibrium conditions. In con-

trast, drawing favors nucleation, so that the number of

nano-crystallites and thus the degree of crystallinity

increase (Fig. 5a).

The influence of drawing on the average orientation of

the crystallites was analyzed using the X-ray pole figure

technique. Figure 6 presents the pole figures of the 002 and

200 reflections. The sharp peak in the 002 pattern reveals

that the reciprocal c*-axis is strongly oriented in the

direction of the normal to the film surface (maximum

intensity at U = W = 0�, compare with Fig. 2b). The pole

figure of the 200 reflection for the film with a drawing ratio

of 1:2 reflects a random distribution of the reciprocal a*-

axis. At higher elongations with drawing ratios of 1:3 and

1:4, two peaks with maximum intensity at U = 66� and

W = ?90� or W = -90� evolve in the pole figures. These

W-values show that the reciprocal a*-axis is strongly ori-

ented transverse to the stretching direction. The value of Uindicates an angle of about 66� between the a*-axis and the

normal (c*-axis). These data are consistent with the

geometry of the monoclinic a-form. According to this

model, the b*-axis is perpendicular to the a*c*-plane, and it

is located in the plane of the film and in the drawing

direction. This means that the crystallographic axis a is in

the transverse direction, the axis b or b* is in the drawing

direction, and the c* axis is in the normal direction: Fig. 7

shows the location of the different crystallographic axes

with respect to the drawing direction (D), the transverse

direction (T), and the normal of the film (N). Remember

that the molecules forming the crystallites are oriented

parallel or anti-parallel to the b or b* axis (Fig. 1a). Thus,

drawing causes the average orientation of the molecule

axes to be parallel to the stretching direction D.

So far we have determined the average orientation of

the nano-crystallites in the stretched films. Next we shall

evaluate the degree of orientation upon drawing. For this

purpose, we calculate average orientation functions

hcos2/i (see Appendix and Table 1) where / denotes the

angle between a crystal axis and a film axis. For hcos2/i = 1,

there is a complete orientation and for hcos2/i = 0.333 a

random orientation, while the axis is oriented perpen-

dicular to a reference when hcos2/i = 0. Most interesting

is the degree of orientation of the b-axis along which the

polyamide molecules orientate (chain direction) with

respect to the stretching direction D. The higher the

drawing ratio, the better the orientation: the values of the

orientation function increase from hcos2/b,Di = 0.52 at a

drawing ratio of (1:2) to hcos2/b,Di = 0.56 at (1:3) to

hcos2/b,Di = 0.6 at (1:4) [see Table 1 where we also list

the values for the other axis]. The equilateral triangles

displayed in Fig. 6 (right) provide an insight into the

orientation of all axes. On the equilateral triangle, the

values of the average orientation function of the c-axis

are located on the line connecting the N-apex (normal)

and the middle point of the TD line for all films. This

means that the c-axis is strongly oriented in the normal

direction of the films. The average orientation functions

of the a-axis for the film of low elongation (1:2) are not

located exactly on a certain axis, because it is randomly

oriented. For the films with higher drawing ratios, the

average orientation values are located nearly on the line

connecting the T-apex (transverse direction) and the

middle point of the DN-line. This indicates a strong

orientation of the a-axis in the transverse direction. It also

implies an orientation in the drawing direction. The

location of the value of the mean orientation functions

of the b-axis on the equilateral triangle is near to the

Fig. 5 The degree of

crystallinity in a and the grain

size in b as a function of the

drawing ratio (1:2, 1:3, and 1:4)

for uniaxially drawn films

J Mater Sci (2014) 49:8074–8083 8077

123

TD line, but it is very far from the N-apex, which means

the b-axis is oriented perpendicular to the N-axis (the

normal) of the film. In Fig. 8, we display the average

orientation of the b-axis (molecule axis) of the polyamide

6 nano-crystals, i.e., the values hcos2/b,Di *100, as a

function of the drawing ratio, in order to visualize the

increasing orientation.

