Multiple Histogram Method and Static Monte Carlo Sampling

Summary: We describe an approach to use multiple‐histogram methods in combination with static, biased Monte Carlo simulations. To illustrate this, we computed the force‐extension curve of an athermal polymer from multiple histograms constructed in a series of static Rosenbluth Monte Carlo simulations. From the complete histogram of the distribution function of the end‐to‐end vectors of the polymer chain, we can efficiently compute the polymer force‐extension curve.

Comparison of the stress‐strain curves for the stress ensemble (symbols) and the strain ensemble (lines). Results obtained for N = 100, 200, 400, and 600. For small x, f(x) = −F′(x) was computed by aproximating F(x) by a second degree polynomial and then taking the derivative. For large x, f(x) = −F′(x) was computed numerically.     

Rheological properties of internal viscosity models with stress symmetry

Internal viscosity models (IVM) for dilute-solution polymer dynamics differ in how they define the deformational force F ^d which includes φ, the IV coefficient, and in how they treat polymer rotational velocity Ω. Here, the handling of angular momentum is shown to be crucial. A torque balance in simple shear flow at shear rate G leads to stress symmetry and specification of Ω(G) which differs greatly from the conventional Ω = G/2. This determines the G dependence of viscosity η and normal stress coefficient ζ. There are also implications of a transition in rotational behavior as φ approaches a critical value. Predictions of η(G), ζ(G), and η^*(ω) are presented for two versions of F^d : one derived recently by the authors and one being most commonly used at present. Limiting cases for high and low φ, and for high and low G and ω, are discussed. Some differences exist between predictions of the two F^d models, but these are surprisingly minor.     

Hosoya polynomials of TUC4C8(R) nanotubes

For a connected graph G, the Hosoya polynomial of G is defined as H(G, x) = ∑_{u,v}?V(G)x^{d(u, v)}, where V(G) is the set of all vertices of G and d(u,v) is the distance between vertices u and v. In this article, we obtain analytical expressions for Hosoya polynomials of TUC₄C₈(R) nanotubes. Furthermore, the Wiener index and the hyper‐Wiener index can be calculated. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

On the validity of the Avrami theorem in the case of decelerating growth

It is argued that the use of the Avrami theorem, S(t) = 1−e^−S_x(t), is in principle not allowed when applied to overlapping diffusion zones or, more generally, in all cases where the phantom nuclei overtake the front of their parent nuclei. Via computer simulations the overtake effect is shown to exist and is found to be surprisingly small. The use of a modified Avrami equation, S(t) = FS_x(t)·[1 −e^−S_x(t)], is suggested for such cases and the function F[S_x(t)] pertaining to diffusion-controlled growth is reported.     

Free Energy of a Non‐Gaussian Polymer Brush

An analytical theory describing layers of polymer chains grafted to a planar surface (i.e. polymer brush) is developed. We consider a brush of chains with finite extensibility (or non‐Gaussian brush) within the framework of molecular field theory. An analytical solution for free energy of the brush and a few other brush characteristics are obtained and studied. Comparison with other known models of a brush is also made.

Chain extensibility E(x, y) for Gaussian model (dashed lines) and BCC model (solid lines) for a few chain end positions y (numbers near curves).     

A Dinuclear Uranium Complex [{UO2F2(NH3)}2(μ-F)2]2− from Reaction of TlF and UO2F2 in Liquid Anhydrous Ammonia and Fluoride Ion Affinities for Some Uranyl(VI) Species [UO2Fx]2−x and [UO2Fx(NH3)5−x]2−x

UO₂F₂ abstracts F⁻ anions from TlF in liquid ammonia solution and the compound [Tl₂(NH₃)₆][{UO₂F₂(NH₃)}₂(μ-F)₂] is formed. The compound has been characterized by single crystal X-ray diffraction, Raman spectroscopy and quantum-chemical calculations for the solid state. Quantum-chemical investigation of the [{UO₂F₂(NH₃)}₂(μ-F)₂]²⁻ anion showed that the U−(μ-F)−U σ-3c-4e-bond is essentially ionic. The [Tl₂(NH₃)₆]²⁺ cation shows a thallophilic Tl⋅⋅⋅Tl interaction. Fluoride ion affinities (FIAs) were calculated for different UO₂²⁺ species [UO₂F_x]^2−x and [UO₂F_x(NH₃)_5−x]^2−x with x=0 to 4.     

