Ultimate tensile properties of elastomers. VII. Effect of crosslink density on time–temperature dependence

Data on tensile strength and elongation at break for a series of Viton A-HV vulcanizates are discussed. The data were obtained at various extension rates at temperatures from ?5 to 230°C (25 ? T — T_g ? 260°C) on seven vulcanizates having crosslink densities v_e (estimated from C₁ in the Mooney-Rivlin equation) from 0.46 × 10^?5 to 24.4 × 10^?5 mole/cm³. At an extension rate of 1 min^?1, an increase in v_e affects the tensile strength σ_b (based on the undeformed cross-sectional area) and the true tensile strength σ_bσ_b (based on the cross-sectional area of a deformed specimen) as follows: σ_b is essentially constant at a low temperature; it passes through a decided maximum at intermediate temperatures; and it increases to a plateau at elevated temperatures. In contrast, λ_bσ_b decreases markedly at all temperatures, an exception being the most lightly crosslinked vulcanizate(s). Application of time—temperature superposition to the ultimate-property data gave log a_T; its temperature dependence is that typical of nonpolar rubbery polymers. Data on the vulcanizates were compared in corresponding temperature states by plotting log 273σ_b/T, log 273λ_bσ_b/T, and (λ_b — 1)/(λ_b — 1)_max against logt_b/(t_b)_max, where t_b is the temperature-reduced time to break and (t_b)_max is the value at which the ultimate extension ratio λ_b attains its maximum, (λ_b)_max. Except for the most lightly crosslink vulcanizate, the comparison shows that 273λ_bσ_b/T and (λ_b — 1)/(λ_b — 1)_max are substantially independent of (or only weakly dependent on) crosslink density, that 273λ_b/T increases with v_e, and that 273λ_b/T ∝? v_e^0.6 and λ_b ∝? v_e^?0.4 at a large value of t_b/(t_b)_max.     

A query on crystallization temperature‐dependent cooling function under nonisothermal condition

The cooling function (κ) in Ozawa model was investigated through theoretic analysis and experimental method. Different from the fact accepted by researchers over past decades that κ(T) depends only on the crystallization temperature (T) and consequently the parameters for nonisothermal crystallization kinetics could be obtained by plotting ln[? ln(1 ? X(T))] versus ln λ at a given T, we found that κ at a given T was also dependent on onset temperature (T₀) of crystallization process. Because T₀ varies with cooling rate (λ) in nonisothermal crystallization, we conclude that κ is a binary function of T and λ, which was validated by our kinetic data from differential scanning calorimetry measurement in a wide λ range from 1 to 80 °C/min. It is suggested that the conventional method for calculating kinetic parameters based on Ozawa model, by plotting ln[? ln(1 ? X(T))] versus ln λ, might not be exact for nonisothermal crystallization. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44:795–800, 2006     

Rayleigh–Schrödinger perturbation expansions for a hydrogen atom in a polynomial perturbation

Rayleigh–Schrouml;dinger (RS ) perturbation expansions for the eigenvalues E(λ) of a hydrogen atom in the general polynomial perturbation V(r) = aλr + b>²r², a, b > 0, are studied. When a² = 2b, the ground state energy is exactly E(λ) = -(1/2) + (3/2)a>, i.e., the RS series is truncated. In the case a² > 2b, the RS series is negative Stieltjes. In general, when λ < 0, a well of depth ω ≈ -a²/(4b²) (note the λ independence) is situated at r_ω = a/(2b|λ|). When a² > 2b/N², and interaction between this well and the hydrogenic state ψ_NLM(λ) is possible, thus creating a pair of asymptotically degenerate eigenstates separated by a “gap” δE(λ). The large order behavior of the RS coefficients E may be computed from the asymptotics of δE(λ), which is, in turn, related to the tunnelling integral. For excited states, stricter inequalities must be obeyed for Stieltjes behavior. The E⁽ⁿ⁾_NLM may be calculated either numerically or in closed form via the “so(4, 2) Lie algebra technology” for such hydrogenic problems.     

