1. Introduction

Polyethylene glycol (PEG), a polymer composed of the -CH₂-CH₂-O- repeating units, is one of the most widely synthetic and inexpensive materials that has medical, chemical, biological and industrial uses ^[1-4]. PEG of different molecular weights are commercially available and are well-known as non-toxic, non-immunogenic, amphiphatic and biocompatible polymers. PEG is utilized as a polymer-protein conjugate^[5] and also known for its ability to mitigate protein adsorption and cell adhesion ^[1]. Thus, it has been successfully attached to biomedical implants, joint replacements and tissue substitutes ^[6-9]. Besides, it is effective for biochips and biosensors ^[10], medical instruments and implants technology ^[11]. With its distinctive physical properties, PEG is chosen for this investigation because it may play an important role in developing radiation shielding and phantom technologies, medical and nuclear applications, and radiation dosimeters. Therefore, knowledge of PEG molecules' radiological parameters; such as mass attenuation and mass energy-absorption coefficients, interaction cross sections, effective atomic numbers, half-value layer and kerma, is vital for understanding their physical properties. In the same context, knowledge of gamma ray interaction with PRG is essential for radiation and nuclear physics and chemistry, radiation protection and dosimetry, and biomedical and technological fields ^[12-16].

A great number of authors reported the mass attenuation coefficients and related parameters for a variety of materials, including dosimeters, metals, polymers, biological and medical materials etc. ^[17-25], but none of the work were dedicated to evaluating the gamma attenuation performance of PEG. Therefore, it is important to study the radiation protection capabilities of this particular polymer against gamma irradiation ^[26]. To achieve this, an investigation of the behavior and performance of PEG of different molecular weights against different gamma-rays has been carried out.

PEG of molecular different weights provides a scope for various application. For example, PEG molecular weight (chain length) affects the long-term stability of biomedical applications and the biofouling performance of PEG ^[27]. The chain length of PEG is a crucial factor affecting its grafting density, the number of hydrogen bonds, and other properties of PEG ^[28]. On the other hand, PEG of low molecular weights was significantly effective as a hydrate inhibitor for designing drilling fluids ^[29]. It is thus of interest to identify the photon energy absorption parameters of PEG of various molecular weights.

In this research, the mass attenuation coefficients (μ_m) for five PEG products were measured at 59.5, 302.9, 356.0, 661.7, 1173.2 and 1332.5 keV by using NaI (Tl) scintillator, and were calculated by using mixture rule. Total atomic, molecular and electronic cross-sections (σ_a, σ_m and σ_e), half-value layer (HVL), effective atomic and electron numbers (Z_eff and N_eff), mass energy-absorption coefficients and kermas relative to air for the PEG samples, were calculated. These radiation interaction data are not tabulated in the literatures but are widely used in the shielding and dosimetry calculations used for medical diagnostic, therapeutic procedures and radiation biophysics. Therefore, the results can hopefully facilitate the use of PEG in specific applications such as gamma-ray shielding effectiveness, phantom technology, and many others.

2. Experimental

PEG of molecular weights of 1000, 10000, 20000, 100000, and 200000 were obtained from Sigma Aldrich (Germany). The common name, molecular formula, molecular weight and mean atomic number of each PEG sample are given in Table 1. The mean atomic number was calculated as: <Z> = ∑ f_i Z_i, where Z_i is the atomic number of the i^th element, f_i = n_i/∑n_i = n_i /n is the fractional abundance of the i^th element with respect to the number of atoms in the compound, n_i is the number of formula units of the i^th constituent element in the compound, and n is the total number of atoms in the molecule.

Each PEG sample was compressed in hydraulic press to thin disk of Φ1 cm×0.2 cm. The PEG samples were exposed to gamma-rays from four standard point sources (10 mCi) of ²¹⁴Am (59.5 keV), ¹³³Ba (302.9 and 356.0 keV), ¹³⁷Cs (661.7 keV), and ⁶⁰Co (1173.2 and 1332.5 keV). A Φ2''×2'' NaI(Tl) scintillation detector (EG&G Ortec, USA) was used. The source-detector distance was 25 cm (Fig. 1), and the detector energy resolution was 8% at 662 keV. A gamma spectrum was collected with a 2048-channel MCA, in data-collection time of 3000 s to ensure good statistics. The gamma spectra were analyzed using the Maestro-ORTEC. Each sample was measured for three times, and the average value was taken for calculating the parameters. The gamma-ray absorption of PEG samples was evaluated by a narrow beam collimated by two lead collimators of a Φ5 mm aperture ^[30]: one in front of the source and the other above the detector (Fig. 1). PEG samples in mass thicknesses of 0.21–0.47 g/cm² were positioned at 15 cm from the source.

