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.     

Effect of surfactant adsorption time on the observation of Newton black film in foam film

Observation of Newton black film (NBF) in foam film is possible only with a certain probability W which depends on the concentration C of surfactant in the solution and on the time t_a during which adsorption of surfactant at the solution/air interface has taken place. In the paper, the W(C,t_a) dependence is derived and used to analyze the effect of t_a on the critical surfactant concentration C_c below which NBF in foam film practically cannot be observed. An expression for the C_c(t_a) function is obtained which reveals that C_c decreases substantially with increasing t_a. This expression is found to describe well experimental C_c(t_a) data for foam films obtained from aqueous solution of the therapeutic surfactant INFASURF.     

Diffusion with discontinuous swelling. V. Type II diffusion into sheets and spheres

Type II diffusion into uniform spheres (radius R) and sheets (thickness 2l) is calculated under the assumption that the glass-gel boundary proceeds at a constant velocity v from the surface towards the interior of the sample, that the diffusion coefficient D_g in the glass is constant and that the diffusion coefficient D_r of the rubbery gel is so much higher than vR or vl that practically no sorbate gradient is needed for the transport through the gel of the sorbate. The diffusion process is completed when this boundary reaches the center of the sample. The concentration profile of the sorbate in the glassy matrix in front of the boundary varies with time and velocity v. It does not, however, influence the boundary propagation velocity. Hence the often observed increase of the rate of the weight gain just at the end of the diffusion process is not considered at all. The relative weight gain of the sample W(t)/W_∞ as a function of time is the only quantity usually measured. From the ordinate intercept A and the initial slope B of the plot of W(t)/t^1/2W_∞ vs. t^1/2, one can calculate the characteristic transport properties, i.e., the diffusion coefficient D_g of the glass and the velocity v of the glass–gel boundary.     

Dynamic Monte Carlo Simulation on the Polymer Chain with One End Grafted on a Flat Surface

A linear polymer chain in good solvent condition with one end grafted on a infinitely large, impenetrable flat surface is investigated using dynamic Monte Carlo simulation on a simple cubic lattice. Chain shape and dimension, angular correlation between the direction of the end‐to‐end vector and that of the longest principal axis of inertia are studied and discussed. Results reveal that the asphericity of end‐grafted polymer chains is greater than that of free ones, the limit ratio 〈L₁²〉 : 〈L₂²〉 : 〈L₃²〉 is about 1 : 3.0 : 14.9. The limit of mean angle 〈θ〉 of end‐grafted chains is about 22°, smaller than that of free chains, indicating angular correlation between the direction of the end‐to‐end vector and that of the longest principal axis is reinforced.     

Convergence of the partial wave expansion of the He ground state

The configuration interaction (CI) method, using a very large Laguerre orbital basis, is applied to the calculation of the He ground state. The largest calculations included a minimum of 35 radial orbitals for each ? ranging from 0 to 12, resulting in basis sets in excess of 400 orbitals. The convergence of the energy and electron–electron δ‐function with respect to J (the maximum angular momenta of the orbitals included in the CI expansion) were investigated in detail. Extrapolations to the limit of infinite angular momentum using expansions of the type ΔX_J = A_X[J + 1/2]^?p + B_X[J + 1/2]^?p?1 + …, gave an energy accurate to 10^?7 Hartree and a value of 〈δ〉 accurate to about 0.5%. Improved estimates of 〈E〉 and 〈δ〉, accurate to 10^?8 Hartree and 0.01%, respectively, were obtained when extrapolations to an infinite radial basis were done prior to the determination of the J → ∞ limit. Round‐off errors were the main impediment to achieving even higher precision, since determination of the radial and angular limits required the manipulation of very small energy and 〈δ〉 differences. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Crystallite orientation in stretched polychloroprene networks. II

