Quantitative Evaluations on Harmonic and Anharmonic Lattice Thermal Capacity of Polymers

Valeri Ligatchev ^*

1. Not Affiliated, Singapore 471748, Singapore

* Correspondence: Valeri Ligatchev

Academic Editor: Changming Fang

Received: December 28, 2024 | Accepted: May 14, 2025 | Published: May 22, 2025

Recent Progress in Materials 2025, Volume 7, Issue 2, doi:10.21926/rpm.2502008

Recommended citation: Ligatchev V. Quantitative Evaluations on Harmonic and Anharmonic Lattice Thermal Capacity of Polymers. Recent Progress in Materials 2025; 7(2): 008; doi:10.21926/rpm.2502008.

© 2025 by the authors. This is an open access article distributed under the conditions of the Creative Commons by Attribution License, which permits unrestricted use, distribution, and reproduction in any medium or format, provided the original work is correctly cited.

1. Introduction

Intensive experimental and theoretical investigations on the characteristics of thermal capacity of polymers emerged as an important branch of material science and engineering since 1945 [1,2,3] (see also references therein). Later, those studies also became an essential research field in medical and biological sciences [4,5,6]. Atomic structures of polymers (macromolecules) customarily comprise multiple (though repetitive) chemical units called ‘monomers’ [6]. In general, polymeric atomic structures may comprise (predominantly) three-dimensional (3D), two-dimensional (2D), or one-dimensional (1D) atomic ‘fragments’, or a combination of those [1,2,3]. Graphical representations of the structural units (monomers) of a few well-known polymers with primarily linear (1D) spatial structures are sketched in Figures 1(a), 1(b), 1(c), and 1(d).

Figure 1 Graphical representation(s) of structural units of (a) ethylene/polyethylene, (b) propylene/polypropylene, (c) butadiene/poly(butadiene) (1,4), and (d) acetylene/polyacetylene, with predominantly linear molecular arrangements.

Grey balls represent carbon atoms, while white ones represent hydrogen atoms in the graphical illustrations of monomers. The MolView (version 2.4) software is used for the creation of graphical representations of the structural units of monomers, while ChemSketch (freeware), version 2.4 (2023), is used for creating sketches of atomic arrangements within all those polymeric structural units.

Otherwise, those polymeric structures might also be classified as well based on spatial connectivity of its atomic ‘network’, as ‘skeletal’ (backbone) fragments, and structural ‘groups’ [2]; the latter ones might incorporate ‘terminal’ C-H bonds, as well as ‘terminal’ CH₃ groups in the atomic structure of polypropylene, see Figure 1(b). Furthermore, the aforementioned polymers and monomers might have (spatially periodic) crystalline and/or (spatially disordered) amorphous atomic structures, or even their mixture [2]; see also references therein. Those (repetitive) molecular monomeric units themselves apparently form one of the key levels in essentially hierarchical atomic structures of polymers, with their first (basic) level formed by the (predominantly carbon and hydrogen) atoms composing those monomers.

Higher levels of these ‘polymeric’ structures could be characterized not only by atomic compositions of the macro-molecules, but also by their topology (i.e., spatial ‘connectivity’ [7] among the adjacent monomers) and morphology-i.e., dimensionality, shapes-or ‘habit’ in terminology of B. Wunderlich [8], -sizes, and crystalline orientation (if any) of molecular fragments (blocks). All these structural, topological, and morphological features of real polymers eventually define their macroscopic characteristics: optical properties, thermal and electrical conductivity, heat capacity, elasticity modulus, etc.

This study is predominantly devoted to comparative descriptions and implementation of several quantitative approaches for simulations on temperature-dependent specific (volumetric) harmonic and anharmonic lattice thermal capacities of a few well-known real polymers. The (isobaric) specific thermal (heat) capacity is routinely defined as follows [8,9]:

\[ \mathrm{C_p=\frac{\partial Q}{m\partial T},} \tag{1} \]

here C_p is the isobaric quantity is evaluated at a constant environmental pressure; ∂Q is the infinitesimally small amount of heat (thermal energy) required to increase the absolute temperature, T, of m grams of the given material by the infinitesimally small ∂T quantity. Similarly, the isochoric heat capacity could be evaluated as well at a constant volume of the material, and this quantity is routinely denoted as C_v. The latter heat capacity is traditionally associated with purely harmonic contributions from fundamental (and essentially single-particle) states of conventional (i.e., longitudinal and transverse) acoustic phonons, while the C_p quantity is commonly expected to take into account fraction(s) of either ‘quasi-harmonic’ or truly anharmonic contributions as well [9]. Aforementioned quasi-harmonic contributions are customarily associated with ‘…effects of thermal expansion against a bulk modulus…’, while truly anharmonic effects ‘…originate from interactions of thermally-excited phonons with other phonons, or from interactions of phonons with electronic excitations…’ [9]. Both the C_p and C_v quantities are temperature-dependent in general; thus, they could be denoted as C_p(T) and/or C_v(T) functions as well. Typical examples of the experimental C_p(T) dependencies, reported for the polyacetylene (PA) in ref. [10] and for polypropylene (PP) in ref. [11], composed predominantly of with linear macromolecules, are plotted in Figures 2(a), 2(b), respectively.

Figure 2 The experimental C_p(T) dependencies (symbols) plotted (a) for the nominally-undoped and iodine-doped samples of polyacetylene (PA) [10] (see the Figure 5 therein), as well as (b) for the amorphous and crystalline modifications of polypro-pylene (PP) [11]; the latter figure plotted based on the data, revealed in the Table 4, Table 5 (respectively) of the ref. [11]. Log-log slope(s) of those dependencies, evaluated using standard least-square-fit procedure in the temperature range of (25 K ≤ T ≤ 300 K) for the panel (a), and in the range of (100 K ≤ T ≤ 150 K) for panel (b), are revealed near to the corresponding data sets in both panels. The temperature range corresponding to the ‘low-temperature anomalies’ is shown in Figure (b) as well. The dashed curve in Figure (b) corresponds to Equation (3) in ref. [11]; see also Equation (2) herein.

It is noteworthy that the experimental data plotted in Figure 2(a) replicates Figure 5 from the ref. [10]; the appropriate data from that figure is obtained (picked up) using Plot Digitizer 2.6.8 freeware, created by Joseph A. Huwaldt in 2015 (see also http://plotdigitizer.sourceforge.net), while all points exhibited in Figure 2(b) are plotted based on the data, tabulated (in Table 4 and Table 5) for the case of crystalline and amorphous modifications (respectively) of the polypropylene in ref. [11]. As it is seen readily from the Figures 2(a), 2(b), the (relatively) low-temperature C_p(T) function(s) of linear PA and PP macromolecules apparently do not follow the well-known Debye’s ‘law’ [12], C_p(T) ∝ T³, which was established firmly for the bulk crystalline and amorphous solids at relatively low temperatures.

Instead, experimental C_p(T) function(s) both for the PA and PP macromolecules rather become fairly close to linear ones, C_p(T) ∝ T, (see Figures 2(a), 2(b)), within the temperature range of (100 K ≤ T ≤ 200 K), which is generally expected for the lattice heat capacity of 1D structures based on the well-known C_p(T) ∝ T^(d/r) ‘rule’ [8], with dimensionality of the system d, and the (integer) factor r in the phonon dispersion dependence: ω ∝ q^r; here q stands for the phonon quasi-wave vector.

Thus, particularities of the Vibrational Density-of-States (VDOS) become of key importance among other factors outlining C_p(T) behavior of the bulk, spatially non-homogeneous and low-dimensional semiconductors and insulators. Indeed, at d = r = 1, C_p(T) ∝ T are anticipated for the 1D (linear) macromolecules with linear (Debye’s) dispersion for the LA and or TA phonons, and the experimental C_p(T) data plotted (re-plotted) in the Figures 2(a), 2(b) follow this ‘trend’ indeed with an accuracy of ±16% in general (or of ±13% for crystalline modifications of PA and PP) within temperature range of (100 K ≤ T ≤ 200 K). Similarly, (almost) linear C_p(T) dependencies have also been reported for the poly-isobutylene and poly-1-bytene elsewhere in ref. [13], as well as for the linear polyethylene [14] (PE, see its structural ‘model’ in Figure 1(a)) for the temperature range of (80 K ≤ T ≤ 200 K), see also Figure 3(a) herein.

Figure 3 The experimental C_p(T) dependencies (symbols) reported for poly-ethylene (PE) in ref. [14]. The Figure (a) shows C_p(T) dependencies obtained for amorphous and crystalline modifications of linear PE (see Table 1 in ref. [14]), while Figure (b) reveals the C_p(T) dependencies, obtained for the ‘high-density’ PE (HDPE) samples with ‘branched’ 3D molecular structure (see Table 2 in ref. [14]). The log-log slope(s) of those dependencies, evaluated using standard least-square-fit procedures in the tem-perature range of (80 K ≤ T ≤ 250 K in Figure (a) and 2 K ≤ T ≤ 20 K in Figure (b)), are revealed near to the corresponding curves in both panels. The temperature range cor-responding to so-called ‘low-temperature anomalies’ is shown in Figure (a) as well.

In general, the C_p(T) ∝ T dependence might also originate from a dominant electronic (hole) contribution to the (low-temperature) specific heat capacity of polymeric samples with significant concentration(s) of nearly-free electrons and holes [8,9]. However, there is no (experimental) data on either the electronic component of the heat capacity nor on electric conductance, chemical potential (Fermi energy), etc., for the particular samples of 1D and 3D polymers, studied elsewhere in refs. [10,11,13,14]. Therefore, direct comparison of the electronic and lattice contributions to the experimentally obtained C_p(T) quantities becomes quite impossible, while ‘indirect’ estimations would be somewhat speculative and quite confusing.

Besides, in general, the electronic contribution to the heat capacity is expected to be significant only for polymeric materials with sufficiently high concentrations of free electrons (holes), like doped polyaniline (PANI). At the same time, the majority of polymers are well-known dielectrics (insulators) [8,9]. Those highly conductive polymers are simply out of focus in this article. Furthermore, even for the iodine-doped (I-doped) PA, exhibited elsewhere in ref. [10], potentially significant contribution to the C_p(T) function from the nearly-free charge carriers, created due to incorporation of the iodine (I) atoms into atomic structure of the polyacetylene, does not amend considerably neither the C_p(T) quantities of I-doped PA, nor its (nearly) linear (in general) character: see features of C_p(T) dependencies plotted in Figure 2(a). This indicates the dominant contribution from the lattice heat capacity to the experimental C_p(T) dependence, reported elsewhere in refs. [11,12,13] for (predominantly) linear macromolecules and other such macromolecules, studied in refs. [15,16,17,18]. Thus, within a certain level of simplification(s) (e.g., ignoring completely the effect of the periodic Born-von Kármán boundary condition [19,20], temperature-dependent lattice heat capacity of linear molecules is widely expected to obey the C_p(T) ∝ T ‘rule’).

It is noteworthy as well that the dashed curve in Figure 2(b) corresponds to the following fitted (within the temperature range of 10 K ≤ T ≤ 100 K) ‘empirical’ Equation (3) in ref. [11]:

\[ C_p(T)=exp[0.241028(\ln T)^3-3.01364(\ln T)^2+13.5529(\ln T)-18.7621], \tag{2} \]

where the C_p(T) quantity is expressed in [J/(mol * K)] units, while the absolute temperature, T, is expressed in Kelvins.

On the other hand, the ‘low-temperature anomalies’ in temperature ranges of (5 K ≤ T ≤ 25 K) emerge clearly as well for both those structural modifications of linear PP (see Figure 2(b)) and PE (see Figure 3(a)). In contrast, ‘branched (i.e., ‘cross-linked’ and predominantly 3D in its atomic network topology) molecular structure of so-called ‘high-density’ PE (HDPE) samples emerges in the ‘canonical’ Debye’s C_p(T) ∝ T³ dependencies for the temperature range (5 K ≤ T ≤ 50 K) [14]: see also Figure 3(b). Thus, an amendment in the structural dimensionality, d, of the polymer would have a direct effect indeed on its vibrational spectrum: compare Figures 4(a) and 4(b). Secondly, as it will be discussed in detail in the next section, alterations in sizes (spatial extents) of macro-molecules might significantly affect as well the low-temperature parts of experimental C_p(T) dependencies of polymeric solids of different spatial dimensionalities.

Figure 4 (color online). Idealized fundamental acoustic phonon spectra of (a) 3D polymeric structure [12], and of (b) polymeric structure composed of 1D and 3D structural fragments (blocks). The latter figure replicates (with some minor amendments) so-called ‘3-band’ model’, depicted elsewhere in Figure 5(b) from ref. [18]. The ℏω_E in this picture denotes an (optical) vibrational mode with an Einsteinian spectrum. In contrast, ℏω₃ and ℏω₁ denote top frequencies of the bands of the thermal acoustic vibrations of the 3D and 1D structural fragments, respectively. Those ℏω₃ and ℏω₁ quantities could also be linked to the corresponding specific temperatures, θ₃ and θ₁, defined as: θ₃ = ℏω₃/k_B; θ₁ = ℏω₁/k_B. See the main text for more details.

As mentioned above in this section, atomic structures of such polymeric solids often comprise simultaneously 1D, 2D, and/or 3D structural fragments. In such a case, appropriate description of their P(ℏω) function(s) might also requires implementation of a (linear) combination (‘super-position’) of P(ℏω) dependencies corresponding to the structural fragments of different spatial dimensionalities [18]; see also reference therein and illustration for this statement in Figure 4(b).

