k-resonant toroidal polyhexes

A toroidal polyhex H(p, q, t) is a cubic bipartite graph embedded on the torus such that each face is a hexagon, which can be described by a string (p, q, t) of three integers (p≥ 1, q≥ 1, 0≤ t≤ p−1). A set of mutually disjoint hexagons of H(p, q, t) is called a resonant pattern if H(p, q, t) has a prefect matching M such that all haxgons in are M-alternating. A toroidal polyhex H(p, q, t) is k-resonant if any i (1 ≤ i ≤ k) mutually disjoint hexagons form a resonant pattern. In [16], Shiu, Lam and Zhang characterized 1, 2 and 3-resonant toroidal polyhexes H(p, q, t) for min(p, q)≥ 2. In this paper, we characterize k-resonant toroidal polyhexes H(p, 1, t). Furthermore, we show that a toroidal polyhex H(p, q, t) is k-resonant (k≥ 3) if and only if it is 3-resonant.      

The forcing number of toroidal polyhexes

The forcing number, denoted by f(G), of a graph G with a perfect matching is the minimum number of independent edges that completely determine the perfect matching of G. In this paper, we consider the forcing number of a toroidal polyhex H(p,q,t) with a torsion t, a cubic graph embedded on torus with every face being a hexagon. We obtain that f(H(p,q,t)) ≥ min{p,q}, and equality holds for p ≤ q or p > q and t∈{ 0,p−q,p−q + 1,..., p−1}. In general, we show that f(H(p,q,t)) is equal to the side length of a maximum triangle on H(p,q,t). Based on this result, we design a linear algorithm to compute the forcing number of H(p,q,t).     

<Emphasis Type="Italic">k</Emphasis>-resonance in Toroidal Polyhexes*

This paper considers the k -resonance of a toroidal polyhex (or toroidal graphitoid) with a string (p, q, t) of three integers (p ≥ 2, q ≥ 2, 0 ≤ t ≤ p − 1). A toroidal polyhex G is said to be k-resonant if, for 1≤ i ≤ k, any i disjoint hexagons are mutually resonant, that is, G has a Kekulé structure (perfect matching) M such that these hexagons are M-alternating (in and off M). Characterizations for 1, 2 and 3-resonant toroidal polyhexes are given respectively in this paper. *This work is supported by FRG, Hong Kong Baptist University; NSFC and TRAPOYT.     

Study of Scaling Exponents of the Surface Evolved in 2 + 1 Dimension through Competitive Growth between Random Deposition and that with Surface Relaxation

Rough surface develops through computer simulation by competition between growth mechanism random deposition (RD) with a probability of occurrence p and growth mechanism random deposition with surface relaxation (RDSR) with a probability of occurrence 1 − p, on L × L square plane for system size L to record the statistical average of time variation of surface roughness W(L, t) and average height H(t) for the model for specific values of L and p. Other than the pure RD model, the entire evolution may be divided into three regions separated by two specific cross-over times t_x and t_sat. The value of interface width at saturation W_sat depends on both L and p. The first growth exponent β₁ increases exponentially with an increase in p and does not depend on L. The values of the second growth exponent β₂, roughness exponent α, dynamic exponent z( = α/β₂ ), and α + z are 0.0234 ± 0.0008, 0.0506 ± 0.0065, 2.1577 ± 0.0073, and 2.2083 ± 0.0138 respectively and they show no dependence on L and p values. Value of the first cross-over time t_x increases exponentially with an increase in p and does not depend on L. Value of the second cross-over time t_sat increases with an increase in both p and L values. The average growth velocity is unity for the model and is independent of both L and p. For the model, the growth velocity is unity and the fractional porosity is zero. The scaling exponents show some deviation from the relevant universality classes and depend on competitive growth probability for this model. No finite-size effect is present in the model.     

