Logarithmic Addition Research Articles

Effect of composition distribution on miscibility and co-crystallization phenomena in the blends of low density polyethylene with conventional and metallocene-based ethylene–butene copolymers

Three ethylene–butene copolymers (two metallocene-based and one conventional) having different composition distributions were blended with low density polyethylene (LDPE) in a wide proportion range. The miscibility of the blend melts was studied with the rheological method. It is found that zero-shear viscosity of all the blends does not obey the logarithmic additivity rule, indicating the immiscible state of the blend melts. The blends were rapidly quenched from the melts and co-crystallization phenomena were investigated by differential scanning calorimetry (DSC). The obtained results show that composition distribution has a great influence on co-crystallization. In the blends of the conventional ethylene–butene copolymer, which has a broad composition distribution, part of the co-crystals may segregate from the crystals of both LDPE and ethylene copolymer and exist as a separated population. This suggests that there exist three phases in the melts. In contrast, the blends of the metallocene-based ethylene–butene copolymers with a narrow composition distribution are only composed of two phases together with some exchange between the two components. It is also observed that broader composition distribution leads to more ethylene–butene copolymer transferred into LDPE.

Microstructure and flow behaviour of liquid water-gelatin-locust bean gum systems

Liquid water-gelatin-locust bean gum (LBG) systems, in the conditions of lowest compatibility (near the isoelectric pH of the protein), were explored using confocal laser scanning microscopy and viscometry. Confocal microscopy observation proved to be a more sensitive method to assess the phase state of the systems than the usual centrifugation or viscometric ones. It showed that in fact the system is already phase-separated well below the apparent binodal line determined by the other methods. When the continuous phase is enriched in LBG, the observed microstructure is of the expected type with spherical droplets of gelatin dispersed in the LBG phase. Bicontinuous systems were observed close to the inversion point, which is clearly detected by viscosity measurements; beyond this point, the gelatin enriched continuous phase contained large inclusions of LBG, apparently due to aggregation of the LBG disperse phase.For systems with a LBG enriched continuous phase, shear thinning behaviour, similar to that of a LBG solution, was observed; the low newtonian viscosity decreased as the volume fraction of the gelatin disperse phase increased, following approximately the usual logarithmic additivity law. In the reverse case, where the continuous phase is enriched in gelatin, flow curves with an apparent yield stress and a negative deviation from the logarithmic additivity law were observed. The variation of the viscosity with the volume fraction of the disperse phase obeyed approximately the equation of Palierne for systems of two viscoelastic phases when the interface is purely elastic.The microstructure of phase-separated mixtures depends on the phase viscosity ratio, which also determines the differences in flow behaviour.

RNS-based logarithmic adder

The operation of logarithmic addition has always been burdened with large volumes of look-up tables. Therefore with a goal of reducing the ROM size required, an elegant and novel technique for logarithmic addition using the residue number system is demonstrated. To formulate this technique, the properties of finite fields and finite rings are exploited. A multiple-base logarithm is defined first, which is then successfully used for the formulation of the proposed technique for logarithmic addition. With this approach, the ROM requirement has been reduced to a bare minimum, thereby reducing the complexity of logarithmic addition, and permitting an elegant and efficient implementation.

Correlation of penetrant transport with polymer free volume: Additional evidence from block copolymers

The transport rates of methanol and water through a series of PEBAX® block copolymers were measured and correlated with the fractional free volume. Excellent agreement between the logarithm of the diffusion coefficient and the inverse of the fractional free volume of the polymer was observed. This provides new evidence of the utility of the free volume theory to describe the transport of highly condensable vapours. Correlation was also quite good when the logarithmic additivity relationship was employed. This relationship has been previously shown to correlate the properties of homogenous blends with copolymer composition. The successful use of this theory for a series of blends that exhibit two glass transition temperature is discussed.

Logarithmic Factors, Critical Temperature, and Zero Temperature Flipping in the 4D Kinetic Ising Model

We determine the critical temperature in the four-dimensional nearest-neighbor Ising model as J/kB Tc=0.149694±0.000002 from kinetic Monte Carlo simulations of up to 5764 spins. Here we assume the critical magnetization to decay with time as (t/ log t)-1/2. However, possible logarithmic additions to this leading scaling behavior could change the estimate beyond these error bars. A reanalyzis of old series expansions for the susceptibility and fourth moment gives 0.149696±0.000004.

Simplification of table structure in logarithmic arithmetic

Logarithmic addition and subtraction, particularly the latter, require a large and complex lookup table. The algorithm presented substantially reduces the size and complexity of the arithmetic, with little additional overhead in time or hardware.

Rheological properties of poly(vinyl chloride)/epoxidized natural rubber blends. Part I: Composition dependence and the effect of compounding conditions

AbstractEpoxidized natural rubber is a recently commercialized modified form of natural rubber. This paper is part of our continuing effort to study the responses of this new material to melt mixing and other shaping processes. The Shimadzu capillary rheometer was used to evaluate the composition dependence of miscibility of polyvinyl chloride/epoxidized natural rubber blends (PVC/ENR blends). The rheometer was also used to evaluate the effect of compounding parameters on the rheological properties of the blends. The results confirmed PVC/ENR blends as miscible systems that show a synergism in apparent shear viscosity highlighted by the positive deviation from the logarithmic additivity rule. These results based on capillary rheometry are also in very good agreement with our earlier attempt to predict optimum mixing from torque rheometry by using the Brabender Plastogram as an indicator.