The orientation of crystallites grown isothermally in several drawn trans-polychloroprene networks is studied as a function of crystallization temperature t_x, degree of crystallinity ω, and elongation ratio α. The orientation distribution is particularly simple for this polymer since the crystallographic c axis (chain axis) orients preferentially along the stretching direction, while a and b are randomly arranged about c. Hence the parameter cos² χ_c adequately characterizes the distribution, where χ_c is the angle between the c axis and the fiber axis, and the average is taken over all crystallites. A treatment due to Krigbaum and Roe is utilized to obtain values of v (the number of statistical segments comprising the crystallization nucleus of critical size) through comparison of the average orientation of crystallites and amorphous statistical segments. The behavior observed falls into two categories. First, if the initial amorphous network is well oriented, 〈cos² χ_c〉 is independent of crystallinity during both crystallization and melting, and v varies with t_z (or the degree of supercooling) as predicted by nucleation theory. If different networks are to have the same crystallite orientation distribution, they must not only be crystallized at the same supercooling, but must also have the same distribution of amorphous segment orientations. Both the relative elongation and the network crosslink density affect the latter distribution. Next, we consider the second category. If the initial amorphous orientation is poor, 〈cos² χ_c〉 decreases linearly during crystallization and increases along approximately the same path during melting. Further, 〈cos² χ_c〉 for a given t_z yields v values which are too large. These two behaviors can be explained if, in the former case, nucleation involves the best oriented statistical segments of all network chains, while in the latter there is a selection according to the chain displacement vector orientation. Thus, if the amorphous orientation is poor, both the orientation and thermodynamic stability of the crystallites decreases with further crystallization. If this decreased stability is reflected in shorter fold lengths, the reversible variation of long period spacing with temperature reported earlier for an oriented polychloroprene network can also be explained as a preferential melting process.     

13C, 13C coupling constants and 13C chemical shifts of aromatic carbonyl compounds. Effects of ortho- and peri-interactions involving the carbonyl substituent

Carbon-13 n.m.r. data have been determined for a series of 26 aromatic carbonyl compounds including benzoyl, naphthoyl and pyrenoyl derivatives ¹³C labelled in the carbonyl group. Doubly labelled anthraquinone has also been included. The compounds investigated comprise non-hindered molecules and molecules in which the carbonyl substituent is subject to ortho- or peri-interactions affecting conjugation of the carbonyl group with the aromatic ring. The dependence of long range ¹³C,¹³C coupling constants involving the carbonyl carbon, in particular ²J and ³J, on steric conditions is discussed, as is the possibility of deciding on the orientation of the carbonyl bond. The following results have emerged. ²J(s-t)>²J(s-c) for ketones and aldehydes, and the reverse is valid for acids and acid derivatives. (s-t and s-c refer to the orientation of the C?O group relative to the aromatic bond in question with respect to the connecting single bond). For ketones ³J(t,s-c)<³J(t,s-t), and both of these ³J(t) values decrease with increasing angle of twist, θ, about the single bond, whereas ³J(c,s-c) increases with θ. For acids and acid derivatives no similar regularity was found. (The initial t and c refer to the geometry of the three-bond coupling path). Generally it is found that ³J(t)>³J(c) and ³J(t)>²J, confirming earlier results. Theoretical calculations on a few model compounds are qualitatively in accordance with the experimental results. Some sign determinations for coupling constants are presented. A short discussion is given of substituent effects on chemical shifts. Observed trends are consistent with earlier results.     

Zur Reaktion von tert-Butyl-phosphaalkin mit Molybdänkomplexen

Reaction of tert -Butyl-phosphaalkyne with Molybdenum Complexes The reaction of tBuC≡P with [(CH₃CN)₃Mo(CO)₃] leads to the complex [Mo(CO)₄〈Mo(CO)₂(η⁴-P₃CtBu){η⁴-P₂(CtBu)₂}〉] 1 as well as to the alkyne complexes [Mo(CO)₄〈{P₃(CtBu)₂}{Mo(CO)₂(CtBu)}{η³-P₂(CtBu)₂}〉] 2 and [Mo(CtBu){η⁴-P₂(CtBu)₂(CO)}{η⁵-P₃(CtBu)₂}] 3 . All compounds are characterized by X-ray structural analysis, by NMR- and IR spectroscopy and by mass spectrometry. In complex 1 a 1,3-diphosphacyclobutadiene and a 1,2,4-triphosphacyclobutadiene are connected by two molybdenum carbonyl centres. In 2 a 1,3-diphosphacyclobutadiene is π- and a novel 1,2,4-triphospholyl ligand is σ-bonded at two Mo centres. A characteristic feature of 3 besides a π co-ordinated 1,2,4-triphospholyl ligand is a 3,4-diphosphacyclopentadienone as ligand, formed via CO insertion during the cyclodimerisation of two phosphaalkynes.     