Spinodal decomposition and phase separation kinetics in nanoclay–biopolymer solutions

Intensity of light, I(q,t), scattered from homogeneous aqueous solutions, of nanoclay (Laponite) and protein (gelatin‐A), was studied to monitor the temporal and spatial evolution of the solution into a phase‐separated nanoclay–protein‐rich dense phase, when the sample temperature was quenched below spinodal temperature, T_s (=311 ± 3 K). The zeta potential data revealed that the dense phase comprised charge‐neutralized intermolecular complexes of nanoclay and protein chains of low surface charge. The early stage, t < 500 s, of phase separation could be described adequately through Cahn‐Hilliard theory of spinodal decomposition where the intensity grows exponentially, I(q, t) = I₀ exp.(2R(q)t). The wave vector, q dependence of the growth parameter, R(q) exhibited a maxima independent of time. Corresponding correlation length, 1/q_c = ξ_c was found to be ≈75 ± 5 nm independent of quench depth. In the intermediate regime, anomalous growth described by I(q, t) ～ t^α with α = 0.1 ± 0.02 independent of q was observed. Rheological studies established that there was a propensity of network structures inside the dense phase. Isochronal temperature sweep studies of the dense phase determined the melting temperature, T_m = 312 ± 4 K, which was comparable with the spinodal temperature. The stress‐diffusion coupling prevailing in the dense phase when analyzed in the Doi‐Onuki model yielded a viscoelastic correlation length, ξ_v determined from low‐frequency storage modulus, G ′₀ ≈ k_B T/ξ, which was ξ_v ≈ 35 ± 3 nm indicating 2ξ_v ≈ ξ_c. It is concluded that the early stage of phase separation in this system was sufficiently described by linear Cahn‐Hilliard theory, but the same was not true in the intermediate stage. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 555–565, 2010     

THE REACTIVITY OF CITRONELLOL and α-THUJENE WITH SINGLET OXYGEN. RATE CONSTANTS OF CHEMICAL REACTION and PHYSICAL QUENCHING*

The quantum yields of Rose Bengal sensitized photooxidation of citronellol and α-thujene have been determined as a function of added acceptor and compared with those of furfuryl alcohol as a standard. The results permitted the calculation of the corresponding rate constants of chemical reaction (k_T) and physical quenching (K_q) of singlet oxygen. The sum (k_T+ k_q) has been verified independently by a Stern-Volmer analysis of the singlet oxygen luminescence quenching. α-Thujene reacts faster with singlet oxygen than citronellol, physical quenching being negligible in both cases.     

Propagation rate coefficients of isobornyl acrylate,tert‐butyl acrylate and 1‐ethoxyethyl acrylate: A high frequency PLP‐SEC study

Pulsed laser polymerization (PLP) coupled to size exclusion chromatography (SEC) is considered to be the most accurate and reliable technique for the determination of absolute propagation rate coefficients, k_p. Herein, k_p data as a function of temperature were determined via PLP‐SEC for three acrylate monomers that are of particular synthetic interest (e.g., for the generation of amphiphilic block copolymers). The high‐T_g monomer isobornyl acrylate (iBoA) as well as the precursor monomers for the synthesis of hydrophilic poly(acrylic acid), tert‐butyl acrylate (tBuA), and 1‐ethoxyethyl acrylate (EEA) were investigated with respect to their propagation rate coefficient in a wide temperature range. By application of a 500 Hz laser repetition rate, data could be obtained up to a temperature of 80 °C. To arrive at absolute values for k_p, the Mark‐Houwink parameters of the polymers have been determined via on‐line light scattering and viscosimetry measurements. These read: K = 5.00 × 10⁵ dL g⁻¹, a = 0.75 (piBoA), K = 19.7 × 10⁵ dL g⁻¹, a = 0.66 (ptBA) and K = 1.53 × 10⁵ dL g⁻¹, a = 0.85 (pEEA). The bulky iBoA monomer shows the lowest propagation rate coefficient among the three monomers, while EEA is the fastest. The activation energies and Arrhenius factors read: (iBoA): log(A/L mol⁻¹ s⁻¹) = 7.05 and E_A = 17.0 kJ mol⁻¹; (tBuA): log(A/L mol⁻¹ s⁻¹) = 7.28 and E_A = 17.5 kJ mol⁻¹ and (EEA): log(A/L mol⁻¹ s⁻¹) = 6.80 and E_A = 13.8 kJ mol⁻¹. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 6641–6654, 2009     

