Nonlinear viscoelasticity of polystyrene solutions. I. Strain-dependent relaxation modulus

Strain-dependent relaxation moduli G(t,s) were measured for polystyrene solutions in diethyl phthalate with a relaxometer of the cone-and-plate type. Ranges of molecular weight M and concentration c were from 1.23 × 10⁶ to 7.62 × 10⁶ and 0.112 to 0.329 g/cm³. Measurements were performed at various magnitudes of shear s ranging from 0.055 to 27.2. The relaxation modulus G(t,s) always decreased with increasing s and the relative amount of decrease (i.e.,–log[G(t,s)/G(t,0)]) increased as t increased. However, the detailed strain dependences of G(t,s) could be classified into two types according to the M and c of the solution. When cM < 10⁶, the plot of log G(t,s) versus log t varied from a convex curve to an S-shaped curve with increasing s. For solutions of cM > 10⁶, the curves were still convex and S-shaped at very small and large s, respectively, but in a certain range of s (approximately 3 < s < 7) log G(t,s) decreased rapidly at short times and then very slowly; a peculiar inflection and a plateau appeared on the plot of log G(t,s) versus log t. The strain-dependent relaxation spectrum exhibited a trough at times corresponding to the plateau of log G(t,s). The longest relaxation time τ₁(s) and the corresponding relaxation strength G₁(s) were evaluated through the “Procedure X” of Tobolsky and Murakami. The relaxation time τ₁(s) was independent of s for all the solutions studied while G₁(s) decreased with s. The reduced relaxation strength G₁(s)/G₁(0) was a simple function of s (The plot of log G₁(s)/G₁(0) against log s was a convex curve) and was approximately independent of M and c in the range of cM <10⁶. This behavior of G₁(s)/G₁(0) was in agreement with that observed for a polyisobutylene solution and seems to have wide applicability to many polymeric systems. On the other hand, log G₁(s)/G₁(0) as a function of log s decreased in two steps and decreased more rapidly when M or c was higher. It was suggested that in the range of cM < 10⁶, a kind of geometrical factor might be responsible for a large part of the nonlinear behavior, while in the range of cM > 10⁶, some “intrinsic” nonlinearity of the entanglement network system might be important.     

The superposition of small deformations on large deformations: Measurements of the incremental relaxation modulus for a polyisobutylene solution

The incremental relaxation modulus ΔG(t) for a concentrated solution of polyisobutylene has been determined from step-shear experiments in which a small deformation Δγ was superimposed on a large deformation γ₁; ΔG(t) was found to decrease with increasing γ₁ and to increase with the time t_e after the imposition of the large deformation. It was also observed that the “apparent relaxation specturm” associated with δG(t) narrows and shifts to shorter times when compared to the spectrum associated with the linear viscoelastic relaxation modulus G(t). The results are well described by the nonlinear constitutive equation of the BKZ elastic fluid theory.     

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.     

Radiolysis of poly(acetaldehyde-co-chloral): A positive E-beam resist

Acetaldehyde and chloral were copolymerized using triethyl aluminum catalyst. The copolymer (ACC) obtained with equimolar monomer feed is not alternating in structure as it was once thought to be; it is comprised of two fractions differing in MW and composition. ACC has good thermal stability which is further improved by endcapping. Radiolysis in vacuo caused depolymerization with a G(M) value (number of monomers produced per 100 eV) of about 4000 to 80% completion. The G(S) value for chain scission is 1.9. These processes are effectively inhibited by benzoquinone. Oxygen markedly increases G(M) to ca. 18,000 and > 97% completion. Addition of tetrabutyl ammonium salt or tetramethyl urea has no effect on the depolymerization, whereas the addition of di-t-butyl-p-cresol causes an induction period after which normal unzipping ensues. Even UV photolysis of ACC in the presence of oxygen produces monomer with a quantum yield of 1.7, but very little photolysis occurs in the absence of oxygen. Gamma radiolysis sensitized by (C₆H₅)₂IPF₆ has G(M) value of 32,700. These results are very similar to the radiolysis and photolysis of the homopolymer of monochloroacetaldehyde and reinforce the mechanisms proposed for them. The E-beam sensitivity of ACC is about 3 × 10^?6 C cm^?2.     

