Computer simulation of brownian motion of a polyelectrolyte molecule (charged bead-spring model)

A Langevin equation of motion for a charged bead-spring statistical chain is written in difference form and the relaxation and equilibrium behavior of the chain is studied by computer simulation. Results are presented for the behavior of end-to-end length h, principal axes of the polymer ellipsoid L₁, L₂, L₃, and chain contour length c in terms of their averages, root mean square values, root mean square fluctuations, orientations, and relaxation strengths and times. The simulation was made with various sets of parameters, bead number N, charge on the bead q, and radius of ion atmosphere around the bead k^?1. It is found that 〈h²〉^1/2 and 〈L₁²〉^1/2 increase more strongly with increasing q and decreasing κ than 〈L₂²〉^1/2, 〈L₃²〉^1/2, and 〈c₁²〉^1/2, indicating that the chain is expanded in three dimensions and at the same time is extended along the end-to-end direction. The relaxation time τ_rot of rotation of the end-to-end vector, which is proportional to N² at q = 0, increases with increasing q and tends to be proportional to N³ for an extended chain, while the relaxation time τ_conf of the magnitude of h is almost independent of q and is always proportional to N². It is concluded that the extended chain possesses a well-defined end-to-end axis and the chain rotates as a whole with a relaxation time τ_rot which is much longer than τ_conf. The complex viscosity of the chain is calculated from the Fourier transform of the time–correlation function of momentum flux and is found to have a frequency spectrum similar to that observed for aqueous solutions of poly(acrylic acid). The dominant mode appearing in the low-frequency range is evidenced to arise from the rotation of the extended chain.     

Thermal Degradation of Adsorbed Bottle-Brush Macromolecules: When Do Strong Covalent Bonds Break Easily?

The scission kinetics of bottle-brush molecules in solution and on an adhesive substrate is modeled by means of Molecular Dynamics simulation with Langevin thermostat. Our macromolecules comprise a long flexible polymer backbone with L segments, consisting of breakable bonds, along with two side chains of length N, tethered to segments of the backbone with grafting density σ_g. In agreement with recent experiments and theoretical predictions, we find that bond cleavage is significantly enhanced on a strongly attractive substrate even though the chemical nature of the bonds remains thereby unchanged. Our simulation results indicate that the mean life time of covalent bonds decreases by more than an order of magnitude upon adsorption even for brush molecules with comparatively short side chains . The distribution of scission probability along the bonds of the backbone is found to change significantly when the length and/or the grafting density of the side chains are varied. The tension, experienced by the covalent bonds is found to grow steadily with increasing σ_g. The mean life time declines with growing contour length L as , and also with growing side chain length N. The probability distribution of fragment lengths at different times is compatible with experimental observations and reveals a two-stage (initially fast, then slow) process with different rates. The variation of the mean length L(t) of the fragments with elapsed time characterizes the thermal degradation process as a first order reaction.     

Properties of Polyvinyl Alcohol Oligomers: A Molecular Dynamics Study

MD studies of liquid isopropyl alcohol and melts of short poly(vinyl alcohol) (PVA) oligomers are described. The specific volume was found to depend inversely on the number N of repeat units. If the chain length is enhanced, the viscosity of the PVA melt increases and the peaks in the radial distribution function become sharper. Additional peaks that appear in melts of PVA chains are of pure intramolecular origin. The calculated radius of gyration was found to depend on the number of formula units via . The orientation correlation functions showed that all molecular vectors of PVA melts with chain lengths N = 1, 2, 3 relax completely within a few nanoseconds. The relaxation times for the O H bond vector as obtained via the Kohlrausch‐Williams‐Watts expression showed an exponential dependence on the number of repeat units.

     

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.     

