Single Composite Curve Research Articles

Fitting Composite Horizontal Curves Using the Total Least-Squares Method

The parameters of horizontal curves are sometimes missing or need to be updated using observed x and y coordinate data. The observed data for existing alignments may be obtained using global positioning systems (GPS) or extracted from digital imagery. Previous methods have addressed the case where only the circular or the transition curves are estimated. This paper presents an optimisation model for composite horizontal curves that simultaneously fits the circular curve, its transition curves and the two tangents using the total least-squares method. The objective function minimises the squared deviations between the observed and predicted values. The model can be applied to fitting single or multiple composite horizontal curves. Since the model is nonlinear and non-convex, a close initial solution is first developed and then used to obtain the global optimal solution. The model is validated and applied using actual data of a horizontal alignment. The proposed model presents an important extension to existing methods for estimating horizontal alignments and therefore should be of interest to surveying professionals.

Effect of fatty acids on the strain‐induced crystallization of natural rubber detected by tear energy measurements

AbstractThe strain‐induced crystallization of natural rubber (NR) was investigated by the measurement of the tear energy of a crosslinked blend consisting of NR and noncrystalline styrene–butadiene rubber (SBR). When NR was dispersed into the SBR matrix, the tear energy of SBR increased at various temperatures and tear rates. After the application of the principle of time–temperature superposition to the tear energy according to the Williams–Landel–Ferry equation, two distinct curves were found for the NR/SBR blend with respect to the reduced tear rate, despite the fact that the tear energy of SBR or the SBR/SBR blend gave its own single composite curve. When the fatty acid in the NR/SBR blend was removed by acetone extraction, the tear energy of the blend drew a single composite curve. The conversion of the two curves into the single composite curve for the NR/SBR blend suggested that the tear energy depended on the strain‐induced crystallization of NR dispersed in the SBR matrix, which was suppressed by the removal of the fatty acid. © 2005 Wiley Periodicals, Inc. J Appl Polym Sci 98: 613–619, 2005

Some Comments on the Analysis of the Third Virial Coefficient for Polymer Chains

An analysis is made of experimental data for the third virial coefficient A3 of atactic oligo- and poly(α-methylstyrene)s (a-PαMS) in the Θ solvent cyclohexane at 30.5 °C and in three good solvents, toluene, 4-tert-butyltoluene, and n-butyl chloride, at 25.0 °C on the basis of the helical wormlike (HW) chain model. It is found that A3 at Θ, which is denoted by A3,Θ, becomes a constant of 5.0×10−4 cm6 mol/g3 independent of the weight-average molecular weight Mw for Mw ≳ 104 and deviates from it for Mw ≲ 104 because of effects of chain ends. The observed dependence of A3,Θ on Mw may be well explained by the HW theory that takes account of the effects. For the three good-solvent systems, the behavior of the factor g defined by A3/[A2(HW)]2Mw is examined as a function of the cubed gyration-radius expansion factor αS3, where A2(HW) is the part of the second virial coefficient A2 without the effects of chain ends. It is found that the data points for a-PαMS samples with Mw ≳ 105 in all the three good solvents form nearly a single-composite curve, as predicted by the HW theory, and that they follow the Stockmayer–Casassa theoretical curve for αS3 ≳ 2 but deviate upward from it for αS3 ≲ 2 because of effects of three-segment interactions. Effects of chain stiffness are of minor importance for an explanation of this deviation. Some literature data for polystyrene are also examined.

Computer-aided bootstrap generation of characteristic curves for radiographic imaging systems.