Thermomechanical behavior of rubberlike materials

A simple form of nonisothermal free energy function A(λ₁, λ₂, λ₃, T) for rubberlike materials results from postulating that the entropy is a separable symmetric function of the extension ratios λ_i along the principal strain directions and considering the fundamental properties of rubberlike materials, i.e., that rubber elasticity is associated primarily with changes in entropy and the variation of elastic tension with changes in temperature is linear. The explicit representation of A is reduced to the Valanis-Landel strain energy function for isothermal cases.     

Lamellar thickening in nascent poly(acrylonitrile) upon annealing

No systematic study has been reported on the lamellar thickening in atactic poly(acrylonitrile) (PAN) upon annealing because PAN, in the form of solution‐cast films or their drawn products, generally shows no small‐angle X‐ray scattering (SAXS) maximum corresponding to the lamellar thickness. In this work, PAN crystals were precipitated during the thermal polymerization of acrylonitrile in solution. The nascent PAN film, obtained by the filtration of the crystal suspension, exhibited a clear SAXS maximum revealing the lamellar structure. The lamellar thickening upon annealing of the nascent PAN films was studied in the temperature range 100–180 °C, where the degradation was minimal, as confirmed by the absence of an IR absorption band at 1605 cm⁻¹ ascribed to the cyclized nitrile groups. Above 190 °C, the degradation of the samples was significant, and the SAXS became too broad to determine the scattering maximum. The long period was significantly affected by the annealing time (t_a) and the temperature (T_a). Depending on t_a, three stages were observed for the lamellar thickening behavior. The lamellar thickness stayed constant in stage I (t_a = 0.5–3 min, depending on T_a), rapidly increased in stage II (t_a = 0.5–8 min), and stayed at a constant value characteristic for each T_a at yet longer t_a's in stage III. The lamellar thickness characteristic for T_a increased rapidly with increasing T_a at 165 °C (or higher), which was 152 °C lower than the estimated melting temperature of PAN (T_m = 317 °C). A possible mechanism for such lamellar thickening in PAN far below the T_m is discussed on the basis of the enhanced chain mobility in the crystalline phase above the crystal/crystal reversible transition at 165–170 °C detected by differential scanning calorimetry and wide‐angle X‐ray diffraction. The structural changes associated with annealing are also discussed. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2571–2579, 2000     

Pulsed NMR observation of ultradrawn polypropylene

Proton spin-spin relaxation times have been measured as a function of temperature for ultradrawn polypropylene with draw ratios λ up to 24. The three relaxation times T_2a (the longest), T_2i (intermediate), and T_2c (the shortest), observed for all the samples, have been ascribed to the relaxations of the amorphous, constrained amorphous, and crystalline components, respectively. T_2i and T_2a, which reflect the changes in structure and mobility in the noncrystalline regions, decrease with increasing λ; T_2i becomes saturated at λ > 9, whereas T_2a shows a substantial decrease up to λ = 24. The continued decrease in T_2a indicates that the constraint on the amorphous segments keeps increasing up to the highest λ. The associated mass fractions F_a, F_i, and F_c also change with λ. At λ < 9, the increasc in F_i with increasing λ is accompanied by a decrease in F_a, with F_c remaining unchanged. At higher λ, however, F_a is almost constant, and stepwise rises in F_c at about λ = 12 and 24 are accompanied by corresponding drops in F_i. It seems that, in this high draw ratio range, some of the taut molecules are fully extended and are in sufficiently good lateral register to transform into crystalline bridges. This conjecture is supported by the similarity in the λ dependence of F_c and the mass-fraction crystallinity obtained from the heat of fusion.     