Development of Entanglements in a Fully Disentangled Polymer Melt

A long‐lived metastable “new melt” state of polymers has been recently reported, where monomers exhibit different mobilities due to an unusual distribution of entanglements. We study the relaxation of (fully disentangled) globules to the entangled state by means of computer simulations, and compare our data to the scenario of de Gennes' explosion upon melting. The entanglement length N_e is measured using the primitive path analysis method. The results show that in the case of relatively short chains (N ≈ 20N_e), the relaxation of the entanglement length is very fast compared to that of the chains' size which slows down as the chain length N exceeds the equilibrium value of N_e.

     

Asymptotic relations between shear stresses and normal stresses in general incompressible fluids

It is shown that in the general theory of incompressible simple fluids with fading memory there are, for several types of nonsteady shearing motions, simple universal asymptotic relations between the shear stress S₁₂ and the first normal stress difference N₁ = S₂₂ – S₁₁. The kinematical situations considered include initiation of steady shearing, rest after steady shearing, and sinusoidal oscillation. In, for example, relaxation following cessation of a steady shearing flow with rate of shear κ, there holds, to within an error O(κ⁴): This and the other derived universal relations between N₁ and S₁₂ are either consequences of, or are closely related to a general asymptotic formula [B. D. Coleman and W. Noll, Revs. Mod. Phys., 33 , 239 (1961), eq. (6.15)] expressing N₁ as an integral of the product of the shear relaxation modulus and the square of the history of the relative shear strain.     

Quenching of N2(A3Σu+) by O2, O,N, and H

Metastable N₂(A³Σ_u⁺), υ = 0, υ = 1, molecules are produced by a pulsed Tesla-type discharge of a dilute N₂/Ar gas mixture. Rate coefficients for quenching these metastable levels by O₂, O, N, and H were obtained by time-resolved emission measurements of the (0, 6) and (1, 5) Vegard–Kaplan bands. In units of cm³/mole · sec at 300°K and with an experimental uncertainty of ±20%, these rate coefficients for N₂(A³Σu⁺) are Within the limits of error these coefficients apply to quenching N₂(A³Σ_u⁺) υ′ = 1 as well.     

Rheology of poly(vinyl chloride) melts. II. Shear rate-dependent properties

Four samples containing 40, 60, 80, and 97 wt-% of poly(vinyl chloride), the rest being plasticizer and stabilizer, were tested by using the Weissenberg Rheogoniometer in the steady-shearing mode at temperatures between 155 and 235°C and rates of shear \documentclass{article}\pagestyle{empty}\begin{document}$ \dot \gamma = 0.01 - 400 $\end{document} sec^?1. The viscosity η versus \documentclass{article}\pagestyle{empty}\begin{document}$ \dot \gamma $\end{document} follows Graessley's theoretical dependence for infinitely entangled system. The primary normal-stress difference coefficient ψ versus \documentclass{article}\pagestyle{empty}\begin{document}$ \dot \gamma $\end{document} is well described by the same theoretical function, used with the square of its argument. The temperature dependence of η₀ and ψ₀ shows discontinuities at T = T_b. The numerical values of T_b can be calculated from the theory of the melting point depression due to diluent. The activation energy of viscous flow E_η below T_b is 5–9 times as large as above this temperature. The activation energy of normal stress is found to be Eψ ≈ 5E_η. The characteristic relaxation times τ_o, ψ_p, calculated from superposition of η versus \documentclass{article}\pagestyle{empty}\begin{document}$ \dot \gamma $\end{document} and ψ versus \documentclass{article}\pagestyle{empty}\begin{document}$ \dot \gamma $\end{document} data, respectively, onto Graessley's master curves, and τ_N, computed from zero shear parameters η₀ and ψ₀, differ in their sensitivity to the melting of microcrystalline regions. It is postulated that in the systems investigated, aggregates with long lifetimes are being formed, increasing the effective molecular weight and introducing changes in the effective polydispersity.     

