Orientation of nano-crystallites and anisotropy of uniaxially drawn α-polyamide 6 films: XRD, FTIR, and microwave measurements

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 resonator to investigate the structure of uniaxially drawn α-polyamide 6 films. The influence of uniaxial drawing on several parameters such as degree of crystallinity, orientation 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 orientation 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.

Appendix: Evaluation of the average orientation function using XRD pole figures

In order to analyze the influence of drawing on the orientation 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 〈cos²ϕ_hkl,q〉, where ϕ_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(ϕ,ψ) 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 ϕ is proportional to the circumference of the circle of radius r, which is given by sinϕ. Therefore, in order to obtain 〈cos²ϕ_hkl,z〉 averaged over the entire surface of the orientation sphere, it is necessary to weight I_hkl(ϕ,ψ) by sinϕ. Thus, 〈cos²ϕ_hkl,q〉 is generally defined as [22]

$$ \left\langle {\cos^{2} \phi_{hkl,q} } \right\rangle = \frac{{\int_{0}^{2\pi } {} \int_{0}^{\pi /2} {I(\phi ,\psi )\cos^{2} \phi \sin \phi d\phi d\psi } }}{{\int_{0}^{2\pi } {} \int_{0}^{\pi /2} {I(\phi ,\psi )\sin \phi d\phi d\psi } }} $$

$$ = \frac{{\int_{0}^{\pi /2} {I(\phi )\cos^{2} \phi \sin \phi d\phi } }}{{\int_{0}^{\pi /2} {I(\phi )\sin \phi d\phi } }}, $$

(6)

where I(ϕ,ψ) 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 ψ and ϕ such that [22]

$$ I(\phi ) = \int_{0}^{2\pi } {I(\phi ,\psi )d\psi } . $$

(7)

Therefore, the measured intensities are first integrated over ψ_N according to Eq. 7, then the integrations over ϕ_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(ϕ_N,ψ_N) which were obtained with the N-axis as a reference 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 ϕ_x, ϕ_y,ψ_x, and ψ_y. Values of I(ϕ_z,ψ_z) are then a read of the original curves. New curves I(ϕ_T,ψ_T) versus ψ_T are constructed, and in the same way curves of I(ϕ_D,ψ_D), a function ψ_D can be produced.

Orientation functions of the monoclinic α form were calculated from the equation of Wilchinsky [23] using the crystallographic data of Holms [10]. In such a way, we obtain the relations

$$ \left\langle {\cos^{2} \phi_{a,q} } \right\rangle = \left\langle {\cos^{2} \phi_{200,q} } \right\rangle , $$

(8a)

$$ \left\langle {\cos^{2} \phi_{b,q} } \right\rangle = 1 - 0.7957\left\langle {\cos^{2} \phi_{200,q} } \right\rangle - 0.6361\left\langle {\cos^{2} \phi_{002,q} } \right\rangle - 0.5678\left\langle {\cos^{2} \phi_{202,q} } \right\rangle , $$

(8b)

$$ \left\langle {\cos^{2} \phi_{c,q} } \right\rangle = 0.63615\left\langle {\cos^{2} \phi_{002,q} } \right\rangle + 0.56782\left\langle {\cos^{2} \phi_{202,q} } \right\rangle - 0.20425\left\langle {\cos^{2} \phi_{200,q} } \right\rangle, $$

(8c)

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. ϕ _hkl,q is the angle made by the hkl plane normal to the q-axis of the film, ϕ _a,q is the angle made by the crystallographic a-axis to the q-axis of the film; ϕ _b,q and ϕ _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 〈cos²ϕ〉 = 1, there is a complete orientation; for 〈cos²ϕ〉 = 0.333, there is a random orientation, while the axis is oriented perpendicular to a reference when 〈cos²ϕ〉 = 0.

The defined values of the orientations 〈cos²ϕ_a,T〉, 〈cos²ϕ_a,D〉, and 〈cos²ϕ_a,N〉 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]:

$$ \left\langle {\cos^{2} \phi_{a,T} } \right\rangle + \left\langle {\cos^{2} \phi_{a,D} } \right\rangle + \left\langle {\cos^{2} \phi_{a,N} } \right\rangle = 1. $$

(9)

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]

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).