Multiscale Modeling of Permittivity of Polymers With Aging: Analysis of Molecular Scale Properties and Their Impact on Electrical Permittivity

This work presents an innovative model for the derivation of permittivity evolution of polyethylene (PE)-based materials with aging. First, the derivation of the microscale contributions to the real permittivity of methylene unit [constitutive repetitive unit (CRU) of PE] and its oxidation products, that is, ketones and hydroperoxides, in the solid state is presented. Then, a chemical kinetic model is recalled predicting the concentration, under proper hypotheses, of the oxidized species created during polymer aging. The proposed model combines the concentrations of methylene unit and its oxidation products with the respective contribution to permittivity, providing the trend of permittivity of polymer with aging. Results depict good agreement with the experimental data, validating the model.

microscale properties of interest are the polarizability and the 31 volume of the molecules the material is made up of [2]. These 32 quantities are usually obtained using molecular simulation 33 software. Due to the high number of chemical units inside 34 the polymer chain, the computational effort is often very high, 35 and supercomputers are typically used for calculation [3], [4]. 36 The recent advances in the study of first-principles calculations 37 via density functional theory (DFT) with periodic boundary 38 conditions resulted in faster calculations for simple polymeric 39 materials, such as polyethylene (PE) [3]- [5]. 40 Clausius and Mossotti paved the way for the discovery of a 41 relationship between molecular microscopical properties and 42 macroscopical permittivity. In diluted systems, for example, 43 gases, the Clausius-Mossotti (CM) equation is valid only if 44 the intermolecular interactions are ignored. When applied to 45 solids, the equation can lead to incorrect estimations of the 46 dielectric constant, since it neglects important properties such 47 as molecular surface size and orientation. Moreover, the CM 48 equation is not explicitly dependent on the internal electric 49 field of the molecules E i due to the introduction of the 50 so-called "uniform polarization hypothesis by Mossotti and 51 Lorentz" [2]. E i is different from the external electric field 52 E, since it represents the action of all the charges on a single 53 dipole, this latter is excluded [3]. It is evident that the internal 54 electric field is not easily derivable from theoretical equations, 55 except for some common geometries (e.g., spherical charge 56 displacement). 57 Furthermore, the properties of polymeric materials may 58 change over time due to aging. This latter, in low-voltage 59 cable systems, is mainly caused by environmental stresses, 60 for example, heat and radiation which may promote various 61 phenomena like oxidation, chain scissions, and crosslinking. 62 The oxidation process results in the formation of new species 63 (e.g., ketones, carboxylic acids, esters, etc.) along the polymer 64 chain, which can significantly alter the polymer properties 65 even at very low concentrations. Up to now, the determination 66 of the permittivity values of degradation species has very 67 limited literature, despite their importance and influence on 68 the macroscopical properties. As a matter of fact, permittivity 69 values reported in previous studies are either based on gas 70 state and diluted solutions containing these species [6] or 71 subject to high uncertainties [7]. This gap can be explained 72 by the impracticality of validating the results generated by 73 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ simulations and models on a fully degraded polymer, as will 74 be discussed in the following. 75 Different experimental techniques, such as transmission 76 Fourier transform infrared (FTIR) spectroscopy, can be applied 77 to measure the degradation of species within aged polymers 78 [8]. On the other side, several researchers formulated kinetic 79 models [8], [9] trying to quantify the increase of concentration 80 of these oxidized species as a function of the aging time and 81 aging conditions. As expected, the higher the accuracy, the 82 bigger the computational effort for the resolution of the kinetic 83 equations. Therefore, semi-empirical models are often used 84 to evaluate these species, but they have narrow applications 85 related to specific polymers and aging conditions.   [8] for PE with the 98 permittivity calculation approach in solids by Natan et al. [3].  In addition, if an electric field is applied to the molecule, the 116 atomic-induced dipole moment may arise as reported in [10].

117
In order to overcome this complexity, various models based 118 on the additivity approach were proposed in the literature.