Polymer chains confined into tubes with attractive walls: A Monte Carlo simulation

A bead-spring off-lattice model of a polymer chain with repulsive interactions among repeating units confined into straight tubes of various cross sections, D_T², is studied by Monte Carlo simulation. We are also varying the chain length from N = 16 to 128 and the strength of a short-range attractive interaction between the repeating units and the walls of the tube. Longitudinal and perpendicular static linear dimensions of the chains are analyzed, as well as the density profile of repeating units across the tube. These data are interpreted in terms of scaling concepts describing the crossover between three-dimensional and quasi-one-dimensional chain conformations and the adsorption transition of chains at flat infinite walls, respectively. We also study the time-dependent mean-square displacements of repeating units and obtain various relaxation times. It is shown that both relaxation times scaling proportional to N² and to N³ play a role in the reptative motion of the chain in the tubes.     

Statistical branching of heterochains

A general theory for the statistical branching of heterochains is proposed on the basis of the random sampling technique. Consider a polymer mixture that consists of N types of chains whose weight fractions are w_i (i = 1, 2, …, N), and number- and weight-average chain lengths are P̄_np,i and P̄_wp,i, respectively. Suppose the transition probability that a branch point on a chain of type i is connected to a chain end of a type j chain is given by p_ij. When the branching density of chains of type i is ρ_i, the weight-average chain length is given by $\bar P_w = W\sum\nolimits_{m = 0}^\infty T ^m \sum\nolimits_{n = 0}^\infty {SU} ^n 1$, where S is the diagonal matrix whose elements are $S_{ii} = \bar P_{wp,i}$, 1 is the column vector whose elements are all unity, U is the transition matrix whose elements are given by $u_{ij} = \rho _i p_{ij} P_{np,j} ,T$ is another transition matrix whose elements are given by t_ij = (w_j/w_i)U_ji, and W is the row vector whose elements are w_i. Simpler expressions of P̄_w are presented for binary systems. In addition to the multicomponent systems, the present equation could also be used such as for free-radical polymerization with long-chain branching, by considering primary chains formed at different times as different types of polymer chains. For the prediction of the full molecular weight distribution, a Monte Carlo simulation method is used to illustrate the resulting distribution profiles.     

Rheology of polystyrene solutions around the coil overlapping concentration: A phenomenological description of stresses in simple shear flow

Linear viscoelasticity behavior is described with the sum of two terms for polystyrene solutions in tricresyl phosphate around the coil overlapping concentration (K. Osaki, T. Inoue, & T. Uematsu, J Polym Sci Part B: Polym Phys 2001, 39, 211). One is a Rouse–Zimm (RZ) term represented by the Zimm theory with arbitrarily chosen values of the hydrodynamic interaction parameter and the longest relaxation time (τ_RZ). The other (the L term) consists of a relaxation mode with a single relaxation time (τ_L > τ_RZ) and a high‐frequency limiting modulus proportional to the square of the concentration. In this study, we describe the viscosity (η) and first normal stress coefficient (Ψ₁) in steady shear with simple formulas. The stress due to the L term is assumed to be given by a Kaye, Bernstein, Kearsley, and Zapas (K‐BKZ) equation with the damping function h(γ) = (1 + 0.2γ²)^?1/2, where γ is the magnitude of shear. Contributions to η and Ψ₁ from the RZ term are derived from the RZ model, in which the relaxation time in steady flow is given by τ_st = τ + (τ_RZ ? τ)/(1 + 0.35τ_RZ γ˙) instead of τ_RZ. Here, γ˙ is the rate of shear, and τ is the τ_RZ value at the infinite dilution limit. η and Ψ₁ at various concentrations for two polystyrene samples (with molecular weights of 2890 and 8420 kg mol^?1) are well described with parameters derived from dynamic viscoelasticity. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1038–1045, 2002     

Reaction of O(1D) with N2O

O(¹D), produced from the photolysis of N₂O at 2139 Å, reacts with N₂O in accord with: We have used the method of chemical difference to obtain an accurate measure of k₂/k₃ = 0.59 ± 0.01. Furthermore, the quantum yield of production of O(³P), either on direct photolysis or on deactivation of O(¹D) by N₂O, is less than 0.02 and probably zero.     