Ionic solvation in water + co-solvent mixtures. Part 8. Total free energies of transfer and free energies of transfer with the “neutral” component removed of single ions from water into water + acetone

The acid dissociation constants of a wide range of acids in water+acetone mixtures have been combined with values for the free energy of transfer of the proton. ΔG⁰_t(H⁺ to calculate values for the free energy of transfer of ions which derive only from the charge on the ion. ΔG⁰_t(i)_c. As the values of ΔG⁰_t(H⁺) have been revised, revised values for the total free energies of transfer of cations and anions, ΔG⁰_t(M⁺) and ΔG^o_t(X^-), are given. New data for ΔG^o_t(MX_n) is also split into values for ΔG⁰_t(Mⁿ⁺) (where n=1 and 2) and ΔG⁰_t(X^?). These free energies of transfer, both total and those deriving from the charge alone, are compared with similar free energies in other mixtures water+co-solvent. Values for ΔG^o_t(i)_c do not conform to a Born-type relationship and show the importance of structural effects in the solvent even when only the transfer of the charge is involved.     

Photon correlation spectroscopy of poly(methyl methacrylate) near the glass transition

Poly(methyl methacrylate) (PMMA) has been studied by photon correlation spectroscopy in the temperature range 120–150°C. The relaxation functions for longitudinal density fluctuations were determined and analyzed using the empirical function ?(t) = exp[?(t/τ)^β]. The average relaxation times were calculated for each temperature and compared to mechanical and dielectric relaxation data. The agreement between the various techniques for the primary glass–rubber relaxation was good. The relaxation function observed by light scattering became increasingly broad as the temperature was lowered. This is similar to the results reported previously for poly(ethyl methacrylate) (PEMA). In fact, the light-scattering relaxation function is dominated by the secondary relaxations in these two polymers. Nevertheless, the average relaxation time 〈τ〉 is dominated by the longest relaxation times associated with the primary glass–rubber relaxation.     

Hydrogen bonding in butadiene copolymers. II. Network relaxation

The mechanical lability of intermolecular hydrogen bonds in stoichiometric mixtures of acidic and basic butadiene copolymers has been studied by both infrared and mechanical methods. Comparison is made between the weight-averaged E(t) and H(τ) spectra of the parent copolymers and those of the mixed copolymers. The results indicate that the maximum contributions of bond-interchange relaxation migrate to shorter times as the concentration of bonding groups increases.     

Stress relaxation and dynamic viscoelastic properties of end-linked poly(dimethyl siloxane) networks containing unattached poly(dimethyl siloxane)

Stress relaxation in uniaxial extension and dynamic shear moduli G′ and G″ have been studied in networks of vinyl-terminated poly(dimethyl siloxane) (PDMS) of five different molecular weights (M _n from 1800 to 29,200) crosslinked with cis-dichlorobis (diethyl sulfide) platinum (II) and containing 10 and 15 wt % of two samples of high-molecular-weight unattached linear hydroxyl-terminated PDMS (M _w 700,000 and 950,000). The M _w/M _n ratio of both the network prepolymers and the unattached linear species was approximately 2. In stress relaxation the stretch ratio was 1.25 or less and the shear relaxation modulus was calculated from the neo-Hookean stress-strain relation. In the dynamic measurements, the strain amplitude was 15% or less; after conversion to the timedependent shear relaxation modulus G(t) the two sets of measurements were combined and the contribution of the unattached species G₁(t) was calculated by difference. After multiplication by (1 − v)⁻¹G/G_e, where v₂ is the volume fraction of network, G is the plateau modulus of the uncrosslinked polymer, and G_e is the equilibrium modulus of the network containing unattached molecules, G₁(t) was compared with G₁₁(t), the relaxation modulus was essentially the same in both environments. The relaxation was slower in the networks than in the uncrosslinked polymer by 1 to 2 orders of magnitude, and it increased gradually with increasing G_e, which is a measure of total to pological obstacles represented by crosslinks plus trapped entanglements. A similar but less striking difference between relaxation in a network and in the homologous environment of a linear polymer was previously observed in end-linked polybutadiene networks and the butadiene phase of a styrene-butadiene-styrene block copolymer. It appears that, in these systems where the topology of the obstacles is fixed, the reptation is severely restricted or else alternative modes of configurational rearrangement which contribute to relaxation in the uncrosslinked polymer are suppressed.     