Strain energy function of styrene butadiene rubber and the effect of temperature

Biaxial and uniaxial tensile stress relaxation tests were made on square sheet specimens of styrene butadiene rubber (SBR), mounted in a universal biaxial tester within a temperature-controlled box, with the object of studying the effect of temperature on the strain energy function. The stress relaxation responses, usually for times up to 10 min, were obtained for various degrees of biaxiality, various extension ratios, and various temperatures within the limits of +25 to ?45°C. The results indicated that if the Valanis–Landel representation of the strain energy function \documentclass{article}\pagestyle{empty}\begin{document}$ W = \sum\nolimits_{i = 1}^3 {w\left( {\lambda _i } \right)} $\end{document} is adopted, then time and strain are factorizable over the indicated temperature range, with time and temperature being related in the usual fashion. That is, changing the temperature does not affect the form of w(λ_i) but only that of G(t/α_T), the temperature-dependent relaxation modulus, a_T being the regular Williams–Landel–Ferry (WLF) shift factor. The results verify the Valanis–Landel theory for various combinations of biaxiality     

Volume dependence of thermal conductivity and isothermal bulk modulus up to 1 GPa for poly(vinyl acetate)

The thermal conductivity λ and heat capacity per unit volume of poly(vinyl acetate) (260 kg mol⁻¹ in weight average molecular weight) have been measured in the temperature range 150–450 K at pressures up to 1 GPa using the transient hot-wire method, which yielded λ = 0.19 W m⁻¹ K⁻¹ at atmospheric pressure and room temperature. The bulk modulus K has been measured in the temperature range 150–353 K up to 1 GPa. At atmospheric pressure and room temperature, K = 4.0 GPa and (∂K/∂p)_T = 8.3. The volume data were used to calculate the volume dependence of λ, $g = - \left( {\frac{{\partial \lambda /\lambda }}{{\partial V/V}}} \right)_T .$ The values for g of the liquid and glassy states were 3.0 and 2.7, respectively, and g of the latter was almost independent of volume and temperature. Theoretical models can predict the value for g of the glassy state to within 25%. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 1451–1463, 1998     

Viscoelasticity of textile fibers at finite strains

Experimental data obtained from stress–strain curves of five different textile fibers, at a series of different, constant strain rates covering a range of 6^1/2 decades have been used to study two methods of nonlinear viscoelastic analysis proposed elsewhere. According to the first of these, time and strain effects are factorizable so that stress σ, strain ε and time t are related by the equation σf₁(ε)/ε = f(t),. This is shown to be unsatisfactory with the present materials, but an empirical modification to σf₁(ε)/ε = f₂(ε) + f(t) is satisfactory. According to the second, general nonlinear viscoelastic behavior can be described by an equation which reduces to the form σ/ε = F₁(t) + εF₂(t) + ε²F₃(t) + when applied to extension at a constant strain rate. This series is shown to be strongly divergent except at fairly small stains. In fact, if it is truncated after about three terms, which are as many as can be estimated with any significance in the present experiments, it is applicable only to strains of about 3–4% and less. Numerical techniques which enable standard statistical procedures to be used have been devised to perform the above analyses and are described in detail.     

A structural model for high-modulus polyethylene derived from entanglement concepts

Starting from the concept that the entanglement network is a controlling factor in polymer deformation, a molecularly based model has been constructed for polyethylene, drawn or extruded to high extension ratios λ. It predicts the experimentally observed form of the increase of Young's modulus E with λ: E^?1 = B + Cλ^?2. The model structure consists of imperfect crystalline microfibrils 10-30 nm in diameter and length αλ², about 1 μm at λ = 30. The microfibrils terminate at clusters of entanglements, and are embedded in a matrix of low modulus. This structure is very similar to that derived from solution-grown shish-kebab material. Available melting-point data for highly extended material fit the structural model well.     

