Dynamic mechanical analysis

Dynamic mechanical analysis (DMA) is used to gain information about first (major), secondary and tertiary transitions in polymers [150]. In DMA, an oscillating force is applied to a specimen and the material response is analyzed. Material properties such as the tendency to flow (viscosity) and the stiffness (modulus) can be determined from the phase lag and the specimen recovery. These properties are related to the ability to lose energy as heat (damping) and the ability to recover from deformation (elasticity). In polymers, information on property changes is obtained by the relaxation of chains or changes in the free volume that occur during DMA [150]. Polymers show viscoelastic behavior. Therefore, stress (load) leads to elastic and plastic response. Although one effect could be dominating, DMA is capable of evaluating both of these mechanical property changes.

The stress (or strain) is applied to the specimen at a specific frequency . Stress σ and strain  curves have sinusoidal shape with the amplitudes σ0 and 0 (Fig. 19). Accordingly, time dependent σ and 

can be described as (Eq. 42 and 43) [145, 150]:

𝜎(𝑡) = 𝜎₀sin 𝜔𝑡 (42)

𝜀(𝑡) = 𝜀₀sin(𝜔𝑡 − 𝛿) (43)

with σ: stress, σ0: stress at time t = 0, : frequency, t: time, : strain, 0: strain at t = 0, δ: phase angle.

The stress and strain curves differ by the phase angle δ (Eq. 43 and Fig. 19).

Figure 19: Schematic representation of periodic sinusoidal applied stress curve with strain curve exhibiting a difference of phase angle δ, after [145, 146, 150].

Stress and strain are related by the complex modulus G* _{which consists of an elastic storage (G’, real}

part of complex) and an imaginary loss (G’’) modulus (Eq. 44):

𝐺∗_{= 𝐺}′_{+ 𝑖𝐺′′ (44).}

G’ is a measure of the ability of a material to store energy (for a perfectly viscous fluid, G’ = 0). G’’ represents a measure of the ability to dissipate energy (for a completely elastic material, G’’ = 0). The ratio of these two moduli is denoted as damping and represented by the value of tan δ (Eq. 45):

𝑡𝑎𝑛(𝛿) =𝐺′′_𝐺′ (45).

Using storage and loss modulus (G‘ and G’’), the complex compliance (J*_{) and the complex}

viscosity (*_{) can be calculated [145, 150].}

DMA can be performed in a range of test modes such as shear, bend, torsion, compression and tension mode [28]. Furthermore, DMA is advantageous due to the capability of measuring a material response over a range of temperatures (temperature sweep) and frequencies (frequency sweep) in a single experiment [145].

In this study, temperature-dependent DMA was performed in shear mode (rheometer of type Ares of TA Instruments, New Castle, Delaware, USA) in a temperature range of -115°C to +120°C. All measurements were conducted using a frequency of 1 Hz and a deflection of 0.05%. DMA specimens were cut from sheets of 1.0±0.3 mm thickness (section 4.1). Specimen dimensions are given in figure 20 and table 1. Three DMA measurements were performed for each PE-HD type.

Figure 20: Schematic depiction of DMA specimen with dimensions. Table 1: Thicknesses of DMA specimens.

PE-HD type (section 4.1) specimen thickness x (Fig. 20) / mm

AGUV 1.23 AGBD 1.10 AQ149 1.26 5021DX 1.18 5831D 1.05

3.1.8. X-ray diffraction

X-ray diffraction (XRD) is a technique to evaluate the structure of crystals [147]. Diffraction occurs when a wave encounters a series of regularly spaced obstacles, which are able to scatter the wave and which have spacings that are comparable in magnitude to the wavelength. Hence, diffraction results from specific phase relationships established between several scattered waves.

X-ray wavelengths usually meet the order of atomic spacings in solids and are scattered by the electrons of atoms or ions lying within the beam path. Constructive interference has to occur for X- ray diffraction by a periodic arrangement of atoms (Fig, 21).

Figure 21: Geometry scheme for interference of an X-ray wave scattered by two planes of atoms separated by spacing d (Bragg’s law), after [147].