Nonlinear viscoelasticity of polystyrene solutions. II. Non-Newtonian viscosity

The steady shear viscosity η(k) and the stress decay function \documentclass{article}\pagestyle{empty}\begin{document}$ \tilde \eta \left({t,k} \right)$\end{document} (the shear stress divided by the rate of shear k after cessation of steady shear flow) were measured for concentrated solutions of polystyrene in diethyl phthalate. Ranges of molecular weight M and concentration c were 7.10 × 10⁵ to 7.62 × 10⁶ and 0.112–0.329 g/cm³, respectively. Measurements were performed with a rheometer of the cone-and-plate type in the range 10^?4 < k < 1 sec^?1. The Cox–Merz relation η(k) = |η^*(ω)|_ω=k was tested with the experimental result (|^*(ω)| is the magnitude of the complex viscosity). It was found to be applicable to solutions of relatively low M or c but not to those of high M and c. For the latter η(k) began to decrease at a lower rate of shear than |η^*(ω)|_ω=k did; the Cox–Merz law underestimated the effect of rate of shear. The stress decay function was assumed to have a functional form \documentclass{article}\pagestyle{empty}\begin{document}$\tilde \eta \left( {t,k} \right) = \sum {\eta _p \left( k \right)e^{ - t/\tau p\left( k \right)} } $\end{document} where τ₁ > τ₂ > …, and the values of τ₁, τ₂ η₁ and η₂ were determined for some solutions. The relaxation times τ₁ and τ₂ were found to be independent of k and equal to the relaxation times of linear viscoelasticity. At the limit of k → 0, η₁ and η₂ were approximately 60 and 20–30%, respectively, of η and the non-Newtonian behavior was due to large decreases of η₁ and η₂ with increasing k. It was shown that η₁(k) may be evaluated from the relaxation strength G₁(s) for the longest relaxation time of the strain-dependent relaxation modulus with a constitutive model for relatively high c–M systems as well as for low c–M systems.     

Can the Theoretical Fitting of the Proton‐Nuclear‐Magnetic‐Relaxation‐Dispersion (Proton NMRD) Curves of Paramagnetic Complexes Be Improved by Independent Measurement of Their Self‐Diffusion Coefficients?

The self‐diffusion (D_c) coefficients of various lanthanum(III) diamagnetic analogues of open‐chain and macrocyclic complexes of gadolinium used as MRI contrast agents were determined in dilute aqueous solutions (3–31 mM ) by pulsed‐field‐gradient (PFG) high‐resolution ¹H‐NMR spectroscopy. The self‐diffusion coefficient of H₂O (D_w) was obtained for the same samples to derive the relative diffusion constant, a parameter involved in the outersphere paramagnetic‐relaxation mechanism. The results agree with an averaged relative diffusion constant of 2.5 (±0.1)×10^?9 and of 3.3 (±0.1)×10^?9 m² s^?1 at 25 and 37°, respectively, for 'small' contrast agents (M_r 500–750 g/mol), and with the value of bulk H₂O (2.2×10^?9 and 2.9×10^?9 m² s^?1 at 25° and at 37°, respectively) for larger complexes. The use of the measured values of D_c for the theoretical fitting of proton NMRD curves of gadolinium complexes shows that the rotational correlation times (τ_R) are very close to those already reported. However, differences in the electronic relaxation time (τ_SO) at very low field and in the correlation time (τ_V) related to electronic relaxation were found.     

