Stress relaxation of polybutadiene at large deformation. Measurements of stress and birefringence in shear and elongation

The rheological behavior of an uncrosslinked polybutadiene on sudden application of finite strain was examined. The shear stress σ, two components of birefringence, and the extinction angle were measured in shear (magnitude of shear γ ≤ 3.5) and tensile stress and the birefringence were measured in uniaxial elongation (elongation ratio λ ≤ 3.8). Measurements were performed at 30°C with a tensile tester equipped with appropriate sample holders. The stress-optical coefficient was 3.01 × 10^?9Pa^?1. The first and second normal-stress differences v₁ and v₂ were separately evaluated with the use of stress-optical law. The Lodge—Meissner relation v₁ = γσ held good. The ratio v₂/v₁ was independent of time and varied from about ?0.3 to ?0.2 with increasing γ in the range of measurements. Each of the stress components was factored into a function of strain and one of time, and the latter was common to all the stress components. Simple formulas were proposed to represent stress components in step deformations.     

Nonlinear relaxation of stress and birefringence in simple extension of 1,2-polybutadiene

Relaxation of stress and birefringence in simple extension has been studied for two samples of 1,2-polybutadiene with 95% and 88% vinyl content and weight-average molecular weight 1.9 and 2.9 × 10⁵, respectively. The extension ratio, λ, ranged from 1.14 to 2.08, temperatures from 0 to 15°C, and times, reduced to 0°C, up to 3 × 10⁵ sec. The stress-optical coefficient C was negative and positive, respectively, for the two samples, the difference being attributable to opposite signs and very different magnitudes of the contributions of the 1,2 and 1,4 moieties to the birefringence. For each polymer, C was independent of time but increased (algebraically) with temperature. For one polymer a very minor dependence of C on λ was observed. At any instant of time, the dependence of both stress and birefringence on λ could be described by equations of the Mooney–Rivlin form with coefficients C₁,C₂ and B₁,B₂, respectively. At short times the contributions of the C₁ and C₂ terms to the stress and of the B₁ and B₂ terms to the birefringence are roughly equal. With increasing time, C₁ and B₁ decrease gradually while C₂ and B₂ remain constant over several decades in time. Finally, C₂ and B₂ decrease rather rapidly. A tentative interpretation of these phenomena in terms of motions of entanglements is given.     

Fourier-transform infrared studies of polypropylene during mechanical deformation

Quantitative Fourier-transform infrared (FTIR) measurements of frequency shifts Δν and absorbance profile asymmetry are reported for various polypropylene samples as a function of uniaxial stress σ. Generally, it was found that the frequency shift coefficient α_χ, defined by Δν = α_χσ, depended on stress rate \documentclass{article}\pagestyle{empty}\begin{document}$\dot \sigma$\end{document}, draw ratio, λ, molecular orientation f, tensile modulus E, and annealing conditions. With annealing, α_χ decreased with increasing shrinkage in the case of highly oriented isotactic PP. The α_χ values for the “helix bands” were less affected than those for the “liquid bands.” With increasing \documentclass{article}\pagestyle{empty}\begin{document}$\dot \sigma$\end{document}, generally α_χ increased to an apparent asymptotic limit. With increasing λ, f, or E, α_χ also increased from α_χ ? 0 for λ = 1 (spherulitic) to maximum values for highly oriented isotactic PP. The observed variations in α_χ can be interpreted in terms of the changes in the peak position and shape of the nonuniform molecular stress distribution. Analogous behavior with x-ray diffraction peaks obtained for polymers under stress is discussed.     

Empirical relations between σI, σoR and taft σ* values of alkyl,acyl, benzoyl and substituted phenyl groups

The X(X) values¹ of the halogens (which resemble the Pauling electronegativities) and of some oxa substituents can be interpreted in terms of the inductive and resonance parameters σ_I and σ^o_R according to the regression equation

and η*_R=η(X)?η(R) it is found that for some substituted methyl, phenyl and benzoyl groups [σ*]^X_R=αη^X_R where α equals ?10.6 and ?10.9 for R = Me and R = Ph, CHO and PhCO respectively. Thus [σ*]^X_Rand η^x_r represent Taft σ* and [σ_I(X)?σ_oR(X)] values relative to that of the parent R group. The hydroxyl frequencies of phenol, and benzoic, acrylic, acetic and formic acids measured in dilute carbon tetrachloride solutions correlate with σ_I(X) and σ^o_R(X) according to the equations v(OH) = ?423.0 σ_I(X) + 3654.7 v(OH) = ?270.0 σ⁰_R(X) + 3586.7 where X = Ph, PhCO, CH₂=CHCO, MeCO and HCO. From these results, it is inferred that the σ* values of substituents having an α sp² hybridized carbon atom are proportional to σ⁰_R according to the equation σ*(X) = 3.97 σ⁰_R(X) + 1 New σ_I σ^o_R and σ* values of some acyi, benzoyl and substituted phenyl groups are presented.     

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.     

