Strain hardening of high density polyethylene

The introduction of true stress strain measurements, at constant strain rate, has promoted the development of empirical or semiempirical models for large deformations in thermoplastics. One such theory, which proposes that the post yield deformation process can be represented by equations derived from the theories of rubber elasticity, has been successfully applied to several glassy polymers. Unexpectedly, it can also model the post yield deformation of many different grades of polyethylene, even when rubber theory is employed in the simplest Gaussian form. Strain hardening is then represented by the single strain hardening coefficient Gp. Examples are given of this equation, which can be modified to give the true engineering or nominal stress σ_n and then be differentiated to give dσ_n/dλ = Gp ? Y₀ / λ² + 2Gp / λ³, where Y₀ is the yield stress and λ the extension ratio. Negative values of this differential then predict the onset of necking in tension and positive values stabilization of the neck. The relation of Gp to molecular weight is then discussed using literature measurements for polyethylenes of differing molecular weight and similar molecular weight distributions. When these results are then plotted, a strong dependency of Gp on molecular weight is observed. Some implications of these measurements are then considered. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 1090–1099, 2007     

Chemical algebra III: Thermochemical approach to completely G-invariant distances

The search for a definition of distances over sets of skeletal analogs (identified to G-Hilbert spaces of vector ligand parameters) is initiated from the algebraic formulation of the constant of stereogenic pairing equilibria (pairing product). A basic definition equation is devised from thermodynamical speculations. The equation is proved to have always a single potential distance solution D_p as soon as the pairing product is discriminating. The equation of D_p is constructed in order to satisfy three consistency requirements: completeG-invariance (arbitrary orientations selected to describe skeletal analogs do not affect the value of D_p); extension properties (D_p coincides with two standard completelyG-invariant distances or with the Euclidean distance in borderline cases); all the distance properties except, perhaps, the triangular inequality. The latter point remains challenging in general, and is computationally verified in some examples.     

Tensile stress and recoverable strain measurements on polystyrene melts. Interpretation of results in terms of rubberlike elasticity

Extensional tests at constant strain rate \documentclass{article}\pagestyle{empty}\begin{document}$ \dot \varepsilon $\end{document} have been carried out on polystyrene melts with different molecular weight distributions at various temperatures and strain rates. The true tensile stress is found to be well approximated by the sum of two contributions: (1) a neo-Hookean expression involving the recoverable strain and (2) a contribution rapidly reaching a steady-state value. Two experimental parameters can be defined: an elasticity modulus \documentclass{article}\pagestyle{empty}\begin{document}$ G(\dot \varepsilon ) $\end{document} from (1) and a viscosity \documentclass{article}\pagestyle{empty}\begin{document}$ \eta _{\rm v} (\dot \varepsilon ) $\end{document} from (2). It is further shown that time-temperature equivalence applies not only to the stress but also to the recoverable strain and to G and η_v. The dependence of G and η_v on strain rate is then discussed. For high strain rates, G is close to the linear viscoelastic plateau modulus of PS melts and decreases with decreasing strain rate. The value of η_v is found to a good approximation to be equal to three times the shear viscosity taken at a shear rate equivalent to the elongational strain rate.     

Relationship between fracture toughness and intrinsic deformation parameters in isotropic and flow‐oriented linear low‐density polyethylene

The fracture toughness of isotropic and flow‐oriented linear low‐density polyethylene (LLDPE) is evaluated by the Essential Work of Fracture (EWF) concept, with a special setup of CCD camera to monitor the process of deformation. Allowing for the molecular orientation, flow‐oriented sample, prepared via melt extrusion drawing, is stretched parallel (oriented‐0°) and perpendicular (oriented‐90°) to its original melt extrusion drawing direction, respectively. The obtained values of specific EFW w_e are 34.6, 10.2, and 4.2 N/mm for the oriented‐0°, isotropic and oriented‐90° sample, respectively. With knowledge of intrinsic deformation parameters deduced from uniaxial tensile tests, moreover, a relationship between specific EFW w_e the ratio of true yield stress to strain hardening modulus σ_ty/G is well established. It means that the fracture toughness of polyethylene is determined by both crystalline and amorphous parts, rather than by one of them. Moreover, the true yield stress seems to be nondecisive factors determining the fracture toughness of polyethylene. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2880–2887, 2006     