Notes on asymptotic series by H0-strictly singular perturbations

Let two strongly continuous one-parameter semigroups of contraction operators, {Z(t); t≧ 0} and {Z₀(t); t ≧ 0}, determine physical evolutions realized in the Hilbert spaces ? and ?₀, respectively. We consider the mappings under Z(t) and Z₀(t), when the infinitesimal generators G and G₀ belong to a product class, properly defined with respect to an H⁰-norm. The inversed transform of each side in the identity R(λ, G) = R(λ, –iH⁰)–iR(λ, –iH⁰)VR(λ, G), which represents a simple algebraic decomposition of the resolvent of G, converges on Ψ if and only if Ψ ∈ ??(G). By iteration an asymptotic series emerges, when t → 0. Numerical considerations of this approximation in its first order may support the form of a deviation from the pure exponential decay when the semigroup is compared with a integral transform corresponding to a certain self-adjoint H₀ in ?₀. The deviation may hardly ever be observed and it is therefore most fruitful to discuss the results inside the framework of the enveloping algebra of the semigroups.     

Entanglement networks of 1,2-polybutadiene crosslinked in states of strain. VII. Stress-birefringence relations

Crosslinks are introduced by γ irradiation into 1,2-polybutadiene while strained in uniaxial extension near T_g with stretch ratio λ₀, thereby trapping a proportion of the entanglements originally present. The stress at any subsequent strain λ is accurately given by the sum σ_N + σ_x, where σ_N is the stress contributed by a trapped entanglement network with λ = 1 as reference and a Mooney–Rivlin stress-strain relation, and σ_x is that contributed by a crosslink network with λ = λ₀ as reference and neo-Hookean stress-strain relation. The birefringence is accurately given as δn = ?_Nσ_N + ?_xσ_x, where the ?'s are the respective stress-optical coefficients. From measurements at λ = λ₀ where σ_x = 0, ?_N can be determined separately. For polymer with 88% 1,2 microstructure, ?_N and ?_x are nearly equal and independent of irradiation dose, though strongly dependent on temperature. For polymer with (95–96)% 1,2, ?_N and ?_x are different (even opposite in sign) and dependent on dose. This behavior is associated with a side reaction of cyclization by the γ irradiation, which is inhibited by the 1,4 moiety in the polymer with lesser 1,2 content. It is responsible for residual birefringence in the state of ease (λ = λ_s) where σ_N = –σ_x and the stress is zero.     

Conformational analysis of asymmetric ethanes : Extraction of acyclic conformational ligand constants from time-averaged geminal fluorine anisochronism

The syntheses and ambient-temperature ¹⁹F NMR data are reported for 27 asymmetric ethanes of the general formula RCF₂CXYZ, including a complete series of 10 compounds with R = Cl and all combinations of the 5 ligands H, F, Cl, Br and Ph. Within the theoretical framework of a previously proposed heuristic mathematical model the geminal ¹⁹F chemical shift differences are fitted to chirality functions X = ?R (λx - λy) (λy - λz) (λz - λx) to yield acyclic conformational ligand constants λ and substituent parameters ?. It is demonstrated that the λ constants already reported for an analogous series of 10 compounds with R = Br are transferable to the Cl series with ?_Cl = 0.63 ± 0.07. Crude first approximations are also reported for the normalised (according to ?_Br = 1) ligand constants λ_Ch3, λ_OH, λ_OCH3 and λ₁ and for the substituent parameter λ_H. It is argued that the λ values thus ext play a role in the conformational analysis of asymmetric ethanes that is conceptually analogous to the conformational free energies in monosubstituted cyclohexanes.     

Heat capacities and thermodynamic functions of the tellurates(IV) of zinc and cadmium

The temperature dependencies of the molar heat capacities of ZnTeO₃, Zn₂Te₃O₈, CdTeO₃ and CdTe₂O₅ are determined. The experimental data are statistically processed using the least squares method to determine the parameters in the equations for the corresponding compounds: C_p,m=a+b(T/K)-c(T/K)^-2. These equations and the standard molar entropies are used to determine Δ^T₀S⁰_m, Δ^T_TH⁰_m and (Φ⁰_m+Δ^T,₀H⁰_m/T) for T'=298.15 K.     