Conformation of nonideal hyperbranched polymer in ABn (n = 2, 4) type polymerization

The conformation of hyperbranched polymers from one pot polymerization with AB_n (n = 2, 4) type monomers, applying the reactive 3D bond fluctuation lattice model, are systematically studied using scaling relation R ～ N^λ, where R is the radius of gyration or the hydrodynamic radius of a hyperbranched polymer with the degree of polymerization N. The exponent λ was calculated at various monomer concentrations and group conversions. When the concentration of monomers with the equal reactivity of B groups increases from 0.1 to 0.9, the exponents λ_g and λ_h (corresponding to the radius of gyration and hydrodynamic radius, respectively) are in the ranges of 0.51–0.37 and 0.41–0.34 at the full conversion of A groups. Especially, we find that λ_g decreases linearly with the reaction conversion increasing. The ratio of z‐average radius, R_gz/R_hz, ranges from 1.08 to 1.32 and indicates that hyperbranched polymer is soft macromolecule with penetrable structure. In the case of AB₂ type monomer with unequal reactivities, λ displays complicated dependence on the reaction conversion and the reactivity ratio. The results of our simulation are consistent with those of experiments and theories, and valuable in better understanding the fundamental properties of hyperbranched polymers. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 610–616, 2010     

Relaxation of shear and normal stresses in step-shear deformation of a polystyrene solution. Comparison with the predictions of the Doi–Edwards theory

The shear stress σ, two components of birefringence, and extinction angle were measured for a concentrated polystyrene solution in step-shear deformation of magnitude of shear 0.3 ≤ γ ≤ 4.0. The stress-optical coefficient did not depend on either γ or time. The first and the second normal-stress differences v₁ and v₂ were evaluated with the use of the stress-optical law. Over a certain range of long times, σ could be factored as σ = γh(γ)G(t) and the quantity h(γ) agreed with the prediction of the Doi–Edwards theory based on the de Gennes tube model of entangled polymer chains. At short times the effect of γ on σ/γ was smaller than at long times. The relaxation spectrum became approximately independent of γ at the short-time end of the rubbery plateau region. The ratios v₁/σ and v₂/v₁ were independent of time and were in quantitative agreement with those predicted by the Doi–Edwards theory: v₁/σ was equal to γ, v₂/v₁ was negative, and |v₂/v₁| decreased with increasing γ.     

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.     

Intrinsic Surface Reaction Constant in 1-pK Model of Mg-Fe Hydrotalcite-like Compounds

The relationship among intrinsic surface reaction constant (K) in 1-pK model, point of zero net charge (PZNC) and structural charge density (σst) for amphoteric solid with structural charges was established in order to investigate the effect of σst on pK. The theoretical analysis based on 1-pK model indicates that the independent PZNC of electrolyte concentration (c) exists for amphoteric solid with structural charges. A common intersection point (CIP) should appear on the acid-base titration curves at different c, and the pH at the CIP is pHPZNC. The pK can be expressed as pK=-pHPZNC log[(1 2αPZNC)/(1-2αPZNC)], where αPZNC≡σst/eNANs, in which e is the elementary charge, NA the Avogadro‘s constant and Ns the total density of surface sites. For solids without structural charges, pK=-pHPZNC. The pK values of hydrotalcite-like compounds (HTlc) with general formula of [Mg1-xFex(OH)2](Cl,OH)x were evaluated. With increasing x, the pK increases, which can be explained based on the affinity of metal cations for H^- or OH^- and the electrostatic interaction between charging surface and H^- or OH^-.     

Fluorescence quenching of para-substitutedstyrenes by tetramethylethylene. Correlationanalysis of the quenching coefficient KSV and thequenching rate constant k_q by the dual-parameter equation with polar and spin-delocalization substituent constants

The fluorescence quenching coefficient (Ksv) and the quenching rate constant kq of ten para-substituted styrenes (1-Ys) have been measured and correlation-analyzed by both the dual-parameter equation (Eq. 1) with (ρxσx+ρ'σ') and the single-parameter equation (Eq. 2) with ρxσx. Ex-cellent results have been obtained for the correlation of KSV against (ρxσmb+ρ'σ'JJ) or (ρxσ+ +ρ'σ'JJ). Our results suggest that, possibly, there might be no need to use excited-state substituent constant for the fluorecence quenching process of excited states of styrenes.     