Crosslinking,filler, or transition constraint of polymer networks. II

The theory of Part I is developed by application to filler reinforcement of NR and SBR. For unswollen but prestretched networks it quantifies entire stress–strain curves and applies new concepts of extensibility and strain hardening. Constraint of swelling is expressed by a constant φ, termed linkage reinforcement, and by an effective hard fraction C_h per cubic centimeter of compound. For rubber–filler swelling v_c the modified Flory functions F(v_c) in part 1 need 3% correction. Then, relative to gum fix points 1/F₀(v_r): where C_h ≤ 1.15C for filler concentrations C per cubic centimeter of compound. The effective C_h comprises the volume fraction C* of bonded particles and 5–10 Å of surface–bound rubber that has been stretched hard by swelling. When needed, the actual crosslink density and intrinsic linkage reinforcement φ₀ can be obtained by dividing by (1 + 2.5C_φ) where C_φ = (C ln φ)/(1 + 2.5C). The case C_h ≤ 1.15C with Graphon or inert fillers is identified and assessed by equations: Results C_h > 1.15C are invalid, but then C_h ≈ 1.15C* ≈ 1.15C, e.g., for carbon blacks. Even Graphon is distinguished from inert fillers at low concentrations C by substantial constraint reinforcement F₀(v_r)/F(v_c) > 1. For prestressed dry rubber a modulus G, network extensibility α_b ? 1, and upturn coefficient μ_h express the whole curve; G and μ_h show identical constraint strength distributions. Network extensibility α_b ? 1 is the microbreaking strain (prestretch); for pure elastomer it is elongation at break. The relation of stress F to extension ratio α is: where C₂* = 0.7 and j = 0.4 from NR/MPC data. Strain-hardening coefficients h are obtained from μ_h by the theory given in Part I. Hard modulus components G_h = 0.7 ln (h/h₀) vanish as h → h₀ (gum) = 110. After high prestresses the residual ln-(h^*/h₀) due to strong carbon-rubber linkages implies G_h* = 0.42 kg/cm², i.e., ca. 10% of the normal cure crosslinks.     

A theory of finite tensile deformation of double-network hydrogels

A continuum damage model was developed to describe the finite tensile deformation of tough double-network (DN) hydrogels synthesized by polymerization of a water-soluble monomer inside a highly crosslinked rigid polyelectrolyte network. Damage evolution in DN hydrogels was characterized by performing loading-unloading tensile tests and oscillatory shear rheometry on DN hydrogels synthesized from 3-sulfopropyl acrylate potassium salt (SAPS) and acrylamide (AAm). The model can explain all the mechanical features of finite tensile deformation of DN hydrogels, including idealized Mullins effect and permanent set observed after unloading, qualitatively and quantitatively. The constitutive equation can describe the finite elasto-plastic tensile behavior of DN hydrogels without resorting to a yield function. It was showed that tensile mechanics of DN hydrogels in the model is controlled by two material parameters which are related to the elastic moduli of first and second networks. In effect, the ratio of these two parameters is a dimensionless number that controls the behavior of material. The model can capture the stable branch of material response during neck propagation where engineering stress becomes constant. Consistent with experimental data, by increasing the elastic modulus of the second network the finite tensile behavior of the DN hydrogel changes from necking to strain hardening.     

Regenerated protein fibers. II. Viscoelastic behavior of silk fibroin solutions

The dynamic viscoelastic behavior of a concentrated solution of silk fibroin dissolved in the “MU” solvent is measured. The dynamic viscosity η′ and dynamic elasticity G′ increase with increasing concentration of silk fibroin at constant frequency; however, the increasing frequency decreases η′ and G′ at a constant concentration of silk fibroin. When the mixing ratio of C₂H₅OH/H₂O in the “MU” solvent is increased at a constant concentration of LiBr·H₂O, η′ and G′ sharply increase at constant frequency. If the LiBr·H₂O concentration is varied in the “MU” solvents whose ratio of C₂H₅OH/H₂O is kept constant at 100 : 0, both η′ and G′ are greater for LiBr·H₂O concentrations of 50% by weight compared to concentrations of 40% by weight. The dependence of η′ on the temperature of the solution can be predicted by Andrade's viscosity equation. Spinnability improves when the SF concentration is increased. © John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 35 : 1955–1959, 1997     