Both Equation (2) above and the mentioned above generic C_p(T) ∝ T^(d/r) ‘rules’ are customarily matching merely for the relatively narrow temperature ranges (usually located well below the Debye’s temperature of the material) of the experimentally evaluated C_p(T) dependencies of (polymeric) solids of different spatial dimensionalities.

More comprehensive description of the C_v(T) (rather than the C_p(T)) dependencies could be obtained using the framework of so-called ‘Tarasov’s Equations ‘(TEs)’ [1,2,3]. For instance, the temperature-dependent C_v(T) function for a ‘chain’ structure is generally expected to obey the 1D version of those TEs [2,15,16,17]:

\[ \frac{C_v(T)}{3R}=D_1\left(\frac{\theta_1}{T}\right), \tag{3} \]

with

\[ D_1\left(\frac{\theta_1}{T}\right)=\left(\frac{T}{\theta_1}\right)\intop_0^{\theta_1/T}\frac{\left(\frac{\theta}{T}\right)^2exp\left(\frac{\theta}{T}\right)}{\left[exp\left(\frac{\theta}{T}\right)-1\right]^2}d\left(\frac{\theta}{T}\right), \tag{3a} \]

where the θ₁ quantity defines highest allowed vibrational temperature (frequency) of the 1D chain (see Figures 4(a), 4(b) and caption to them), while the 3D version of TE reads:

\[ \frac{C_v(T)}{3R}=D_3\left(\frac{\theta_3}{T}\right), \tag{4} \]

where

\[ D_3\left(\frac{\theta_3}{T}\right)=3\left(\frac{T}{\theta_3}\right)^3\intop_0^{\theta_3/T}\frac{\left(\frac{\theta}{T}\right)^4exp\left(\frac{\theta}{T}\right)}{\left[exp\left(\frac{\theta}{T}\right)-1\right]^2}d\left(\frac{\theta}{T}\right). \tag{4a} \]

Here D₁(θ₁, T) and D₃(θ₃, T) represent in fact (slightly re-formulated) one- and three-dimensional ‘Debye functions’ [15,16,17], while the θ₁, and θ₃ quantities are their respective ‘frequency’ limits (or characteristic temperatures; see Figures 4(a), 4(b), and caption to them) [2,17], whereas R is the ‘universal gas constant’. Equation (4a) represent well-known (though re-formulated in terms of actual and ‘critical’ temperatures only) Debye’s equation for the heat capacity of 3D bulk solids [12], and ‘covers’ quite a wide temperature range, corresponding to both ‘incremental’ and ‘saturating’ parts of entire C_v(T) dependencies. Similar equations have also been re-formulated for different spatial dimensionalities: see Equations (3, 4c) herein and refs. [2,15,16,17]. In particular, the following 2D TE had been proposed for the ‘layer structures’ [2,17] with the appropriate 2D ‘Debye function’:

\[ D_2\left(\frac{\theta_2}{T}\right)=2\left(\frac{T}{\theta_2}\right)^2\intop_0^{\theta_2/T}\frac{\left(\frac{\theta}{T}\right)^3exp\left(\frac{\theta}{T}\right)}{\left[exp\left(\frac{\theta}{T}\right)-1\right]^2}d\left(\frac{\theta}{T}\right). \tag{4b} \]

It is noteworthy that synthesis and applications of 2D polymers recently became really ‘hot topics’ in physics and chemistry of polymers and low-dimensional solids in general due to their unique (sometime-truly exceptional) mechanical, thermal, and optical properties [21,22]. However, the latter TE takes in to account only contributions from ‘in-plane’ TA and LA modes of (polymeric) 2D layers. In contrast, so-called ‘out-of-plane’ ones are not taken into account by Equation (4b), though they make a dominant impact(s) on the low-temperature C_p(T) and C_v(T) functions of 2D materials [23,24,25]; see also references therein. Yet another vital term within the TE framework is its Einstein’s term [17]:

\[ \frac{C_{v,E}(T)}{N_ER}=\left(\frac{\theta_U}{\theta_U-\theta_L}\right)\left[D_1\left(\frac{\theta_U}{T}\right)-\frac{\theta_L}{\theta_U}D_1\left(\frac{\theta_L}{T}\right)\right], \tag{5} \]

here θ_L and θ_U are ‘low’ and ‘upper’ temperature (frequency) limits of distributed uniformly (or ‘boxed’ [8]) vibrational density-of-states, while N_E stands for the total number of atoms involved in the Einstein’s mode vibrations [17]. For the Einstein term, θ_L → θ_U = θ_E has to be assumed; see also refs. [2,17] for the comprehensive list of Tarasov’s equations of different spatial dimensionalities. It is important, that different forms of TEs (see, for instance, the Equations (4a, 5) above) allow-in principal-the following significant simplifications [26]:

\[ C_{\nu,E}(T)=N_ER\frac{\left(\frac{\theta_E}{T}\right)^2\exp\left(\frac{\theta_E}{T}\right)}{\left[\exp\left(\frac{\theta_E}{T}\right)-1\right]^2}=\frac{N_ER}{4}\left(\frac{\theta_E}{T}\right)^2csch^2\left(\frac{\theta_E}{2T}\right), \tag{5a} \]

It is noteworthy as well, that all TEs, expressed by the Equations (3-5a) above could be formulated as well for a more specific situation(s), which, for instance, might comprise of a case when the spectrum of linear (chain) molecules does not commences from zero frequency, but rather from some positive θ_L quantity, forming a ‘rectangular’ band(s) (or ‘boxes’ [8]), within a temperature range limited (again) by the θ_L and θ_U quantities [15,16,17].

Such situation is actually illustrated in Figure 4(b), (which depicts essential features of the ‘3-band model’ [18]) where θ_L = θ₃ = ℏω₃/k_B, and θ_U = θ₁ = ℏω₁/k_B, are assumed for 1D vibrations. In such a case, contribution (to the lattice heat capacity) from 1D vibration(s) is expected to be express by the following form of TEs [15,16,17]:

\[ \frac{C_v(T)}{3R}=D_1\left(\frac{\theta_1}{T}\right)-\left(\frac{\theta_3}{\theta_1}\right)\left[D_1\left(\frac{\theta_3}{T}\right)-D_1\left(\frac{\theta_1}{T}\right)\right], \tag{6} \]

with the D₁(θ₁, T) and D₁(θ₃, T) terms expressed by the Equation (3a).

\[ D_n\left(\frac{\theta}{T}\right)=n\left(\frac{T}{\theta}\right)^n\intop_0^{\theta/T}\frac{x^{n+1}exp(x)}{[exp(x)-1]^2}dx=n\left(\frac{T}{\theta}\right)^n\intop_0^{\theta/T}\frac{x^{n+1}}{4}csch^2\left(\frac{x}{2}\right)dx. \tag{7} \]

The squared hyperbolic cosecant (0.25 * csch(x/2)) function(s) [27] in Equations (5a, 7) replaces the combination of the following terms: exp(x)/[exp(x) - 1]², within the TEs expressed by Equations (3-5). Based on the following well-known identity: csch(x/2) = 1/sinh(x/2) [27], even further simplification becomes viable, with x = (ℏω/k_BT); the ℏω standing here for vibrational (phonon) energy, and k_B for the Boltzmann constant. Due to those substitutions (s), the volumetric specific heat of a solid from a single vibrational mode (of the frequency of ℏω_E) might be expressed as follows [26]:

\[ C_\nu(\hbar\omega_E,T)=k_B\frac{\left(\frac{\hbar\omega_E}{2k_BT}\right)^2}{\left[\sinh\left(\frac{\hbar\omega_E}{2k_BT}\right)\right]^2}. \tag{8} \]

This generic term replaces the appropriate TE for the single-mode contribution to C_v(T) quantity, see, in particular, Equations (5, 5a) above. The [csch(x/2)] and [1/sinh(x/2)] hyperbolic functions in Equations (5a, 7, 8) diverge as their argument, x = (ℏω/k_BT), approaches zero (i.e., in the x → 0 limit) [26]. In the context of evaluation on temperature dependencies of the heat capacity of solids of different spatial dimensionalities, the vanishing x quantities might emerge either:

(a) in the T → ∞ limit (which is of no physical significance for real solids), and/or

(b) at infinitesimally small vibrational energy (i.e., in the ℏω → 0 limit).

The latter situation (strictly speaking) might emerge within the TEs formalism for the bulk solids of (formally) infinite spatial extension(s) (sizes) of different spatial dimensionalities. In particular, the {exp(x)/[exp(x) - 1]² → 0.25^*csch²(x/2)} substitution does not really work for the 1D, and 2D solids of infinite spatial extent(s). This eventually implies that the majority of the listed above TEs would not allow such substitution due to divergence in the csch(x/2) and [1/sinh(x/2)] functions in the ℏω → 0 limit. Nonetheless, for bulk 3D solids, the aforementioned substitution emerges in physically meaningful results. Indeed, within the framework of Debye’s (classical) approximation, the P(ℏω) ∝ (ℏω)² is generally expected for the long-wavelength vibrations within bulk 3D solids [12]. Thus, since csch²(x/2) ∝ (2/x)², for infinitesimally small x quantities, the [P(ℏω) csch²(ℏω/2)] product would eventually yield just a finite (low-temperature) C_v(T) quantity in the ℏω → 0 limit. However, since the P(ℏω) ∝ (ℏω) is expected for 2D solids within the long-wavelength limit, and the P(ℏω) function is expected to be phonon-energy-independent in this limit for 1D solids, their product with the csch²(x/2) function would diverge in the x → 0 limit: the [P(ℏω) csch²(ℏω/2)] function would diverge as ~(1/ℏω) for 2D solids at ℏω → 0 limit, and as ~(1/ℏω)² for 1D ones of (formally) infinite spatial extent in the same limit. On the other hand, the aforementioned divergence problem might be avoided naturally, due to the phonon confinement phenomenon and related to it ‘infra-red’ gap in the phonon spectra, for the polycrystalline, microcrystalline, and especially for nano-crystalline solids of different spatial dimensionalities, as well as for their amorphous and/or amorphous-crystalline counterparts. Nevertheless, even the framework of 3D TEs customarily does not depict yet another significant feature of experimental low-temperature C_p(T) dependencies, obtained for different kinds of real polymers of finite spatial extents. In particular, so-called ‘low-temperature anomalies’ emerge clearly for experimental C_p(T) dependencies, obtained for 1D, 2D, and 3D spatially non-homogeneous solids; typical cases of such ‘anomalies’ are shown in Figure 2(b) and Figure 3(a).

Indeed, C_p(T) dependencies in these figures diminish rapidly with the temperature decrement below T ≅ 30 K for both amorphous and crystalline modifications of PP. These ‘anomalies’ are intimately related to the specific features (i.e., infra-red ‘cut-offs’) of the phonon spectra of those linear macromolecules. For instance, the (1D-in this particular case) phonon confinement effect would originate ‘discretization’ in the allowed energies of those phonons, and open the ‘gap’ as well in the ‘bottom’ of the spectra of the confined LA and TA phonons [23,24,25,28]. Thus, briefly discussed above in this section quasi-continuous P(ℏω) function(s) (see Figures 4(a), 4(b) for illustration), which is customarily obeying the P(ℏω) ∝ (ℏω)^d-1 ‘rule’ for the vibrational spectra of polymeric solids have to be replaced with its ‘discretized’ counterpart (as it is illustrated schema-tically in Figure 5 for the ‘3-band model’ case), when the ‘critical’ spatial extent (size) of the solid becomes well comparable with its (average) interatomic distance. Therefore, instead of horizontal line, representing the vibrational spectrum of a 1D structural fragment (of formally infinite length) in Figure 4(b), we should rather expect a set of vertical lines (representing Dirac’s δ-functions), separated on the horizontal scale by the (phonon) energy gap(s) of ΔE_1D ≡ E_min = (2hc_s/L_z); here E_min stands for the ‘minimal’ vibrational energy of spatially confined 1D vibrations [28], see right-hand side of Figure 5. At c_s = 15000 m/s and L_z = 100 nm this equation yields: ΔE_1D ≅ 1.24 meV.

Figure 5 More realistic representation (though not to scale) of fundamental acoustic phonon spectra of a polymeric structure composed of 1D and 3D structural fragments (blocks) of limited spatial extents. The quantization effects of acoustic modes confined within those structural blocks originate separation(s) between the phonon energy levels, which apparently become ‘discrete’ in such a case, with the separation (gap) energies of ΔE_1D, and ΔE_3D, respectively. See for comparison the original ‘3-band’ model [18] depicted in Figure 4(b) above and other details in the main text.

Such energy gap(s) diminish in vibrational spectra of 1D MWs (inversely proportionally) with the L_z enlargement and formally completely vanish in the L_z → ∞ limit. Similarly, the separation (gap) among neighboring vibration levels of the rectangular 2D and parallelepipedal 3D structures would be defined by appropriate algebraic combinations of the (effective) sound velocities of the material(s), and (orthogonal) spatial extents of those structures [25,29]. Further discussion(s) on the physical origin(s) and particular features of those P(ℏω) and C_v(T) functions within the relatively low phonon energy and temperature ranges for polymeric solids of different morphologies and spatial dimensionalities would be unveiled in the next section.