Laser-induced kinetics: Arrhenius parameters for t-C4H9 + XI → i-C4H9X + I (X = H,D) and the Heat of Formation of the t-butyl radical

The metathesis reaction of DI with t-C₄H₉ generated by 351-nm photolysis of 2,2′-azoisopropane was studied in a low-pressure reactor (VLP? Knudsen cell) in the temperature range of 302–411 K. The data obeyed the following Arrhenius relation when combined with recent data by Rossi and Golden gathered by the same technique (t-C₄H₉ by thermal decomposition of 2,2′-azoisobutane): log k₂^D(M^?1s^?1) = 9.60 – 1.90/θ, where θ = 2.303RT kcal/mol for 302 K < T > 722 K. The metathesis reaction of HI with t-C₄H₉ was studied at 301 K and resulted in k₂^H(M^?1·s^?1) = (3.20 ± 0.62) × 10⁸. An analogous Arrhenius relation was calculated for the protiated system if the small primary isotope effect k₂^H/k₂^D was assumed to be √2 at 700 K. It was of the following form: log k₂^H(M^?1·s^?1) = 9.73 – 1.68/θ. Preliminary data of Bracey and Walsh indicate that earlier Arrhenius parameters determined for the reverse reaction are somewhat in error. Their value of log k₁(M^?1·s^?1) = 11.5 – 23.8/θ yields 7delta;H_f,300⁰(t-butyl) = 9.2 kcal/mol and S₃₀₀⁰(t-butyl) = 74.2 cal/mol7°K when taken in conjuction with this study.     

Charge transport in polyaniline doped with 3-nitro-1,2,4-triazol-5(4H)-one

Polyaniline (PANI) base was protonated in aqueous solutions of an organic acid, 3-nitro-1,2,4-triazol-5(4H)-one (NTO). The temperature dependence of DC conductivity of PANI-NTO seems to correspond to the theory of variable range hopping (VRH) in three dimensions. The frequency dependence of AC conductivity also reflects the hopping nature of mobile charges. The activation energy for the polymers with protonation degree above 0.12 remains constant with increasing dopant concentration and DC conductivity. The value of this constant may correspond to the energy needed for the ionization of dopant counterion. The fit of the electric relaxation function to the stretched exponential function ϕ(t) = exp[−(t/τ)^β] gives the stretch parameter β about 0.35, which shows that the distribution of relaxation times is broad and indicates a high inhomogeneity in the distribution of a dopant.     

Interpretation of differences in temperature and pressure dependences of density and concentration fluctuations in amorphous poly(phenylmethyl siloxane)

Differences in the temperature and pressure dependences of the relaxation times of a slow diffusional process and the α structural relaxation pose an interesting problem. This feature, observed by dynamic light scattering in amorphous poly(phenylmethyl siloxane), is related to another basic feature of lack of thermorheological simplicity discovered by Plazek in polystyrene, poly(vinyl acetate), and amorphous polypropylene. A quantitative explanation based on the predictions of a general coupling theory of relaxations has been found. The coupling theory also predicts the Kohlrausch fractional exponential time correlation function exp[?(tτ^*)^1?n] at long times, as observed by photon correlation spectroscopy, and crossover to an exponential time dependence exp–(t/τ₀) at short times, as frequently assumed in Brillouin scattering. An additional relation between τ^* and τ₀ predicted by the theory is confirmed also by the experimental data.     

Concentration dependence of the viscoelastic properties of polystyrene-tricresyl phosphate solutions

Based on creep and creep-recovery measurements, the viscoelastic functions [J(t), J_r(t), J′(ω), J″(ω), G′(ω), G″(ω), and L(lnτ)] are presented for solutions of a narrow molecular-weight-distribution polystyrene in tri-m-tolyl phosphate in the concentration range of 1% to 100% polymer. For concentrations of 25% polymer and above, two maxima are exhibited by the retardation spectrum, L(lnτ). In the neighborhood of each of the maxima the retardation spectra of the more concentrated solutions can be superimposed by translations along both the logL and logτ axes. Reflecting the increasing width of the rubbery plateau with increasing polymer concentration, the dependence of the concentration time-scale shift factors is greater for the terminal region of response. The response of the solvent is seen at the lower concentrations and it is a less sensitive function of the concentration than that of the polystyrene. This behavior is associated with the previously reported observation of two glass-transition temperatures in the middle concentration range. For the higher concentrations, both the steady-state and rubbery-plateau compliances are inversely proportional to the square of the concentration.     

