A Flory–Huggins thermodynamic approach for predicting sorption equilibrium in ternary polymer systems

The Flory–Huggins theory as modified by Pouchlý has been applied to calculate preferential (λ) and total (Y) sorption coefficients for a ternary polymer system. The ternary interaction function (?₁?₂?₃G_T(u₁, ?₃)) is described as the product of three independent binary functions. This expression allows prediction of λ and Y from binary interaction parameters χ, χ, g, g, and g₁₂(?₁₀). Three ternary polymer systems are used to check the validity of the expression. Moreover for polymer systems in which the parameters g and/or g are unknown, a procedure to evaluate them has been developed and verified on systems for which sufficient experimental information is available.     

Melt‐state polymer chain dimensions as a function of temperature

The unperturbed chain dimensions (〈R²〉_o/M) of cis/trans‐1,4‐polyisoprene, a near‐atactic poly(methyl methacrylate), and atactic polyolefins were measured as a function of temperature in the melt state via small‐angle neutron scattering (SANS). The polyolefinic materials were derived from polydienes or polystyrene via hydrogenation or deuteration and represent structures not encountered commercially. The parent polymers were prepared via lithium‐based anionic polymerizations in cyclohexane with, in some cases, a polymer microstructure modifier present. The polyolefins retained the near‐monodisperse molecular weight distributions exhibited by the precursor materials. The melt SANS‐based chain dimension data allowed the evaluation of the temperature coefficients [dln 〈R²〉_o/dT(κ)] for these polymers. The evaluated polymers obeyed the packing length (p)‐based expressions of the plateau modulus, G = kT/np³ (MPa), and the entanglement molecular weight, M_e = ρN_anp³ (g mol^?1), where n_t denotes the number (～21) of entanglement strands in a cube with the dimensions of the reptation tube diameter (d_t) and ρ is the chain density. The product np³ is the displaced volume (V_e) of an entanglement that is also expressible as pd or kT/G. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1768–1776, 2002     

Thermodynamics of adsorption of uranyl ions onto amidoximated poly(acrylonitrile)/poly(N‐vinyl 2‐pyrrolidone) interpenetrating polymer networks

The interaction of uranyl ions (UO) with interpenetrating polymer networks (IPNs) based on amidoximated poly(acrylonitrile)/poly(N‐vinyl 2‐pyrrolidone) was examined. The adsorption capacity of IPN hydrogels as well as the adsorption kinetics and the effect of temperature on UO ion adsorption were investigated. Thermodynamic quantities and kinetic parameters were calculated with adsorption isotherm data. The initial adsorption‐rate values for each temperature were calculated, and the corresponding rate constants decreased with increasing temperature. The adsorption enthalpy, entropy, and free energy of the UO ion with amidoximated IPN hydrogels were calculated from basic thermodynamic relations. It was assessed that adsorption occurred by strong electrostatic interactions with an adsorption enthalpy of ?31.5 kJ/mol. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 986–993, 2004     

Quantitative evaluation of stress distribution in bulk polymer samples through the comparison of mechanical behaviors between giant single‐crystal and semicrystalline samples of poly(trans‐1,4‐diethyl muconate)

The Raman shift and crystallite modulus were measured under the application of tensile force for a giant single crystal and a series of uniaxially oriented semicrystalline samples of poly(trans‐1,4‐diethyl muconate) (polyEMU). The apparent Raman shift factor α^app or a vibrational frequency shift per 1 GPa tensile stress was higher for the semicrystalline samples with lower crystallinity or lower bulk modulus. The apparent crystallite modulus E or Young's modulus along the chain axis in the crystalline region was not constant but varied remarkably between the giant single crystal and semicrystalline samples. A systematic change in α^app and E among the polyEMU samples with different preparation history could be interpreted quantitatively on the basis of a mechanical series parallel model consisting of crystalline and amorphous phases. The origin of different E and α^app was speculated to be a stress concentration on the taut‐tie chain contained as a parallel crystalline component in the mechanical model. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 444–453, 2003     