DETERMINATION OF THE ACRYLAMIDE QUENCHING CONSTANT FOR PROTEIN AND MODEL INDOLE TRIPLETS

Abstract— The decay of the indole triplet of single tryptophan-containing proteins and model compounds can be readily measured at room temperature in aqueous solution by monitoring the triplet-triplet absorption or phosphorescence emission following a 265 nm exciting laser pulse. The quenching action of acrylamide on the triplet excited state of indole side chains was studied in an analogous fashion to that previously done at the singlet level (Eftink and Ghiron, 1977). The acrylamide triplet quenching constant (^tk_q) ranged from a high of 7.8 times 10⁸M^-1 s^-1 for the exterior indole of corticotropin (ACTH) to a low of 2 times 10⁵ Af^-1 s^-1 for the interior indole of ribonuclease T, (RNase T,). The ratio (7) of these values with their respective acrylamide singlet quenching constants (^tk_q),(γ=^tk_q⁸K_q) ranged from a high of 0.22 for ACTH to a low of 0.001 for RNase T₁,. Acrylamide is also an inefficient quencher of model indoles in various solvents (i.e. it has a γ less than 1). The magnitude of γ varied from a high of 0.3 in H₂0 to a low of 0.02 in acetonitrile, but did not correlate with viscosity, dielectric constant or polarity. The lower efficiency observed for internal indole groups can not be explained by that class of models which predict the presence of static quenching at the triplet level, since none was observed. The present results confirm the observation of Calhoun et al. of a large discrepancy between acrylamide's singlet and triplet quenching constants for buried indole side chains, but suggest that it may be largely explained by the fact that acrylamide is an inefficient quencher of the indole triplet state (1983). The magnitude of this inefficiency is probably determined by specific microenvironmental factors. Thus, unlike ⁸K_q, the environmentally sensitive ^lk_H cannot be easily used to characterize the dynamics of proteins.     

Benchmarking and validating algorithms that estimate p<Emphasis Type="Italic">K</Emphasis><Subscript>a</Subscript> values of drugs based on their molecular structures

The REGDIA regression diagnostics algorithm in S-Plus is introduced in order to examine the accuracy of pK _a predictions made with four updated programs: PALLAS, MARVIN, ACD/pKa and SPARC. This report reviews the current status of computational tools for predicting the pK _a values of organic drug-like compounds. Outlier predicted pK _a values correspond to molecules that are poorly characterized by the pK _a prediction program concerned. The statistical detection of outliers can fail because of masking and swamping effects. The Williams graph was selected to give the most reliable detection of outliers. Six statistical characteristics (F _exp, R ², , MEP, AIC, and s(e) in pK _a units) of the results obtained when four selected pK _a prediction algorithms were applied to three datasets were examined. The highest values of F _exp, R ², , the lowest values of MEP and s(e), and the most negative AIC were found using the ACD/pK _a algorithm for pK _a prediction, so this algorithm achieves the best predictive power and the most accurate results. The proposed accuracy test performed by the REGDIA program can also be applied to test the accuracy of other predicted values, such as log P, log D, aqueous solubility or certain physicochemical properties of drug molecules.     

Kinetics and Mechanism of Oxidation of S2O32− by a Co‐Bound μ‐Amido‐μ‐Superoxo Complex

In acetate buffer media (pH 4.5–5.4) thiosulfate ion (S₂O₃^2?) reduces the bridged superoxo complex, [(NH₃)₄Co^III(μ‐NH₂,μ‐O₂)Co^III(NH₃)₄]⁴⁺ ( 1 ) to its corresponding μ‐peroxo product, [(NH₃)₄Co^III(μ‐NH₂,μ‐O₂)Co^III(NH₃)₄]³⁺ ( 2 ) and along a parallel reaction path, simultaneously S₂O₃^2? reacts with 1 to produce the substituted μ‐thiosulfato‐μ‐superoxo complex, [(NH₃)₄Co^III(μ‐S₂O₃,μ‐O₂)Co^III(NH₃)₄]³⁺ ( 3 ). The formation of μ‐thiosulfato‐μ‐superoxo complex ( 3 ) appears as a precipitate which on being subjected to FTIR shows absorption peaks that support the presence of Co(III)‐bound S‐coordinated S₂O₃^2? group. In reaction media, 3 readily dissolves to further react with S₂O₃^2? to produce μ‐thiosulfato‐μ‐peroxo product, [(NH₃)₄Co^III(μ‐S₂O₃,μ‐O₂)Co^III(NH₃)₄]²⁺ ( 4 ). The observed rate (k₀) increases with an increase in [T_Thio] ([T_Thio] is the analytical concentration of S₂O₃^2?) and temperature (T), but it decreases with an increase in [H⁺] and the ionic strength (I). Analysis of the log A_t versus time data (A is the absorbance of 1 at time t) reveals that overall the reaction follows a biphasic consecutive reaction path with rate constants k₁ and k₂ and the change of absorbance is equal to {a₁ exp(–k₁t) + a₂ exp(–k₂t)}, where k₁ > k₂.     