Melting transitions in metallic elements correlated with shear modulus and atomic volume

We show that two quite recent treatments of dislocation-mediated melting transitions result in the thermal energy associated with the melting temperature, T _m, being expressed as a product of a volume factor and a combination of elastic constants times a lattice structure-dependent factor. We further show that the result for the latent heat of fusion L _m obtained in one of these studies leads to the ratio L _m/GΩ, where G is the shear modulus at melting and Ω the atomic volume, being a constant. Since the ratio of the vacancy formation energy to GΩ is also found to be roughly constant, we suggest that the factor GΩ at melting is crucial in determining the melting temperature, the latent heat of fusion and the vacancy formation energy and we comment on the reasons why this should be so.     

Crosslinking,filler, or transition constraint of polymer networks. I

The basic theory of modulus/swelling is developed to allow for limited extensibility, filler reinforcement or transition effects, and steric hindrance of aligned segments by extended chains or filler particles. Filler forms an effective hard fraction C_h per cubic centimeter of compound with v_c a new (compound) index of swelling. For 1/M_c + σ fix points having ratio φ to gum values 1/F₀(v_r) and with F(v_c) replacing the Flory function F(v_r): where σ denotes entanglement. Linkage reinforcement φ does not vary with sulfur crosslinking of SBR. Vacuoles invalidate φ from mass-increment F₀(v_r)/F(v_r) for inert fillers. Then, or for Graphon, with negligible φ ≈ 1: The effective C_h includes rubber stretched hard on Graphon by swelling or trapped inside hard aggregates. Only the right-hand equation fits normal blacks. In theory, C_h can always be obtained from swollen moduli G by linear slopes (1 + 1.4C_h) relating F(v_c) and (1 ? C)ρRT/G. For filler fractions C ≥ 0 cm^?3 and low strains α = 1.5?2.0 below prestretch the modulus G is given a new basic definition: Here C₂* ≈ 0.7 corresponds to Mooney-Rivlin C₂ and the effective crosslinking 1/[M_c] = (ρRT)^?1G is equal to (1 ? C)(1/M_c + σ) for unswollen prestretched rubber (v_r = 1). For higher strains a hypothesis of strain hardening is proposed. This is distinct and opposite in character to the initial prestretch softening (Mullins effect). Nonlinear effects of crosslinks are expressed by a fractional stress-upturn Ω (1/M_c + σ), effective mesh wieght (1/M_c + σ)^?1 ? Ω, and hard fraction Ω(1/M_c + σ). For μ_h characterizing strain hardening up to the prestretch (α_h ? 1) their contribution is: The sixth-power refinement has J = j(α_b ? 1)^1/2 with j ≈ 0.4. The hard phase is augmented by filler and grows with increasing strain up to the prestretch.     