Determination of thermodynamic properties of macroporous glycidyl methacrylate‐based copolymers by inverse gas chromatography

Macroporous crosslinked poly(glycidyl methacrylate‐co‐ethylene glycol dimethacrylate) (PGME) was synthesized by suspension copolymerization and modified by ring‐opening reaction of the pendant epoxy groups with ethylene diamine (EDA). Inverse gas chromatography (IGC) at infinite dilution was applied to determine the thermodynamic interactions of PGME and modified copolymer, PGME‐en. The specific surface areas of the initial and modified copolymer samples were determined by the BET method, from low‐temperature nitrogen adsorption isotherms. The specific retention volumes, V, of 10 organic compounds of different chemical nature and polarity (nonpolar, donor, or acceptor) were determined in the temperature range 333–413 K. The weight fraction activity coefficients of test sorbates, , and Flory–Huggins interaction parameters, , were calculated and discussed in terms of interactions of sorbates with PGME and PGME‐en. Also, the partial molar free energy, , partial molar heat of mixing, , sorption molar free energy, ΔG, sorption enthalpy ΔH, and sorption entropy, ΔS, were calculated. Glass transitions in PGME and PGME‐en, determined from IGC data, were observed in the temperature range 373–393 K and 363–373 K, respectively. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 2524–2533, 2005     

Determination of molecular weight and its distribution of rigid‐rod polymers determined by phase‐modulated flow birefringence technique

The phase‐modulated flow birefringence (PMFB) method is widely accepted as one of the most sensitive and accurate techniques suitable for experimental tests on the molecular theory of polymer solutions. The objective of this study is to develop a systematic method to determine molecular weight and distribution of rigid‐rod polymers by the PMFB technique. Using molecular theory for rigid polymers, birefringence Δn and orientation angle χ have been expressed as a function of molecular weight and distribution. Δn has been shown to be proportional to Σc_iM, and cot 2χ turned out to have a linear relationship with Σc_iM/Σc_iM. From the experimental results for PBLG solutions, birefringence and orientation angle data were in some degree matched with the theory presented. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 509–515, 2000     

Thermal,morphology, and NMR characterizations on phase behavior and miscibility in blends of isotactic polystyrene with poly(cyclohexyl methacrylate)

The miscibility and phase behavior in a binary blend of isotactic polystyrene (iPS) and poly(cyclohexyl methacrylate) (PCHMA) were investigated by differential scanning calorimetry, optical microscopy (OM), and solid‐state ¹³C cross‐polarity/magic‐angle spinning NMR. The iPS/PCHMA blend was miscible when all compositions showed a single composition‐dependent glass‐transition temperature (T_g) and when the blend went through a thermodynamic phase transition upon heating to above the lower critical solution temperature as determined by OM measurements. The ¹H NMR spin‐relaxation times in the laboratory frame (T) and in the rotating frame (T) for iPS/PCHMA blends with various compositions and neat components were directly measured through solid‐state¹³C NMR. The results of T indicated that the blends are homogeneous, at least on a scale of 75–85 nm, confirming the miscibility of the system. The single decay and composition‐dependent T values for each blend further demonstrated the blends are homogeneous on a scale of 2.5–3.5 nm. The results suggested that iPS and PCHMA are intimately mixed at the molecular level within the blends at all compositions. The tacticity of polystyrene does not seem to adversely influence the miscibility in blends of iPS/PCHMA. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 772–784, 2003     

Molecular orientation in biaxially oriented sheets of poly(ethylene terephthalate). I. Characterization of orientation and comparison with models

A study of molecular orientation in a series of five one-way-drawn sheets of poly(ethylene terephthalate) is reported. Five orientation averages P which characterize the benzene-ring orientation were determined directly by Raman spectroscopy. Two further averages P and P were determined from a combination of Raman spectroscopy and refractive index measurements on the basis of two different conformational models. As in a previous publication, the seven averages P were used to estimate all the P up to fourth order for the benzene rings on the assumption that in each case the actual distribution of orientations is close to the most probable distribution. The seven orientation averages P for each drawn sheet were also compared with those calculated on the basis of two simple models for the development of molecular orientation during drawing, the pseudoaffine deformation scheme and the rubber network model. It was shown that the orientation of the chain axes is closer to that predicted by the rubber network model, the pseudoaffine deformation scheme greatly overestimating the degree of chain biaxiality. The results also show that the benzene ring planes are preferentially oriented toward the planes of the sheets. The chain axis orientation is, however, nearly uniaxial and similar to that observed for uniaxially drawn samples, suggesting that there is no direct connection between the processes of chain axis orientation and preferential orientation of the benzene-ring planes.     