Modeling Termination Kinetics of Non‐Stationary Free‐Radical Polymerizations

Pulsed‐laser induced polymerization is modeled via an approach presented in a previous paper.^[1] An equation for the time dependence of free‐radical concentration is derived. It is shown that the termination rate coefficient may vary significantly as a function of time after applying the laser pulse despite of the fact that the change in monomer concentration during one experiment is negligible. For the limiting case of t ≫ c^–1 (k_pM)^–1, where c is a dimensionless chain‐transfer constant, k_p the propagation rate coefficient and M the monomer concentration, an analytical expression for k_t is derived. It is also shown that time‐resolved single pulse‐laser polymerization (SP–PLP) experiments can yield the parameters that allow the modeling of k_t in quasi‐stationary polymerization. The influence of inhibitors is also considered. The conditions are analyzed under which M (t) curves recorded at different extents of laser‐induced photo‐initiator decomposition intersect. It is shown that such type of behavior is associated with a chain‐length dependence of k_t.     

On the Nitroxide Quasi‐Equilibrium in the Alkoxyamine‐Mediated Radical Polymerization of Styrene

Summary: The range of validity of two popular versions of the nitroxide quasi‐equilibrium (NQE) approximation used in the theory of kinetics of alkoxyamine mediated styrene polymerization, are systematically tested by simulation comparing the approximate and exact solutions of the equations describing the system. The validity of the different versions of the NQE approximation is analyzed in terms of the relative magnitude of (dN/dt)/(dP/dt). The approximation with a rigorous NQE, k_c[P][N] = k_d[P N], where P, N and P N are living, nitroxide radicals and dormant species respectively, with kinetic constants k_c and k_d, is found valid only for small values of the equilibrium constant K (10⁻¹¹–10⁻¹² mol · L⁻¹) and its validity is found to depend strongly of the value of K. On the other hand, the relaxed NQE approximation of Fischer and Fukuda, k_c[P][N] = k_d[P N]₀ was found to be remarkably good up to values of K around 10⁻⁸ mol · L⁻¹. This upper bound is numerically found to be 2–3 orders of magnitude smaller than the theoretical one given by Fischer. The relaxed NQE is a better one due to the fact that it never completely neglects dN/dt. It is found that the difference between these approximations lies essentially in the number of significant figures taken for the approximation; still this subtle difference results in dramatic changes in the predicted course of the reaction. Some results confirm previous findings, but a deeper understanding of the physico‐chemical phenomena and their mathematical representation and another viewpoint of the theory is offered. Additionally, experiments and simulations indicate that polymerization rate data alone are not reliable to estimate the value of K, as recently suggested.

Validity of the rigorous nitroxide quasi‐equilibrium assumption as a function of the nitroxide equilibrium constant.     

On the Chirality of Torus Curves and Knots

As is well known, a (p, q) torus knot is topologically equivalent to a (q, p) torus knot. The sign of the writhe number, which characterizes the topological chirality, must evidently be the same in both cases. We here show by an analytic criterion related to the torsion that a (p, q) torus curve and a (q, p) torus curve have opposite chirality, although they are not enantiomers. An erratum to this article can be found at     

Extremal hexagonal chains concerning k-matchings and k-independent sets

Denote by _n the set of the hexagonal chains with n hexagons. For any B _{n n} , let m _k (B _n ) and i _k (B _n ) be the numbers of k-matchings and k-independent sets of B _n , respectively. In the paper, we show that for any hexagonal chain B _{n n} and for any k0, m _k (L _n )m _k (B _n )m _k (Z _n ) and i _k (L _n )i _k (B _n )i _k (Z _n ), with left equalities holding for all k only if B _n =L _n , and the right equalities holding for all k only if B _n =Z _n , where L _n and Z _n are the linear chain and the zig-zag chain, respectively. These generalize some related results known before.     

Vinyl polymerization: Kinetics of Ce(IV)–acetophenone-initiated polymerization of acrylonitrile in acetic–sulfuric acid mixtures

The polymerization of acrylonitrile (M) initiated by the Ce(IV)–acetophenone (AP) redox pair has been studied in acetic–sulfuric acid mixtures in a nitrogen atmosphere. The rate of polymerization is proportional to [M]^3/2, [AP]^1/2 and [Ce(IV)]^1/2. The rate of disappearance of ceric ion,–R_Ce, is proportional to [AP], [M], and [Ce(IV)]. The effect of certain salts, solvent, acid and temperature on both the rates have been investigated. A suitable kinetic scheme has been proposed, and the composite rate constants k_p ²(k/k/_t) and k₀/k_i are reported.     