Determination of the characteristic curve is essential for quantitative evaluation of digital as well as screen-film radiographic imaging systems. When it is not practical to generate the entire curve through variation of a single exposure parameter, bootstrap methods can be used. For the bootstrap method used here, curve segments are generated by varying one exposure parameter and multiple segments are produced at different exposure levels by varying a second parameter. The segments then are joined to form a single composite characteristic curve. If the second parameter is one for which the sensitivity of the receptor is not constant (such as x-ray tube potential or beam filtration), the shift of each segment along the log relative-exposure axis needed to join the overlapping curve sections is not known and some form of segment matching must be employed. A spline interpolation method and a polynomial fit integral method were developed for automatic segment matching and compared. For both methods, the shifts between successive segment pairs which result in optimal overlap are determined and the cumulative shifts to obtain the complete composite curve are calculated for all segments. The two methods are evaluated for several image receptors and with a known curve generated from an analytic function. Both methods closely agreed in the shifts determined for the image receptors and provided a good visual matching of the curve segments. The spline interpolation method more accurately determined the appropriate shifts for the mathematically generated curve segments. The bootstrap methods can provide complete, accurate characteristic curves, and the segment joining program makes the process fast and precise.

Viscoelastic Properties of Low Temperature Thermotropic Liquid Crystalline Polymers

Abstract Main-chain type thermotropic liquid crystalline polymers HPC-P having lower liquid crystalline temperature were obtained by esterification of hydoxypropyl cellulose. Two different samples were obtained, the one is HPC-P(l) having DS = 2.4 and another one is HPC-P(2) having DS = 2.8. Logarithmic curves of dynamic modulus(G') and loss modulus(G') against frequency obtained at different temperatures within liquid crystalline temperature range can be superposed to obtain a single composite curve for both samples. The composite curves for G' show lower frequency side plateau(Ca. 300 dyne/cm2). The log G' against temperature curves for both samples decrease with increasing temperature upto transition temperature and then increase upto TC and then decrease again with increasing temperature above TC showing maxima at around Tc.

Dynamic viscoelasticity of end-linking α,ω-dimethyl silyl poly(propylene oxide) solutions near the gel point

Dynamic viscoelasticity of α,ω-dimethyl silyl poly(propylene oxide) solutions was studied through the gelation process with end-linking. The gel point was determined as the reaction time, t = t c, at which the storage and the loss shear moduli, G′( ω) and G″( ω), respectively, both became proportional to ω n over the whole ω range measured, where ω is the angular frequency and n is a constant. Effects of polymer molecular weight M and concentration c on the steady-state viscosity η before the gel point and the equilibrium modulus G eq after the gel point were examined in terms of critical exponents, k and z, defined by power laws, η ∼ ( t − t c | t c ) −k ∼ ε −k and G eq ∼ ε z , respectively (ε being the relative distance from the gel point). The three exponents n, k and z were found to take universal values of n = 0.66 ± 0.02, k = 1.0 ± 0.1 and z = 2.0 ± 0.1 irrespective of M and c for unentangled systems. In these systems, the reduced plot of G eq (ε) G ∞ versus ε gave a single composite curve in the range of 0.1 < ε < 0.7, where G ∞ was the modulus for the ideal network in which all functional groups reacted. The presence of entanglement coupling in the prepolymer solutions made the critical behaviour of η and G eq obscure and affected the exponent values. The results were briefly compared with theoretical predictions.

Rheological properties of concentrated polymer solution: Polybutadiene in good and θ solvents