119
These methods are, in general, successful in reproducing the 120 molecular mean polarizability [11]. Nonetheless, the deviation 121 of simulated data obtained through pure addition of atomic 122 polarizability from experimental ones is usually below 10%.

123
As an example, in the case of urea molecules, the deviation 124 is reported to be ∼8% [10]. This value tends to reach zero quantified the minimum distance between molecules beyond 131 which intramolecular interaction may be considered negligible. 132 However, it is not possible to obtain a unique value valid for 133 all the species, since it is deeply influenced by the considered 134 molecular (e.g., dipolar momentum) and matrix (e.g., density) 135 properties.

136
In the case of dense matter, for example, solids, 137 Natan et al. [3] found very good accordance between simple 138 addition and different simulation methods as DFT for polar-139 izability calculations of aromatic molecules. These species 140 are apolar molecules, hence the intermolecular forces may 141 be reduced to the matrix properties of the considered matter. 142 As a confirmation, the same authors claim that the addition 143 approach was successfully applied to the apolar alkyl chain 144 monolayers which can be considered a good structural model 145 for PE chains. 2) Simulation Approach: In this work, the additional 147 approach is used as it is considered a good approximation 148 for the calculation of polarizability of dense solid matter. 149 The values of polarizability α are calculated through the 150 chemical simulation software ChemAxon Marvin v.21.8 and 151 Avogadro [12]. As known, PE chains may have different 152 lengths and arrange themselves into ordered structures, that 153 is, crystals. As a first attempt, the authors neglect the mor-154 phological arrangement of these species, considering that the 155 PE chain polarizability is given by the sum of polarizabilities 156 of the constitutive repetitive units (CRUs) (e.g., methylene 157 unit -CH 2 -in the case of unaged PE) only. The negligence 158 of the contribution of the methyl termination groups (-CH 3 ) 159 is acceptable in the case of long chains, due to the reduced 160 impact of the additional hydrogen atom on the global polar-161 izability, as will be discussed in the following. Moreover, 162 these macromolecules are considered stand-alone so that the 163 intermolecular interactions can be neglected, and the additive 164 approach may be properly applied.     Under the hypotheses presented in the previous section, 209 it is possible to consider the contribution of the degradation 210 products to the polymer polarizability through a parametric 211 study. As a result, constitutive CRU polarizabilities and dipole 212 moments, obtained through the Marvin Chemaxon platform, 213 for the different degradation groups are reported in Table I.

214
From this table, it is possible to notice that the presence 215 of oxygen molecules raises all the analyzed electrical micro-216 scopical quantities. In particular, as polar species, oxygen 217 increases the polarizability of the molecule and, consequently, 218 the electrical response of the resulting chain once subjected 219 to an external electrical field. Indeed, the modification of the 220 repetitive unit structure, that is, the introduction of -OH or 221 multiple bonds, raises the atomic polarizability of the carbon 222 atom and, consequently, the polarizability of the global mole-223 cule. As an example, in the case of ketones, the presence of 224 the double bond with oxygen causes the atomic polarizability 225 of carbon to raise to 1.36 Å 3 [see

227
The increase in polarizability is not the same among the 228 considered species. As expected, molecular polarizability and 229  TABLE II  COMPARISON TABLE BETWEEN POLARIZABILITIES OBTAINED THOUGH  SIMULATION SOFTWARE AND THE Table II).

232
Let us consider two of the most common degradation  Table II the macromolecular structure arrangement inside the polymer. 264 As an example, the density of semicrystalline polymers is 265 placed in between the values of density related to the cor-266 responding crystalline (denser) and amorphous (less dense) 267 phases.

268
Thus, it is evident that molecular volumes obtained through 269 calculations involving density are different from, usually lower 270 than, the theoretical van der Waals volumes.