Yet another important feature of experimental C_p(T) dependencies of real polymeric solids is (potentially) significant contributions from anharmonic effects, especially at elevated temperatures. The temperature-dependent (in general) difference among the C_p and C_v quantities of 3D materials could be (formally) linked to the coefficient of the volumetric thermal expansion, α_v, [24,30]:

\[ C_p^0-C_v^0=V_0B_TT\left[\alpha_v^2+\left(\intop_0^T\alpha_vdT\right)\left(\frac{d\alpha_v}{dT}\right)\right], \tag{9a} \]

where C_p⁰ is an isobaric heat capacity of a solid at a zero pressure, C_v⁰ is its isochoric (harmonic) counterpart, measured at zero temperature, V₀ is a ‘molar volume’ of a solid at T = 0, while B_T is its so-called isothermal ‘bulk modulus’, B_T = -V₀ * (∂P/∂V₀)|_T [23,29,30]. The ratio of the B_T quantity to its isochoric counterpart, B_v, is routinely defined via the following ratio: [C_p(T)/C_v(T)]; which is very close (within accuracy of ~0.1%) to unity for majority of solids at relatively low temperature, T. Therefore, the B_T quantity is expected to be (roughly) equal to the B_v one.

For cubic crystals (of diamond atomic structure), the B_v quantity is routinely defined as follows: B_v = (c₁₁ + 2c₁₂)/3, where c₁₁ and c₁₂ denote two (of four independent) elastic constants (components of elastic tensor) of those crystal(s) [29,30], see also references therein. For a 3D material with a temperature-independent volume expansion coefficient, α_v, the latter integral equation could be simplified to the following algebraic one [29,30]:

\[ C_p^0-C_v^0=V_0B_TT\alpha_v^2. \tag{9b} \]

It is noteworthy that Equations (9a, b) are based essentially on the assumption of validity of ‘quasi-harmonic’ approximation for a given 3D solid [8,29,30]. Therefore Equations (9a, b) predict linear enlargement of the in C_p(T) quantities with the temperature at the constant (temperature-independent) α_v quantity, C_p(T) ∝ T, for the temperatures, T, well exceeding the Debye’s temperature of the 3D solid, where the C_v(T) dependence usually ‘saturates’ based on prediction of the well-known Debye’s model [12] for the specific (volumetric) thermal capacity of bulk solids.

For low-dimensional (i.e., 1D, 2D) solids, the coefficient of the volume thermal expansion has to be replaced with its low-dimensional counterpart(s): i.e., with the linear and ‘areal’ thermal expansion coefficient(s), respectively. Such a replacement would cause amendments in the Equations (9a, b). However, this kind of amendment(s) would require using the appropriate low-dimensional ‘bulk modulus’ of 1D and 2D materials [31,32,33,34]. Such elastic modulus (i.e., their ‘low-dimensional’ counterparts) is not well-defined for the 2D and especially for 1D materials. However, a more detailed discussion on this issue is given elsewhere in the refs. [35,36,37]; see also references therein. Therefore, implementations of Equations (9a, b) at quantitative evaluations on anharmonic and/or quasi-harmonic fractions of (lattice) thermal capacity of low-dimensional solids become infeasible.

There is also some controversy regarding appropriate computational methodology, even at quantitative evaluations on the harmonic thermal capacity of oligomers: solid and (especially) liquid polymers comprising just a few ‘repetitive’ units [38]. As it was suggested in ref. [38]. The specific heat of a solid comprising the only vibrational mode might be evaluated appropriately based on Equation (8) above in this section; see also Equations (3) in ref. [38] and ref. [39] for more details. For relatively small structural units of oligomers, the quantum confinement effect is generally expected to be of apparent significance for 1D, 2D, and 3D cases. In particular, it would eliminate all potential divergence problems with the integrand of appropriate TEs—nevertheless, the essentially classical (i.e., obtained without considering quantum effects) approach presented in refs. [38,39] is based on (classical version of) molecular dynamics (MD) simulation methodology and ‘mass-weighted velocity’ autocorrelation function [39], and it has been implemented elsewhere in ref. [38] at simulations on temperature dependencies of lattice thermal capacities of oligomers. Such methodology requires quantum corrections due to the presence of hydrogen atoms and/or ‘terminal groups’ in the atomic structures of polymers: (thermal) motions of such atoms have to be treated in an essentially quantum manner, which customarily is not incorporated into the ‘classical’ MD methodology [38]. Quantum nature of those nuclear degrees of freedom might be treated within the ‘path-integral’ framework [38]. However, this approach is computationally demanding [38,40].

More practical ways of computational corrections might be based on the so-called ‘Wigner-Kirkwood expansion’ [41,42]. However, it is essentially ‘classical’ in its nature, as well as still computationally challenging, and valid for a limited temperature range only, which does not cover absolute temperatures well below Debye’s one [38,41,42], and-in particular-the temperature range of aforementioned ‘low-temperature anomalies’. Furthermore, this technique does not capture the aforementioned generic quantum mechanical, and especially anharmonicity effect(s) [38]. Herein, the 1D, 2D and 3D versions of so-called ‘Generalized Skettrup Model’ (GSM) [23,24,25,28,29] (see also references therein), representing ‘alternative’ (as compared to approximations, mentioned above in this section) and fully quantized computational approaches will be implemented at simulations on temperature dependencies of lattice thermal capacities of few polymers of different spatial dimensionalities and morphological characteristics. However, before it, the fundamental equations of the 1D, 2D and 3D versions of the GSM will be re-introduced briefly in the next section.

2. Basic Equations of GSMs

The isochoric lattice heat capacity of 3D bulk polymeric solids of (formally) unlimited spatial extents (sizes) is customarily defined by the one-dimensional integration over their quasi-continuous static (and usually temperature-independent) vibrational (phonon) spectrum, P(ℏω), ‘weighted’ by the (temperature-dependent) Bose-Einstein occupation factor for those vibrational states, and by the (appropriate) phonon energy, ℏω [12]. Within the framework of the (original) Debye’s model, all (static) states of the longitudinal and transverse (LA, TA) acoustic phonons are presumed to obey linear (classical) dispersion(s) [12].

Spatial extents of ‘morphological units’ (e.g. crystallites of various shapes, cylindrical, spherical and conical non-homogeneities, etc.) of 3D polycrystalline, spatially non-homogeneous amorphous [23,43] and especially those of the nano-structured solids [24] are limited by definition for all of them. Under (Casimir’s) assumption on insignificant phonon-phonon interactions (scattering) within volumes of those non-homogeneities (morphological units), propagation length (mean free path) of their LA and TA phonons would be defined by the sizes (spatial extents) of those morphological units [24,29] rather than by phonon-phonon interactions. This allows one to treat volume(s) of the single morphological unit (3D cavity) as the actual volume of spatial confinement (coherence) of the (static) LA and TA vibrational states [24,29]. Since neighboring energy levels of confined LA and TA phonons are generally expected to be separated from each other [43], alteration in the morphological characteristics of the polycrystalline and spatially non-homogeneous amorphous solids might originate considerable re-distribution in the energies (spectrum) of the confined LA and TA vibrations [24,29]. Therefore, spectral and statistical characteristics of those LA and TA phonon states confined within the 3D units (cavities) are generally expected to be affected by their morphology i.e., shapes (or ‘habit’ [8]): e.g., parallelepipedal, cylindrical, conical, etc., of those non-homogeneities, as well as by their sizes and (dominant) crystalline orientation(s)-if any [24,29,43]. Thus, in general, the temperature-dependent lattice thermal capacities of spatially non-homogeneous and especially of the low-dimensional solids are expected to be affected by their morphology [24].

2.1 Basic Equations of 1D GSM

Within framework of 1D version of the ‘Generalized Skettrup Model’ (GSM) the temperature-dependent lattice thermal capacities of infinitesimally thin (in both spatial directions, orthogonal to the MW elongation) 1D ‘Molecular Wires’ (MWs) of a finite length, L_z, is defined by the following expression [28,44]:

\[ C_p^{1DGSM}(T)\cong\frac{1}{k_BT^2}\sum_{N=1}^{N_M}\sum_{p=1}^{Np_{max}}\frac{exp\left(p\frac{E_{min}}{k_BT}\right)}{\left[exp\left(\frac{pE_{min}}{k_BT}\right)-1\right]^2}\frac{(pE_{min})^2}{N!\left(Z_{1D}^N\right)}\left[floor\left(\frac{L_z}{a_lN}\right)\right]^N, \tag{10} \]

where the function floor(x) returns an integer part of its rational argument, a_l is an (average) interatomic distance (‘lattice constant’) of the MW material, k_B is the Boltzmann constant, E_min is the ‘minimal’ energy of the phonons, confined within 1D MW, Z_1D^N is the N-phonon ‘statistical sum’ (N-particle partition function), while c_eff is an effective sound velocity, routinely defined as follows: 3/c_eff = (1/c_l + 1/c_t1 + 1/c_t2) [23], where c_l stands for the longitudinal sound velocity, while c_t1 and c_t2 denote two (different in general) transverse sound velocities of the MW material. The aforementioned E_min and Z_1D^N quantities are customarily defined as follows [28,44]:

\[ E_{min}=\frac{2hc_{eff}}{L_z}, \tag{10a} \]

\[ Z_{1D}^N=\left[L_z\left(\frac{k_B\theta_D}{hc_{eff}}\right)\right]^N. \tag{10b} \]

The p_max quantity in Equation (10) is defined as: p_max = floor(k_Bθ_D/E_min); where θ_D is Debye’s temperature of the MW material [27,43]. The Equation (10) allows the following simplification, discussed (to some extent) in the previous section: {exp(x)/[exp(x) - 1]²} → 0.25 csch²(x/2) [26], with the dimensionless x = (ℏω/k_BT) = (pE_min/k_BT) quantity. Such substitution simplifies Equation (10) till its following form:

\[ C_p^{1DGSM}(T)\cong4k_B\sum_{N=1}^{N_M}\sum_{p=1}^{Np_{max}}\frac{csch^2\left(\frac{pE_{min}}{2k_BT}\right)}{N!Z_{1D}^N}\left(\frac{pE_{min}}{2k_BT}\right)^2\left[floor\left(\frac{L_z}{a_lN}\right)\right]^N. \tag{10c} \]

The basic equation(s) of the 1D GSM version, formalized by the Equations (10-10c), generally predict linear C_v(T) enlargement with the absolute temperature, T, in its low-temperature range (where T << θ_D holds) for sufficiently long MWs (see simulation results and detailed discussion on this issue in the next section), though the ‘low-temperature’ anomalies also emerged clearly for the C_v(T) curves, obtained for relatively short MWs: see, in particular, Figure 2 in ref. [28] as well as Figure 2 in ref. [44]. It is noteworthy as well, that Equations (10, 10c) incorporates not only harmonic lattice capacity of MWs (evaluated based on those equation at N ≡ 1), but also its anaharmonic fraction(s), evaluated at N = 2, 3, 4…. Further analysis and discussion on features of C_v(T) and C_p(T) dependencies, evaluated based on Equations (10-10c) for linear polymers (chains) and oligomers will be presented in next section.

2.2 Basic Equations of 2D GSM

For instance, the lattice thermal capacity of the rectangular 2D nano-ribbons, it is essentially defined by the two following ‘components’ [22,23,24,25]. The ‘in-plane’ and ‘out-of-plane’ ones [25]. The former one reads [25]:

\[ C_p^{2DGSM}(T)\cong\frac{2}{k_BT^2}\sum_{N=1}^{N_M}\sum_{p=1}^{Np_{max}}\frac{exp\left(p\frac{E_{min}}{k_BT}\right)}{\left[exp\left(\frac{pE_{min}}{k_BT}\right)-1\right]^2}\frac{(pE_{min})^2}{N!Z_{2D}^N}\left[\frac{2\sqrt{2}\kappa pF_2\left(L_x,L_y\right)}{N\sqrt{1+\left(L_x,L_y\right)^2}}\right]^N, \tag{11a} \]

with ‘form-factor’, F₂(L_x, L_y), equals to F₂(L_x, L_y) = (π/2), for the square flakes [25], but varying significantly with the (L_x/L_y) ratio for rectangular ribbons: see Figure 1 in ref. [25], while the term κ = (p_max/2)[1 + (p_max/4)] takes into account contribution(s) from permutation(s) of appropriated single-phonon state into statistical weigh of the given multi-phonon ones [25]. The ‘out-of-plane’ contribution reads [25]:

\[ C_{ZA}^{2DGSM}(T)\cong\frac{1}{k_BT^2}\sum_{N=1}^{N_M}\sum_{q=1}^{q_{max}}\frac{\exp\left(q\frac{E_{min}}{k_BT}\right)}{\left[\exp\left(\frac{qE_{min}}{k_BT}\right)-1\right]^2}\frac{(qE_{min})^2}{Z_{ZA}}\left(\frac{L_xL_yqE_{min}}{\pi\hbar\xi_{ZA}}\right), \tag{11b} \]

with q_max = p_max = floor(k_Bθ_D/E_min), and,

\[ E_{min}=\frac{2hc_{eff}}{\left[\left(L_x^2+L_y^2\right)/2\right]^{1/2}}, \tag{11c} \]

while the ‘statistical sums’, Z_2D^N, and Z_ZA, read [23,25,45]:

\[ Z_{2D}^N=\left[\frac{L_xL_y}{2}\left(\frac{k_B\theta_D}{hc_{eff}}\right)^2\right]^N, \tag{11d} \]

\[ Z_{ZA}=\left[\frac{L_xL_y}{2}\left(\frac{\hbar\omega_{ZA}^M}{\pi h\zeta_{ZA}}\right)^2\right]. \tag{11e} \]

However, for the rectangular ribbons with larger (‘microscopic’) sizes (spatial extents), the basic equation of a ‘quasi-continuous’ version of the 2D GSM could implemented as well [23,24,25]:

\[ C_p^{2DGSM}(T)\cong4k_BC_2\sum_{N=1}^{N_M}\intop_{E_{min}}^{Nk_B\theta_D}\frac{csch^2\left(\frac{E_T}{2k_BT}\right)}{N!Z_{2D}^N}\frac{E_T^2}{(2k_BT)^2}\left[\frac{2L_xF_2\left(L_x,L_y\right)E_T}{Nhc_{eff}}\right]^NdE_T, \tag{11f} \]

with its size-dependent ‘normalizing coefficient’, C₂, defined as follows [23,24,25]:

\[ C_2=\left[\frac{(k_B\theta_D)^2}{L_xL_y}\right]\left[\frac{L_xL_y}{2\pi^2\left(hc_{eff}\right)^2}\right]^{3/2}. \tag{11g} \]

Mind the {exp(x)/[exp(x) - 1]²} → 0.25 csch²(x/2) replacement [26] in Equation (11f); with x = (E_T/k_BT). Some results of implementations of the basic equations of 2D GSM at simulations on temperature-dependent lattice thermal capacity of 2D polymers have been discussed elsewhere in the refs. [23,24,25].