Concentration dependence of the global and anisotropic dimensions of confined macromolecules

The chain dimensions 〈R²〉 of nondilute polymer solutions confined to a slit of the width D were studied using lattice simulations. It was found that the chain compression induced in good solvents by the concentration ϕ is enhanced in a slit relative to the bulk. The global dimensions of chains also change with ϕ in confined and unconfined theta solutions. At intermediate slit widths, a region was noted where coils are squeezed along all three axes. This region is manifested as a channel on the three‐dimensional surface 〈R²〉(D,ϕ) in both good and theta solvents. The coil anisotropy, given by the ratio of the parallel and perpendicular components of the chain dimensions 〈R_y²〉/〈R_x²〉, reaches high values at strong confinements, where coils form quasi‐two‐dimensional pancakes. The concentration‐induced reduction of the global chain dimensions in good solvents is almost fully transmitted to the parallel component 〈R_y²〉. The computed effects of concentration and confinement were compared with the predictions of mean‐field and scaling theories, and implications of the results to ultrathin films and layered nanocomposites were discussed. In addition, the distribution functions of the components of the end‐to‐end distance R perpendicular and parallel to the plates, W (R_x) and W (R_y), were calculated. The function W (R_x) combined with the concentration profile ϕ (x) along the pore provided details of the chain structure close to walls. A marked difference in the pace of the filling up of the depletion layer was noticed between chains in theta and good solvents. From the distribution functions W (R_x) and W (R_y), the highly anisotropic force‐elongation relations imply the deformation of chains in confined solutions and ultrathin bulk films.     

Rheological Behavior of Dope Solutions of Poly(acrylonitrile‐co‐itaconic acid) in N,N‐dimethylformamide: Effect of Polymer Molar Mass

The rheological behavior of dope solutions of poly(acrylonitrile‐co‐itaconic acid) or poly(AN‐co‐IA) is important from the point of view of deriving the spinning conditions for good quality special acrylic fibers. The viscosity of the resin dope is dictated by the polymer concentration, molar mass, temperature and shear force. The dynamic shear rheology of concentrated poly(AN‐co‐IA) polymer dope solutions in N, N‐dimethylformamide, in the molar mass (M¯_v) range of 1×10⁵ to 1×10⁶ g/mol, was investigated in the shear rate (γ′) range of 1×10¹ to 5×10⁴ min^?1. An empirical relation between η and M¯_v was found to exist at constant shear rate. The dope viscosity was dependent on the molar mass and the shear rate at a given temperature (T) and concentration. The polymer molar mass index of dope viscosity (m) was calculated as functions of concentration (c), shear rate and temperature. The m values increased with shear rate and temperature. A master equation relating m, with shear rate and temperature was derived for a given dope concentration. At higher shear rates, m tends to the value of 3.4, which is close to the molar mass index of viscosity reported for molten thermoplastics. m increased significantly with shear rate and nominally with temperature, while an increase in concentration decreased it. The onset of shear thinning of the dope shifted to a lower shear rate regime with an increase in polymer concentration and the molar mass. For a given value of molar mass, the increase in viscosity of the dope solution with polymer concentration was dependent on the shear rate.     

Compression of Polymer Brushes Under a Lateral Force

Summary: The influence of a lateral force (or lateral shear) acting on chains in a polymer brush is investigated theoretically. Brushes consisting of chains with temperature dependent anisotropic interactions between monomers (main‐chain mesogenic groups) are considered. It is shown that a lateral force applied to polymer brush induces its compression. In contrast to a conventional brush, the compression of brush, capable of forming a liquid crystalline (LC) state, can be caused by comparatively small shear forces. Moreover, such shear forces can induce a phase transition of a brush into the tilted LC state with a several‐fold decrease in brush thickness. These results allow us to predict a possibility to observe a decrease in brush thickness in a real experiment with reasonably values of shear rate.

Model of a chain in a polymer brush under an influence of lateral force p.     

Polydispersity effects on diffusion in polymers: Concentration profiles of d-polystyrene measured by forward recoil spectrometry

Forward recoil spectrometry is shown to be a useful technique for measuring diffusion of d-polymer chains in h-polymer melts. Concentration profiles of a deuterated diffusing species may be determined with a depth resolution of 80 nm and a sensitivity of 0.1 vol % d-polymer in h-polymer. Consequently diffusion coefficients as small as 10^?16 cm²/s can be readily measured. If polymer chains diffuse by a reptation mechanism, the concentration profile ø(x) of diffusing polydisperse polymer should be quite different from ø_m(x), the Fickian solution, which one obtains for monodisperse polymer. This idea was tested by measuring ø(x) of polydisperse d-polystyrene (d-PS) diffusing into h-PS. The results are in excellent agreement with the ø(x) predicted from the reptation model and the experimentally determined molecular weight distribution.     