Infrared spectroscopy

Figure 9 shows the IR spectra of a polyamide 6 film with a

drawing ratio of 1:4 for different polarization angles H (see

Fig. 3). The spectra are governed by the peaks of the CO–NH-

vibrations at 930 and at 960 cm-1. The amplitudes of the peaks

change in a complementary way. At a polarization angle of 0�,

002 reflection 200 reflection Equilateral triangle

1:2

1:3

1:4

Fig. 6 Pole figures and their equivalent equilateral triangles of the

002 and 200 reflections for polyamide 6 films drawn uniaxially with

different ratios (1:2, 1:3, and 1:4). For the coordinates, we refer to

Fig. 1b. N is the normal to the surface of the film, D the drawing, and

T the transverse direction

8078 J Mater Sci (2014) 49:8074–8083

123

when the electric field E is parallel to the drawing direction D

and thus to the chain axis b (see Figs. 3, 7, 10), the intensity is

maximal at a wave number of 930 cm-1 and minimal at a wave

number of 960 cm-1. Increasing the polarization angle to 90�,

where E is parallel to the a-axis and thus parallel to the inter-

molecular hydrogen bonds between amide groups (see Fig. 10),

will decrease the intensity at wave number of 930 cm-1 to its

minimum and increase the intensity at 960 cm-1 to its maxi-

mum value. Increasing the polarization angle further up to 180�causes the intensity to return to its initial value. This behavior is

due to the transition moment of the CO–NH band, labeled as

parallel to the polarization direction of the light at 930 cm-1

and as perpendicular at 960 cm-1 according to Sandeman and

Keller [21]. In order to show more clearly the complementary

excitation of the vibrations, we evaluate the relative intensity

a = F930/(F930 ? F960), where F930 and F960 represent the area

under the peak for the given wave number (after subtracting a

baseline and fitting the remaining spectra as a superposition of 4

peaks). In Fig. 11, we plot a as a function of the polarization

angle for the different drawing ratios. We fit the curves with the

function

aðhÞ ¼ F930

F930 þ F960

¼ A � sin ðh� hFTIRÞð :p

1800Þ2 þ C; ð1Þ

where A and C are constants, and h is the polarization

angle. This yields the orientation angle hFTIR, which is the

angle at which the relative intensity is minimal. We found

that hFTIR is -5.89�, -2.78�, and -2.84� for a drawing

ratio of 1:2, 1:3, and 1:4, respectively. Note that the

Fig. 7 A sketch illustrating the direction of crystallographic axis a

and of the reciprocal axis a*-, b*-, and c* with respect to the drawing

and transverse directions in uniaxially drawn films. N is the normal to

the surface of the film, D the drawing, and T the transverse direction.

Since b* k b holds, the long molecules are oriented parallel to the

stretching direction (compare with Fig. 1)

Table 1 Values of the orientation parameters for a-, b-, and c- axes in the three principal coordinates (T, D, and N) of the film that have been

used to produce the equilateral triangles in Fig. 6

a-axis b-axis c-axis

cos2 /a;T

� �cos2 /a;D

� �cos2 /a;N

� �cos2 /b;T

� �cos2 /b;D

� �cos2 /b;N

� �cos2 /c;T

� �cos2 /c;D

� �cos2 /c;N

� �

1:2 0.33 0.25 0.43 0.37 0.52 0.11 0.29 0.23 0.50

1:3 0.39 0.21 0.39 0.33 0.56 0.11 0.28 0.22 0.49

1:4 0.42 0.18 0.39 0.30 0.60 0.10 0.27 0.22 0.46

Boldface indicates the importance of these values; the main (long) molecules are aligned in the direction of the b-axis