2.3 Basic Equations of 3D GSM

The (quasi-continuous though essentially anisotropic) version of the basic equation of the 3D GSM reads:

\[ C_p^{*GSM}(T)\cong12k_BC_3\sum_{N=1}^{N_M}\intop_{E_{min}}^{Nk_B\theta_D}\frac{E_T^2csch^2\left(\frac{E_T}{2k_BT}\right)}{N!Z_N(k_BT)^2}\left[\frac{2L_xL_y\mathbf{F}\left[\left(\frac{L_x}{c_{efx}}\right),\left(\frac{L_y}{c_{efy}}\right),\left(\frac{L_z}{c_{efz}}\right)\right]E_T^2}{Nh^2c_{efx}c_{efy}}\right]^NdE_T, \tag{12} \]

with the ‘orthogonal’ components, c_efx, c_efy, and c_efz, of anisotropic effective sound velocity, for the phonons (thermal vibrations), confined within the parallelepipedal volume (crystallite) of the orthogonal rib lengths of L_x, L_y, L_z, and the dimensionless function, F[(L_x/c_efx), (L_y/c_efy), (L_z/c_efz)], depends solely on the (L_x/c_efx), (L_y/c_efy), and (L_z/c_efz) ratios; see also Equation (26b) in ref. [29], and Equation (1) in ref. [46].

Quantitative evaluation on the c_efx, c_efy, and c_efy, values could be, for instance, carried out based on so-called ‘Christoffel Matrix’ (CM) formalism [24,29,46]; see also references therein and further discussion on this issue in the next section. Other key parameters of the Equation (12) are defined as follows [29,46]:

\[ E_{min}=\frac{2h[\left(c_{efx}^2+c_{efy}^2+c_{efz}^2\right)/3]^{1/2}}{\left[\left(L_x^2+L_y^2+L_z^2\right)/3\right]^{1/2}}, \tag{12a} \]

\[ Z_M=\left[\frac{L_xL_yL_z}{3}\left(\frac{k_B\theta_D}{hc_l^{avr}}\right)^3\right]^M, \tag{12b} \]

\[ C_3=\left[\frac{(k_B\theta_D)^3}{L_xL_yL_z}\right]\left[\frac{L_xL_yL_z}{2\pi^2\left(hc_{ef}^{avr}\right)^3}\right]^{4/3}, \tag{12c} \]

with c_ef^avr = {[(c_efx)² + (c_efy)² + (c_efz)²]/3}^1/2 [46]. These variant(s) of the basic equations of the anisotropic version of the 3D GSM could be, for instance, implemented at quantitative evaluations on temperature-dependent lattice thermal capacity of polyethylene, with significant anisotropy in its single-particle vibrational DOS function and sound velocities, see next section for details. It is note-worthy, that statistical characteristics of many-particle states of the LA and TA phonons embedded in all mentioned above versions of the GSM are obtained based essentially on concept of many-particle vibrational density-of-states (VDOS), introduced by the author of this article elsewhere in 1995 [47].

2.4 Entropy Calculations

Listed above (just in the previous sub-section) basic equations of the 1D, 2D and 3D versions of the GSM have obtained based entirely on quantitative evaluation on the fundamental vibrational energy of acoustic phonons, confined within 1D, 2D, and/or 3D structural fragments of solid polymers, as well as of its many-particle counterparts. Another crucially important thermodynamic characteristic of the solid polymers is their entropy [8,9]. Indeed, as it was stated by Professor Brent Fultz elsewhere in ref. [9], ‘Without entropy to complement energy, thermodynamics would have the impact of one hand clapping.’; please see beginning of the section I.1 of the ref. [9]. In general, the entropy might be evaluated based on the Boltzmann’s definition: S_B = k_B lnΩ [9], which is mainly determined by the ‘…number which counts the way of finding of the internal coordinates for thermodynamically-equivalent macroscopic states.’, Ω Since the Ω quantity (as well as its natural logarithm) is apparently dimensionless, physical dimension of the Boltzmann’s entropy is of J/K; i.e., it is the same as that of the Boltzmann’s constant and of heat capacity (when measured for a unit mass of the substance).

However, more practical approach to evaluation on the temperature-dependent and size-dependent entropy of polymers could be based on an additional ‘treatment’ of preliminary simulated temperature dependencies of the isobaric lattice heat capacity:

\[ S_T-S_0=\intop_0^T\frac{C_p(\vartheta)}{\vartheta}d\vartheta, \tag{13} \]

here S₀ stands for the zero-temperature (Nernst’s) entropy. It is easy to verify that, based on Equation (13), the entropy dimension would be of J/(mol * K) again. Based on the so-called ‘third law’ of thermodynamics, S₀ ≡ 0 is postulated for the crystalline solids with perfectly ordered atomic structures, while for amorphous (as well as for majority of polymeric) solids, positive S₀ quantities are generally expected [9]. Therefore, Equation (13) would rather yield relative changes in the S_T quantities with the temperature. On the other hand, in combination with the Equations (10a-10c, 11a-11f, 12-12c), it might be implemented straightforwardly at quantitative evaluation on the entropy of 1D, 2D, and 3D structural fragments (respectively) of polymeric solids. Practical examples of such evaluation(s) for the polymeric materials with clearly dominant 1D and 3D structural fragments would be exhibited and discussed in the next section of the article. At the same time, the C_p(T) quantities taken into account in Equation (13) should not include electronic contribution(s): such a contribution is not related to the entropy of the polymeric lattice!

As mentioned above, atomic structure(s) and morphologies (habitats) of real polymers might comprise fragments of different spatial dimensionalities. When (rigid) bounds of such structural fragments are of simple and (essentially) ‘convex’ geometrical shape (e.g., square, rectangular, cubical, parallelepipedal, etc.), an appropriate combination of the fundamental equations of the 1D, 2D and 3D versions of the GSM (see, in particular, Equations (10-10c, 11a-11d, and 12-12c) above in this section) might be implemented readily at quantitative evaluations on temperature dependencies of thermal capacities of real polymers. However, the composition of fragments of different spatial dimensionalities might result in more complicated morphologies of those polymers. For instance, even ‘linear’ (chain) polymeric structures of PA and PP might comprise ‘MWs’ of different lengths. The realistic sketch of a thinkable polymeric 2D morphology might be thought as (approximately) rectangular 2D ’blocks’ (of the rib length of L_x^* and L_y^*), though with ‘non-rigid’ ‘corrugated’ (e.g., -‘concave’-see an example in Figure 6 below) and even ‘fractal’ borders of the ribbon(s).

Figure 6 A sketch of an evocative rectangular (ribbon) polymeric 2D morphology with a combination of ‘convex’ and ‘concave’ ribbon borders. The (appropriately) averaged (orthogonal) lengths of this ribbon are denoted as L_x^* and L_y^*. In principle, this particular shape could be represented as a ‘superposition’ of conventional rectangular fragments with ‘convex’ borders: see dashed lines in the figure. See also more details in the main text.

In such a case, the ‘resonant’ conditions for (standing) acoustic waves (phonons) might be (somewhat) different (non-equivalent) for various parts of the 2D polymeric fragment. Thus, both eigenenergies of the confined vibrational states as well as separation among the neighboring vibrational level(s) spatially confined within a 2D fragment (ribbon) might vary sizably even for different parts of the atomic structures located within such a ribbon. Therefore, appropriate evaluation(s) on the (practical) spatial extents (sizes) and consecutive implementation of basic Equations (11a-11g) at quantitative simulations on the temperature-dependent lattice thermal fragment would require some kind of ‘averaging’ of the geometrical parameters (sizes) of those 2D fragments with ‘concave’ ribbon borders.

Alternatively, an ‘effective’ spatial extent(s) of the phonon confinement could be approximated using the following equation [48]:

\[ \Lambda_{eff}=\frac{1+p_r}{1-p_r}\Lambda_{Cs}, \tag{14} \]

Here, Ʌ_eff is the ‘effective’ mean-free-path (MFP) of phonons, p_r is the probability of a phonon to be specularly reflected on the edge of the 2D (and/or 3D) structural fragment of the polymeric structure. Ʌ_cs is the so-called ‘Casimir’ MFP, evaluated based on the assumption of dominance of phonon scattering on the borders of the ribbons (non-homogeneities) as well as on the absence of significant phonon-phonon interactions within each ribbon [48].

The p_r = 0 corresponds to the ‘purely diffusive’ case, while p_r = 1 corresponds to the ‘purely specular one [48]; see also comment to Equation (2) in ref. [48]. Equation (14) had been initially proposed for evaluation on thermal transport parameters of silicon nano-wires (SiNWs) with ‘corrugated’ shapes. Though features of phonon confinement within SiNWs of cylindrical morphology in general is expected to be defined by its 3D geometry, axial symmetry of SiNW might reduce an effective spatial ‘dimensionality’ of its phonon confinement and diminish and log-log slopes of the C_v(T) and/or C_p(T) dependencies within the ‘low-temperature’ range(s) [28,44]. However, such a dimensionality reduction and the slope diminishment might not occur for SiNWs with ‘corrugated’ shapes, where the cylindrical symmetry might be ‘distorted’ considerably. Similarly, an actual shape (habit) of 3D fragments of real polymeric morphology (not shown herein) could be even far more complicated than compared to presumed (within the framework of 3D GSM) parallelepipedal ones. Some predictions on temperature dependencies of the lattice thermal capacity of more realistic structural and morphological models of polymeric materials obtained using an appropriate combination(s) of the basic equations of 1D and 3D versions of the GSM are discussed in the next section.

3. Simulation Results and Discussion

Effects of alteration (variation) in the lengths of MWs (presumably) composing a PP-based polymeric solid of (predominantly) 1D-linked atomic structure on temperature dependence of its isochoric (harmonic) lattice thermal capacity, C_v(T), are illustrated in Figures 7(a) and 7(b). All simulated C_v(T) dependencies exhibited in these figures are obtained based on combination(s) of Equation (10c), implemented for the case of PolyPropelene (PP), with c_eff = 2740 m/s [49], a_l = 20.78 Å [8], θ_D = 600 K, and presumably composed of two MWs of different lengths. In particular, the PP samples with C_v(T) dependencies, revealed in the Figure 7(a) is (presumably) composed of two MWs of L_z = 4 nm and L_z = 6 nm, incorporated into the same PP sample with equal volume fractions of 50% for each type (lengths) of the MW.

Figure 7 (color online). Curves: simulated temperature dependencies of isochoric lattice thermal capacity of PP samples ‘composed’ of MWs of two essentially different lengths: (a) of 4 nm and 6 nm (with volume fractions of those MWS of 50% to 50%, respectively) (b) of 2 nm and 8 nm (with volume fractions of those MWs of 5% to 95%, respectively). Blue curves in both panels represent linearly ‘averaged’ (measurable) C_v(T) dependencies for those PP samples, evaluated based on the volume fractions of the two MWs. The dashed black curve in the left panel corresponds to C_v(T) dependence, simulated using Equation (10c) for ‘geometrically averaged’ MW length; see main text for details. Open symbols: experimental C_p(T) data set, replicated (with some ‘truncation’) from ref. [11], see also Figure 2(b) above. Two grey rectangular(s) on the right panel highlight two distinguishable ‘low-temperature anomalies’, emerging into the simulated blue curve for the PP samples composed of two MWs of distinctly different lengths. Log-log slopes of a couple of curves, evaluated within a temperature range of (32 K ≤ T ≤ 72 K), are indicated in the close vicinity of those curves.

In contrast, their counterparts revealed in Figure 7(a) are obtained for PP samples (presumably) composed of two MWs of L_z = 2 nm and L_z = 8 nm, incorporated into the PP sample with the volume fractions of 95% and 5%, respectively. Thereafter, the ‘averaged’ (measurable) C_v(T) dependencies (represented with the blue curves in Figures 7(a), 7(b)) are obtained through the linear combination of two ‘individual’ C_v(T) curves, ‘weighted’ by their volume fractions. As it seen clearly from comparison of C_v(T) dependencies, plotted in Figures 7(a), 7(b), alteration in lengths of MWs might originate considerable differences in temperature behaviors of those ‘averaged’ C_v(T) curves, especially in the temperature ranges, corresponding to that of ‘low-temperature anomalies’. Indeed, mixture of two MWs with their length of 4 nm and 6 nm would yield the ‘averaged’ C_v(T) curve with a behavior, very similar to that of individual C_v(T) dependencies, simulated independently for the PP samples (presumably) composed entirely of MWs of the same lengths: 4.0 nm (red curve in the Figure 7(a)) and 6.0 nm (green curve in the Figure 7(a)).