Rigidity percolation model of polymer fracture

A theory of the fracture of polymers with network microstructure was developed that was based on the vector, or rigidity percolation (RP) model of Kantor and Webman, in which the modulus, E, is related to the lattice bond fraction p, via E ～ [p ? p_c]^τ. The Hamiltonian for the lattice was replaced by the strain energy density function of the bulk polymer, U = σ²/2E, where σ is the applied stress and p was expressed in terms of the lattice perfection via the bond density ν, with the entanglement molecular weight, ν = ρ/M_e and appropriate measures of crosslink density for rubber, thermosets, and carbon nanotubes. The stored mechanical energy, U, was released by the random fracture of νD_o[p ? p_c] over stressed hot bonds of energy D_o ≈ 330 kJ/mol. The polymer fractured critically when p approached the percolation threshold p_c, and the net solution was obtained as σ = (2EνD_o [p ? p_c])^1/2 with a fracture energy, G_1c ～ [p ? p_c]. The fracture strength of amorphous and semicrystalline polymers in the bulk was well described by, σ = [ED_oρ/16 M_e]^1/2, or σ ≈ 4.6 GPa/M_e^1/2. Fracture by disentanglement was found to occur in a finite molecular weight range, M_c < M < M*, where M*/M_c ≈ 8, such that the critical draw ratio, λ_c = (M/M_c)^1/2, gave the molecular weight dependence of the fracture as G_1c ～ [(M/M_c)^1/2 ? 1]². The critical entanglement molecular weight, M_c, is related to the percolation threshold, p_c, via M_c = M_e/(1 ? p_c). Fracture by bond rupture was in accord with Flory's suggestion, G/G* = [1 ? M_c/M], where G* is the maximum fracture energy. Fracture of an ideal rubber with p = 1 was determined not to occur without strain hardening at λ > 4, such that the maximum stress, σ = E (λ ? 1/λ) = 3.75E. The fracture properties of rubber were found to behave as σ ～ ν, σ ～ E, and G_1c ～ ν. For highly crosslinked thermosets, it was predicted that σ ～ (Eν)^1/2, σ ～ (X ? X_c)^1/2, and G_1c ～ ν^?1/2, where X is the degree of reaction of the crosslinking groups and X_c defines the gelation point. When applied to carbon nanotubes (SWNT and MWNT) of diameter d and hexagonal bond density ν = j/b², the nominal stress as a function of diameter is σ(d) = [16 ED_o(p ? p_c) j/b]^1/2/d ≈ 211/d (GPa.nm) and the critical force, F_c(d) ≈ 166 d (nN/nm), in which j = 1.15, b = 0.142 nm, E ≈ 1 Tpa, and D_o = 518 kJ/mol. For polymer interfaces with Σ chains per unit area of length L and width X ～ L^1/2, G_1c is then ～ [p ? p_c], where p ～ ΣL/X. The results predicted by the RP fracture model were in good agreement with a considerable body of fracture data for linear polymers, rubbers, thermosets, and carbon nanotubes. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 168–183, 2005     

Synthesis and Thermal Decomposition Kinetics of the Complex of Samarium p-Methylbenzoate with 1 ,10-Phenanthroline

The complex [Sm（p-MBA）3phen]2 （p-MBA, p-methylbenzoate; phen, 1,10-phenanthroline） was prepared and characterized by elemental analysis, IR and UV spectra. The thermal decomposition process of [Sm（pMBA）3phen]2 was studied under a static air atmosphere by TG-DTG and IR techniques. Thermal decomposition kinetics was investigated employing a newly proposed method, together with the integral isoconversional non-finear method. Meanwhile, the thermodynamic parameters （AH#, △G# and AS#） were also calculated. The lifetime equation at mass-loss of 10% was deduced as In r=-24.7825＋ 18070.43/T by isothermal thermogravimetric analysis.     

Effect of finite extensibility on the viscoelastic properties of a styrene–butadiene rubber vulcanizate in simple tensile deformations up to rupture

Stress–strain and rupture data were determined on an unfilled styrene–butadiene vulcanizate at temperatures from ?45 to 35°C and at extension rates from 0.0096 to 9.6 min^?1. The data were represented by four functions: (1) the well-known temperature function (shift factor) a_T; (2) the constant strain rate modulus, F(t,T), reduced to temperature T₀ and time t/a_T, i.e., T₀F(t/a_T)/T; (3) the time-dependent maximum extensibility, λ_m(t/a_T); and (4) a function Ω(χ) where χ = (λ ? 1)λ_m⁰/λ_m, in which λ is the extension ratio and λ_m⁰ is the maximum extensibility under equilibrium conditions. The constant strain rate modulus characterizes the stress–time response to a constant extension rate at small strains, within the range of linear response; λ_m is a material parameter needed to represent the response at large λ; and Ω(χ) represents the stress–strain curve of the material in a reference state of unit modulus and λ_m = λ_m. The shift factor a_T was found to be sensibly independent of extension. At all values of t/a_T for which the maximum extensibility is time-independent, the relaxation rate was also found to be independent of λ. These observations indicate that the monomeric friction coefficient is strain-independent over the ranges of T and λ covered in the present study. It was found that λ_m⁰ = 8.6 and that the largest extension ratio at break, (λ_b)_max, is 7.3. Thus, rupture always occurs before the network is fully extended.     