Molecular orientation in poly(vinyl chloride) studied by Raman spectroscopy and birefringence measurements

The ratios of the intensities of Raman scattering in the C? CI stretching region for eight combinations of sample orientation and directions of polarization of incident and scattered light have been measured for 15 samples of poly(vinyl chloride) (PVC) containing 0, 5, 10, 15 or 20 pph dioctyl sebacate (DOS) plasticizer which had been drawn uniaxially at 22, 70, 75, 80, or 90°C to draw ratios in the range 1–4.5. The birefringences of the samples were also measured. The Raman data were analyzed to give 〈P₂(cosθ)〉_cryst and 〈P₄(cosθ)〉_cryst, the values of the second- and fourth-order Legendre polynomials in cosθ averaged over the distribution of orientations of the crystallites, where θ is the angle between the c axis of a typical crystallite and the draw direction. Comparison of 〈P₂(cosθ)〉_cryst with the birefrigence showed that the crystallites are more highly oriented than the noncrystalline material in samples containing the higher amounts of plasticizer drawn at the higher temperatures. A value of 13.0 × 10^?3 was deduced for the birefringence of fully oriented PVC. The values of 〈P₄(cosθ)〉_cryst for a given 〈P₂(cosθ)〉_cryst were found to be higher than predicted by calculations based on two simple models, the pseudoaffine rigid-rod rotation model and the affine rubber elasticity model.     

On exact and approximate methods of calculating an overall termination rate coefficient from chain length dependent termination rate coefficients

In this paper is tackled the problem of calculating the overall termination rate coefficient 〈k_t〉 which follows from values of k_t^{i, j}, by which is denoted the rate coefficient for termination between two free radicals of degrees of polymerization i and j, respectively. The significance of this problem is that polymerization experiments yield 〈k_t〉 values, whereas microscopic models predict K_t^i,j values. An assumption-free method is presented for computing the steady state 〈k_t〉 corresponding to a set of K_t^i,j values. Parallel to this, approximate methods for calculating steady state 〈k_t〉 values are developed: the so-called short-long approximation is used, and coarse graining of the radical chain length distribution is applied. Calculations are firstly carried out using a microscopic termination model which describes an intermediate conversion polymerization system, and then a set of calculations are performed with low conversion conditions in mind. Comparison of the completely exact solutions with the various approximate solutions reveals which of the approximate and therefore more tractable models is suitable for accurate, microscopic modeling of polymerization kinetics at the different physical conditions envisaged. In this respect the credentials of one of the coarse grain approaches are found to be persuasive. Also executed are calculations probing how values of 〈k_t〉 depend on factors, such as the rate of propagation and of chain transfer to monomer, not traditionally regarded as having anything to do with termination rates; interesting results emerge. Because calculations are carried out with real systems very much in mind, the latter (and other) results are felt to be genuinely relevant to the mechanism of actual free radical polymerizations.     

Atomic level picture of stress relaxation in polymer melts

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

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.     

The effective direct correlation function,an approach to the theory of liquid solutions: A new definition of the effective solute potential

A Mayer graph analysis of the solute effective direct correlation function c₀₀^eff(1,2) for a binary fluid mixture is presented. The function c₀₀^eff(1,2) is defined in terms of the solute pair correlation function h₀₀(1,2) by a one-component effective Ornstein-Zernike relation. The graphical series for c₀₀^eff(1,2) is shown to be identical to the series for the direct correlation function of a pure fluid except that pure fluid f-bonds are replaced by effective f-bonds f₀₀^eff(1,2) which exactly include the effects of solvent-solute interaction. An effective solute-solute potential W₀₀(1,2) is defined by the relation f₀₀^eff(1,2) = exp[-βW₀₀(1,2)] - 1. A graphical series for W₀₀(1,2) is presented. This single effective solute pair potential plays a role in the present formulation analogous to the role the infinite set of n-particle solute potentials plays in the McMillan-Mayer theory.     