Pressure and volume dependence of thermal conductivity and isothermal bulk modulus up to 1 GPa for poly(isobutylene)

The thermal conductivity λ and heat capacity per unit volume ρc_p of poly(isobutylene)s, one 2.8 in weight average molecular weight and one 85 kg mol⁻¹ in viscosity average molecular weight (PIB-2800 and PIB-85000), have been measured in the temperature range 170–450 K at pressures up to 2 GPa using the transient hot-wire method. At 297 K and atmospheric pressure, λ = 0.115 W m⁻¹ K⁻¹ for PIB-2800 and λ = 0.120 W m⁻¹ K⁻¹ for PIB-85000. The bulk modulus B_T has been measured in the temperature range 170–297 K up to 1 GPa. At atmospheric pressure, the room temperature bulk moduli B_T are 2.0 GPa for PIB-2800 and 2.5 GPa for PIB-85000 with dB_T/dp = 10 for both. These data were used to calculate the volume dependence of λ, At room temperature and atmospheric pressure (liquid phase) we find g = 3.4 for PIB-2800 and g = 3.9 for PIB-85000, but g depends strongly on temperature for both molecular weights. The difference in g between the glassy state and liquid phase is small and just outside the inaccuracy of g of about 8%. The best predictions for g are given by the theoretical model of Horrocks and McLaughlin. We have found that PIB exhibits two relaxations, where one is associated with the glass transition. The value for dT_g/dp at atmospheric pressure (for the main glass transition) is about 0.21 K MPa⁻¹ for both molecular weights. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 1781–1792, 1998     

Low-angle electron diffraction from high temperature polystyrene crazes

Low-angle electron diffraction (LAED) was used to study the microstructure of crazes produced at different temperatures T and strain rates in thin films of monodisperse polystyrene (PS). At a slow strain rate of 4.1 × 10^?6 s^?1 both the fibril diameter D and the fibril spacing D₀ of crazes in 1800k molecular weight PS remained constant with temperature up to T ≈ 70°C and then sharply increased as T approaches T_g. At a higher strain rate of ～ 10^?2 s^?1, both D and D₀ increase only slightly with T. The values of D and D₀ over a range of temperature are in very good agreement with those values obtained in bulk samples using small-angle x-ray scattering. The crazing stress was measured as a function of temperature in the thin films of the 1800k molecular weight PS strained at the same slow strain rate used for the LAED measurements. These measurements were analyzed using a simple model of craze growth to reveal the temperature and strain rate dependence of the craze surface energy Γ. At room temperature Γ ≈ 0.076 J/m² (versus Γ ≈ 0.087 J/m² predicted) and was observed to remain constant up to T ≈ 70°C and then decrease by approximately a factor of two at T = 90°C. This decrease in Γ is believed to result from chain disentanglement to form fibril surfaces at sufficiently high temperatures and occurs in the same temperature range in which the craze fibril extension ratio λ was observed to increase.     

Verteilung und Valenz der Kationen in Spinellsystemen mit Eisen und Chrom. III. Gitterkonstanten,Mößbauer-Spektren und Thermokraft der Mischkristalle ZnFeCrO4?Fe2CrO4

Distribution and Valence of the Cations in Spinel Systems with Iron and Chromium. III. Lattice Constants, Mössbauer Spectra, and Seebeck Coefficients of the Solid Solution ZnFeCrO₄? Fe₂CrO₄ For the spinel system Zn_1–x²⁺Fe_x–λ²⁺Fe_λ³⁺(Fe_λ²⁺ · Fe_1?λ³⁺ Cr³⁺)O₄ λ has been determined by lattice constants and ionic distances: λ = 0 in the region 0 ? x ? 0.3; in the region 0.3 < x ? 1 λ increases linearly to 0.44. Mössbauer spectra between x = 0 and x = 0.6 confirm this distribution. All spinels are n-type hopping conductors mainly conducting on the octahedral sites.     