Estimation of the agglomeration structure for conductive particles and fiber‐filled high‐density polyethylene through dynamic rheological measurements

A study on the correlation between electrical percolation and viscoelastic percolation for carbon black (CB) and carbon fiber (CF) filled high‐density polyethylene (HDPE) conductive composites was carried out through an examination of the filler concentration (?) dependence of the volume resistivity (ρ) and dynamic viscoelastic functions. For CB/HDPE composites, when ? was higher than the modulus percolation threshold (?_G ～ 15 vol %), the dynamic storage modulus (G′) reached a plateau at low frequencies. The relationship between ρ and the normalized dynamic storage modulus (G_c^′/G_p^′, where G_c^′ and G_p^′ are the dynamic storage moduli of the composites and the polymer matrix, respectively) was studied. When ? approached a critical value (?_r), a characteristic change in G_c^′/G_p^′ appeared. The critical value (G_c^′/G^′_p)c was 9.80, and the corresponding ?_r value was 10 vol %. There also existed a ? dependence of the dynamic loss tangent (tan δ) and a peak in a plot of tan δ versus the frequency when ? approached a loss‐angle percolation (?_δ = 9 vol %). With parameter K substituted for A, a modified Kerner–Nielson equation was obtained and used to analyze the formation of the network structure. The viscoelastic percolation for CB/HDPE composites could be verified on the basis of the modified equation, whereas no similar percolation was found for CF/HDPE composites. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1199–1205, 2004     

Strain induced nanocavitation and crystallization in natural rubber probed by real time small and wide angle X‐ray scattering

The concomitant appearance of crystallites and nanocavities under uniaxial strain is investigated by X‐ray scattering in a model natural rubber system. The nanocavities appear after crystallization and only when the true stress is above a critical cavitation stress σ_Cav. The presence of crystallites alone does not influence the calculation of the void volume fraction ?_void. The nanocavities formed are 20–50 nm in size with a constant aspect ratio. The presence of filler shifts the critical crystallization extension ratio λ_Cry, λ_Cav, and σ_Cav to lower values. The clear correlation between σ_Cav and the crystallinity at the onset of cavitation χ_C(λ_Cav) implies that the crystallites take most of the mechanical loading thus delaying the cavitation in the amorphous phase. Under cyclic loading, nanocavitation is significant only in the first loading and in the successive loadings if the extension ratio is above its maximum historical value. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2013, 51, 1125–1138     

Changes in superstructure and mechanical properties of poly(ethylene‐2,6‐naphthalate) fibers with critical necking tension

Poly(ethylene‐2,6‐naphthalate) fibers were zone‐drawn under a critical necking tension (σ_c) defined as the minimum tension needed to generate a necking at a given drawing temperature (T_d). In the zone drawing under σ_c, the neck was observed from 110 to 160 °C. The superstructure in a neck zone induced at each T_d was studied. The σ_c value decreased exponentially with increasing T_d and dropped to a low level at a higher T_d. The draw ratio increased rapidly with T_d increasing above 90 °C, but the birefringence and degree of crystallinity decreased gradually. To study the molecular orientation in the neck zone, we measured a dichroic ratio (A_‖/A_?) of a C? O band at 1256 cm^?1 along a drawing direction in the neck zone with a Fourier transform infrared microscope. A_‖/A_? at T_d = 110 °C increased rapidly in the narrow neck zone, and that at T_d = 140 °C increased in the edge of the wide neck zone. Wide‐angle X‐ray diffraction patterns of the fibers obtained at T_d = 130 °C and lower showed three reflections due to an α form, but those at T_d = 140 and 150 °C had reflections due to the α form and a β form. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 1629–1637, 2001     

Equilibration and Deformation of Amorphous Polystyrene: Scale‐jumping Simulational Approach

A polymer sample‐preparation method (extended‐chain condensation, ECC) based solely on molecular‐dynamics simulations has been compared to a connectivity‐altering Monte Carlo method (coarse‐grained end‐bridging, CGEB). Since the characteristic ratio for the CGEB samples is closer to the experimental value, ECC results in polymer structures that are too compact. The stress–strain relations are different in the strain‐hardening regime. For CGEB samples, a stronger strain hardening is observed and the strain‐hardening modulus is more realistic; for the CGEB polystyrene (PS) sample G_R = 9 ± 1 MPa is found versus G_R = 4 ± 2 MPa for the ECC samples. These differences have to be attributed to a steeper increase in the contributions to the total stress from bond‐ and dihedral angles for CGEB than for ECC samples.