New simple method of measuring the surface glass transition temperature of polymers

A lap‐shear joint mechanical testing method has been probed to measure the surface glass transition temperature (T) of the thick bulk films of high‐molecular‐weight polymers. As T, the temperature transition “occurrence of autoadhesion–nonoccurrence of autoadhesion” has been proposed. The influence of chain flexibility, of molecular architecture, of polymer morphology, and of chain ends concentration on the T has been investigated. The correlation between the reduction in T with respect to the glass transition temperature of the bulk (T) and the intensity of the intermolecular interaction in the polymer bulk in amorphous polymers has been found. The effect of surface roughness on T has been discussed. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 2012–2021, 2010     

Determination of the crystallinity of polyethylene/α‐olefin copolymers by thermal analysis: Relationship of the heat of fusion of 100% polyethylene crystal and the density

The heat of fusion measured with differential scanning calorimetry (DSC) is typically divided by a constant value of the heat of fusion of 100% polyethylene (PE) crystal (ΔH) for the estimation of the fraction crystallinity of PE copolymers, regardless of the density [i.e., the short‐chain branching (SCB) concentration]. In this work, values of ΔH of about 288 J/g were determined with a combined DSC and X‐ray diffraction (XRD) method for a series of PE copolymers containing SCB from 0 to 50 Br/1000 C (density = 0.965–0.865 g/cc). There was no systematic change in ΔH observed across this density range. This result supports the suitability of determining the fraction crystallinity of PE of any density by the simple division of the observed heat of fusion determined by DSC by a constant value of ΔH. This DSC method yielded values of PE crystallinity in good agreement with corresponding values determined by XRD for a series of PE copolymers. The determination of ΔH involved a small precision error for higher density (lower SCB) PEs, but the precision error increased for lower density (i.e., higher SCB) PEs. This was due to the difficulty in measuring the heat of fusion for lower density PEs, which exhibited low values of the heat of fusion and melted only slightly above room temperature, and due to the difficulty of measuring lower values of crystallinity by XRD. The crystal thickness measured by small‐angle X‐ray scattering for this series of PE copolymers decreased exponentially from about 280 to 6 Å. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1637–1643, 2002     

Nonlinear rheology of highly entangled polymer solutions in start‐up and steady shear flow

Orientation angle and stress‐relaxation dynamics of entangled polystyrene (PS)/diethyl phthalate solutions were investigated in steady and step shear flows. Concentrated (19 vol %) solutions of 0.995, 1.81, and 3.84 million molecular weight (MW) PS and a semidilute (6.4 vol %) solution of 20.6 million MW PS were used to study the effects of entanglement loss on dynamics. A phase‐modulated flow birefringence apparatus was developed to facilitate measurements of time‐dependent changes in optical equivalents of shear stress (n₁₂ ≈ Cσ) and first normal stress differences (n₁ = n₁₁ ? n₂₂ ≈ CN₁) in a planar‐Couette shear‐flow geometry. Flow birefringence results were supplemented with cone‐and‐plate mechanical rheometry measurements to extend the range of shear rates over which entangled polymer dynamics are studied. In slow (τ > ) steady shear‐flow experiments using the ultrahigh MW polymer sample (20.6 × 10⁶ MW PS), steady‐state n₁₂ and n₁ results manifest unusual power‐law dependencies on shear rate [n_12,ss ～ ^0.4 and n_1,ss ～ ^0.8]. At shear rates in the range τ < < τ, steady‐state orientation angles χ_SS are found to be nearly independent of shear rate for all but the most weakly entangled materials investigated. For solutions containing the highest MW PS, an approximate plateau orientation angle χ_p in the range 20–24° is observed; χ_p values ranging from 14 to 16° are found for the other materials. In the start‐up of fast steady shear flow (γ˙ ≥ τ), transient undershoots in orientation angle are also reported. The molecular origins of these observations were examined with the help of a tube model theory that accommodates changes in polymer entanglement density during flow. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2275–2289, 2001     