Atomic level picture of stress relaxation in polymer melts

We have been developing a physical picture on the atomic level of stress relaxation in polymer melts by means of computer simulation of the process in model systems. In this article we treat a melt of freely jointed chains, each with N = 200 bonds and with excluded-volume interactions between all nonbonded atoms, that has been subjected to an initial constant-volume uniaxial extension. We consider both the stress relaxation history σ(t) based on atomic interactions, and the stress history σ_e(t; N_R) based on subdividing the chain into segments with N_R bonds each, with each segment regarded as an entropic spring. It is found that at early times σ(t) > σ_e(t; N_R) for all N_R, and that, for the remainder of the simulation, there is no value of N_R for which σ(t) = σ_e(t; N_R) for an extended period; by the end of the simulation σ(t) has fallen just below the value σ_e(t; 50). The decay of segment orientation, 〈P₂(t; N_R)〉, and of bond orientation 〈P₂(t; 1)〉, is computed during the simulation. It is found that the decay of the atom-based stress σ(t) is closely related to that of 〈P₂(t; 1)〉. This result may be understood through the concept of steric shielding. The change in local structure of the polymer melt during relaxation is also studied. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36 : 143–154, 1998     

Stress overshoot of polymer solutions at high rates of shear

Overshoot of shear stress, σ, and the first normal stress difference, N₁, in shear flow were investigated for polystyrene solutions. The magnitudes of shear corresponding to these stresses, γ_σm and γ_Nm, for entangled as well as nonentangled solutions were universal functions of γ˙τ_eq, respectively, and γ_Nm was approximately equal to 2γ_σm at any rate of shear, γ˙. Here τ_eq = τ_R for nonentangled systems and τ_eq = 2τ_R for entangled systems, where τ_R is the longest Rouse relaxation time evaluated from the dynamic viscoelasticity at high frequencies. Only concentrated solutions exhibited stress overshoot at low reduced rates of shear, γ˙τ_eq < 1. The behavior at very low rates, γ˙τ_eq < 0.2, was consistent with the Doi–Edwards tube model theory for entangled polymers. At high rates, γ˙τ_eq > 1, γ_σm and γ_Nm were approximately proportional to γ˙τ_eq. At very high rates of shear, the peak of σ is located at t = τ_R, possibly indicating that the polymer chain shrinks with a characteristic time τ_R in dilute solutions. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1917–1925, 2000     

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.     

Mechanics of peeling of rubbery materials. II. Initiation and propagation of peeling

The peel strength of a joint with flexible materials bonded by an elastic adhesive was evaluated in relation to the fracture mechanism. It was found that initiation and propagation of peeling are governed by different mechanisms. Initiation (the formation of an initial crack) occurs when the maximum stress in the adhesive layer reaches a definite value. In this case, the strength f_i in a trousers-type peeling is given by 2f_i = y₀σ_b?_b, where y₀ is the half-thickness of the adhesive layer, σ_b is the tensile strength, and ?_b is the tensile elongation of the adhesive. On the other hand, propagation is governed by the surface energy of the adhesive. In this case, the peeling strength f_s is determined by a balance of energies. For trousers-type peeling it is given by 2f_s = Γ, where Γ is twice the surface energy. These results were verified experimentally.     

Application of a new structural theory to polymers. I. Uniaxial stress in crosslinked rubbers

A continuum rheological theory, endowed with generalized structural significance, has recently been developed. Based on nonequilibrium thermodynamics, it relates stress σ, strain rate \documentclass{article}\pagestyle{empty}\begin{document}$\dot \varepsilon$\end{document} and temperature in terms of material evolution through a series of structural states. The theory has previously had success in dealing with crystalline metals and surface physics, and here it is applied to crosslinked rubbery polymers in the nominally amorphous condition. Structure is believed to be related to interchain associations, chain entanglements, chain ends, and other defects in the hypothetical ideal network which by itself would lead to neo-Hookean predictions in uniaxial deformation, σ^nH ∝ λ² — λ^?1, where λ is the stretch ratio. Predictions are made for σ(λ) in both tension and compression and shown to be more compatible with data than either σ^nH(λ) or the Mooney—Rivlin expression σ^MR(λ). Only two parameters are required, moduli G_o (reflecting initial structure) and G_s (the steady-state condition), and rate effects are incorporated through G_o(\documentclass{article}\pagestyle{empty}\begin{document}$\dot \varepsilon$\end{document}) and G_s(\documentclass{article}\pagestyle{empty}\begin{document}$\dot \varepsilon$\end{document}). The phenomena of yielding and stress softening in cyclic tensile loading are also predicted, suggesting advantages to this approach relative to conventional viscoelastic continuum models.     

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.     

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     

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.     

Self-diffusion in melts of statistical copolymers: The effect of changes in microstructural composition

We have used nuclear reaction analysis to measure diffusion coefficients D in couples consisting of hydrogenated polybutadienes of structure (C₂H₃(C₂H₅))_x(C₄H₈)_1?x and their partly deuterated counterparts. The 1,2- and 1,4-olefinic isomers are randomly distributed along the chains and the mean vinyl fraction x varies between 0.38 and 0.94. We find that the effective monomeric mobility D₀ [defined by D = D₀(N_e/N²) for each copolymer, where N is the backbone length and N_e the entanglement spacing] decreases monotonically with increasing vinyl content x. Over the range of microstructures and temperatures T (?14?40°C) investigated we find log(D₀/T) varies smoothly with (T ? T_g), where T_g is the glass transition temperature of the respective melts. An analysis of our data in terms of a simple activated rate process model suggests that D₀ is controlled by thermally activated hopping of segments whose effective volume is close to that of the respective statistical segment lengths of the copolymeric chains. ©1995 John Wiley & Sons, Inc.