Kinetics of methyl methacrylate and n‐butyl acrylate copolymerization mediated by 2‐cyanoprop‐2‐yl dithiobenzoate as a RAFT agent

The RAFT (co)polymerization kinetics of methyl methacrylate (MMA) and n‐butyl acrylate (BA) mediated by 2‐cyanoprop‐2‐yl dithiobenzoate was studied with various RAFT concentrations and monomer compositions. The homopolymerization of MMA gave the highest rate. Increasing the BA fraction f_BA dramatically decreased the copolymerization rate. The rate reached the lowest point at f_MMA ～ 0.2. This observation is in sharp contrast to the conventional RAFT‐free copolymerization, where BA homopolymerization gave the highest rate and the copolymerization rate decreased monotonously with increasing f_MMA. This peculiar phenomenon can be explained by the RAFT retardation effect. The RAFT copolymerization rate can be described by 〈R_p〉/〈R_p〉₀ = (1 + 2(〈k_c〉/〈k_t〉)〈K〉)[RAFT]₀)^?0.5, where 〈R_p〉₀ is the RAFT‐free copolymerization rate and 〈K〉 is the apparent addition–fragmentation equilibrium coefficient. A theoretical expression of 〈K〉 based on a terminal model of addition and fragmentation reactions was derived and successfully applied to predict the RAFT copolymerization kinetics with the rate parameters obtained from the homopolymerization systems. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 3098–3111, 2007     

Mononukleare und mehrfach verbrückte dinukleare Phthalocyaninate(1–/2–) des Yttriums durch Solvenz‐kontrollierte Kondensation; kleine Solvenz‐Cluster als Liganden

Mononuclear and Multiply Bridged Dinuclear Phthalocyaninates(1–/2–) of Yttrium by Solvent Controlled Condensation; Small Solvent Clusters as Ligands Green chlorophthalocyaninato(2–)yttrium(III), [Y(Cl)pc^2–] forms when yttrium chloride is heated with o‐phthalonitrile in 1‐chloronaphthalene. Black cis‐di(chloro)phthalocyaninato(1‐)yttrium(III), ^cis[Y(Cl)₂pc^–] is obtained as a stable intermediate by partial reduction. Both complexes are soluble in many O‐donor solvents and pyridine. The solubility in water is remarkable: [Y(Cl)pc^2–] dissolves with green, ^cis[Y(Cl)₂pc^–] with red‐violet color. Typical absorptions of the pc^2– ligand are observed at 14800 and 29700 cm^–1. A solvent dependent monomer‐dimer equilibrium is found for the pc^– radical. The monomer with absorptions at 12100 and 19900 cm^–1 is favored in non‐polar solvents, while in polar solvents the dimer with absorptions at 8700, 13200 and 18600 cm^–1 is preferred. cis‐Tri(dimethylformamide)chlorophthalocyaninato(2–)yttrium(III) etherate ( 1 ) crystallises from a solution of [Y(Cl)pc^2–] in MeOH/dmf, cis‐tetra(dimethylsulfoxide)phthalocyaninato(2–)yttrium(III) chloride etherate methanol disolvate ( 2 ) from thf/dmso, μ‐di(chloro)‐μ‐di〈di(pyridine)(μ‐water)〉di(phthalocyaninato(2–)‐ yttrium(III)) ( 5 ) from py, and cis‐(chloro)pyridine(triphenylphosphine oxide)phthalocyaninato(2–)yttrium(III) semi‐etherate ( 3 ) is obtained from a solution of [Y(Cl)pc^2–] and triphenylphosphine oxide in py. 1 condenses in MeOH yielding a (1 : 1)‐mixture ( 4 ) of μ‐di(chloro)di(〈trans‐(diwaterdimethanol)〉〈dimethanol〉phthalocyaninato(2–)yttrium(III)) ( 4 a ) and μ‐di(chloro)di(dimethylformamide〈dimethanol〉phthalocyaninato(2–)yttrium(III)) ( 4 b ); co‐ordinatively bound solvent clusters are in brakets. The structures of 1 – 5 have been established by X‐ray crystallography. Apart from 3 with hepta‐co‐ordinated yttrium, the metal ion prefers octa‐co‐ordination, and the bond arrangement around Y³⁺ is always a distorted quadratic antiprism. In the dinuclear complexes obtained by solvent controlled condensation both antiprisms share an edge by two μ‐Cl atoms in 4 , while in 5 the antiprisms are face‐shared by two trans positioned μ‐Cl atoms and μ‐O atoms, respectively. In 5 , the bent ^b〈{py}₂(μ‐H₂O)〉 cluster is stabilised by a combined interplanar bonding of pyridine by short N…H–O bonds (d(N…O) = 2.664(7) Å; 2.81(2) Å) and strong van‐der‐Waals interactions with the ecliptic pc^2– ligands. 4 a and 4 b contain the dimeric methanol cluster 〈(MeOH)₂〉, and 4 a in addition the cyclic heterotetrameric trans‐diwaterdimethanol cluster, trans^c〈(H₂O)₂(MeOH)₂〉. The neutral clusters co‐ordinatively bound to the Y atom are compared with structurally established cluster‐anions of type 〈(OMe)(MeOH)〉^–, linear ^l〈(OMe)(MeOH)₂〉^–, cyclic ^c〈(OH)₃(H₂O)₃〉^3–, ^b〈{H₂O}₂(μ‐O)〉^2–, and ^b{H₂O}₂(μ‐F)〉^–.     