Intrinsic birefringence of poly(trimethylene terephthalate)

Highly oriented poly(trimethylene terephthalate) (PTT) fiber has a low birefringence that is unexpected for an aromatic polyester with a high refractive index. To explain this observation, the intrinsic birefringence Δn of PTT crystal was calculated from its bond polarizabilities to be 0.029. This Δn is almost an order of magnitude smaller than poly(ethylene terephthalate)'s value at 0.22, although both polymers have nearly identical crystal refractive indices. The small Δn is due to the arrangement of PTT's methylene groups in gauche conformations, causing the chain‐repeating unit to be tilted ～53° away from the c axis toward the basal plane. Because of the small Δn, the crystalline‐phase orientation made only a small contribution to the overall birefringence despite the fiber's high crystallinity and orientation. To understand the effect of the number of methylene groups on polyester optical anisotropy, the Δn's of a series of poly(m‐alkylene terephthalates) with m = 2–5 were compared and correlated with ψ: an angle made by the normal of the benzene ring with the crystal's axis. As ψ′ decreases, Δn of the polyesters diminishes. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1513–1520, 2002     

Interaction parameters of crystalline/crystalline polypropylene/poly(butene‐1) blends: Effect of molecular fractionation

Compatibility of crystalline/crystalline polypropylene (PP)/poly(butene‐1) (PB‐1) blends was investigated via the method of equilibrium melting temperature depression followed by determining the polymer–polymer interaction parameter (χ) using the Nishi–Wang equation. The composition variation of the equilibrium melting temperatures of blends (T) was determined with the Hoffman–Weeks plot. The T and its variation with the blend composition depended on the crystallization temperature range. The morphological effect of the blend composition was not a contribution factor for the T depressions of PP and PB‐1 in the blends. The interplay of the dilution effect and molecular fractionation effect of the amorphous component on crystallization of the crystalline component in the blends governed the relation of T with the blend composition. The calculated χ values were negative depending on the blend composition. The negative χ values suggested that PP and PB‐1 in the amorphous region were compatible. The composition variation of the χ values was attributed to the molecular fractionation effect during crystallization. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 638–648, 2002; DOI 10.1002/polb.10125     

A procedure for predicting sorption equilibrium in ternary polymer systems from Flory–Huggins binary interaction parameters and the inversion point of preferential solvation

A procedure has been developed, based on the Flory–Huggins theory as generalized by Pouchlý, which permits the calculation of preferential (λ) and total (Y) sorption coefficients from previous information on the binary interaction parameters, χ, χ, and g₁₂(?₁₀) and on the mixture composition at which the sign of λ inverts. The expressions obtained were applied to 10 cosolvent polymer systems for which experimental values of λ and Y are known. Practically in all the studied systems, the theoretical predictions are in fair accordance with the experimental data.     

Spin probe mobility in relation to free volume and relaxation dynamics of glass‐formers: A series of spin probes in poly(isobutylene)