Effect of the structure of latex particles on adhesion. Part II: Analogy between peel adhesion and rheological properties of acrylic copolymers

This article, the second part of this series, concerns the development of an analogy between the peel behavior of pressure-sensitive adhesives and the dynamic mechanical properties of the corresponding copolymers. The adhesive copolymers used were synthesized by emulsion polymerization processes. Their physical and dynamic mechanical properties were characterized and presented in Part I of this series. In this study, an analogy was built up between the force in a peel test as a function of peel velocity, F_p(v_p), and the loss modulus of the adhesive as a function of the angular frequency in a dynamic mechanical experiment, G″(ω). This was done by superimposing the curves of F_p versus v_p and those of G″ versus ωβ₀/β, where β₀/β is a shift factor with β being a parameter in the Kaelble theory and β₀ being some reference value of the Kaelble parameter. When the curves of F_p ～ v_p and those of G″ ～ ωβ₀ were plotted together, they followed the same trend of variation. This analogy between G″(ωβ₀/β) and F_p(v_p) was further confirmed by the fact that the apparent activation energies of the primary glass transition for G″(ω) and F_p(v_p) are virtually the same, suggesting that the analogy between G″(ω) and F_p(v_p) is dictated by the glass transition. The existence of the above-mentioned analogy between G″(ω) and F_p(v_p) shows that the performance of an adhesive can be evaluated or predicted from the dynamical loss modulus of the corresponding (co)polymer. ©1995 John Wiley & Sons, Inc.     

Drag on a metallic or nonmetallic particle exposed to a rarefied plasma flow

Drag force on a metallic or nonmetallic spherical particle exposed to a plasma flow is studied for the extreme case of a free-molecule regime. Analytical expressions are derived for the drag components due to, respectively, atoms, ions, and electrons and for the total drag on the whole sphere due to all the gas species. It has been shown that the drag is proportional to the square of the particle radius or the drag coefficient is independent of the particle radius. At low gas temperatures with a negligible degree of ionization, the drag is caused mainly by atoms and could be predicted by using the well-known drag expression given in ordinary-temperature rarefied gas dynamics. On the other hand, the drag is caused mainly by ions at high plasma temperatures with a great degree of ionization. The contribution of electrons to the total drag is always negligible. Ignoring gas ionization at high plasma temperatures would overestimate the particle drag. There is a little difference between metallic and nonmetallic spheres in their total drag forces, with a slightly higher value for a metallic sphere at high plasma temperatures, but usually such a small difference could be neglected in engineering calculations. The drag increases rapidly with increasing gas pressure or oncoming speed ratio. For a two-temperature plasma, the drag increases at low electron temperatures but decreases at high electron temperatures with the increase in the electron/heavy-particle temperature ratio.Nomenclature C _d Drag coefficient - e Elementary charge - f _D,F _D Local and total drag (N/m ² andN) - f ^– Velocity distribution function for incident gas particles - f ⁺ Velocity distribution function for reflected gas particles - k Boltzmann's constant - m Gas particle mass (kg) - n Number density of gas species (m ^–3) - P ^–,P ⁺ Surface pressure due to incident and reflected gas particles - R ₀ Sphere radius (m) - S Speed ratio,S _j=U/(2kT _j/m_j)^1/2 - T _e,T _h Electron and heavy-particle (atom, ion) temperature - T _w Wall temperature - U Oncoming plasma flow velocity - v _x, v_y, v_z Velocity components of gas particles in thex, y, andz directions (m/sec) - v Thermal motion speed of gas particles,v _j =(8kT _j /m _j )^1/2 - v _ze Smallestv _z of electrons which could reach the sphere surface,v _ze=(2e/m _e)^1/2 (m/sec) - v _zw Value ofv _z of ions or electrons as arriving at the sphere surface (m/sec) - Center angle - Gas density (kg/m³) - Shear stress (N/m²) - Absolute value of the floating potential (V) - , Local and total particle fluxes incident to the surface - a Atoms - e Electrons - h Heavy particles - i Ions - j jth gas species - m Metallic sphere - mn Nonmetallic sphere A preliminary version of this paper was presented at the Eighth International Symposium on Plasma Chemistry held in Tokyo, September 1987.     

Dynamics of a polymer attached to a surface: Bead and spring model

The Smoluchowski formalism is used to solve the problem of a bead of frictional resistance β attached to a surface with a spring of force constant _k over which a linear shear field of strenght α flows. The power dissipation is given by βα²kT/_k. k and T have their usual meanings. The result is generalized to an n-bead polymer. It is found that the power dissipation of a Rouse model polymer attached to a surface at one end is twice that of an identical polymer flowing freely in solution. If the force constant _k arises from an entropy force, then, because of the effect of the surface on the number of polymer configurations, there is an additional factor of two. The same relationship is expected to also hold for the frequency-dependent power dissipation. It is argued that a net circulation exists in the beads above the surface and that the magnitude of the circulation is roughly comparable to that which exists in a polymer freely rotating in solution under a shear field of the same magnitude.     