On the Escape Transition of a Tethered Gaussian Chain; Exact Results in Two Conjugate Ensembles

Summary: Upon compression between two pistons an end-tethered polymer chain undergoes an abrupt transition from a confined coil state to an inhomogeneous flower-like conformation that is partially escaped from the gap. In the thermodynamic limit the system demonstrates a first-order phase transition. A rigorous analytical theory of this phenomenon for a Gaussian chain is presented in two ensembles: a) the H-ensemble, in which the distance H between pistons plays the role of the control parameter, and b) the conjugate f-ensemble in which the external compression force f is the independent parameter. A loop region for 〈f(H)〉 with negative compressibility exists in the H-ensemble, while in the f-ensemble 〈H(f)〉 is strictly monotonic. The average lateral forces taken as functions of H (or 〈H〉, respectively) have distinctly different behavior in the two ensembles. This result is a clear counterexample of the main principles of statistical mechanics stating that all ensembles are equivalent in the thermodynamic limit. Another theorem states that the thermodynamic potential as a function of volume must be concave everywhere. We demonstrated that the exact free energy in the H-ensemble contradicts this statement. Inapplicability of these fundamental theorems to a macromolecule undergoing the escape transition is clearly related to the fact that phase coexistence in the present system is strictly impossible. This is a direct consequence of the tethering and the absence of global translational degrees of freedom of the polymer chain.     

Zymomonas mobilis as catalyst for the biotechnological production of sorbitol and gluconic acid

The conversion of glucose and fructose into gluconic acid (GA) and sorbitol (SOR) was conducted in a batch reactor with free (CTAB-treated or not) or immobilized cells of Zymomonas mobilis. High yields (more than 90%) of gluconic acid and sorbitol were attained at initial substrate concentration of 600 g/L (glucose plus fructose at 1:1 ratio), using cells with glucose-fructose-oxidoreductase activity of 75 U/L. The concentration of the products varied hyperbolically with time according to the equations (GA)=t(GA)_max/(W_GA +t), (SOR)=t (SOR)_max/(W_Sor+t), v_GA=[W_GA (GA)_max]/(W_GA+t)² and V_SOR=[W_SOR (SOR)_max]/(W_SOR+t)². Taking the test carried out with free CTAB-treated cells as an example, the constant parameters were (GA)_max= 541 g/L, (SOR)_max=552 g/L, W_GA=4.8h, W_SOR=4.9h, υ_GA=112.7 g/L· and υ_SOR=112.7 g/L·.     