Parallel planes of atoms (A and B in Fig. 21) are separated by the interplanar spacings d. When parallel, monochromatic and coherent X-ray beams of wavelength λ incide on these planes at an angle , several rays (1 and 2 in Fig. 21) are scattered by the atoms. The interplanar spacing d gives

rise to different path lengths of X-rays that are scattered from different planes. Constructive interference (1’ and 2’) occurs at an angle  to the planes, if path length difference δ is equal to an integer (n) of wavelengths. The condition for diffraction is given by Bragg’s law (Eq. 46) [145, 147, 151]:

𝑛𝜆 = 2𝑑𝑠𝑖𝑛Θ (46)

with n: order of reflection, which is any integer consistent with sin  not exceeding unity. If Bragg’s law is unsatisfied, interference will be nonconstructive and result in a very low-intensity diffracted beam. Therefore, X-ray spectra can be obtained by plotting the measured scatter intensity in dependence on diffraction angle . The location and the shape of peaks in these spectra provide information on the crystal structures. Perfect crystals exhibit sharp, imperfect crystals show broadened peaks. Completely amorphous materials exhibit no peaks. The diffuse spectra of semi- crystalline PE-HD consist of broadened peaks due to the presence of an amorphous phase [152]. Two different types of X-ray scattering techniques are distinguished. When the experiment is performed using a scattering angle of 6° and higher, the technique is termed wide angle X-ray scattering (WAXS) [153]. WAXS spectra provide information on the overall crystallinity Xc,XRD,

which is calculated by integrating the peak area and subtracting the scattering area of the amorphous phase [145]. The WAXS crystallinity Xc,XRD may differ from the crystallinity obtained from DSC

analysis, because structural characteristics affect the measurement parameters differently.

In contrast, small angle X-ray scattering (SAXS) is performed with scattering angles  lower than 6°. Usually, SAXS experiments start from diffraction angles less than 1° [154, 155]. Since structures of a size between 1 nm and 100 nm can be depicted, the fine molecular structures such as the lamella structure of PE are analyzed with this technique. Furthermore, the periodic structure within a material can be obtained by SAXS. The thickness of the periodic structure is denoted as long period Lp. In PE, Lp equals the thickness of the sum of a crystalline lamella and an amorphous

layer. To obtain Lp, Bragg’s law is modified by considering small diffraction angles (sin  =  = 𝜖)

and it simplifies to (Eq. 47) [145]:

𝐿𝑝𝜖 = 𝑛𝜆 (47)

with Lp: long period, 𝜖: Bragg angle of intensity maximum, n: level of scattering, λ: wavelength of X-

rays.

Since Lp was found to be proportional to the polymer crystallinity Xc,SAXS and since Lp consists of one

crystalline and one amorphous layer, lamella thickness of a polymer can be calculated. In turn,

Xc,SAXS can also be obtained from lamella thickness measurements (Eq. 48):

𝑙𝑎𝑚𝑒𝑙𝑙𝑎 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 = 𝑋𝑐,𝑆𝐴𝑋𝑆 [%] 𝐿𝑝 ⇒ 𝑋𝑐,𝑆𝐴𝑋𝑆[%] =𝑙𝑎𝑚𝑒𝑙𝑙𝑎 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠_𝐿 (48).

In SAXS, using the scattering vector q as the difference of the wave numbers of incident and scattered wave (𝑞 = 𝑘𝑓− 𝑘𝑖) and using the assumption for small angles |𝑘𝑓| ≈ |𝑘𝑖| =2𝜋_𝜆,

Bragg’s law can be considered as (Eq. 49):

|𝑞| =4𝜋_𝜆 𝑠𝑖𝑛(Θ) (49)

with q: scattering vector, λ: wavelength, : scattering angle. The scattering cross-section per volume element is applied for further analysis, which is defined in one dimension (z-direction) for an isotropic two-phase system according to (Eq. 50):

∑ 𝑞 =_4𝜋𝑞22𝑟𝑒2(2𝜋)2∫ 𝑒𝑥𝑝(−𝑖𝑞𝑟) ∞

−∞ 𝐾(𝑧)𝑑𝑧 (50)

with the correlation function K(z). re is the classic electron radius obtained from the vacuum

permeability (‘magnetic constant’) µ0, the electric charge of an electron e, the mass of an electron me

and equation 51: 𝑟_𝑒= 𝜇0𝑒2

4𝜋𝑚𝑒≈ 2.818 ∙ 10

−15_{𝑚 (51).}

Applying Fourier transform, the correlation function K(z) can be derived from ∑ 𝑞 (Eq. 50) by (Eq. 52): 𝐾(𝑧) = 1 𝑟𝑒2(2𝜋)3∫ 4𝜋𝑞 2 _{Σ𝑞 cos(𝑞𝑧)} ∞ 𝑞=0 𝑑𝑞 (52).

From the characteristic K(z), the long period Lp (Eq. 47) can be determined graphically [146].

In this study, X-ray scattering experiments (XRD) were performed (D8 Advance, Bruker, Billerica Massachusetts, USA) ranging 2 scattering angles from 3° to 50°. The diffractometer was equipped either with a molecular-metrology setup (WAXS) consisting of a copper anode (focus point, 40 kV, 55 mA) and a 2D detector (q-range: 0.08 to 2.80 nm-1_{) or a slid-collimated Kratky-compact camera}

(SAXS, copper anode with 40 KV, 40 mA, line focus, point detector, q-range of 0.13 to 5.40 nm-1_).