We present the dynamics of a series of three paramagnetic molecules of different volume, mass, and shape in amorphous glass‐forming polymer poly(isobutylene) (PIB) as investigated by means of electron spin resonance (ESR) technique. The reorientation behavior of spin probes is related to the ortho‐positronium (o‐Ps) annihilation in PIB from positron annihilation lifetime spectroscopy (PALS) and the extracted free volume information. It is also related to the dynamic data of PIB from broadband dielectric spectroscopy (BDS), neutron scattering (NS), and nuclear magnetic resonance (NMR) spectroscopy from literature. In the case of the smallest spin probe, 2,2,6,6‐tetramethyl‐1‐piperidinyloxy (TEMPO), a discontinuous course of the spectral parameter 2A_zz′ versus T dependence was observed and the subsequent phenomenological model‐free analyses of the spectral parameter, 2A_zz′ versus T, as well as of the correlation time, τ_c, versus 1/T plots provided the characteristic ESR temperatures ( , T_50G, ) and (T, T, T). These characteristic ESR temperatures were found to be consistent with the characteristic PALS temperatures: T, T = T from temperature dependences of the mean o‐Ps lifetime, τ₃, or the width of o‐Ps lifetime distribution, σ₃, respectively. In addition, the relationships between the spin probe size, V, and the free volume hole size distributions g_n(V_h) at the characteristic ESR temperatures indicate the significant influence of the free volume fluctuation at the crossover from slow to rapid regime as well as within the rapid motional regime. On the other hand, the two larger spin probes exhibit a rather continuous 2A_zz′–T plots with the respective T_50G's lying in the vicinity of T independently of their volume, mass and shape, suggesting the common origin of underlying process controlling this T_50G transition. Finally, these mutual PALS and ESR findings were compared with the known dynamic behavior of PIB which suggest that the dynamics of the TEMPO and the larger spin probes are related to free volume fluctuation associated with primary α ‐ and secondary β processes, respectively. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 1058–1068, 2009     

Solid‐state nuclear magnetic resonance relaxation times in crosslinked macroporous polymer particles of divinylbenzene homopolymers

Monodisperse porous particles of poly(divinylbenzene) prepared by the activated swelling method have been investigated by solid‐state ¹³C crosspolarization magic‐angle spinning (CPMAS) nuclear magnetic resonance (NMR) relaxation measurements. Homopolymeric combinations of two porogens (toluene and 2‐ethylhexanoic acid) and two monomers (meta‐ and para‐divinylbenzene) were studied. Residual vinyl groups were systematically reacted with increasing amounts of bromine, producing 20 different polymers samples for which we measured crosspolarization times, T_CH, proton rotating frame spin‐lattice relaxation, T, ¹³C spin‐lattice relaxation, T, and proton spin‐lattice relaxation, T. These parameters were chosen to reflect expected changes in a wide range of frequencies of motion as a function of structure. Relative differences in the molecular mobility of the major functional groups (aromatic, vinyl and aliphatic) is related to initial reactants used, vinyl concentration, relative reactivity of vinyl groups, distribution of vinyl groups, pore structure, and degree of crosslinking. Variable temperature ¹H combined rotation and multiple pulse NMR (CRAMPS) was used to derive activation energies for selected samples via measurement of the proton spin‐lattice relaxation time, T. Irreversible thermal effects were observed in ambient temperature relaxation after heating to temperatures in the range of 393–418 K. Simple univariate statistical analyses failed to reveal consistent correlations among the known variables. However, the application of more sophisticated multivariate and neural network analyses allowed excellent structure–property predictions to be made from the relaxation time data. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 1307–1328, 1999     

Evaluation of two‐center,three‐ and four‐electron integrals over Slater‐type orbitals in elliptical coordinates

An algorithm for evaluation of two‐center, three‐electron integrals with the correlation factors of the type rr and rrr as well as four‐electron integrals with the correlation factors rrr and rrr in the Slater basis is presented. This problem has been solved here in elliptical coordinates, using the generalized and modified form of the Neumann expansion of the interelectronic distance function r for k ≥ ?1. Some numerical results are also included. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Entanglement friction and dynamics in blends of starlike and linear polymer molecules