271
Nonetheless, the values obtained through density parameters 272 showed to be more realistic delivering good results for the cal-273 culation of the real permittivity, as will be seen in Section III. 274 Moreover, as density varies during aging [14], [15], the 275 derivation of density values for oxidized polymers may bring 276 to the definition of molecular volumes of degradation species. 277 The procedure for obtaining molecular volumes is based 278 on simple chemical calculations [see the diagram in Fig. 7 279 and (1)] involving molecular weight M W , density ρ, and the 280 Avogadro number N A , as in the following equation: Similar to what was reported for the calculation of polar-283 izability, we consider the unaged PE as made up of the 284 same CRU (-CH 2 -), which has a molecular weight equal to 285 M CH2 = 14 g/mol.

286
The density ρ is usually well known in the case of common 287 polymers, such as PE (ρ∼0.9 g/cm 3 ). On the contrary, the 288 density values of the degradation species coming from polymer 289 aging are not a priori defined and further analysis is required, 290 as reported in [14]. 291 2) Determination of the Density Changes With Aging: In the 292 case of semicrystalline polymers, it is possible to consider the 293 relationship reported in (2). The density ρ can be expressed as 294 a function of the densities of its amorphous ρ a and crystalline 295 phases ρ C as follows: where V C is the volume fraction of crystals.

298
According to (2), two causes may be responsible for an 299 increase in ρ during aging. 1) The incorporation of "heavy" atoms such as oxygen into 301 a polymer structure initially contains many "light" atoms 302 (i.e., carbon and hydrogen) [14].

303
2) The integration of short fragments, coming from chain 304 scission phenomena to crystalline lamellae. This induces 305 a chemicrystallization, that is, thickening of crystalline 306 lamellae and an increase in the crystallinity ratios 307 (i.e., V C ), as experimentally observed elsewhere [14]. 308 Thus, (2) relates the increase of density due to aging with the 309 integration of oxidized species along with the macromolecular 310 structure modification. The density variation given by the introduction of degrada-312 tion products inside the PE matrix can be written as products formed in the XLPE matrix (see Table III).

325
The second proportionality constant can be empirically  it is possible to obtain the density of the degradation product 334 inside a polyethylene matrix by the simple addition: are substituted with the oxidative groups reported in Table III. 347 Obviously, such a material is not obtainable in real conditions,  As a result, the calculated molecular volumes are lower 359 (down to half) than the values related to the neat PE group. 360 This result is unexpected since the substitution of a small 361 atom, as hydrogen in PE, with a bigger atom, as oxygen, 362 should increase the volume of the considered molecule. This 363 is the case of the van der Waals volumes obtained through the 364 molecular simulation software. On the other hand, as partially 365 described above, the theoretical increase of volume is coun-366 terbalanced by, though minor than the effect of, the higher 367 density inside the reference volume. This brings to stronger 368 interaction forces among molecules, which result to be more 369 packed and squeezed, leading to a reduction of their volume 370 in comparison with the theoretical van der Waals one. Similar 371 results may also be seen in the work by Krevelen [7]. The interest in the calculation of the dielectric constant 375 of solid materials encouraged the formulation of innovative 376 methods and approaches. Among those, Natan et al. [3] 377 analyzed the properties of polarizability and dielectric constant 378 of nanoscale molecular layers by comparing the calculations 379 coming from the DFT with phenomenological models based 380 on polarizable dipolar arrays.

381
It is worth recalling that polarization P, that is, the 382 induced dipole per unit volume (6) is given by the product 383 between susceptibility χ and the average internal field in 384 the material E i For monolayers made up of finite-length monomers, it is 387 possible to write the average internal field as where E is the applied external electric field.

390
It is possible to approximate that the internal electric field 391 is almost equal to the external one if the distances in-between 392 molecules are big enough to neglect the effect of the electric 393 field related to the polarization of the adjacent molecules. 394 This is the case of, for example, gases and nonviscous liquids 395 (CM model). Then, combining (7) and (8), we obtain In the case of solids, the term −4πP in (7) The values of dielectric permittivity given by (11) were  Table V.

423
It is worth highlighting that, as expected, the oxidized  Table V.   Literature reports several approaches for the calculation 451 of the global permittivity for composite materials, usually 452 considering the volume fraction occupied by the inclusions 453 (e.g., fillers) in the lattice. One of the most common is the 454 Maxwell-Garnett equation [16]. However, all these approaches 455 consider the substitution of one species (e.g., polymer) with 456 another one (e.g., fillers), not allowing the use of multiple 457 inclusions.