Yields of O2(1Σg+) and O2(1Δg) in the H + O2 reaction system,and the quenching of O2(1Σg+) by atomic hydrogen

The generation of metastable O₂(¹Σg⁺) and O₂(¹Δg) in the H + O₂ system of reactions was studied by the flow discharge chemiluminescence detection method. In addition to the O₂(¹Σg⁺) and O₂(¹Δg) emissions, strong OH(v = 2) → OH(v = 0), OH(v = 3) → OH(v = 1), HO₂(²A′₀₀₀) → HO₂(²A″₀₀₀), HO₂(²A′₀₀₁) → HO₂(²A″₀₀₀), and H O₂(²A″₂₀₀) → HO₂(²A″₀₀₀) emissions were detected in the H + O₂ system. The rate constants for the quenching of O₂(¹Σg⁺) by H and H₂ were determined to be (5.1 ± 1.4) × 10^?13 and (7.1 ± 0.1) × 10^?13 cm³ s^?1, respectively. An upper limit for the branching ratio to produce O₂(¹Σg⁺) by the H + HO₂ reaction was calculated to be 2.1%. The contributions from other reactions producing singlet oxygen were investigated.     

Identification of unknown diffusion coefficient in pure diffusive linear model of chronoamperometry. I. The theory

Coefficient identification problem for diffusion equation u _t (x, t) = (D(x)u _x (x, t)) _x arising in chronoamperometry is studied. The adjoint problem approach is developed for the case when the output measured data is given in the form of left/right flux. Analytical formulas for determination of the values D(0), D(L) at the endpoints x = 0; L, of the unknown coefficient D(x), via the solution v(x, t) of the constant coefficient equation v _t (x, t) = D v _xx (x, t) is obtained. The integral identity relating solutions of the forward and corresponding adjoint problems is derived. This integral identity permits one to prove the monotonicity and invertibility of input-output map, as well as formulate the gradient of the cost functional via the solutions of the direct and adjoint problems.     

Comparison of Characteristic Structural Features among the Triade of Tris(cyclopentadienyl)(Group-4 Metal) Complex Cations: a Combined Theoretical and Experimental Study

A density functional theory computational chemistry study has revealed a fundamental structural difference between [Ti(Cp)₃]⁺ and its congeners [Zr(Cp)₃]⁺ and [Hf(Cp)₃]⁺/(Cp=cyclopentadienyl). Whereas the latter two are found to contain three uniformely η⁵-coordinated Cp ligands (3η⁵-structural type), [Ti(Cp)₃]⁺ is shown to prefer a 2η⁵η² structure. [Ti(Cp)₃]⁺[B(C₆F₅)₃(Me)]⁻ ( 10 ⋅[B(C₆F₅)₃(Me)]⁻) was experimentally generated by treatment of [Ti(Cp)₃(Me)] ( 7a ) with B(C₆F₅)₃ (Scheme 3). Low-temperature ¹H-NMR spectroscopy in CDFCl₂ (143 K, 600 MHz; Fig. 8) showed a splitting of the Cp resonance into five lines in a 2 : 5 : 2 : 5 : 1 ratio which would be in accord with the theoretically predicted 2η⁵η²-type structure of [Ti(Cp)₃]⁺. The precursor [Ti(Cp)₃(Me)] ( 7a ) exhibits two ¹H-NMR Cp resonances in a 10 : 5 ratio in CD₂Cl₂ at 223 K. Treatment of [HfCl(Cp)₂(Me)] ( 6c ) with sodium cyclopentadienide gave [Hf(Cp)₃(Me)] ( 7c ) (Scheme 1). Its reaction with B(C₆F₅)₃ furnished the salt [Hf(Cp)₃]⁺[B(C₆F₅)₃(Me)]⁻ ( 8 ⋅[B(C₆F₅)₃(Me)]⁻), which reacted with tert-butyl isocyanide to give the cationic complex [Hf(Cp)₃(C=N−CMe₃)]⁺ ( 9a ; with counterion [B(C₆F₅)₃(Me)]⁻ (Scheme 2). Complex cation 9a was characterized by X-ray diffraction (Fig. 7). Its Hf(Cp₃) moiety is of the 3η⁵-type. The structure is distorted trigonal-pyramidal with an average D−Hf−D angle of 118.8° and an average D−Hf−C(1) angle of 96.5° (D denotes the centroids of the Cp rings; Table 6). Cation 9a is a typical d⁰-isocyanide complex exhibiting structural parameters of the C≡N−CMe₃ group (d(C(1)−N(2))=1.146 (5) Å; IR: v˜(C≡N) 2211 cm⁻¹) very similar to free uncomplexed isonitrile. Analogous treatment of 8 with carbon monoxide yielded the carbonyl (d⁰-group-4-metal) complex [Hf(Cp)₃(CO)]⁺ ( 9b ; with counterion [B(C₆F₅)₃(Me)]⁻) (Scheme 2) that was also characterized by X-ray crystal-structure analysis (Fig. 6). Complex 9b is also of the 3η⁵-structural type, similar to the peviously described cationic complex [Zr(Cp)₃(CO)]⁺, and exhibits properties of the CO ligand (d(C−O)=1.11 (2) Å; IR: v˜(C≡O) 2137 cm⁻¹) very similar to the free carbon monoxide molecule.     