AbstractThe zero shear viscosity, η° of three polybutadiene samples having different molecular weights over a wide range of concentration (1.0–35.0% polymer) in good and θ solvents has been studied. Superposition of viscosity data has been made to give a single composite curve for each solvent by shifting them vertically by a factor (M°/M)3.4, where M° represents the molecular weight of the reference sample. The shift factor is found to be proportional to M3.4 in the region of higher concentration, which indicates that the 3.4‐power law is valid for the data of polybutadiene. The double‐logarithmic plots of relative viscosity η°r as a function of c5M3.4 yielded a single composite curve approximating a straight line with slope of unity at the higher values of the variables. The results indicate that over a considerable range of the variables (molecular weight and concentration) at a constant temperature, the relative viscosity is a single function of c5M3.4. The results for double‐logarithmic plots of zero shear specific viscosity η°csp as a function of concentration confirmed those observed in polycholoroprene samples studied earlier that the η0sp values in θ solvents at higher concentration region are found to be higher than those found in good solvents, whereas in the moderately concentrated region the values are just opposite in θ and good solvents. The viscosity crossover in θ solvents is not as sharp as is found in case of polychloroprene samples and that crossover, too, has taken place in the range of concentration of 11.7–31.6% polymer, which is comparatively higher than that of polychloroprene samples (6.06–21.0% polymer). The results indicate some relation between viscosity crossover and polymer polarity, supporting the idea of enhanced intermolecular association in poor solvents. To correlatethe viscosity data obtained in good and poor solvents, two methods, one given by Graessley and the other given by Dreval and coworkers involving the correlating variable c[η], were considered. The plots of relative viscosity η°, versus the correlating variable c[η] in benzene (good solvent) yielded one curve, but in the case of θ solvents (dioxane and isobutyl acetate), the same plots yielded three separate curves instead of a single curve, which is rather unusual. The appropriate correction on the correlating variable for chain contraction in the concentrated region in a good solvent moved the data to a common curve, especially in lower concentration region, but at the higher concentration region a slight overestimation of data seems to have been effected. On the other hand, the plots of log η as a function of correlating variable c[η] yielded a single curve for three samples in the good solvent benzene, but in poor solvents (diozane and isobutyl acetate) the same plots yielded three separate curves for three samples instead of a single curve, the reason for which is not known at present. However, the normalization of the correlating variable c[η] with Martin constant KM reduced all experimental data of the polymer samples to a common curve. The correlation of the viscosity data by either of the two methods seems to be possible in the case of the nonpolar flexible polymer, polybutadiene.

An analysis of excluded‐volume effects in a polystyrene‐cyclohexane system

AbstractAn attempt was made to analyse Miyaki's data for the radius of gyration &lt;s2&gt; and the intrinsic viscosity [η] of very high molecular weight samples of polystyrene in cyclohexane. The excluded‐volume variable z was estimated by adopting the temperature dependence of the excluded‐volume integral as predicted by Eichinger. It was found that the volume expansion as given by the ratio of &lt;s2&gt; to &lt;s2&gt;0 under the unperturned state was a function of z only, as required by two‐parameter theories. These data were in good agreement with the Domb‐Barrett and the Suzuki equations. On the other hand, the ratio of [η] to [η] 0 under the unperturbed state also gave a single composite curve when plotted against z. However, it was found that the observed hydrodynamic behaviour was not satisfactorily explained by two‐parameter theories of [η]. A brief comment was made on the validity of two‐parameter theories from the experimental point of view.

Dynamic Viscoelastic Properties of Gelatin Gels in Glycerol‐Water Mixtures

The storage shear modulus (G′) and loss tangent (tan δ) have been measured for gelatin gels in water and glycerol‐water mixtures containing 60, 73, and 91% glycerol by weight. The concentration of gelatin (molecular weight 35,000) was 15.0% and the temperatures were 5, 12, and 20°C. Measurements were made from 0.08 to 12.5 Hz in the Plazek torsion pendulum and (for one gel) from 0.05 to 500 Hz in the modified Birnboim transducer apparatus. Single composite curves extending over five logarithmic decades of frequency were obtained for most of the data by plotting logarithmically tan δ and G′Gq0/Gq against ωηsMc2/ηs0Mc02, where Gq is (approximately) G′ at the lowest frequency of measurement, ω the radian frequency, ηs the solvent viscosity, and Mc the average molecular weight of a network strand (estimated from Gq by the statistical theory of Pearson and Graessley); the subscript 0 refers to a reference state of water at 12.0°C. The relaxation spectrum H(τ), calculated from G′ and G′′=G′ tan δ by standard ap...

Excluded-Volume Effects in Dilute Polymer Solutions. V. Butyl Rubber in Various Solvents

Abstract It is a fundamental requirement of the two-parameter theory of linear homopolymers in dilute solution that the expansion factor α, and the interpenetration function Ψ depend only on the excluded-volume parameter z and that these functions are universal in the sense that their forms are independent of the chemical nature of the polymer and solvent as well as of the temperature. It follows immediately from the above requirement that the values of Ψ for different combinations of polymer, solvent, and temperature should form a single composite curve when plotted against αs.