Rubber networks containing unattached macromolecules. IV. Stress relaxation in end-linked polybutadiene with unattached styrene-butadiene copolymer

Stress relaxation has been studied in networks of dihydroxy-terminated polybutadiene (mostly cis:trans:vinyl = 34:40:26) crosslinked by triphenyl methane-4,4′,4″-triisocyanate and containing about 9.5% by weight of unattached linear random styrene-butadiene copolymer with various molecular weights (from 1.4 to 3.3 × 10⁵) and with styrene content and butadiene microstructure chosen to match the average solubility parameter of the end-linked network. Stress relaxation measurements were made also on networks containing no unattached species and containing 9.3% hydrocarbon oil, and on the various uncrosslinked linear polymers. The stretch ratio was 1.25 and the Young's relaxation modulus was calculated from the neo-Hookean stress-strain relation. For the uncrosslinked linear polymers, the relaxation modulus E₁₁(t) corresponds to a rather narrow distribution of relaxation times whose magnitudes were approximately proportional to the 3.4 power of viscosity-average or weight-average molecular weight; for one polymer, the time dependence agreed closely with the prediction of the Doi-Edwards theory modified for a small degree of molecular weight distribution. The disengagement times calculated from the Doi-Edwards theory as modified by Graessley appeared to be of the correct order of magnitude. The contribution of the unattached species in the networks E₁(t) was calculated by difference; after multiplication by (1?v)^?1, where v₂ is the volume fraction of network, and correction for the difference in monomeric friction coefficient associated with the difference in fractional free volume in the two environments, E₁(t) was compared with E₁₁(t) for each linear polymer. The relaxation was slower in the network than in the uncrosslinked polymer by about an order of magnitude, but the form of the relaxation modulus was similar in both environments except for two linear polymers for which the relaxation in the network became very much slower at long times. This behavior appeared to be correlated with a broader molecular weight distribution.     

Singlet-triplet splitting of divalent five-membered fused rings C<Subscript>12</Subscript>H<Subscript>8</Subscript>M (M = C,Si, Ge,Sn, and Pb): Application in nanocomplex synthesis

Energy differences, ΔX _s−t (X = E, H, and G) (ΔX _s−t = X(singlet) − X(triplet)) between singlet (s) and triplet (t) states of C₁₂H₈M were calculated at B3LYP/6-311+G*. The DFT calculations indicated that the ΔG _s−t between singlet (s) and triplet (t) states of C₁₂H₈M were increased from M = C to M = Pb. The ΔG _s−t of C₁₂H₈M was compared with its analogue C₄H₄M through replacement of heavy atoms from M = C to M = Pb. Configurations of the electrons in orbitals (σ² or π²) for the singlet state of C₁₂H₈M were discussed.     

Dynamic mechanical properties of moderately concentrated polystyrene solutions

The storage (G′) and loss (G″) shear moduli have been measured in the frequency range from 0.04 to 630 Hz for solutions of narrow distribution polystyrenes with molecular weights (M) 19,800 to 860,000, and a few of poly(vinyl acetate), M = 240,000. The concentration (c) range was 0.014–0.40 g/ml and the viscosities of the solvents (diethyl phthalate and chlorinated diphenyls) ranged from 0.12 to 70 poise. Data at different temperatures (0–40°C) were combined by the method of reduced variables. Two types of behavior departing from the usual frequency dependence describable by the Rouse-Zimm-Tschoegl theories were observed. First, for M ? 20,000, the ratio (G″ ? ωη_s)/G′ in the neighborhood of ωτ₁ = 1 was abnormally large and the steady-state compliance J was abnormally small, especially at the lowest concentrations studied. Here ω is circular frequency, η_s solvent viscosity, and τ₁ terminal relaxation time. Related anomalies have been observed by others in undiluted polymers at still lower molecular weights. Second, at the highest concentrations and molecular weights, a “crossover” region of the logarithmic frequency scale appeared in which G″ ? ωη_s < G′. The width of this region is a linear function of log c; the frequency dependence under these conditions can be represented by a sequence of Rouse relaxation times grafted on to a sequence of Zimm relaxation times. For each molecular weight, the terminal relaxation time τ₁ was approximately a single function of c for different solvents of widely different η_s. At lower concentrations, τ₁ was close to the Rouse prediction of 6ηM/π²cRT, where η is the steady-flow viscosity; but at higher concentrations, τ₁ was proportional to η/c² and corresponded, according to a recent theory of Graessley, to an average molecular weight of 20,000 between entanglement coupling points in the undiluted polymer.     