Control of Luminescence by Tuning of Crystal Symmetry and Local Structure in Mn4+‐Activated Narrow Band Fluoride Phosphors

Mn⁴⁺‐doped fluoride phosphors have been widely used in wide‐gamut backlighting devices because of their extremely narrow emission band. Solid solutions of Na₂(Si_xGe_1?x)F₆:Mn⁴⁺ and Na₂(Ge_yTi_1?y)F₆:Mn⁴⁺ were successfully synthesized to elucidate the behavior of the zero‐phonon line (ZPL) in different structures. The ratio between ZPL and the highest emission intensity υ₆ phonon sideband exhibits a strong relationship with luminescent decay rate. First‐principles calculations are conducted to model the variation in the structural and electronic properties of the prepared solid solutions as a function of the composition. To compensate for the limitations of the Rietveld refinement, electron paramagnetic resonance and high‐resolution steady‐state emission spectra are used to confirm the diverse local environment for Mn⁴⁺ in the structure. Finally, the spectral luminous efficacy of radiation (LER) is used to reveal the important role of ZPL in practical applications.     

Liquid–liquid phase separation in multicomponent polymer solutions. IX. Concentration-dependent pair interaction parameter from critical miscibility data on the system polystyrene–cyclohexane

Critical miscibility data obtained from measurements of phase-volume ratios have been used to calculate the concentration dependence of the pair interaction parameter for the system polystyrene–cyclohexane. The measured temperature and concentration ranges are 11–30°C and 4–18% polymer by weight, respectively. With the Gibbs free energy of mixing expressed in polymer segment mole fractions, x^*, the pair interaction parameter is g(x^*, T) = 0.4961 + 71.92/T + 0.2312x^* + 0.0750x^*2. In a polymer volume fraction formulation the parameter is g(φ, T) = 0.4099 + 90.65/T + 0.2064 φ + 0.0518 φ², which approximates to χ(φ, T) = 0.2035 + 90.65/T + 0.3092 φ + 0.1554 φ². Comparison of the temperature and concentration dependence with that obtained by other authors shows very good agreement, even when extensive extrapolations in temperature and concentration are applied. The present function is believed to be the most accurate. Solutions of mixtures of two narrow-distribution polystyrenes in cyclohexane show separation into three liquid phases under the exact conditions predicted by theoretical calculation with the present pair-interaction function.     

Trifluoromethylation of [AuF3(SIMes)]: Preparation and Characterization of [Au(CF3)xF3−x(SIMes)] (x=1–3) Complexes

Trifluoromethylation of [AuF₃(SIMes)] with the Ruppert–Prakash reagent TMSCF₃ in the presence of CsF yields the product series [Au(CF₃)_xF_3−x(SIMes)] (x=1–3). The degree of trifluoromethylation is solvent dependent and the ratio of the species can be controlled by varying the stoichiometry of the reaction, as evidenced from the ¹⁹F NMR spectra of the corresponding reaction mixtures. The molecular structures in the solid state of trans-[Au(CF₃)F₂(SIMes)] and [Au(CF₃)₃(SIMes)] are presented, together with a selective route for the synthesis of the latter complex. Correlation of the calculated SIMes affinity with the carbene carbon chemical shift in the ¹³C NMR spectrum reveals that trans-[Au(CF₃)F₂(SIMes)] and [Au(CF₃)₃(SIMes)] nicely follow the trend in Lewis acidities of related organo gold(III) complexes. Furthermore, a new correlation between the Au−C_carbene bond length of the molecular structure in the solid state and the chemical shift of the carbene carbon in the ¹³C NMR spectrum is presented.