Flow Properties of Moderately Concentrated Solutions of Polystyrene in Chlorinated Diphenyl

Flow properties such as complex moduli, complex moduli with superimposed steady shear and non-Newtonian viscosity were measured on moderately concentrated solutions of polystyrene in chlorinated diphenyl. The range of molecular weight M and of concentration c was 5.6×105–2.5×106 and 1–4.5 g/dl, respectively. The real part G′ and the imaginary part G″ of the complex modulus of solutions with various molecular weights and concentrations were superimposed to give single composite curves, respectively, if G′M and G″M were plotted against ωηM, where ω is the angular frequency and η is the zero-shear viscocity. Thus the steady shear compliance Je is proportional to M and is independent of c. The values of the real part G′ and the imaginary part G″ of the complex moduli with superimposed shear decreased when the rate of superimposed shear κ increased. They decreased more rapidly in the range of low frequency than in the range of high frequency. The degree of decrease was larger for the real part than for the imaginary part. This effect of the superimposed steady shear on the complex moduli, together with the molecular weight and concentration dependence of the complex moduli, were qualitatively interpreted by assuming molecular aggregates in moderately concentrated polymer solutions.

Effects of molecular weight and strain rate on the flexural properties of polycarbonate

AbstractThe effects of changes in molecular weight (7000 to 22,000) and strain rate (0.0001 to 4 min.−1) on the flexural properties of polycarbonate have been examined in detail with the use of speciments of different molecular weight prepared by high‐energy electron irradiation. The results have been plotted as surfaces which show the dependence of both stress and strain on molecular weight and strain rate, and these surfaces have been described in terms of brittle, transitional, and ductile regions. The relationships between stress or strain and molecular weight in the brittle region have been shown to be hyperbolic. A single failure locus has been found to include all the corresponding stress and strain data obtained at the various molecular weights and strain rates. In the low strength region this locus exhibits a proportionality between stress and strain, while at high strength values, strain becomes a logarithmic function of stress. Stress–molecular weight data obtained at the various rates have been superimposed to form a single composite curve, and the corresponding crossplots of stress–log rate have been treated similarly. It is concluded from these superpositions that an equivalence exists between changes in both molecular weight and strain rate such that a tenfold change in strain rate corresponds approximately to a change of 1000 in molecular weight. Strain‐strain rate data obtained at the various molecular weights have also been superimposed in a similar manner. Modulus is shown to increase slowly with decrease in molecular weight and appears to be relatively insensitive to changes in strain rate.

Solution Properties of Synthetic Polypeptides. III. Viscosity Behavior of Poly-γ-Benzyl-L-Glutamate in the Helix—Coil Transition Region

Limiting viscosity numbers [η] and specific rotations [α]578 of five samples of poly-γ-benzyl-L-glutamate (PBLG) (ranging in weight-average molecular weight M̄w from 1.9×104 to 4.0×105) in mixtures of dichloroacetic acid (DCA) and ethylene dichloride (EDC) of various compositions were measured as functions of temperature in the range where the PBLG molecule underwent a thermal transition from a disordered, random-coil state to an ordered helical state. For one sample the measurements of optical rotatory dispersion were made at various solvent compositions and temperatures, and the data were analyzed in terms of the Moffitt plot. When the M̄w of a given sample was above a certain critical value, [η] first decreased (or was kept constant), passed through a shallow minimum, and then increased sharply to a limiting value as the temperature was raised. Below the critical value, the change in [η] with temperature was a monotonic decrease. Irrespective of the molecular weight of the sample, [α]578 increased sharply in a narrow range near 30°C and then gradually leveled off as the temperature was raised. This narrow region coincided with the region where [η] underwent a marked increase. It was found that the helical content fH of a given sample can be evaluated from the reduced mean residue rotation measured at 578 mμ. When the values of [η] obtained under different solvent compositions and temperatures are plotted against the fH values so calculated, the plotted points formed a single composite curve over the entire range of fH. This curve exhibits a quite similar character to the theoretical curves for 〈S2〉 (mean-square radius of gyration) vs fH calculated by Nagai on the statistical-mechanical treatment of helix-forming polypeptides.