Partitioning technique,perturbation theory,and rational approximations

Instead of the Schródinger equation ??Ψ = EΨ subject to the boundary condition 〈φ|Ψ〉 = 1, where φ is a normalized reference function in the Hilbert space, one studies the inhomogeneous equation (?? ? ?)Ψ_? = aφ, where ? is a complex variable, with the same boundary condition, which gives a = 〈φ|??|Ψ_?〉 ? ? = ?₁ ? ?. Introducing the projector P = 1 ? |φ〉〈φ| for the complement to O = |φ〉〈φ|, one finds easily the explicit solution Ψ_? = (1 ? P??/?)^?1φ = (1 + T_???)φ, where T_? = (? ? P??)^?1P = P(? ? P??P)^?1P is the reduced resolvent associated with the auxiliary Hamiltonian H? = P??P. The existence of these operators is discussed. It is shown that, if the parameter ? is real in the “discrete part” of the spectrum to ??, then ? and ?₁ = 〈φ|??|Ψ_?〉 = 〈φ|?? + ??T_???|Φ〉 ≡f(?) bracket a true eigenvalue E satisfying the relation E = f(E). The Newton-Raphson solution to the equation F(?) = ? ? f(?) = 0 is related to the variation principle. Putting ?? = ??₀ + V and expanding the inverse (? ? P??₀ ? PV)^?1 in terms of powers of V or (V ? α), one gets various expansions relating to finite-order perturbation theory. Exact expressions for the ordinary wave and reaction operators are obtained. If A is an arbitrary nonsingular operator and h = {h₁,h₂,…,h_n} is a linearly independent set, the inner projection Á_n = | h 〉 〈 h |A^?1| h 〉^?1〈 h | is a “rational approximation” to the operator A which converges toward A when n→∞ and the set h becomes complete. If A is positive (or has a finite negative part) the convergence is from below. Applying this principle to the partitioning technique, one gets rational perturbation approximations instead of the standard power series, similar to the Padé approximants but derived in a different way with the remainder term under control. The method has been used to calculate lower bounds to eigenvalues.     

Neutral polymer slow mode and its rheological correlate

Quasi‐elastic light scattering spectroscopy intensity–intensity autocorrelation functions [S(k,t)] and static light scattering intensities of 1 MDa hydroxypropylcellulose in aqueous solutions were measured. With increasing polymer concentration, over a narrow concentration range, S(k,t) gained a slow relaxation. The transition concentration for the appearance of the slow mode (c^t) was also the transition concentration for the solution‐like/melt‐like rheological transition (c⁺) at which the solution shear viscosity [η_p(c)] passed over from a stretched exponential to a power‐law concentration dependence. To a good approximation, we found c^t[η] ≈ c⁺[η] ≈ 4, [η] being the intrinsic viscosity. The appearance of the slow mode did not change the light scattering intensity (I): from a concentration lower than c^t to a concentration greater than c^t, I/c fell uniformly with increasing concentration. The slow mode thus did not arise from the formation of compact aggregates of polymer chains. If the polymer slow mode arose from long‐lived structures that were not concentration fluctuations, the structures involved much of the dissolved polymer. At 25 °C, the mean relaxation rate of the slow mode approximately matched the relaxation rate for the diffusion of 0.2‐μm‐diameter optical probes observed with the same scattering vector. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 323–333, 2005     

Variation of crystallite orientation with degree of crystallinity: Isotactic polystyrene

The orientation of the crystallographic c axis (chain axis) was followed by x-ray diffraction during the crystallization of four samples of isotactic polystyrene differing in elongation ratio. The crystallite orientation can be expressed by 〈cos² χ_c〉, where χ_c is the angle between the c axis and the stretching direction. The degrees of crystallinity w were estimated from the diffraction data by using density for calibration. It was found that 〈cos² χ_c〉 decreases in a linear manner with crystallinity, the rate of decrease being very small when the elongation ratio α is 5, but becoming progressively larger as α is decreased toward unity. A qualitative measure suggests that amorphous orientation decreases during crystallization at a rate which is nearly independent of α. The variation of 〈cos² χ_c〉 with w is therefore governed by the orientation of the statistical chain segments prior to crystallization. If the elongation ratio is small, the supply of well oriented statistical segments is limited, and 〈cos² χ_c〉 will decrease at a rapid rate during crystallization. A treatment due to Krigbaum and Roe permits evaluation of the ratio, ν/N, where ν and N are the average numbers of statistical segments in the crystallization nucleus of critical size, and in a network chain, respectively. Our polystyrene samples were not crosslinked, so chain entanglements must serve as junction points. Values of ν could not be obtained, since N was unknown. However, the (ν/N) ratio for isotactic polystyrene decreases slowly with α, and the values agree reasonably well with those obtained in a previous study of oriented polychloroprene networks. After nearly complete crystallization (ω ca. 0.30), the long period spacing measured by low angle diffraction was approximately 135 Å, and varied only slightly with elongation ratio in the range α = 1 to 5. It therefore appears that chain folded lamellae are present in both drawn and undrawn samples of isotactic polystyrene.