Crazing and shear deformation in crosslinked polystyrene

Thin films of polystyrene (PS) are bonded to copper grids and crosslinked with electron irradiation. When the films are strained in tension regions of local plastic deformation, either crazed or plane stress deformation zones (DZs), nucleate and grow from dust particles. the nature of the local deformation, as well as the local extension ration λ, is determined by transmission electron microscopy. The behavior of the PS glass is consistent with its being a network of molecular strands of total density v = v_E + v_X, where v_E is the entangled strand density inferred from melt elasticity measurements of uncrosslinked PS and v_X is the density of crosslinked strands determined from the ratio of the applied electron dose to the electron dose for gelation. when v is less than 4 × 10²⁵ m^?3 (<1.3v_E), only crazes are observed whose microstructure is similar to those in uncrosslinked PS. As v increases from 4 × 10²⁵ to 8 × 10²⁵ m^?3 (from 1.3v_E to 2.5v_E) shear deformation begins to compete with crazing. As v increases above 8 × 10²⁵ m^?3, only shear DZs are observed, the strain in which becomes progressively more diffuse as v increases. The λ in the crazes and DZs correlate well with λ_max, the maximum extension ratio of a strand in a network of density v computed using the Porod×Kratky model. For crazes ln(λ) ? 0.9 ln(λ_max) and for DZs ln(λ) ? 0.55 ln(λ_max). The strain at which crack nucleation is first observed increases as v increases from <5% in uncrosslinked PS with v = 3.3 × 10²⁵ m^?3 to >20% in PS with v = 33 × 10²⁵ m^?3 (v = 10v_E); crosslinking to still higher crosslink densities, e.g., v = 14v_E, results in cracks which propagate in a catastrophic manner at low applied strains. An optimum v thus exists, one not too high to suppress local shear ductility but high enough to suppress crazes which can act as crack nucleation sites. these results are compared with previous results on a variety of linear homopolymers, copolymers, and polymer blends that are characterized by a wide range of v (v = v_E). The transitions from crazing to crazing plus shear and from crazing plus shear to shear only take place at almost identical values of v. In addition the correlation between λ in the crazes and DZs and λ_max for a single network strand is the same for both classes of polymers. This agreement implies that chain scission is the major mechanism by which strands in the entanglement network are removed in forming fibril surfaces. Craze suppression, by either increasing v in the crosslinked polymer or v_E in the uncrosslinked ones, is due to the extra energy required to break more main-chain bonds to form these surfaces.     

Effect of molecular entanglements on craze microstructure in glassy polymers

Thin films of ten glassy polymers are bonded to copper grids and strained in tension to produce crazes, which are then examined in the transmission electron microscope. The average craze fibril extension ratio λ for each polymer is determined from microdensitometer measurements of the mass thickness contrast of the crazes. The extension ratio λ is found to increase approximately linearly with the chain contour length l_e between entanglements, as determined from melt elasticity measurements of the entanglement molecular weight of these polymers. These results are analyzed by comparing them with λ_max, the maximum extension ratio of an entanglement network in which polymer chains neither break nor reptate (i.e., permanent entanglement crosslinks are assumed). The values of λ_max are given by l_e/d where d, the entanglement mesh spacing in the unoriented glass, is computed from d = k(M_e)^1/2 with k determined either from small-angle neutron scattering results on isolated chains in the glass or from coil size measurements in dilute solutions of a θ solvent. The craze extension ratios fall somewhat below λ_max at low λ but increase to well above λ_max for polymers with high l_e. This comparison suggests a significant contribution due to chain breakage (or reptation) in the higher-λ crazes of large-l_e polymers, which may arise from the higher true stresses in the craze fibrils (which for a given applied stress increase proportionally to λ). The results also imply that a useful way to increase the “brittle” fracture stress and decrease the ductile-to-brittle transition temperature of a glassy polymer is to decrease its entanglement contour length l_e.