Fig. 8 The degree of orientation as a function of the drawing ratio

(1:2, 1:3, and 1:4) for uniaxially drawn films. This is the average

orientation of the b-axis (molecule axis) of polyamide 6 nano-crystals,

i.e., the values hcos2/b,Di * 100 as determined by XRD pole figures

Fig. 9 FTIR absorption spectrum for a polyamide 6 film with a

drawing ratio of (1:4). The intensity is plotted as a function of the

wave number with the polarization angle as parameter

J Mater Sci (2014) 49:8074–8083 8079

123

subtraction of a baseline just amplifies the variation of a,

but does affect the value of hFTIR. Taking into account

errors and local variations in the film properties, we con-

clude that hFTIR & 0 holds. IR measurements thus allow us

to easily determine the orientation of the crystallites b-axis

(corresponding to the drawing direction for uniaxially

stretched films): this is the direction, where the intensity of

peak at 930 cm-1 reaches its maximum. Next we want to

evaluate the degree of orientation of the crystallites, i.e.,

the fraction of crystallites that are oriented in the drawing

direction. Let N be the number of these oriented crystallites

and M the number of those being randomly oriented. Then

the intensity of the IR-peak is

I930 ¼ c1 �XNþM

i¼1

cos2 Hi

!

¼ c1 � N � cos2ðH�HFTIRÞ þ 0:5 �M� �

ð2Þ

with c1 being a constant. A polarization dependent mea-

surement as a function of h then yields a maximum value of

Imax930 = c1�(N ? 0.5�M) and a minimum value of Imin

930 =

c1�0.5�M. The fraction of oriented crystallites is thus the

degree of orientation,

N

N þM¼ c1 � N

c1 � ðN þMÞ ¼Imax930 � Imin

930

Imax930 þ Imin

930

: ð3Þ

Note that in contrast to the empirical subtraction of a

baseline for the evaluation of a (Eq. 1 and Fig. 11), the

background is now important, since it is proportional to the

number of randomly oriented crystals. We thus directly use

the measured intensities at a wave number of 930 cm-1

without any fitting of peak areas or background elimina-

tion. In Fig. 12, we display the degree of orientation. It

increases from 34 % at a drawing ratio of (1:2) to 47 % at

(1:3) to 54 % at (1:4). These values are a little bit lower

than those obtained with the XRD pole figure technique

(52, 56, and 60 %, see Fig. 8) but reflect the same drawing

dependence. Probably it would be necessary to subtract a

small contribution due to the amorphous polymer phase

from the measured spectra in order to improve the result.

On the other hand, it is much easier to perform IR-mea-

surement than to produce and evaluate XRD pole figures.

At least at higher drawing ratios, the IR analysis gives a

good estimate of the degree of orientation.

Fig. 10 A sketch illustrating the direction of the transition moment of

CO–NH band with respect to the field E at a wave number of

930 cm-1

Fig. 11 Relative IR intensity (peak areas after subtraction of a

background; see text) according to Eq. 1 as a function of the

polarization angle for polyamide 6 films with different drawing ratios

(1:2, 1:3, 1:4)

Fig. 12 Degree of orientation as determined via IR measurements

according to Eq. 3 as a function of the drawing ratio (compare with

Fig. 8)

8080 J Mater Sci (2014) 49:8074–8083

123

Microwave resonator measurements

The dielectric measurements are presented in Fig. 13,

where the real part of the permittivity e is plotted as a

function of the samples rotation angle, i.e., the angle

between the electric field and the drawing direction (see

Fig. 4a). The imaginary part of the permittivity is very

small compared to the real part, i.e., at 4 GHz, dielectric

losses are negligible. The data for the different drawing

ratios were fitted with the following equation:

e hÞð ¼ emax � cos2 h�ð horÞ þ emin � sin2 h� horð Þ: ð4Þ

Hor denotes the angle where the largest permittivity is

measured. We find values of -2.7� and 9.7� for drawing

ratios of 1:3 and 1:4, respectively. Keeping in mind the very

small amplitude of the e(h)-variation, that is of the order of

0.06 (just 2 % of the mean value) and unavoidable mea-

surement errors, it is obvious that the data in Fig. 13 are also

well described by Eq.4 with a fixed value of Hor = 0. An

angle Hor = 0 means that the permittivity is maximum when

the electric filed is parallel to the drawing direction and thus

to the polyamide 6 chains (the b-axis, see Fig. 7). Accord-

ingly, the minimum permittivity is obtained for a perpen-

dicular orientation of the sample, i.e., when the electric field

is parallel to the a-axis (see Fig. 7). Figure 14a shows that the

minimum values of permittivity emin decrease with increas-

ing drawing ratio. The dielectric anisotropy, Adiel, is defined

as the quotient of the difference of the extreme values, emax

and emin, over their average value, (emax ? emin)/2 i.e., as

Adiel ¼ 2 � emax � emin

emax þ emin

: ð5Þ

It is plotted as a function of the drawing ratio in

Fig. 14b: the higher the drawing ratio, the stronger the

dielectric anisotropy. Summarizing, resonator measure-

ments give access to a macroscopically averaged quantity,

the permittivity, to which both the crystalline and the

amorphous phase of the polymer contribute. Dielectric

anisotropy reflects the orientation of the nano-crystallites.

Thus, dielectric measurements are a fast way to analyze

both the strength and the direction of orientation in the

polymer films.

Conclusions

We have shown that the use of different techniques such as

XRD, FTIR, and dielectric measurements yields a consis-

tent picture of structure formation in uniaxially stretched

polyamide films. The XRD analysis is the most detailed

and sensitive one giving all the microscopic information

about crystallinity, size of nano-crystallites, crystal struc-

ture, and average orientation of the crystal axis upon

drawing. But once this basic knowledge has been achieved,

it is not in all cases necessary to perform these sophisti-

cated and time consuming measurements to characterize

Fig. 13 The real part of the permittivity versus the rotation angle of

the sample with the drawing ratio as a parameter

Fig. 14 The minimum values of permittivity (left) and dielectric anisotropy (right) as a function of the drawing ratio

J Mater Sci (2014) 49:8074–8083 8081

123

polyamide films. When sufficiently big changes between

drawing ratios are studied (as we did), it is much easier and

faster to use IR spectroscopy and dielectric measurements,

at least to determine the direction and degree of the crys-

tallites orientation.

We have shown that degree of crystallinity and orien-

tation as well as dielectric anisotropy increase proportional

to the drawing ratio. The orientation of the c-axis is always

in the normal direction of the films, while the main axis (b-

axis) is oriented in the drawing direction. This means that

the long molecules align in the drawing direction, while the

intermolecular hydrogen bonds lie in the film plane and are

aligned perpendicularly to the drawing direction.

Acknowledgements H. Shanak gratefully acknowledges the finan-

cial support from ‘‘Deutscher Akademischer Austauschd-

ienst’’(DAAD), Bonn, Germany.

Appendix: Evaluation of the average orientation

function using XRD pole figures

In order to analyze the influence of drawing on the orien-

tation of crystallites in the different films, it is necessary to

determine the position of the crystallographic axes (a, b

and c) with respect to the axes of the film which are

equivalent to transverse direction (T), drawing direction

(D), and the normal to the surface (N), respectively. This

will be done using a pole figure analysis. The quantitative

analysis can be done by evaluating the orientation function

hcos2/hkl,qi, where /hkl,q is the angle made by the hkl plane

normal to the q-axis of the film (q = T, D or N-axis). The

analysis of the experimental data consists in evaluating the

distribution of the plane normal in an appropriate pole

figure or directly from the intensity distribution I(/,w)