Photon correlation spectroscopy of polystyrene as a function of temperature and pressure

Photon correlation spectroscopy is employed to study the slowly relaxing density and anisotropy fluctuations in bulk atactic polystyrene as a function of temperature from 100 to 160°C and pressure from 1 to 1330 bar. The light-scattering relaxation function is well described by the empirical function ?(t) = exp[?(t/τ)^β], where for polystyrene β = 0.34. The average relaxation time is determined at each temperature and pressure according to 〈τ〉 = (τ/β)Γ(1/β) where Γ(x) is the gamma function. The data can be described by the empirical relation 〈τ〉 = 〈τ〉₀ exp[(A + BP)/R(T ? T₀)] where R is the gas constant and T₀ is the ideal glass transition temperature. The empirical constant A/R is in good agreement with that determined from the viscosity or the dielectric relaxation data (1934 K). The empirical constant B can be interpreted as the activation volume for the fundamental unit involved in the relaxation and is found to be comparable to one styrene subunit (100 mL/mol). The quantity B appears to be a weak function of temperature. The use of pressure as a tool in the study of light scattering near the glass transition now has been established.     

Modulation of the Lifetime of Water Bound to Lanthanide Metal Ions in Complexes with Ligands Derived from 1,4,7,10‐Tetraazacyclododecane Tetraacetate (DOTA)

A series of di‐ and tetraamide derivatives of DOTA were synthesized, and their lanthanide(III) complexes were examined by multinuclear ¹H‐, ¹³C‐, and ¹⁷O‐NMR spectroscopy, and compared with literature data of similar, known complexes (Table). All ligands formed structures similar to the parent [Ln^III(DOTA)]^? complexes, with four N‐atoms and four O‐atoms from DOTA and one O‐atom from the inner‐sphere water molecules. Interestingly, the lifetimes τ_M of the inner‐sphere, metal‐bound water molecules vary widely, ranging from nano‐ to milliseconds, depending on the identity of the pendent amide side chains. In general, positively charged [Ln^III(DOTA‐tetraamide)]³⁺ complexes display the longest residence times (high τ_M values), while complexes with additional charged functional groups on the extended amides display much smaller τ_M values, even when the side groups are not directly coordinated to the central Ln³⁺ ions. The design of novel [Ln^III(DOTA‐tetraamide)]³⁺ complexes with a wide, tunable range of τ_M values is of prime importance for the application of fast‐responding, paramagnetic chemical‐exchange‐saturation‐transfer (PARACEST) imaging agents used for the study of physiological and metabolic processes.     

The influence of monomer chemistry on radical formation and secondary reactions during electron-beam polymerization

The vast majority of industrial electron-beam (EB) polymerizations are initiated via radical mechanism. Because radicals drive EB polymerization, understanding their formation and secondary reactions can provide insight into polymerization kinetics and property development. Primary and initiating radicals were quantified by measuring G(R^•) and G(M^•) , respectively, for various acrylate and methacrylate monomers. Monomer chemistry was shown to impact primary radical formation; however, increased primary radical concentration did not necessarily correlate to increased conversion. Despite exhibiting high values of G(R^•) , methacrylates achieved very little conversion and had G(M^•) values near zero. Acrylates achieved much higher G(M^•) values and rates of polymerization compared to their methacrylate counterparts. Additionally, the efficiency of primary radicals converting to initiating radicals, f(M^•) , for each acrylate monomer was shown to be a good predictor of the amount of gel fraction formed during polymerization. Understanding radical formation and secondary radical reactions can help guide the structure/processing conditions/properties relationships that are currently underdeveloped for EB reactions.