Effect of concentration of the crosslinking agent and diluent during polymerization on the viscoelastic spectra of poly(2-hydroxyethyl methacrylate) networks

The effect of concentration of the crosslinking agent (ethylene dimethacrylate) and diluent (water) during the crosslinking copolymerization on the shape and position of retardation spectra in the dry state has been investigated for poly(2-hydroxyethyl methacrylate) networks. With increasing water content during network formation, the maxima of the retardation spectra, L_m, increase and the position of the spectra is shifted toward shorter retardation times, τ. The results are in quantitative agreement with the modified Rouse–Mooney (R–M) molecular theory and suggest the influence of deformation due to the diluent during network formation on the viscoelastic behavior. With increasing content of the crosslinking agent, the retardation spectra are shifted toward longer times. At a constant reference temperature T₀ = 115°C the retardation time, τ_m, at the maxima of the spectra increases with increasing content of effective chains in the network, ν_e. However, after a correction for the effect of the monomeric frictional coefficient, ξ, τ_m/ξ decreases with increasing v_e at a rate which agrees quantitatively with the R–M theory. The slope of the retardation spectra in the main transition region and the value of their maxima decrease with increasing v_e; a comparison of these dependences with theory leads to the most probable distribution of submolecules in the chains. The contribution of long retardation times to the equilibrium compliance, J_e, of the systems under investigation was estimated; it was shown that the application of the Thirion–Chasset extrapolation method for the determination of J_e of loose networks requires a certain type of dependence of the retardation or relaxation spectra on τ.     

Isometric and Isotensional Force‐Length Profiles in Polymethylene Chains

Summary: The force‐length curves and related thermodynamic quantities of single polymethylene (PM) chains in the high‐force region were calculated by the statistical mechanics. Two statistical mechanics ensembles (isometric and isotensional) were used to represent the chain stretching under the conditions, respectively, of the fixed length L or of the fixed force F in single‐chain experiments by AFM and related techniques. The input deformation potentials of highly extended conformations of PM chains were obtained from the molecular‐mechanics calculations. Variations of the energy, entropy, and Helmholtz energy with the end‐to‐end length L of chains were computed under the condition of fixed length. The ensuing isometric profile of the mean force 〈F〉(L) is non‐monotonic, featuring a sawtooth‐like pattern of ascending peaks. In contrast, the mechanical equation under isotensional conditions, given by the variation of the mean length 〈L〉(F), shows a conventional monotonous shape with a distinct plateau region at low temperatures. The considerable difference in the shapes of 〈F〉(L) and 〈L〉(F) curves arises from the large fluctuations of mean values due to the small number of conformers present in molecules at high strains. The computed data should be relevant to the AFM stretching experiments on short polyethylenes or on other soft matter materials involving linear paraffinic chains. It is argued that the dual character of elastic response described by the conjugated force profiles is a universal feature of mechanochemistry of chain molecules whenever the chain length discontinuously increases at a transition.

Force‐length curves of short polymethylene chains at 300 K.     

Relativistic theory of binding energies of heavy positive ions

The self-consistent relativistic Thomas–Fermi theory of heavy positive ions with N electrons and nuclear charge Ze is shown to lead to a chemical potential μ which has the scaling property with ? = α²Z², α being the fine structure constant. Combining this with the Layzer–Bahcall expansion for the total energy E(Z, N), namely, it is proved that the coefficients E_nm (N) at large N have the asymptotic behavior N^n–2m/3#1/3. The corresponding result for the scaling of the relativistic Thomas–Fermi energy is Scaling properties of the higher order terms in E_nm (N) and E(Z, N) are also proposed.     