Shear-Rate Dependence of the Intrinsic Viscosity of Flexible Linear Macromolecules

Shear-rate dependence of the intrinsic viscosity has been studied on polystyrene in four different solvents by using a Zimm—Crothers rotational viscometer and Ubbelohde viscometers. The former viscometer was employed particularly for measuring zero-shear intrinsic viscosity [η]0. The results are summarized in terms of the [η]/[η]0 versus generalized rate of shear β=([η]0ηsM/RT)G, where ηs is the solvent viscosity, M the solute molecular weight, G the rate of shear, and others have usual significance. It has been found that, (i) at the values of β below 0.1, [η] is practically independent of rate of shear (or shear stress); (ii) the non-Newtonian effect does exist even in θ conditions; and (iii) at the values of β below 2, the [η]/[η]0-vs-β curves of the systems with different molecular weights and in different solvents give a single composite curve, which fairly well resembles to the curve of a prolate ellipsoid with the axial ratio p=3.

Rheo-Optical Studies of High Polymer X Time Dependent Birefringence for Low Density Polyethylene

An equipment has been designed to provide a record of the birefringence change accompanying a stress-strain measurement or a stress relaxation measurement. The equipment consists of Instron type tensile tester and an optical system to measure the birefringence which has been designed to be easily mounted on or removed from the tensile tester. The optical system has a conventional arrangement consisting of a light source, filter, crossed polaroids and so on, and the birefringence can be recorded continuously by the intensity method with a photomultiplier.The change in the birefringence with time during the stress relaxation has been measured for low density polyethylene with this equipment. The birefringence vs. time curves at different strain levels show a similar shape, being consistent with the data obtained by Strella et al. There can be found straight-line relationship between the birefringence and the strain at any time during the relaxation under the strain level lower than about 5%. The time dependence of the strain-optical coefficient depends highly upon temperature, and a transition of the strain-optical coefficient can be observed at the temperature region covered by this experiment. By shifting the curves at different temperatures horizontally, a single composite curve was obtained. The annealed and quenched films of low density polyethylene show definitely different birefringence, but the temperature dependence of the shift factor log aT for these films gives the same straight line when it is plotted against reciprocal absolute temperature. This can be understood qualitatively, at least, on the basis that the relaxation mechanism which is responsible for the birefringence of both the annealed and quenched polyethylene films in such a temperature region is the same. The optical relaxation spectrum for the annealed and quenched polyethylene has been obtained by the first approximation method. The longest relaxation time for the annealed material is almost two decades longer than that for the quenched one.

Viscoelastic Properties of Glucose Glass near Its Transition Temperature

Storage (G′) and loss (G″) shear moduli of supercooled glucose have been measured at thirteen temperatures from 27.4 to 41.1°C in the frequency range from 0.3 to 1.8 cps. Shear creep has been measured at four temperatures from 41.1 to 47.6°C. The temperature dependence of the storage moduli and the creep compliance was described by reducing the time and frequency scales with shift factors calculated from the WLF equation, the constants corresponding to a fractional free volume of 0.026 at 41.1°C and a free volume expansion coefficient of 3.6×10−4 deg−1. The loss moduli plotted against reduced frequency did not give a single composite curve, however. Conversion of the dynamic measurements to creep compliance by approximation methods provided the latter function over ten decades of logarithmic time except for a gap of three decades in the middle of the range. At long times, the creep was indistinguishable from viscous flow; the viscosity at 41.1°C was 1.82×1010 poise. At short times, the limiting creep compliance appeared to be 0.32×10−16 cm2/dyne, and there was an additional time-dependent compliance of at least this magnitude with a broad distribution of retardation times. The relaxation and retardation spectra were calculated for this region.

Stress relaxation in nylon and terylene: Influence of strain, temperature, and humidity

AbstractStress relaxation data are presented for nylon 66 and Terylene yarns held at constant extensions from ½% to 15% for dry, 65% R.H. and wet conditions all at 25°C. and in the dry condition at 25, 50, and 75°C. A single composite relaxation curve fitted the data for Terylene at all strains at 25°C. A theory of stress relaxation gives a good fit for the experimental results of dry nylon at all temperatures between 25°C. and 75°C. In moist nylon, there are two relaxation processes superimposed on each other, one due to thermal agitation and the other due to absorption of extra moisture as the fiber is stretched. It is shown that the relaxation caused by the extra moisture absorbed is consistent with the change in moisture regain with strain at constant humidity calculated thermodynamically.