Fig. 1:Full-screen View

Experimental setup for measuring γ-ray attenuation of PEG samples.

The μ_m for different PEG samples and energies were calculated by Eq.(1) ^[31]:

where I₀ and I denote the incoming and attenuated photon intensities, respectively; μ_m (cm²/g)= μ_ℓ/ρ, with μ_ℓ (cm⁻¹) being the linear attenuation coefficient; ρ (g/cm³) is the mass density; and x (g/cm²) is mass thickness of the sample. For a material composed of more than one element, μ_m is calculated by WinXcom ^[32] based on the mixture rule ^[33] as:

where (μ_m)_i is μ_m of the i^th element in the composite and w_i = n_i A_i /∑ n_j A_j is the proportion by weight of the i^th element, where A_i is the atomic weight of that element. The mixture rule gives the attenuation coefficients of any substance as the sum of the appropriately weighted contributions from the individual atoms. The WinXcom can generate cross sections or attenuation coefficients on a standard energy grid, spaced approximately logarithmically, on a grid specified by the user, or for a mix of both grids. This program provides total cross sections and attenuation coefficients as well as partial cross sections for incoherent and coherent scattering, photoelectric absorption, and pair production ^[34].

The maximum uncertainties (Δμ_m) in μ coefficients were calculated by Eq.(3):

where ΔI₀, ΔI, and Δx are the uncertainties in the intensities I₀ and I and the mass thickness x, respectively. For composite materials, the effective atomic number (Z_eff) is used to describe the gamma-ray interaction processes ^[35,36], calculated by ^[39]:

where σ_a is the atomic cross-section ^[37] and σ_e is the total electronic cross-section ^[38]:

The mass attenuation coefficient is used to calculate the total molecular cross section (σ_m):

where M = ∑ n_i A_i is the molecular weight of the compound. The effective electron number is calculated as:

where N_A is the Avogadro's number and A_r = ∑n_i A_i /∑n_i = M /n is the relative atomic mass of the compound.

The half-value layer, the material width required to reduce the air kerma of an X- or γ-ray to half its value, is defined as:

Kerma (kinetic energy released per unit mass) in a material, is feasible to uncharged particles and photons, and is related to energy fluence and mass energy-absorption coefficient (μ_en/ρ). Kerma of PEG material relative to air is calculated as:

where the (μ_en/ρ) = ∑ w_j (μ_en/ρ)_i for PEG or air ^[40,41]. For air, the (μ_en/ρ)_i coefficients are taken from Ref. [42].

3. Results and discussions

The μ_m values for PEG 100 000 measured and calculated at the selected photon energies are shown in Fig. 2, while the insert shows μ_m values for the other PEG samples. Experimental uncertainty in μ_m coefficient was estimated at ≤3%, being mainly due to uncertainty in measuring the mass density and thickness, and counting the incident and transmitted gamma intensities. The μ_m depends evidently on the photon energy and the chemical composition of PEG. At low energy region, the μ_m coefficient decreases sharply with increasing energy, as photoelectric absorption is the main interaction between gamma-rays and PEG, which is characterized by the importance of atomic binding. Compton scattering process at intermediate energies (302.9, 356.0 and 661.7 keV), and pair production process at high energies (1173.2 and 1332.5 keV), dominate over photoelectric absorption process. Thus, μ_m coefficients show a less energy-dependent behavior and gradual decrease with increasing energy. Fig. 2 and its insert, the measured μ_m for almost all samples is slightly lower than the calculated values. However, they agree well with each other within the experimental uncertainty. The observed discrepancy between measured and calculated values of μ_m may be ascribed to potential existence of trace amounts of impurities in the PEG materials. Other factors that may include possible deviation of the experimental setup from perfect-narrowness, causing systematic uncertainty in the measured values of μ_m ^[43].