Kinetics and mechanism of the reaction between N-and N,N′-methyl ethylenediamines and palladium(II) chloro complexes in aqueous acid medium

The substitution of N-alkyl substituted ethylenediamines for chloride ions in the rapidly equilibrating system has been investigated in aqueous acid medium. The kinetic data can be accommodated by the general rate law where n = 0, 1, or 2 and m = 0, 1, or 2, depending on whether none, one, or two methyl groups are attached to the two nitrogen atoms of ethylenediamine. Reaction with the most heavily substituted ethylenediamine, namely, N₂N₂en discloses a change of the mentioned rate law to on going from a lower to a higher chloride ion concentration range. This change in the mathematical form of the rate law can be explained in terms of an ion-pair association of N₂N₂enH⁺ and free chloride ions.     

Driven Chain Macromolecule in a Heterogeneous Membrane-Like Medium

Monte Carlo simulations are performed to study the conformational relaxation of a large polymer chain driven into a heterogeneous (membranelike) substrate on a discrete lattice. Chains are created on trails of constrained self-avoiding walks (SAW) on the lattice. Kink–jump, crank–shaft, and reptation moves are used to move segments of chains. Short chains of length L _sc are driven by a field E ₁ toward an impenetrable substrate to design a membrane medium with mobile chain segments. A long chain of length L _lc is then driven by a field E ₂ into the membrane medium and is subsequently allowed to relax in a field E ₃. Radius of gyration R _g and end-to-end distance R _e of the long chain are examined. The relaxation of the conformation of the long chain and its magnitude is found to depend on the initial (predeposition) conformation of the chain, i.e., on E ₂. For a relatively relaxed initial conformation (at E ₂ = 0.1), the longitudinal component of the radius of gyration (R _gz ) is found to decay with the driving field E ₃ with a power law, R _gz E ₃ where 0.1 at low field (E ₃ 0.1) and 1/3 at high field E ₃ 0.1.     

The kinetics and the mechanism of the thermal gas-phase reaction between bis(trifluoromethyl)trioxide,CF3O3CF3 and 1,1-dichlorodifluoroethylene,CF2CCI2

The thermal addition of CF₃O₃CF₃(T) to CF₂CCl₂(E) has been investigated between 49.6 and 69.5°C. The initial pressure of CF₃O₃CF₃ was varied between 7 and 240 torr and that of CF₂CCl₂ between 4 and 600 torr. Four products of formula CF₃O(E)_j OOCF₃, where j = 1 → 4 are formed. The sum of the products Σ CF₃O(E)_jOOCF₃ is equal to the amount of trioxide decomposed. The reaction is homogeneous. Its rate is not affected by the total pressure and the presence of inert gas. It is a free radical telomerization with four basic steps: thermal decomposition of CF₃O₃CF₃ into CF₃O^. and CF₃O₂^., chain initiation by addition of CF₃O^. to olefin incorporated in, and telomeric radicals termination. The consumption of alkene is well represented by the equation: where (d[E]/d[T]) = is the mean chain length of telomerization. varies from 1.45 at 1.5 torr of E to 3.3 at 400 torr of E. Above this pressure E has no influence on . The estimated value of the constant for the addition of telomeric radicals to alkene is:      

Theory of polydisperse reacting polymer systems

The kinetics of irreversible reactions between polymer chains of different molecular weights are studied, with emphasis on the case of highly reactive end groups. We calculate the rate constant k(N, M) for reaction between chains of lengths N and M respectively, in dilute and semi-dilute solutions and in the melt. In all cases, k(N, M) is dominated by the shortest chain: the limit k(N) ≡ k(N, ∞) is well-defined and scales as if both chains were of length N. In dilute solutions k(N, M) obeys mean field theory, being proportional to the equilibrium reactive group contact probability. For melts and concentrated solutions, k(N, M) follows diffusion-controlled laws: k(N, M) ≈ (R/τ_N)ƒ(M/N) where R_N and τ_N are the coil size and relaxation time of the shortest chain N, and ƒ(M/N) is a cross-over function describing the approach to the asymptotic form k(N) for M/N ≫ 1. We calculate the leading contributions to this cross-over function, which has universal forms depending on the concentration regime. The implications of these results for high-conversion free-radical polymerization are discussed.