     

An experimental investigation of fracture by cavitation of model elastomeric networks

A new methodology to investigate the failure of elastomers in a confined geometry has been developed and applied to model end-linked polyurethane elastomers. The experimental in situ observations show that the elastomers fail by the growth of a single cavity nucleated in the region of maximum hydrostatic stress. Tests carried out at different temperatures for the same elastomer show that the critical stress at which this crack grows is not proportional to the Young's modulus E but depends mainly on the ratio between the mode I fracture energy G_IC and E. A reasonable fit of the data can be obtained with a model of cavity expansion by irreversible fracture calculating the energy release rate by finite elements with a strain hardening constitutive equation. Comparison between different elastomers shows that the material containing both entanglements and crosslinks is both tougher in mode I and more resistant to cavitation relative to its elastic modulus. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48:1409–1422, 2010     

Transient effects in the crystallization of polyethylene

A specimen of linear polyethylene was subjected to isothermal secondary crystallization at a series of temperatures below the primary isothermal crystallization temperature, the melting and primary crystallization stages being held constant throughout the investigation. Dilatometric measurements exhibit an S-character at low values of undercooling T_p – T_s, where T_p and T_s are, respectively, the primary and secondary crystallization temperatures, whereas at larger undercooling, an initial very rapid crystallization is followed by a very slow stage. When corrected for thermal contraction of the polymer, the net degree of secondary transformation is seen to peak at a temperature about 5°C below T_p. The S-character of the isotherms and the peaked temperature variation of degree of transformation lead to the conclusion that a large portion of the secondary crystallization consists of the nucleation and growth of the new crystallites. Johnson-Mehl-Avrami analysis leads to a model of heterogeneous nucleation within the remaining amorphous zones, followed by one-dimensional, diffusion-controlled growth.     

The study of alanine oscillating reaction catalyzed by tetraazamacrocyclic nickel(II) complex

A closed oscillation system comprised of alanine, KBrO₃, H₂SO₄ and acetone catalyzed by tetraazamacrocyclic nickel(II) complex is introduced, and quantitatively characterized with kinetic parameters, namely the rate constant (k _in, k _p), the apparent activation energy (E _in, E _p) and pre-exponential constant (A _in, A _p) and thermodynamic functions (ΔH _in, ΔG _in, ΔS _in and ΔH _p, ΔG _p, ΔS _p), where indexes “in” and “p” mean “induction period” and “oscillation period,” respectively. The results indicate that tetraazamacrocyclic nickel(II) complex can catalyze alanine oscillating reaction and the reaction corresponds exactly to the feature of irreversible thermodynamics as the entropy of system is negative.     

Thermo‐Viscoelastic Response of Polycarbonate Reinforced with Short Glass Fibers

Observations are reported for oscillatory torsion tests at several temperatures ranging from room temperature to 100 °C on a polymer composite consisting of a polycarbonate matrix reinforced with short glass fibers. Constitutive equations are derived for the linear viscoelastic behavior of the polymer composite, which is treated as an equivalent heterogeneous network of chains bridged by junctions (entanglements and glass fibers). The network is thought of as an ensemble of meso‐regions with arbitrary shapes and sizes. With reference to the concept of cooperative relaxation, the time‐dependent response of an ensemble is associated with the rearrangement of meso‐domains. The rearrangement events occur at random times as meso‐regions are agitated by thermal fluctuations. Stress–strain relations for isothermal deformation of an ensemble of meso‐domains are derived by using the laws of thermodynamics. The governing equations are determined by five adjustable parameters that are found by fitting the experimental data. The effects of temperature and filler content on the material parameters are studied in detail.

The shear modulus G GPa versus the content of short glass fibers ν wt.‐%. Symbols: treatment of observations in oscillatory torsion tests at T = 25 (unfilled circles) and T = 100 °C (filled circles). Solid lines: approximation of the experimental data by Equation (27). Curve 1: G₀ = 1.05, G₁ = 3.83 × 10⁻². Curve 2: G₀ = 0.91, G₁ = 3.65 × 10⁻².