Fig. 2:Full-screen View

The μ_m values versus photon energy for PEG 100 000. The inset shows the μ_m for other PEG samples. The confidence level is 95%.

The σ_a, σ_m and σ_e values versus photon energy is similar to that of μ_m values, as shown in Fig. 3 for sample PEG 200,000. The behavior of σ_a, σ_m and σ_e values for all PEG samples is nearly identical.

Fig. 3:Full-screen View

The σ_a, σ_e and σ_m versus photon energy for sample PEG 200,000. Uncertainty bars indicate the 95% confidence level.

The μ_m values, measured and calculated, were used to calculate the effective atomic number (Z_eff) and the electron density (N_eff) for PEG samples. The Z_eff and N_eff for PEG 200 000 are listed in Table 2. The percentage differences between the calculation and experimental results are below 1% for the PEG samples (and for other PEG samples) at the designated energies. From Table 2, the Z_eff values of PEG lie within the range of the atomic numbers of its constitute elements (1<Z_eff <8), which is consistent with others findings for low Z constituents' materials ^[44].

The Z_eff and N_eff versus photon energy for sample PEG 20 000 are shown in Fig. 4, and the insert in Fig.4(a) shows the Z_eff versus for the other PEG samples. The behavior of Z_eff and N_eff with energy for all PEG samples is almost the same and the measurement and calculation results agree well with each other. These confirm that the effective atomic number depends on the energy and interaction processes involved^[41]. Although Z_eff has its highest value at lower energy region where photoelectric effect is dominant ^[16], it can be observed from Fig. 4(a) that Z_eff values for PEG samples do not vary appreciably over the considered energy range. This observation agrees with the results in Ref. [38] that the effective atomic number tends to remain constant with increasing photon energy in materials containing carbon, hydrogen and oxygen. Also, from the inset in Fig. 4(b), N_eff increases linearly with Z_eff for the PEG samples.

Fig. 4:Full-screen View

A typical plot of (a) Z_eff and (b) N_eff versus photon energy for sample PEG 20,000. The inset to Fig. 4 (b) shows Z_eff versus N_eff for sample PEG 20,000. Uncertainty bars indicate the 95% confidence level.

The energy dependence of the HVLvalues for all PEG samples are given in Table 3. The HVL increases with energy. All PEG samples showed similar behavior indicating that PEG with various molecular weights are equivalent in attenuating gamma radiation as their HVL values are almost the same.

The variation of theoretical μ_en/ρ coefficient and kerma relative to air versus photon energy are given in Table 4 for all PEG samples. Regarding the values of μ_en/ρ coefficient and kerma relative to air, two distinct energy regions can be observed in Table 4. A strong energy dependence in the low energy region, at which the main interaction mechanism is photoelectric effect, and a less energy dependence in the higher energy regions, at which Compton scattering and pair production processes are the predominant interaction mechanisms.

4. Conclusion

The mass attenuation (μ_m) and mass energy-absorption (μ_en/ρ) coefficients, total atomic and electronic cross-sections (σ_a and σ_e), half-value layer (HVL), effective atomic (Z_eff) and electron (N_eff) numbers, and kermas relative to air for PEG in molecular weights of 1 000−200 000, have been investigated at 59.5, 302.9, 356.0, 661.7, 1173.2 and 1332.5 keV. The experiment and calculation results agree well with each other. The photon energy and compositional dependence of the values of μ_m, σ_a, and σ_e are remarkable in the low energy range due to the predominant photoelectric absorption mechanism. Z_eff and N_eff behave with photon energy in a similar manner for all PEG samples and therefore they were linearly related. HVL values increase with photon energy. The energy dependence of μ_en/ρ coefficients and kermas relative to air show two distinct energy regions in which they behave quite differently. To our best knowledge, μ_m coefficients and the related radiation energy absorption parameters of PEG are not available in the literature. Therefore, our results can be useful in scientific and industrial fields of radiation and nuclear physics and chemistry, radiation protection and dosimetry, biomedical and technological applications.