The dashed black curve in Figure 7(a) is obtained based on Equation (10c) for the case of ‘geometrically averaged’ MW length, L_z^*, with L_z^* = (L_z¹L_z²)^0.5, which yields L_z^* ≅ 4.9 nm at L_z¹ = 4.0 nm and L_z² = 6.0 nm. When, however, lengths difference of the MW ‘components’ becomes much more distinct (4 times-to be exact), two ‘low-temperature’ anomalies of such PP sample might emerge in principle at a certain combination of concentrations of MWs, composing the PP sample: see temperature sub-ranges for the ‘averaged’ (blue) curve, highlighted with two grey rectangu-lar(s) in the Figure 7(b). At higher temperatures (i.e., at T > 30 K), the log-log slopes of the simula-ted C_v(T) curves in Figures 7(a), 7(b), are close to unity (with an accuracy of ±12%), which is generally expected for the temperature-dependent C_p(T) function of the 1D solids with linear phonon disperseon(s) based on the well-known C_p(T) ∝ T^(d/r) ‘rule’, discussed briefly in the introductory section.

The temperature dependencies of the entropy, S_T(T), evaluated for the one-dimensional (linear) MWs of polypropylene based on the combination of Equations (10a-10c, 13), are plotted in Figure 8. It is noteworthy, that the simulated C_v(T) dependencies plotted in this figure are obtained based on the isochoric C_v(T) curves (i.e., similar to those plotted for the PP samples in Figures 7(a), 7(b)), rather than on their isobaric counterparts, C_p(T), embedded into Eq. (13). Such replacement is predominantly enthused by the fact, that the difference among the C_p(T) and C_v(T) curves becomes rather negligible at relatively low temperatures (at T << θ_D), where amendments in the lengths of the PP MWs effects the simulated S_T(T) dependencies mostly.

Figure 8 (color online). Curves: simulated (at different lengths of MWs) temperature dependencies of the relative entropy of linear PPs. All exhibited simulation results are obtained based on combinations of Equations (10a-10c, 13) in the previous section. The S_T(T) dependencies plotted in this figure are actually obtained based on simulated ishochoric, C_v(T), dependencies, obtained based on Eq. (10c) for the PPs, rather than on their isobaric counterparts, C_P(T), stated in the Eq. (13). Mind double-logarithm scales of the plot and close to linear behaviour of its upper curve in the temperature range of (10 K ≤ T ≤ 100 K). The S_T(T) data tabulated in Table 4 of the ref. [11] For crys-talline PP, plotted for comparison with open stars. See the main text for more details.

Indeed, as it is seen readily from Figure 8, alteration in the lengths of PP MWs affects strongly shape of the S_T(T) dependencies within their (5 K ≤ T ≤ 30 K) sub-ranges, where their shape is affected predominantly by the ‘low-temperature anomalies’ of the simulated C_v(T) dependencies: see, for instance, Figures 7(a), 7(b). It is noteworthy that the simulated S_T(T) curves match fairly well with their experimental counterpart at L_z quantities ranging from 5 to 7 nm: see Figure 8. However, enlargement in the length of the PP MWs weakens anomalous behavior both for the C_v(T) and S_T(T) curves in Figure 7(a) and Figure 8, and aforementioned ‘anomalies’ (almost) vanish when the MW length exceeds 30 nm. In such a case, the C_v(T) dependence is expected to be close to a linear one at relatively low temperatures, and, based on Eq. (13), we also should anticipate the (nearly) linear S_T(T) curve. Indeed, for the linear C_v(T) dependence (which is generally expected for the 1D semiconductors and insulators at the relatively low temperatures-see detailed discussion on this issue in the introductory section), a constant [C_v(θ)/θ] ‘integrand’ in Eq. (13) emerges-and, consequently-the linear S_T(T) curve is apparently anticipated customarily for such a case.

The significant diminishment(s) in the S_T(T) quantities at relatively low temperatures in Figure 8 could also be explained based on Boltzmann’s definition of entropy. Indeed, as it was already mentioned above, the ‘low-temperature anomalies’ are eventually originated from a ‘gap’ in the low-energy part(s) of the vibrational spectrum of nano-materials. This implies straightforwardly, that the ‘number of ways’ in which the ‘thermodynamically-equivalent’ microscopic state(s) of the nano-material could be achieved at the given temperature would be reduced sharply as compared to the similar ‘number of ways’ for a ‘low-temperature’ state of its bulk counterpart [9]. This would eventually diminish as well the low-temperature Boltzmann’s entropy of the nano-structured solids and the measure of their thermally-induced structural disorder. However, at higher absolute temperatures, rises of both C_v(T) and S_T(T) curves are slowing down (see Figures 7(a), 7(b), Figure 8)-regardless to the MW length(s), because of the temperature-independent numbers of microscopic states occupied at elevated temperatures-which eventually yield deviation from the linear S_T(T) dependencies in Figure 8.

Thus, implementation of the basic equation(s) of the 1D GSM at quantitative evaluation(s) on the temperature dependencies of the isochoric lattice thermal capacity of real polymeric solid composed predominantly of 1D MWs of different lengths might provide useful inside into effects of 1D spatial confinement within those MWs on the behavior of the ‘averaged’ C_v(T) dependencies of the entire PP samples. Moreover, the same equation of the 1D GSM might also be implemented at realistic simulations on the experimental C_v(T) dependencies, obtained from real PP samples, and in combination with Eq. (13) of their temperature-dependent entropy: see Figure 8. As shown in Figure 9(a), the implementation of the first term of Equation (10c) from the previous section might replicate convincingly the ‘recommended’ (in ref. [11]) isobaric experimental data for the crystalline modification of PP at L_z ≅ 4.5 nm. Similarly, simulated S_T(T) dependences for PP approximate reasonably well its experimental counterpart at L_z = 5 ÷ 7 nm, see Figure 8. As it is seen readily from Figure 9(a), within the temperature range (10 K ≤ T ≤ 200 K), the simulated C_v(T) curve follows closely the experimental data for the crystalline modification of PP, reported in ref. [11]. Though it apparently deviates from the data at higher temperatures. In particular, above this temperature range, the simulated C_v(T) (or C_H(T)) dependence underestimates the experimental trend, obtained for the crystalline modification of PP in ref. [11].

Figure 9 (color online). The experimental (symbols, see also Figure 2(b) above) and simulated (curves) temperature dependence of the isobaric and isochoric lattice thermal capacity of amorphous and crystalline modifications of linear polypropylene (PP). Figure (b) just ‘stretches’ the part of Figure (a), highlighted with the cyan rectangular area. All simulated results are obtained based on Equation (10c) in the previous section, both for harmonic (at N ≡ 1) and dominant anharmonic (at N = 2) fractions of the lattice thermal capacity of 1D (linear) PP. The ‘glass transition’ temperature, T_g = 260 K [11], of ‘atactic’ and amorphous modification(s) of PP is indicated with the dashed vertical line in both panels. An average log-log-slope for the crystalline modification of the PP, evaluated above the T_g temperature is indicated as well in Figures (a) and (b). Mind double-logarithmic axes (scales) in Figure (a) and linear ones in Figure (b). See more details in the main text.

This deviation has to be attributed predominantly to the fact that the very first term of Equation (10c) evaluates only the harmonic (isochoric) contribution to temperature-dependent lattice thermal capacity of PP, while the impact from the anharmonic fraction of the isobaric heat capacity might be significant at the elevated temperature.

In order to take into account the aforementioned anharmonic contribution(s) to the experimental isobaric C_p(T) quantities, the high-order term(s) of Equation (10c) have to be evaluated as well. In particular, the contribution from the dominant anharmonic term (obtained from Equation (10c) at N = 2) becomes significant at elevated temperatures, exceeding 200 K in this particular case; see also simulation results in Figure 2 in ref. [28] and in Figure 2 in ref. [44] for an alternative set of material (simulation) parameters. In general, basic eqation(s) of the 1D GSM predict (almost) linear increament with the temperature for both harmonic and anharmonic fractions of the lattice thermal capacity of MWs within quite a wide temperature range(s) below Debye’s temperature of the MW material [28,44]. However, such a range for the anharmonic fraction usually becomes considerably wider due to higher (in general, being proportional to the integer N quantity) summation limits for the anharmonic terms in Equations (10, 10c). Therefore, an appropriate sum of the harmonic and dominant anharmonic terms, evaluated based on Equation (10c) for the PP case, eventually yields a reasonably good approximation for the experimental C_p(T) dependence for the crystalline modification of PP: see the solid curve in Figure 9(b). It is noteworthy that the contribution from the anharmonic fraction to the total (isobaric) lattice thermal capacity of MWs is not reflected (plotted) in Figure 9(a).

Similarly, implementation of the first and second tems of Equation (10c) at simulation(s) on the tempearture dependencies of harmonic and anharmonic thermal capacity of amorphous modification of linear polyethylene (PE) yield fairly good approximations at c_eff = 2142 m/s, a_l = 2.6 Å [50], θ_D = 900 K, and L_z = 5.8 nm for experimental C_v(T) and/or C_p(T) data, reported elswhere in ref. [14], in quite a wide temperature range: see Figures 10(a), 10(b). Indeed, the first term of Equation (10c) yields a convincing approximation for the experimental C_p(T)/C_v(T) dependence reported for the linear PE in ref. [14] Within the temperature range up to 150 K (see Figure 10(a)), while at higher temperatures, the contribution from the anharmonic term to the lattice heat capacity of PE MWs becomes significant as well, see Figure 10(b).

Figure 10 (color online). The experimental (symbols, see also tabulated data in ref. [14], and Figure 3(a) herein) and simulated (curves) temperature dependence of the isobaric and isochoric lattice thermal capacity of amorphous modification of linear polyethylene (PE). Figure (b) actually just ‘stretches’ the part of Figure (a), highlighted with the cyan rectangle. All simulated results are obtained based on Equation (10c) in the previous section both for harmonic (simulated at N ≡ 1) and dominant anharmonic (simulated at N = 2) fractions of the lattice thermal capacity of 1D (linear) PE. Material parameters for linear PE are chosen based on the data revealed in refs. [8,50]. An average log-log slope for the amorphous modification of the linear PE, evaluated within temperature range (80 K ≤ T ≤ 250 K) is indicated as well Figure (a), while similar slope in the Figure (b) is evaluated for the temperature range of (250 K ≤ T ≤ 325 K). Mind double-logarithmic axes (scales) in Figure (a) and linear ones in Figure (b). See more details in the main text.

Hence, comparison of the experimental C_p(T) and S_T(T) data reported for both crystalline and amorphous modifications of linear PP and PE elsweher in refs. [11,14] (respectivelly), and replicated herein in Figures 8, 9(a)-10(b), with the C_p(T)/C_v(T) simulation(s) carried out based on the first (harmonic) and second (dominat anharmonic) terms of Equation (10c),-in combination with Eq.(13) for S_T(T) evaluations-proves validy of those equations for quite realistic approxi-mation of temperature-dependent harmonic and anharmonic fractions of the lattice thermal capa-city the essentially 1D polymeric materials (composed mainly of MWs). However, further enlargement in the absolute temperature of those 1D polymeric materials causes significant deviations even of the simulated C_p(T) and S_T(T) dependencies from their experimental counter-parts. Indeed, as it is seen readily from the Figure 9(a), and Figure 10(b), high-temperature log-log slope(s) of experi-mental C_p(T) dpendencies deviate significantly from the unity (which is generally expected for the polymers with the linear molecular structures and MWs [8]): such slope becomes equal to ~1.32 for the crystalline modification of PP above T ≅ 260 K (see Figure 9(a)), while it approximately equlas to ~1.44 for the amorphous modification of PE above T ≅ 250 K (see Figure 10(b)). Such deviations of the experimental C_p(T) results from the simulated curves obtained based on Equation (10c) could be attributed both to implementation of a ‘truncated’ (by its just two dominant terms) version of this equation, as well as by (somewhat) simplified version of the entire Equation (10c) itself. Indeed, in general, as it was mentioned elsewhere in ref. [25], complete set of the confined vibrational many-particle (anharmonic) states of LA and TA phonons have to incorporate essentially permutations of their fundamental states (or so-called ‘Fock states’ [51]) with all available (for the given system) quantities of single-phonon energy, while all anharmonic states incorporated into Equation (10c) are ‘composed’ merely on (algebraic) products of their single-phonon states of the same energy [28,44]. However, such modifications of the Equation (10c) would not be introduced and analyzed herein, though similar modifications are reflected in the basic equation of the 2D GSM: see Equation (3) of ref. [25]; and Equation (11a) herein.

Nevertheless, in this study, we do not implement the aforementioned 2D equations for any kind of simulations on the temperature-dependent thermal capacity of 2D polymeric materials, predominantly due to the lack of reliable experimental C_p(T) data for those 2D polymers. On the other hand, earlier versions of (quasi-continuous and/or ‘discretized’) basic equations of 2D GSM have already been used at simulation(s) on the C_v(T) and C_p(T) functions for square flakes and rectangular ribbons off graphene, hexagonal boron nitride, molybdenum and tungsten disulfides [23,24,25]. Those simulations were carried out taking into account explicitly contributions from both ‘in-plane’ and ‘out-of-plane’ components of the harmonic and/or anharmonic fractions of confined 2D lattice thermal vibrations [23,24,25].