Stress Relaxation of Thermo-Reversible Gels. Polyvinyl Alcohol–Congo Red–Water System

Abstract Stress relaxation curves were obtained for the gels of the polyvinyl alcohol-conga red-water system in the temperature range, 15∼60°C, by means of a chainomatic relaxometer. The molecular weight of polyvinyl alcohol was 8.5×104, and gels of 15.0 and 17.5% were used, where the concentration of congo red was adjusted to be 1/10 of that of polyvinyl alcohol. Master relaxation curves were obtained for each concentration over the range, 10−2∼109hr, and relaxation spectra were calculated by the second order approximation. The following results were obtained. 1) The logaT vs. T relation gives a single curve for two concentrations, and does not obey the equation of WLF form, Also it has characteristics similar to that of agar-agar gels. 2) The apparent activation energy is kept constant to be 42 kcal. below about 43°C, and above this temperature it sharply increases with temperature. 3) The reduction on concentration to a single composite relaxation curve seems to be possible in rough approximation.

Mechanical properties of poly- n-octyl methacrylate at low frequencies and in creep; entanglements in methacrylate polymers

The storage modulus and loss tangent have been measured between 0.06 and 1.6 cycles/sec. in the temperature range from 54° to 140°C. for a fractionated poly- n-octyl methacrylate with molecular weight 3.62 × 10 6. Creep was measured at 120°C. over a time interval up to 2000 sec. The usual method of reduced variables, using a T factors determined in an earlier study at higher frequencies, failed to give single composite curves for data at different temperatures; the residual anomaly appears most clearly in the loss compliance, the maximum of which shifts upward and to lower frequencies with increasing frequency, as previously observed in the n-butyl and n-hexyl methacrylate polymers. The data for all three polymers in this region can be brought into coincidence, however, by an additional reduction which is formally equivalent to assuming a dissociation of entanglement points with increasing temperature. The apparent heat of dissociation is 2.3 kcal./mole for the butyl and hexyl polymers and 1.6 for the octyl. In the plateau zone of the time scale, both the retardation and the relaxation spectra of poly- n-octyl methacrylate pass through minima at about the same point; the sharpness of curvature makes it necessary to use higher approximations than usual to calculate the spectra. From the width of the plateau zone, the average degree of polymerization between entanglement points at 100°C, is estimated to be 240.

Dynamic Mechanical Properties of the System Polystyrene-Decalin

The dynamic rigidities and viscosities of solutions of polystyrene (number-average molecular weight 197 000) in decalin have been measured by three experimental methods over ranges of concentration from 10 to 62 percent, of temperature from −5° to 50°C, and of frequency from 30 to 2000 cps. Similar measurements were made on the undiluted polystyrene from 115° to 133°C and 30 to 2800 cps. The data for the solutions, when reduced to a reference state of unit density and viscosity at a standard temperature, provide single composite curves for dynamic rigidity and viscosity. The distribution function of relaxation times calculated from these curves overlaps that previously derived from stress relaxation measurements on this system, and the combined function covers a range of time scale of ten powers of ten. It shows the three characteristic regions previously identified in the relaxation distribution function of polyisobutylene; there are no obvious anomalies associated with the fact that decalin is a very poor solvent for polystyrene. The data for the undiluted polymer, when reduced to a standard temperature, provide single composite curves for dynamic rigidity and viscosity. The temperature reduction factors provide apparent energies of activation for relaxation which agree with those for viscous flow obtained by Fox and Flory. The relaxation distribution function of the undiluted polymer is similar in shape to, but somewhat sharper than, those for other polymers in the transition from soft to glassy consistency. When reduced to a reference state of unit density and viscosity, the distribution function of the undiluted polymer lies near that reduced from data on solutions, but is considerably sharper than the latter.