from which the pole figure was derived (Fig. 2). For an

axial orientation with respect to N (see Fig. 2), the total

number of hkl plane normals oriented at a given colatitude

/ is proportional to the circumference of the circle of

radius r, which is given by sin/. Therefore, in order to

obtain hcos2/hkl,zi averaged over the entire surface of the

orientation sphere, it is necessary to weight Ihkl(/,w) by

sin/. Thus, hcos2/hkl,qi is generally defined as [22]

cos2 /hkl;q

� �¼R 2p

0

R p=2

0Ið/;wÞ cos2 / sin /d/dw

R 2p0

R p=2

0Ið/;wÞ sin /d/dw

¼R p=2

0Ið/Þ cos2 / sin /d/R p=2

0Ið/Þ sin /d/

; ð6Þ

where I(/,w) is the pole concentration, i.e., the measured

intensity of the diffraction peak at these coordinates [22,

23]. It represents the relative amount of crystalline material

having a plane normal in the direction of w and / such that

[22]

Ið/Þ ¼Z 2p

0

Ið/;wÞdw: ð7Þ

Therefore, the measured intensities are first integrated

over wN according to Eq. 7, then the integrations over /N

are carried out according to Eq. 6 in order to evaluate the

average orientation function of the crystallographic axes [a,

b, or c that correspond to a (hkl) plane normal] with respect

to a reference direction (N).

It is possible to cross-plot the pole figure original data

I(/N,wN) which were obtained with the N-axis as a refer-

ence in order to produce the distribution of the poles with

respect to the D- and T- axis. A coordinate transformation

[23] can be used to produce the values of /x, /y,wx, and

wy. Values of I(/z,wz) are then a read of the original

curves. New curves I(/T,wT) versus wT are constructed,

and in the same way curves of I(/D,wD), a function wD can

be produced.

Orientation functions of the monoclinic a form were

calculated from the equation of Wilchinsky [23] using the

crystallographic data of Holms [10]. In such a way, we

obtain the relations

cos2 /a;q

� �¼ cos2 /200;q

� �; ð8aÞ

cos2 /b;q

� �¼ 1� 0:7957 cos2 /200;q

� �

� 0:6361 cos2 /002;q

� �

� 0:5678 cos2 /202;q

� �; ð8bÞ

Fig. 15 Sketch of an equilateral triangle showing several limiting

cases [14]: point 1 means perfect orientation parallel to T. Point 2

lying on the N-axis means axial orientation about N. Point 3 (exactly

in the center) means random orientation. Point 4 (on one side) means

orientation in the D, N plane and perpendicular to the T direction [14]

8082 J Mater Sci (2014) 49:8074–8083

123

cos2 /c;q

� �¼ 0:63615 cos2 /002;q

� �þ 0:56782 cos2 /202;q

� �

� 0:20425 cos2 /200;q

� �;

ð8cÞ

where a, b, and c are the crystallographic axes of the unit

cell, and q is equivalent to the T, D, or N-axis of the film.

/hkl,q is the angle made by the hkl plane normal to the

q-axis of the film, /a,q is the angle made by the crystal-

lographic a-axis to the q-axis of the film; /b,q and /c,q are

defined analogously.

Each value of the average orientation function delivers

information about the orientation of a certain axis with

respect to a reference, e.g., for hcos2/i = 1, there is a

complete orientation; for hcos2/i = 0.333, there is a ran-

dom orientation, while the axis is oriented perpendicular to

a reference when hcos2/i = 0.

The defined values of the orientations hcos2/a,Ti,hcos2/a,Di, and hcos2/a,Ni can be used to specify a location

of a point on an equilateral triangle diagram as shown in

Fig. 15. Two of the three quantities are independent, while

the third quantity is fixed by the orthogonally relation [14]:

cos2 /a;T

� �þ cos2 /a;D

� �þ cos2 /a;N

� �¼ 1: ð9Þ

In this way, it is possible to plot the state of orientation

of all the crystallographic axes (a, b and c) on an equilateral

ternary diagram. Several limiting cases are shown in

Fig. 15 (for explanations see the figure caption).

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