We investigate relaxation dynamics in a series of six‐arm star/linear 1,4‐polybutadiene blends with mechanical rheometry measurements. Blend systems are formulated to systematically probe constraint release and arm relaxation dynamics. Zero shear viscosity and terminal relaxation times of star/linear polymer blends with fixed star arm molecular weights (M_a) and compositions (?_S) are found to follow nonmonotonic dependencies on the linear polymer molecular weight (M_L). At low values of ?_S, at least two scaling regimes are apparent from the data (ξ₀ ～ M and ξ₀ ～ M), where ξ₀ refers to the zero shear viscosity or terminal relaxation time of the blend. The two regimes are separated by a critical linear polymer molecular weight M^* that is more than 20 times larger than the critical molecular weight for entanglements. When the linear polymer contribution to blend properties is removed, a clear transition from dilution dynamics, ξ₀ ～ M, to Rouse‐like constraint‐release dynamics, ξ₀ ～ M, is apparent at low values of ?_S. At higher ?_S values, a new activated constraint‐release dynamic regime is evident in which ξ₀ ～ M and ξ₀ ～ ?, where α changes continuously from approximately 2 to 0.5 as ?_S increases and β varies from 2.0 to 1.0 as M_L increases. The experimental results are compared with theoretical predictions based on a drag coupling model for entangled polymer liquids. All features observed experimentally are captured by this model, including the value of M^* for the transition from dilution to Rouse constraint‐release dynamics. Predictions of the drag coupling model are also compared with published data for the zero shear viscosity and terminal relaxation time in bidisperse linear polymer blends and pure entangled starlike molecules. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2501–2518, 2001     

Crystallization mechanism of regioregular poly(3‐alkyl thiophene)s

The crystallization properties of three regioregular poly(3‐alkyl thiophene)s (P3ATs) are studied: poly(3‐hexyl thiophene) (P3HT), poly(3‐octyl thiophene) (P3OT), and poly(3‐dodecyl thiophene) (P3DDT). The morphology of the isothermally crystallized samples is a whisker type. The values of the enthalpy of fusion of ideal crystals (ΔH), determined from the melting point depression in the polymer–diluent system, are 99, 73.6, and 52 J/g for P3HT, P3OT, and P3DDT, respectively. The values of the equilibrium melting point (T), determined from the Hoffman–Weeks extrapolation procedure, are 300, 230, and 180 °C for P3HT, P3OT, and P3DDT, respectively. From the linear extrapolation of the P3AT data, the T and ΔH values of unsubstituted polythiophene are predicted to be 400 °C and 139 J/g, respectively. The crystallization kinetics of these polymers are studied with differential scanning calorimetry, and the Avrami exponents vary between 0.6 and 1.4, indicating one‐dimensional heterogeneous nucleation with linear growth. As the P3AT whiskers are produced from the chain‐folding process, the Lauritzen–Hoffman growth rate theory is applied to analyze the temperature coefficient of the crystallization rate data. Graphical plots indicate a transition from regime I to regime II during isothermal crystallization for all the P3ATs studied. The fold surface energy and the work of chain folding calculated from the slopes of the graphical plots decrease with an increase in the number of carbon atoms of the side chain. The primary crystallization process of the side‐chain crystallization is very fast and is attributed to the zipping effect of the main‐chain crystals. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 2073–2085, 2002     

Preparation of exfoliated polyacrylic clay nanocomposites with high loading: An investigation into the intercalation of ammonium‐terminated polyacrylic acid and polyacrylates

Low‐molecular‐weight polyacrylic acid with amine chain end used as a macromolecular intercalating agent was synthesized by radical polymerization using 2‐aminoethanethiol hydrochloride as chain transfer agent. Three polyacrylates (sodium polyacrylate‐t‐NH, calcium polyacrylate‐t‐NH, and zinc polyacrylate‐t‐NH) were prepared by neutralization from this polyacrylic acid using sodium hydroxide, zinc oxide, and calcium hydroxide as alkalies. The intercalation of ammonium‐terminated polyacrylic acid and polyacrylate was investigated by viscosity measurement, XRD, and TEM. Using this ammonium‐terminated polyacrylic acid as the intercalating agent, exfoliated polyacrylic acid/clay and polyacrylate/clay composites with a clay loading of ～30 and 20 wt %, respectively, were prepared through the evaporation of solvent from their clay suspensions. The thermal degradation of polyacrylic acid/clay and polyacrylate/clay composites was also studied by TGA. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 2335–2340, 2008