As for 3D anisotropic version of the basic equation(s) of the GSM, exhibited in the previous section (see also references therein), they might be implemented at simulation(s) on C_v(T) and C_p(T) dependencies for ‘cross-linked’ polymers with substantially 3D topology of molecular network. As it was argued elsewhere in ref. [52], The spatial characteristic of thermal transport in so-called ‘high-density’ polyethylene (HDPE) is essentially anisotropic, which implies the necessity of implementing the anisotropic version of the basic equation of the 3D GSM at appropriate simulations on isochoric and isobaric lattice thermal capacity of this polymeric material. However, that basic equation requires the orthogonal components, c_efx, c_efx, and c_efx, of the effective sound speed of PE as equation parameters. Since reference literature data on this quantities in PE is quite controversial (in particular the group and/or phase sound velocities of PE may vary in the range from approximately of 1.35 km/s to approximately of 17 km/s [53]; see also references therein), it would be more appropriate to evaluate it in a more reliable way based on so-called ‘Christoffel Matrix’ (CM) formalism [28,44,54].

The CM, Γ_ij, for the non-piezoelectric solid reads [54]:

\[ \Gamma_{ij}=\frac{l_{iK}c_{KL}l_{Lj}}{\rho_M}, \tag{15} \]

Here, l_KL is the ‘propagation direction’ matrix, c_KL is the elastic stiffness constants matrix, while ρ_M stands for mass density of (polymeric) solid. The c_KL matrix manifests a compact representation of the fourth-rank ‘stiffness tensor’ [54]. In general, the elastic stiffness constants matrix is of the sizes 6 × 6, and comprises of 36 elements, and could be obtained for crystalline solids with atomic structures of every known symmetry class (Bravais lattice) [54]. The CM approach is based essentially on simplifying assumptions on linear dispersion dependencies of LA and TA phonon branches, even at relatively high quasi-wave-vectors.

Atomic structure of the ‘cross-linked’ PE crystallizes ‘…in an orthorhombic (Pnam) phase…at room temperature and atmospheric pressure…’ [39,50]. The c_KL matrix for the orthorhombic crystals comprises 9 independent non-zero elements [55,56]:

\[ c_{IK}=\begin{bmatrix}c_{11}&c_{12}&c_{13}&0&0&0\\c_{12}&c_{22}&c_{23}&0&0&0\\c_{13}&c_{23}&c_{33}&0&0&0\\0&0&0&c_{44}&0&0\\0&0&0&0&c_{55}&0\\0&0&0&0&0&c_{66}\end{bmatrix} \tag{16a} \]

The following quantities of the elastic modulus for the orthorhombic PE have been evaluated elsewhere in ref. [56]: c₁₁ = 15.95 GPa, c₁₂ = 6.37 GPa, c₁₃ = 2.40 GPa, c₂₂ = 15.53 GPa, c₂₃ = 5.99 GPa, c₃₃ = 362.51 GPa, c₄₄ = 6.16 GPa, c₅₅ = 3.38 GPa, c₆₆ = 6.02 GPa. Numerical ‘diagonalization’ (using an utility created based on the Octave 3.6.4 freeware) of the Equations (15, 16a) with the listed just above elastic constants and at ρ_M = 950 kg/m³ [50] yields the following ‘components’ of the sound velocities of the orthorhombic PE: c_t1 = 1886.24 m/s, c_t2 = 2517.31 m/s, and c_l = 4097.50 m/s (with the c_eff = 2560.93 m/s) for the <100>-oriented ‘cross-linked’ crystalline PE; thus, obtained c_t1x, c_t2, and c_eff quantities are not deviating vastly from the c_s = 2142 ± 2 m/s reported elsewhere in ref. [50], and implemented at simulations on C_v(T) and C_p(T) dependencies of the linear PE, revealed in the Figures 10(a), 10(b).

However, for the <111>-oriented ‘cross-linked’ PE, c_t1 = 2219.59 m/s, c_t2 = 3674.46 m/s, and c_l = 11433.86 m/s (with c_eff = 3703.05 m/s) are obtained via the numerical ‘diagonalization’. Thus, an alteration in the crystalline orientation of orthorhombic crystalline PE causes quite significant changes in its longitudinal, transverse(s), and effective sound velocities. It is noteworthy that the CM expressed by Equation (16a) is actually obtained under the assumption of essential (spatial) uniformity of the single-crystalline media within the entire volume of the (non-disrupted) LA and TA phonon propagation (coherence).

However, real atomic structure and morphology of the samples of ‘cross-linked’ PE might deviate considerably from those of the ‘idealized’ (i.e., essentially single-crystalline in this particular case) orthorhombic phase of the PE [57]. Indeed, the atomic structure of real samples of PE would rather be ‘composed’ of a set of uniform 3D fragments (morphological units) of the orthorhombic crystal structure, separated by their grain boundaries, amorphous inclusions (‘tissues’), voids, dislocations, etc. [57,58]. Furthermore, even crystalline orientation(s) of the neighboring 3D fragments (morphological units) might ‘fluctuate’ considerably [57,58]. In such a case, the so-called ‘orthotropic’ representation of atomic structure(s) of real spatially non-homogeneous materials (including polymeric ones) is often used rather than the approximation of just (uniformly) anisotropic media [57]. The ‘orthotropic’ approximation presumes that actual properties of (really anisotropic and-essentially-spatially non-homogeneous) materials might be represented systematically by just three mutually orthogonal spatial directions. In contrast, those properties of anisotropic materials may generally vary in different spatial directions, which are not necessarily mutually orthogonal, as in the case of orthorhombic and hexagonal crystals [28,44,46].

Nevertheless, the CM formalism still could be used for quantitative evaluation(s) on the effective sound velocities of those solids in just 3 mutually orthogonal spatial directions [57,59]. However, the structure of the stiffness (elasticity) matrix, expressed by Equation (16a) for the case of the orthorhombic crystals [55,56], would be (somewhat) different, with just 5 independent non-zero elasticity moduli (constants) [58]:

\[ c_{IK}=\begin{bmatrix}c_{11}&c_{11}-2c_{66}&c_{13}&0&0&0\\c_{11}-2c_{66}&c_{11}&c_{13}&0&0&0\\c_{13}&c_{13}&c_{33}&0&0&0\\0&0&0&c_{44}&0&0\\0&0&0&0&c_{44}&0\\0&0&0&0&0&c_{66}\end{bmatrix} \tag{16b} \]

Latter CM corresponds to so-called ‘transversely isotropic media’, which is (presumably) axisymmetric about z axis only [57,59]. In general, elastic modulus (constants) of the CM, suitable for the ‘orthotropic’ approximation-as well as those incorporated into Equation (16b) in particular-are somewhat different as compared to their counterparts, customarily used for the CM of spatially uniform (though anisotropic) solids even when they are (almost always) denoted by the same symbols [57,58,59]. Nevertheless, as a ‘zero-order’ approximation, we may presume that the quantities of all 5 elasticity modules embedded in Equation (16b) are just (pair-wise) equal to their (appropriate) counterparts, incorporated into Equation (16a). Consequently, numerical ‘diagonalization’ of the ‘orthotropic’ CM, with the elasticity (stiffness) matrix, expressed by the Equation (16b) yields: c_t1 = 2517.31 m/s, c_t2 = 2546.41 m/s, and c_l = 4097.50 m/s (with c_eff = 2901.32 m/s) for the <100>-oriented ‘cross-linked’ crystalline PE. Thus, only c_t1 and c_eff quantities are changed for the orthotropic case as compared to the sound velocities, evaluated via implementation of the Equations (15, 16a) for the <100>-oriented orthorhombic crystal within framework of such approximation, while the c_t1 and c_l quantities remain the same for both orthorhombic and orthotropic approximations (cases).

Now, the C_v(T) and C_p(T) dependencies might be simulated based on Equation (12) for both orthorhombic and orthotropic modifications of 3D PE. The Figures 11(a), 11(b), compares results of implementation of the basic anisotropic equation of the 3D GSM for simulation on the temperature-dependent harmonic (in Figure 11(a)) and anharmonic (Figure 11(b)) fractions of the lattice thermal capacity of the ‘high-density polyethylene’ (HDPE) with the experimental C_p(T) dependencies reported elsewhere in refs. [14,60].

Figure 11 (color online). The experimental data were replicated from Refs. [14,60] (open symbols) and simulated (curves)-based on Equation (12) from the previous section-temperature dependences of the (a) harmonic and (b) ratio(s) of the anharmonic to the harmonic fractions of the lattice thermal capacities of HDPE, evaluated for different crystalline orientations using sound velocities for 3 mutually orthogonal spatial directions, obtained via implementation of the orthorhombic (OR) and orthotropic (OT) CMs: see Equations (15-16b). The contribution from group vibrations (optical phonons) is taken into account as well: see green curve in the left panel. The log-log slopes of some (low-temperature) C_H(T) and [C_AH(T)/C_H(T)] dependencies have been evaluated using standard least-squares procedures and indicated with the blue symbols near the appropriate curves in Figures (a) and (b). Vertical arrow in the right panel indicates the melting temperature of the crystalline PE (T_M = 414.6 ± 0.5 K), reported elsewhere in ref. [60]. See the main text for more details.

All simulation results are obtained at the same (fixed) sizes of the cubical (in this particular case) phonon confinement volume, i.e., at L_x = L_y = L_z = 1.0 μm, but for 3 different sets of the (an-isotropic in general) sound velocities, evaluated (based on Equations (15-16b)) and listed above in this section of the cases of <100>- and <111>-oriented orthorhombic PE, as well as of <100>-oriented ‘orthotropic’ PE. Contribution(s) from the group vibrations (optical phonons) are also taken into account in Figure 11(a). The latter contributions are evaluated based on a ‘truncated version’ of the fundamental equation 3D GSM: i.e., Equation (12), in the previous section. In particular, this version corresponds merely to the N = 1 and E_T = ℏω_grp = ℏω_opt (here ℏω_grp and ℏω_opt stand for the energies of the ‘group vibrations’ and optical phonon, respectively); thus, neither summation over different numbers of quasi-particles nor integration over single-particle vibrational spectrum of the acoustic phonons is required for this ‘truncated’ version of Equation (12). The fixed θ_D = 289 K [53] and θ_E = 1800 K [8] quantities are used in the simulations on the C_p(T) dependencies, revealed in Figure 11(a). The latter temperature matches fairly well with that of the ‘wagging’ vibrations of the CH₂ group of polyethylene (ranging from 1698.3 K to 1976.6 K), suggested elsewhere in ref. [2]. It is noteworthy that all our simulation results remain significant only for the temperatures below the melting temperature of crystalline PE, which is indicated with the vertical arrow in Figure 11(b). Indeed, as it is demonstrated in Figures 9(a), 9(b) for the case of polypropylene (PP), behavior of the experimental C_p(T) dependencies obtained both for amorphous and crystalline modifications of PP reveal abrupt changes at the ‘glass transition’ temperature, T_g, of this material, implying apparent inter-relation among structural characteristics of PP and its lattice thermal capacity at elevated temperatures. Therefore, thermal properties of PE at temperatures exceeding its melting temperature, T_M, might deviate significantly from their counterpart for the crystalline PE. Significant alteration might also be expected for the anharmonic fraction of the lattice thermal capacity of PE in the vicinity of its ‘melting point’. Temperature dependence of this fraction has been evaluated solely for the many-particle acoustic components of confined thermal vibrations: it is obtained from Equation (12) at N = 2, …, 25.

In other words, the temperature dependence of the C_AH(T)/C_H(T) ratio plotted in Figure 11(b) has been evaluated without taking into consideration any contribution from the group vibrations (optical phonons), even to the C_H(T) quantities, for this particular case. As it is seen readily from Figures 11(a) and 11(b), amendments in the (anisotropic) sound velocities of the HDPE instead affect the magnitude of the simulated C_H(T) and/or C_AH(T) dependencies than their generic behavior. Indeed, at the relatively low temperatures (T < 30 K), both experimental C_p(T) and simulated C_H(T) dependencies follow closely to the well-known ‘Debye’s law’, C_p(T) ∝ T³ [12] for the 3D bulk solids, see Figure 11(a), though being evaluated in this particular case based on the very first term of Equation (12), which takes into account both quantum and anisotropic (orientation-dependent) effects. Since sizes of the (cubical) confinement volume presumed the same (fixed) at all our simulations on thermal properties of solid polymers, the mentioned above alteration in the C_p(T) magnitudes for PE samples with different crystalline orientations have to be attributed entirely to the affect(s) of anisotropic sound velocities. As it is seen clearly from Figure 11(a), the larger (averaged) sound velocities routinely yield lower C_p(T) magnitudes for the low-temperature ranges. However, at elevated temperatures (T > 320 K), the experimental C_p(T) dependencies, reported elsewhere in refs. [14,60] and re-plotted in Figure 11(a), apparently deviate from the ‘saturation’ behavior (or purely classical Dulong-Petit law), predicted by the well-known Debye’s theory [12] for the temperature range, exceeding Debye’s temperature of the material. Indeed, the C_p(T) ‘bumps’ emerge clearly at temperatures exceeding 400 K for the case of HDPE, Figure 11(a). Appearance of such ‘bumps’ is routinely attributed to impact from the (terminal) group vibrations with the Einsteinian (of δ-like-function) vibrational spectrum [8], similar to that of non-dispersive optical phonons. Indeed, as it is shown in the Figure 11(a), combined effect of the canonical C_H(T) dependence, evaluated based on purely ‘acoustic contribution’ from the Equation (12), and an optical ‘fraction’ (shown with the green curve in this figure), yields fairly reasonable approximation for the experimental C_p(T) dependencies, reported elsewhere in refs. [14,60]. Moreover, all simulated C_H(T) dependencies plotted in Figure 11(a) reveal (almost) linear C_p enlargement with the temperature within the temperature range closely surrounding ~100 K, as it is ‘highlighted’ with the cyan solid straight lines in this figure. Such kinds of linear C_p(T) dependencies are quite common for 3D polymeric solids [8], and customarily attributed to the combined impact of ‘skeletal’ and ‘group’ vibrations on the experimental C_p(T) dependencies of those solids [8].