Quasielastic light scattering study of chemically crosslinked gelatin solutions and gels

Quasielastic light scattering measurements are reported for experiments performed on mixtures of gelatin and glutaraldehyde (GA) in the aqueous phase, where the gelatin concentration was fixed at 5 (w/v) and the GA concentration was varied from 1×10⁻⁵ to 1×10⁻³ (w/v). The dynamic structure factor, S(q,t), was deduced from the measured intensity autocorrelation function, g ₂(τ), with appropriate allowance for heterodyning detection in the gel phase. The S(q,t) data could be fitted to S(q,t)=Aexp(−D _f q ² t)+Bexp(−t/τ_c)^β, both in the sol (50 and 60 ^∘C) and gel states (25 and 40 ^∘C). The fast-mode diffusion coefficient, D _f showed almost negligible dependence on the concentration of the crosslinker GA; however, the resultant mesh size, ξ, of the crosslinked network exhibited strong temperature dependence, ξ∼(0.5−χ)^1/5exp(−A/RT) implying shrinkage of the network as the gel phase was approached. The slow-mode relaxation was characterized by the stretched exponential factor exp(−t/τ_c)^β. β was found to be independent of GA concentration but strongly dependent on the temperature as β=β₀+β₁ T+β₂ T ². The slow-mode relaxation time, τ_c, exhibited a maximum GA concentration dependence in the gel phase and at a given temperature we found τ_c(c)=τ₀+τ₁ c+τ₂ c ². Our results agree with the predictions of the Zimm model in the gel case but differ significantly for the sol state. Received: 25 May 1999 /Accepted in revised form: 27 July 1999     

The use of chebyshev polynomials orthogonal on a finite arbitrary system of points for interpolating changes in nematic-isotropic liquid phase transition temperatures in homologous series

Chebyshev polynomials Ψ _q (x) orthogonal on a finite arbitrary system of points x _i (i = 1−N) are used to interpolate changes in nematic-isotropic liquid phase transition temperatures t _c(x) in homologous series of liquid crystals (x = 1/n, where n is the number of the homologue). The expansion of the t _c(x) function into a series in Ψ _q (x) polynomials was found to be very effective. Already at q ≤ 3, this series describes the known types of the t _c(x) dependences with high accuracy and very small root-mean-square deviations for mesogenic molecules of various chemical structures and dimensions. The dependence of the limiting t _l = t _c(0) value on the form of X-shaped molecules and linear dimensions of N-mers with N rigid aromatic fragments linked with each other by flexible spacer chains was studied.

     

On the (im)possibility of evaluating correct individual rate constants of chain propagation kp and chain termination kt by combining kp2/kt and kp/kt data for chain‐length dependent termination

On the basis of simulated data two ways of evaluating individual rate constants by combining k_p²/k_t and k_p /k_t (k_p , k_t = rate constants of chain propagation and termination, respectively) were checked considering the chain‐length dependence of k_t. The first way tried to make use of the fact that pseudostationary polymerization yields data for k_p²/k_t as well as for k_p /k_t referring to the very same experiment, in the second way k_p²/k_t (from steady state experiments) and k_p/k_t data referring to the same mean length of the terminating radical chains were compared. In the first case no meaningful data at all could be obtained because different averages of k_t are operative in the expressions for k_p /k_t and k_p²/k_t. In spite of the comparatively small difference between these two averages (≈15% only) this makes the method collapse. The second way, which can be regarded as an intelligent modification of the “classical” method of determining individual rate constants, at least succeeded in reproducing the correct order of magnitude of the individual rate constants. However, although stationary and pseudostationary experiments independently could be shown to return the same k_t for the same average chain‐length of terminating radicals within extremely narrow limits no reasonable chain‐length dependence of k_t could be derived in this way. The reason is an extreme sensitivity of the pair of equations for k_p/k_t and k_p²/k_t towards small errors and inconsistencies which renders the method unsuccessful even for the high quality simulation data and most probably makes it even collapse for real data. This casts a characteristic light on the unsatisfactory situation with respect to individual rate constants determined in the classical way, regardless of a chain‐length dependence of termination. As a consequence, all efforts of establishing the chain‐length dependence of k_t are recommended to avoid this way and should rather resort to methods based on inserting a directly determined k_p into the equations characteristic of k_p²/k_t or k_p/k_t, properly considering the chain‐length dependent character of k_t.