458
In this work, the authors propose a new approach that relates 459 the concentration of the different species (e.g., PE matrix, 460 degradation products) and the corresponding permittivities 461 with the global permittivity of the polymeric compound. The 462 obtained results should be then validated by comparing them 463 with experimental ones. As a first attempt, we could consider 464 a linear dependence between the two parameters, namely where Y (t) i is the molar fraction of the considered species, 467 calculated through the kinetic model [8], and ε i is the real part 468 of permittivity of the species (as in Table V).

469
Values of the real part of permittivity given by solving 470 (12) for different aging times result to be significantly lower 471 than the one obtained through experimental tests at lab-472 scale measurable frequencies. The reason for that can be 473 related to the permittivity dependence on frequency. Indeed, 474 the hypotheses considered for the calculation of polarizabil-475 ity (Section II-A) and permittivity (Section III) neglect the 476 contribution of the temperature-dependent polarizations (i.e., 477 dipolar and interfacial polarization), occurring at frequencies 478 lower than ∼10 10 Hz. Nonetheless, the contributions of these 479 polarization mechanisms to the real permittivity are minor in 480 the case of nonpolar and nonfilled materials, for example, 481 plain PE where the dipolar momentum is equal to 0 D and 482 the interfaces given by its semicrystalline structure are very 483 few. On the contrary, the dipolar and interfacial polarization 484 mechanisms are prevailing if we introduce polar species inside 485 the polymeric compound, such as during polymer aging. 486 As reported in Section II-A, simulation results claim a very 487 high value of dipolar momentum, up to 3 D (see Table I), 488 which may completely rule the permittivity trend in the dipolar 489 polarization frequency region, raising the ε value. At the same 490 time, the introduction of different oxidative groups in the 491 polymer matrix enhances the interfacial polarization response. 492 To consider the impact of the degradation species in the 493 permittivity calculation, we introduce a dipolar enhancement 494  The frequency chosen for the evaluation of the aging status 501 of polymer is 100 kHz, which was found in our previous works 502 to successfully follow the increase of the degradation products 503 due to aging stresses [15], [17]- [19]. At this stage, η is chosen 504 to best fit the experimental data. In the case of the investigated 505 PE, a good tentative value for η was found to be equal to 3.

509
The schematic summarizing the proposed modeling approach 510 is reported in Fig. 8. In order to validate the proposed model, simulation results are 528 compared to experimental tests performed on a plain XLPE. 529 In this work, we considered a PE crosslinked through the 530 condensation of silanol side groups (Si-XLPE) subjected to 531 three different aging conditions. To accelerate the degradative 532 effects, aging was obtained through the combination of gamma 533 radiation and temperature, as reported in Table VI. Radiation 534 aging was performed by UJV (Rez, Czech Republic) through 535 a 60 Co irradiation source. The permittivity of these materials 536 is evaluated by means of a Novocontrol Alpha Dielectric 537 Analyzer v2.2 operating in the frequency region 10 −2 -10 6 Hz 538 at 3 V rms . 539 Fig. 9 reports the results obtained by the model and the 540 experimental data. Solid curves, one per each aging condition 541 considered, represent the results acquired by the application 542 of (13) for the different aging conditions and durations con-543 sidered. Scatter points refer to the experimental data coming 544 from the dielectric analyzer. As it can be seen, very good cor-545 respondence between the model and the experimental results 546 is obtained, claiming the effectiveness of the proposed formula 547 for coupling the species concentrations with the corresponding 548 simulated permittivities.

549
It is worth noting that the model curves (see Fig. 9) follow 550 the expected kinetics trend. Indeed, permittivity values exhibit 551 a steep increase during the first aging period due to the rise 552 of all the degradation species. Then, once the hydroperoxides 553 reach their plateau, the real part of permittivity is ruled by 554 the carboxyl acid and ketone kinetics only. This brings to the 555 change of the increasing pace of ε with aging time.