As for the temperature-dependent ratios of the anharmonic to harmonic fraction(s) of the lattice thermal capacity of HDPE, their low-temperature log-log slope emerged to be very close to 2 (see Figure 11(b)), which is generally expected for 3D amorphous and crystalline solids based on predictions of the 3D GSM (see, for instance, some discussion on this issue in the section 5.2 in ref. the [24]). However, there is no apparent ‘low-temperature anomalies’ revealed in Figures 11(a), (b), neither for the experimental C_p(T) dependencies, nor for their experimental counterparts. The obvious reason for the absence of such anomalies for those simulated curves is the extremely low E_min quantity, evaluated based on Equation (12a): E_min ≅ 2 * 10^-5 eV (20 μeV) emerges at L_x = L_y = L_z = 1.0 μm. Therefore, based on the familiar E_min = k_BT relation, the temperature range where the ‘low-temperature anomalies’ might be (in principle) observed, is limited (from the top) by the T ≅ 2.3 * 10^-4 K (0.23 mK!), which is more than 3 orders of the magnitude below the left margin of the horizontal axes in the Figures 11(a), 11(b). In other words, though based on ‘fully quantum’ basic equation of the 3D GSM, the ‘low-temperature’ anomaly is always expected to emerge for the C_p(T) dependence, evaluated for real 3D solid of limited spatial extents, temperature range of its appearance might be ‘shifted’ towards extremely low temperatures for the parallelepipedal confinement volume of micrometer-sized spatial extents. It is noteworthy that the [C_AH(T)/C_H(T)] ratio saturates in Figure 11(b) at significantly higher (approximately by 200 K) temperature as compared to the temperature of saturation of the C_H(T) dependencies, shown in Figure 11(a). Again, this difference in the saturation temperature(s) of the simulated curves, plotted in Figures 11(a) and 11(b), has to be attributed to higher integration limits for the many-particle (anharmonic) terms of Equation (12). As it is mentioned above, contributions from the ‘group vibrations’ (optical phonons) are not taken into account in evaluations on the C_AH(T) quantities and the [C_AH(T)/C_H(T)] ratio(s). In general, such a contribution might significantly amend the ‘high-temperature’ behavior of the latter ratio(s), and, in principle, might make it even decline with the temperature at T > 300 K for the case of ‘cross-linked’ PE.

As for quantitative evaluations on the temperature-dependent entropy of the linear (i.e., essentially 1D) and branched (3D) polyethylene(s), PEs, such simulations could be carried out readily using appropriate combinations of Equations (10a-10c, 13) for 1D case(s), and that of Equations (12-12c, 13) for 3D ones. Figures 12(a) and 12(b) compare experimental (open symbols) and simulated (curves) S_T(T) dependencies obtained for both the 1D PE and 3D HDPE. As it is seen readily from those figures, the combinations of Equations (10a-10c, 13) yields fairly reasonable approximation at L_z = 5.8 nm for the experimental S_T(T) dependence, tabulated for linear PE elsewhere in the Table 3 of ref. [14]: see the lowest simulated curve in the Figure 12(a). This implies straightforwardly, that presence of fairly strong spatial confinement, ‘discovered’ via the comparison of the experimental and simulated C_v(T) curve for linear PE in Figure 10(a) above, is ‘re-confirmed’ again via the comparison of the experimental and simulated S_T(T) dependencies, shown in the Figure 12(a). Indeed, the aforementioned reasonable approximation for the tabulated S_T(T) dependence is obtained under the assumption that the dominant length of the PE MW in the samples studied elsewhere in ref. [14]. It is just under 6 nm! However enlargement in the length of the of the PE MWs would affect considerably the entire shape of the simulated S_T(T) dependencies: the ‘low-temperature anomalies’ apparently become less ‘articulated’ with the L_z enlargement, while entire S_T(T) curve approaching a straight line for PE MWs of L_z = 100 nm: see Figure 12(a). Such behavior is very similar to that of the simulated S_T(T) dependencies, plotted for the (almost) linear ‘chains’ of polypropylene, see Figure 8 above. However, the ‘standard entropy’ of PP is ~2.82 times that of PE. As for the temperature-dependent entropy, S_T(T), of the branched HDPE, it rises with the temperature much faster than that of the linear one: compare Figures 12(a) and 12(b). On the other hand, it increases, but not as fast as the C_p(T) function of 3D solids: compare Figure 11(a) and Figure 12(b).

Figure 12 (color online). The experimental S_T(T) dependencies (open symbols) are plotted based on data from Tables 3 and 10, given in refs. [14,60] (respectively), and simulated based on combinations of equations (10a-10c, 12-12c, 13) from the previous section (curves) for a (a) linear PE [14] at different MW lengths, and (b) ‘branched’ HDPE [60]. The log-log slope(s) of S_T(T) dependencies, evaluated in the temperature range(s) of 100 K ≤ T ≤ 300 K (for panel (a)) and of 0.1 K ≤ T ≤ 5 K (for panel (b)) are indicated as well near appropriate curves. See the main text for more details.

In other words, the temperature-dependent entropy, S_T(T) ∝ T^~2.8, in Figure 12(b) does not follows Debye’s ‘law’, C_v(T) ∝ T³, well-established for the heat capacity of 3D solids [12]. This deviation apparently originates from the linear (in temperature) term embedded in the denomi-nator of the ‘integrand’ of the Eq. (13). In contrast to the results revealed in the Figure 12(a), where both simulated S_T(T) and C_v(T) dependencies have been re-calculated (using appropriate Octave 3.6.4 utility) for every given length of PE MW based on the combinations of Equations (10a-10c, 13), the simulated curve in the Figure 12(b) is rather obtained by the appropriate ‘treatment’ of the experimental C_v(T) data reported elsewhere in ref. [60], and re-plotted herein in Figure 11(a). In particular, the [C_v(T)/T] ratio evaluated numerically for that data set has been further integrated numerically using Origin 8.5 software package, installed on my home computer. As it is seen readily from comparison of the experimental dependencies, plotted with the open symbols in the Figure 12(b), with its ‘simulated’ counterpart (plotted with the red curve), both those dependencies are matching fairly well within the temperature range limited (on its top) with T ≅ 32 K. However, above this temperature, the ‘simulated’ S_T(T) curve rises significantly faster with the temperature than its experimental counterpart does. The most evident reason for such behavior is that the original (experimental) C_v(T) dependence, plotted in the Figure 11(a), and thereafter ‘treated’ (in the explained just above way) to obtain the ‘simulated’ S_T(T) curve, is impacted significantly at relatively high temperatures by contribution(s) from the vibration of dangling groups (see green curve in the Figure 11(a) and comments to it). This high-temperature deviation among the experimental and ‘simulated’ S_T(T) dependencies implies that those ‘group vibrations’ are not contributing significantly to the entropy of essentially 3D (orthorhombic) lattice of HDPE (see Figure 12(b)), though yield apparent impact to the C_p(T) dependencies in the Figure 11(a). This discrepancy might suggest dominant concentration (location) of those relatively intermittent dangling ‘groups’ on the periphery of the 3D structural blocks (‘crystals’) of the HDPE.

At relatively ‘low-temperature’ (e.g., at T ≅ 10 K), the entropy of the 1D (linear) PE is much higher (up to the order of the magnitude!) than that of its 3D HDPE counterpart: compare Figures 12(a), 12(b). This difference probably has to be attributed mainly to the fact that the orthorhombic 3D lattice of the HDPE is far more ‘rigid’ as compared to its 1D counterpart, and the temperature-induced structural disorder is diminished in the 3D spatial arrangement as compared to that in its 1D counterpart. On the other hand, manifestation of the spatial confinement and directly related to it ‘low-temperature anomalies’ in the 1D polymers with MWs of limited (by just few nano-meters!) spatial extent reduces significantly the entropy of 1D lattice at temperatures well below 1 K (where its entropy also emerges well below than that of the low-temperature 3D counterpart) due to essentially ‘discretized’ vibrational spectrum of 1D polymeric ‘chains’, and significantly ‘suppressed’ contribution to the 1D entropy from their vibrational levels of relatively high energy.

Thus, implementation of different versions of the fundamental equations of the GSM, revealed in the previous section, allows one to carry out realistic quantitative simulations on the temperature-dependent isochoric (harmonic) and isobaric lattice thermal capacity, as well as on entropy of solid but spatially non-homogeneous polymeric materials with different dominant spatial dimensionalities, topology of their atomic ‘networks’ and morphology: i.e., for 1D, 2D and 3D ‘versions’ of those materials-within wide temperature ranges, limited, however, by the temperatures of phase transitions for such solid materials. Alteration (s) in the spatial dimensionality of atomic networking of polymers caused by their phase transitions would affects directly crucial features of the experimental and simulated C_v(T) and C_p(T) dependencies. Since all GSM equations include explicitly key morphological parameters of solid(s) (polymers) of different spatial dimensionalities and extents, as well as those of their atomic structures, which allows one to evaluate quantitatively effect of alteration(s) in the sizes (volumes) of the phonon confinement area(s) and crystalline orientation of those (if any) on the key features of their temperature-dependent lattice heat capacity and entropy of linear and ‘cross-linked’ polymers and oligomers composed of structural units of spatially different extent(s), varying in the range from just few nano-meters to (potentially) tenths micro-meters. Therefore, abrupt alterations in the phonon confinement sizes caused (for instance) by the phase transition(s) are generally expected to cause drastic changes in the thermal characteristics of polymers. On the other hand, strong phonon confinement within relatively small (in sizes) linear molecular chains of solid polymers and oligomers emerges in an appearance of well-articulated ‘low-temperature’ anomalies in the C_p(T) dependencies when the chain length (size) diminishes till 5-10 nm, depending on other parameters (molecular weight, Debye temperature, etc.) of polymers: see simulation results and experimental data plotted in Figures 7(a)-12(a). Features of those ‘low-temperature’ anomalies are linked intimately to the spatial extents (sizes) of the phonon confinement within the polymeric structures and connected directly by quantization (‘discretization’) of their vibrational spectra. Thus, the appearance of such anomalies becomes unambiguous experimental evidence on the manifestation of phonon confinement(s). When, however, the spatial extent(s) of the confinement rises till micro-meters (and above), those ‘low temperature anomalies’ might not be manifested in the C_p(T) and S_T(T) dependencies within the entire temperature range, available for ‘routine’ measurements: see, for instance, Figures 11(a), 11(b). This, however, does not imply any diminishment in the physical essences of the phonon confinement phenomenon for thermal characteristics of (solid) polymers with an arbitrary spatial extent. Therefore, the phonon confinement is taken into account explicitly in all versions (s) of the basic equation(s) of the GSM, regardless of their dimensionalities and (anticipated) spatial extents: see details in the previous section. On the other hand, the absence of visible ‘low-temperature anomalies’ might instead encourage additional experimental investigation(s) on essential features of atomic structure and morphology of real polymeric material, to obtain decisive information regarding spatial characteristics (sizes, orientations, etc.) of the thermal vibration of the polymeric solids.

An appropriate combination of fundamental (single-particle) vibrational spectra of the basic equations of the 1D and 3D GSMs might yield a fairly reasonable approximation for the quantized (‘discretized’) and anisotropic (if any-for the 3D case) version(s) of the ‘three-band’ model [18] (see also Figure 4(b) and Figure 5 in the introductory section of this article), which was successfully implemented for decades at simulations on the vibrational characteristics and temperature dependencies of the lattice thermal capacities of various polymeric materials [8,18]; see also references therein. However, there are some significant differences in the physical essences of the original version of the ‘three-band’ model [18] (see also Figure 4(b) and caption to it for illustration) and a (linear) combination of the fundamental equations of the 1D and 3D version(s) of the GSM. First, the original ‘three-band’ model presumes essentially the quasi-continuous spectral distributions for the vibrational DOS function of LA and TA phonons, confined within both 1D and 3D structural fragments of the polymeric material [18]. Secondly, the non-overlapping energy ranges corresponding to the acoustic vibrations confined within the 1D and 3D structural fragments are presumed essentially within the framework of the original version of the ‘three-band’ model [18]; see also references therein and Figure 4(b) herein for an illustration.

However, the letter assumption implies straightforwardly an extremely (and-to be honest-unrealistically) high ‘onset’ energy (or corresponding temperature, θ₃,-see Figure 4(b) and caption to it) for the 1D vibrational spectra of the original ‘three-band’ model, while a combination of the basic equations of the 1D and 3D GSMs rather corresponds to the (significantly) overlapped-in general-but ‘discretized’ spectra of thermal vibrations (phonons), confined within the 1D and 3D structural fragments. Consequently, the lowest energies (denoted as E_min in the previous section) of the two overlapping components (bands) of the entire ‘discretized’ vibrational spectrum within framework of the GSM(s) would be defined via just a linear combination(s) of the Equations (10a, 12a) for the 1D and 3D structural fragments of the polymeric material, respectively furthermore, since such spectra are essentially ‘discretized’ (see Figure 5), even the overlapping area of the vibrational spectra still expected to yield a non-uniformly ‘discretized’ vibrational spectra for entire fundamental spectra of polymeric material, though with some local ‘densification’ of quantized vibrational level within the ‘overlapped’ energy range. In contrast, any overlapping among 1D and 3D continuous spectra of the original ‘tree-band’ model would yield a ‘quadratic’ overall P(ℏω) function, shifted upward by the constant (energy-independent) quantity of the 1D component of the vibrational DOS, see Figure 4(b). Similar amendments are expected for overlapping ranges of the vibrational energies, corresponding to Tarasov’s equations of different spatial dimensionalities. This would imply straightforwardly an interaction among acoustic vibrations of 1D, 2D, and 3D structural fragments within the overlapping energy range. This brings us well beyond the framework of the ‘canonical’ three-band model [18].

On the other hand, due to essentially ‘discretized’ vibrational spectra of basic equations of both 1D and 3D version of the GSM, even existence of overlapping vibrational ranges (bands) of the 1D and 3D (and customarily spatially separated) structural fragments would not imply essential interaction(s) among them, except of relatively rare cases of perfect coincidence (resonance) among their vibrational frequencies. Indeed, the quantized (discrete) vibrational frequencies of the confined 1D and 3D vibrations are defined apparently by essentially different morphologies of those (spatially separated) 1D and 3D structural fragments, as well as by significantly different Equations (10-10c) for the 1D fragments, and Equations (12-12c) for the 3D ones. Therefore, cases of ‘perfect coincidence’ (resonance) among those vibrational frequencies are expected to be relatively infrequent. Subsequently, qualitative impression on essential features of the C_v(T) and C_p(T) functions of-for instance-polyethylene with its atomic structure (morphology) composed of spatially separated micro-meter-sized 3D (amorphous and/or crystalline) ‘blocks’, but linked with the nano-meter-long 1D ‘chains’ (MWs) might be achieved indeed based on a (linear) combination(s) of the C_H(T) dependencies, exhibited in Figures 9(a)-11(a) above in this section. Furthermore, those simulated C_H(T) dependencies might be used as well in quantitative evaluations on the temperature-dependent entropy of polymeric materials: please see Figures 8, 12(a), 12(b) and comments to them above. As it is seen from the Figures 8, 9(a)-12(b), within this temperature range, the ‘low-temperature’ anomalies might emerge clearly (see also Figures 2(b), 3(a), and 3(b) for experimental evidence on existence of such ‘anomalies’) for polymeric structures of different spatial dimensionality, and such features of the experimental C_v(T), C_p(T) and S_T(T) dependencies might be replicated convincingly based on an appropriate combination of equations of the different versions of the GSM, and the well-known ‘generic’ Equation (13). It is noteworthy as well, that corrected version(s) of (‘boxed’ [8]) 1D, 2D and 3D Tarasov’s equation(s), and those of the ‘three-band’ model might predict appearance of the ‘low-temperature anomalies’ as well, when the ‘onset’ frequencies of the acoustic phonons are chosen to be positive (non-zeroth). However, those ‘onset’ frequencies for the different versions of the Tarasov equations are often chosen (or adjusted) ‘arbitrary’, which would yield unrealistic behavior of the simulated ‘anomalous’ C_v(T) dependencies, while similar frequencies of all basic equations of (different versions) of the GSM are linked straightforwardly to the actual morphological features (sizes of phonon confinement, crystalline orientation of the confinement volume-if any, etc.) of the spatially non-homogeneous and low-dimensional amorphous and crystalline polymeric solids. Furthermore, an appropriate combination of the many-particle terms of Equations (10a-10c, 12) would allow one to simulate temperature dependencies of the anharmonic fraction of the lattice thermal capacities of various polymers composed of structural blocks of different spatial dimensionalities, and-in combination with the ‘generic’ Equation (13)-would yield valuable information on the temperature-dependent entropy of those polymeric samples. In contrast, neither (different versions) of Tarasov’s equations, nor those of the ‘mass-weighted velocity’ autocorrelation function formalism [39], ‘Wigner-Kirkwood expansion’ [41,42], etc., would allow one to evaluate the (truly) anharmonic contributions to the lattice thermal capacity quantitatively.

It is noteworthy that the isochoric (harmonic) lattice thermal capacity and entropy of 1D and 3D polymers are evaluated entirely based on single-particle (fundamental) states of the confined LA, TA, and optical phonons, depicted by the very first term(s) of the appropriate basic equation of the GSM. In a certain respect, the aforementioned contributions from the fundamental vibrational states of the GSM replicate their counterparts from the ‘skeletal’ and ‘group’ thermal vibrations of polymeric structures to their temperature-dependent lattice thermal capacity. Statistical characteristics of many-particle (and essentially anharmonic) states of the LA and TA phonons are obtained based on the concept of many-particle vibrational density-of-states, introduced by the author of this article in 1995 [47]. This concept might be further generalized using the famous ‘Fock space formalism’, pioneered elsewhere in ref. [51]. The statistical characteristics of those many-particle states define features of the temperature dependencies of the anharmonic lattice capacities of 1D, 2D, and 3D versions of the GSM. Thus, implementation of different versions of the GSM equations brings essentially new insight to understanding as well as more appropriate ways of quantitative evaluations on crucial (quantized) features of the single-particle and many-particle density-of-states states of phonons spatially confined within structural fragments of polymeric materials of different spatial dimensionalities, as well as on those of harmonic and especially anharmonic thermal capacity, and even entropy of spatially non-homogeneous and low-dimensional polymeric solids. It is important as well that those anharmonic fractions of the lattice thermal capacity are evaluated within frameworks of the GSM in profoundly non-perturbative ways, via appropriate evaluations of contributions from the many-phonon states and corresponding many-particle vibrational DOS functions. In contrast, the ‘first-principles’ calculations implemented widely nowadays at quantitative evaluations on the thermal-transport-related properties of bulk and spatially non-homogeneous (though primarily crystalline) solids within frameworks of so-called ‘Density Functional Theory’ (DFT) and/or ‘Density Functional Perturbation Theory’ (DFPT) one are based essentially on ‘perturbative’ evaluations of the force constants and other key ‘internal’ parameters of crystalline materials [61,62,63].

Furthermore, such evaluations are still extremely computationally demanding (even for elementary solids with large numbers of atoms within their unit cells, like beta-modification of boron, which comprises of 106 atoms (!) in its unit cell), while non-perturbative approach developed within framework of (different versions) of the GSM allows one to carry out computationally ‘non-expensive’ and fairly rapid quantitative evaluations on the many-particle VDOS functions and closely related to them anharmonic fractions of the lattice capacities of spatially non-homogeneous and low-dimensional solids [23,24,25,28,44]. The non-perturbative approach is generally expected to be especially meaningful for 1D cases, like evaluations on vibrational and thermal properties of ’molecular wires’. Moreover, the DFT/DFPT evaluations on vibrational and thermal properties of ‘industrial’ polymeric materials with a (often not known with a sufficient accuracy) combination of structural and/or morphological features of fragments of different spatial dimensionalities could be simply computationally ineffective, though such evaluations might give very useful inside on specific vibrational and thermal characteristics of unique polymeric ‘species’, like ‘fullerenes’, ‘cages’, etc. Thus, in general, predictions of the (different versions) of the basic (and essentially non-perturbative) equations of the GSM goes well beyond prediction(s) of the well-known Tarasov’s equations, those of the ‘three-band’ model as well as of other theoretical approximation(s), reviewed in the introductory section.

4. Conclusions

In spite of intensive and lasting experimental and theoretical investigations on the features of temperature-dependent thermal capacity of polymers, some crucially important questions remain unanswered. First of all, it is not established so far the physical origin(s) behind sharp decline in the heat capacity of some linear polymers with the temperature diminishment, or-in other words-actual reasons for appearance of so-called ‘low-temperature anomalies’: see discussion with examples on this issue in the introductory section of this article. Secondly, there is no satisfactory approach to quantitative evaluations of the truly anharmonic contributions to the thermal capacity of polymers, especially of one-dimensional (1D), linear, and two-dimensional (2D) ones. To resolve these issues, the ‘discretized’ version(s) of so-called ‘Generalized Skettrup Model(s)’ (GSMs) of different (1D, 2D, 3D) spatial dimensionalities are described (see the second section for details) and implemented in the third section of the article at quantitative evaluations on temperature-dependent harmonic and anharmonic fractions of lattice thermal capacity as well as of the entropy of linear polypropylene and polyethylene as well as of those of ‘cross-linked’ polyethylene with dominant crystalline and/or amorphous atomic structures. Basic equations of all aforementioned versions of the GSM take into account explicitly the spatial confinement (within the 1D, 2D and/or 3D structural fragments of those polymers) of conventional longitudinal acoustic (LA), transverse acoustic (TA), and optical phonons and closely related to it quantization effects of their single-particle (fundamental) and many-particle vibrational states, see details in the second section. On the other hand, those equations also imply the absence of significant phonon-phonon interactions (scattering) within every particular phonon confinement volume, which is, in other words, equivalent to a presumption on the Casimir limit for the mean free path of those confined LA, TA, and optical phonons. It is noteworthy that none of the well-known Tarasov’s equation(s) (even those with ‘boxed’ vibrational spectra) take into account appropriately the spatial confinement effect. A convincing coincidence with appropriate experimental data reported for aforementioned solid polymeric materials is achieved based on fairly reasonable sets of thermal and acoustic parameters for those 1D and 3D polymers. For instance, experimentally identified ‘low-temperature anomalies’ of the lattice thermal capacity and temperature-dependent entropy of linear polyethylene and polypropylene are replicated for such polymeric chains (oligomers) at their chain length(s) varying in the range from 2 nm to 10 nm, see simulation results plotted in the previous section. Such ‘anomalies’ manifest the unambiguously existing strong spatial confinement for acoustic and optical vibrations within real solid polymeric materials.

On the other hand, essential features of different versions of the GSM are discussed as well in comparison with those of Tarasov’s equations, the well-known ‘three-band’ model, as well as those of other theoretical approximations, reviewed briefly in the introductory section. Anisotropic effects in the orthorhombic atomic structure of ‘cross-linked’ polyethylene are taken into account via evaluation of anisotropic sound velocities of conventional thermal waves (acoustic phonons) with linear dispersions, confined within its 3D (poly)crystalline fragments. Such evaluations have been carried out quantitatively via implementation of the Christoffel Matrix (CM) formalism, see details in the second section. similar simulation results were obtained earlier via implementation of the GSM(s) for 1D MWs [28,44], 2D square flakes [23] and/or rectangular nano-ribbons [25], as well as that of 3D parallelepipedal crystallites [24] with ‘convex’ borders and (presumably) well-established spatial extents (sizes) of all those spatial non-homogeneities. However, the simulation results revealed in the previous section of this article are obtained essentially for polymeric structures of well-known spatial dimensionalities, but generally unknown actual morphological parameters. In a such situation, meaningful description of ensembles of confined acoustic phonons might be obtained via an appropriate ‘averaging’ of those characteristics for structural fragments of the same dimensionalities (e.g., MWs of the different lengths) as well as thermal characteristics of structural fragments of different spatial dimensiona-lities, i.e., in the ‘spirit’ of an ‘ideology’ of the well-known since the canonical ‘three-band’ model: its basic features are discussed to some extent in the introductory section herein. Thus, in distin-ction from simulation approaches revealed in previously published articles and books, in general, an appropriate combination(s) of the fundamental equations of different versions of the GSM would require for realistic description of statistical characteristics of ensembles of acoustic and optical phonons, confined within structural (atomic) networks of real polymeric materials, as well as at simulations on their temperature-dependent lattice thermal capacities an entropy.

On the other hand, experimentally substantiated information on features of 1D, 2D, and 3D structural and morphological fragments (and/or combination of those) of real polymers would be crucially important for the most straightforward implementation(s) of different versions of the GSM. Such information might be obtained readily using several well-established techniques for state-of-the-art structural and morphological characterization of real polymeric materials, and via continuing efforts in the appropriate implementation of those techniques. These would diminish greatly existing uncertainties in the vital information on actual structural and morphological features of the well-known and newly synthesized polymeric materials. Such information would be critical for appropriately implementing the different versions of the GSM, described in this article.

One of the main disadvantages of the current basic equation(s) of the different versions (s) of the GSM is the essential implementation of (apparently oversimplified) linear phonon dispersion relations for all polymeric materials studied in this article. Though, in principle, incorporation of a more realistic non-linear (at elevated quasi-wavevectors of phonons) dispersion(s) and corresponding vibrational DOS functions into framework of GSM could be fulfilled readily, such kind of correction(s) might not be carried out based entirely on the parameters of the GSM alone, but would rather require a ‘complementary’ knowledge on appropriate phonon dispersion parameters, evaluated, for instance, via implementation of the DFT and/or DFPT simulations.

So far, the basic equations of different versions of the GSM as well as those of the CM formalism have been successfully implemented herein at simulations on both harmonic and anharmonic thermal characteristics of ‘industrial’ polymeric materials: polyethylene and polypropylene. Convincing coincidence achieved for the simulated dependencies with their experimental counterpart(s) inspires implementation of those equations at evaluations on lattice heat capacities of biopolymers and other key organic and inorganic materials for biomedical and many other vitally essential applications.

Acknowledgments

All simulation results revealed in this article are obtained with utilities created by its author using Octave 3.6.4 free software: Copyright©1996-2011 John W. Eaton jwe@octave.org; see also http://www.octave.org.

Author Contributions

Authors of this article wrote individually main text of its every section, created ‘quantized’ versions of all basic equations of the different versions of the ‘Generalized Skettrup Model’ (GSM), and implemented them at evaluations on the harmonic and anharmonic lattice thermal capacities of polymeric structures, designed and programmed utilities for simulations on the temperature dependencies of harmonic and anharmonic lattice thermal capacities based on those equations of the GSMs, and carried out all aforementioned simulations for the one-dimensional and three-dimensional polymers, as well as plotted all simulation results exhibited in the article and created captions to every figure (plot).

Funding

None reported.

Competing Interests

The author declares NO potential conflict of interests.