Rheological properties of poly(vinyl chloride)/epoxidized natural rubber blends. Part I: Composition dependence and the effect of compounding conditions

Epoxidized natural rubber is a recently commercialized modified form of natural rubber. This paper is part of our continuing effort to study the responses of this new material to melt mixing and other shaping processes. The Shimadzu capillary rheometer was used to evaluate the composition dependence of miscibility of polyvinyl chloride/epoxidized natural rubber blends (PVC/ENR blends). The rheometer was also used to evaluate the effect of compounding parameters on the rheological properties of the blends. The results confirmed PVC/ENR blends as miscible systems that show a synergism in apparent shear viscosity highlighted by the positive deviation from the logarithmic additivity rule. These results based on capillary rheometry are also in very good agreement with our earlier attempt to predict optimum mixing from torque rheometry by using the Brabender Plastogram as an indicator.

On The Parametric Model of Western Boundary Outflow

A simple model equation for western boundary outflow in the Stommel model of the large scale ocean circulation is obtained by evaluating the potential vorticity equation at the western boundary. A series solution to this model equation demonstrates similar behavior to the boundary layer solution of the potential vorticity equation, in particular that “resonances” are present at a discrete series of parameter values which necessitate the addition of logarithms to the series; these resonances occur because the model equation has a logarithmic branch point at these values.

Viscoelasticity of melts of polyethylene blends. 2

1. The activation energies of viscoelastic relaxation and viscous flow of melts of LDPE/HDPE blends of the same composition are the same. 2. The viscosity of melts of blends of linear and branched polyethylenes with low shear stresses is higher and corresponds to the value predicted by the rule of logarithmic additivity for high shear stresses. 3. The shear rate corresponding to the appearance of elastic turbulence increases exponentially with an increase in the proportion of HDPE in the polyethylene blend. 4. The equilibrium shear modulus of the melt can be regulated within broader limits by mixing HDPE and LDPE than by altering the molecular characteristics of the polymer. 5. The Havriliak-Negami equation in principle cannot describe the ratio of the rates of accumulation and dissipation of energy in steady-state shear flow whenPw/2ġgη0 > 0.5 (not the experimentally observed viscosity superanomaly is predicted, i.e.,dσ12/dġg < 0). 6. The first difference of normal stresses in shear flow, calculated with the die swell with the Bagley equation, is close to the experimental results for HDPE melts and HDPE/LDPE blends if the concentration of HDPE > 60%; for LDPE melts and blends containing HDPE < 60%, the best agreement of the predicted and experimental values is attained in calculations with the Tanner dependence.

Fast optical binary multiplication using a sign/logarithm number system

A new fast binary multiplication scheme based on a nonholographic optical content-addressable memory (CAM) is presented. By using a sign/logarithm number (SLN) system, the multiplication operation is performed through a binary logarithmic addition. A three-stage CAM-based multiplication scheme is proposed in which the first CAM converts input binary numbers to SLN's, the second CAM performs a fixed-point binary addition, and the third CAM converts this SLN result back to the binary number system. The design and experimental demonstration of a 7-bit nonholographic optoelectronic CAM-based multiplier are presented.

Rheological Properties and Morphological Structure of Blended and Filled Nylon-6

Abstract Rheological properties and morphological structure of PA 6/NBR and PA 6/EVA blends with and without CaCO3 fillers are studied. It is found that viscosities of samples investigated can be described approximately by the power law. The rheological behaviour of PA 6/NBR blends slightly deviates from the so-called logarithmic additivity law of the relation between viscosity and composition. The viscosity-composition curve of PA 6/EVA blends shows a positive deviation at the range of low shear rate and changes to the manner of negative deviation after the shear rate is greater than a certaine value. All the samples fabricated in the experiment perform very satisfactory in their processing. An unique morphological structure is found for them, which may be presented by the model of “sea and island”.

An architecture for addition and subtraction of long word length numbers in the logarithmic number system

An architecture is described for performing addition and subtraction of numbers in the logarithmic number system using small lookup tables. Previous implementations require approximately 4*2/sup F/ words for F bits of precision in the fraction. The author shows how to reduce the size of the lookup table to fewer than (18+F)*2/sup F/2/ words. The key to this is the use of a linear approximation of the nonlinear functions stored in the lookup tables. The functions involved are highly nonlinear in some regions, so variable size regions are used for the approximation. The replacement of the F exponential dependence on the number of words by F/2 allows roughly 50% more bits of fractional precision to be obtained for a given amount of ROM. The architecture is mathematically analyzed, yielding explicit expressions for all design parameters. The approach is illustrated with an example logarithmic addition and subtraction unit using 32-b words with 30-b exponents containing 22 fractional bits. A factor of 118 reduction in table size compared to previous techniques is achieved for this example. >

Second-generation RISC floating point with multiply-add fused

A 440000-transistor second-generation RISC (reduced instruction set computer) floating-point chip is described. The pipeline latency is only two cycles, and a double-precision result is produced every cycle. System throughput and accuracy are increased by using a floating-point multiply-add-fused unit, which carries out a double-precision accumulate as a two-cycle pipelined execution with only one rounding error. While the cycle time (40 ns) is competitive with other CMOS RISC systems, the floating-point performance stretches to the range of bipolar RISC systems (7.4-13 MFLOPS LINPACK). Leading zero anticipation makes the two-cycle pipeline possible by nearly eliminating the additional postnormalization time, and it allows for reduced overall system latency. Partial decode shifters allow complete time sharing for the multiply and data alignment. Improved design techniques for logarithmic addition and higher order counters for multiplication complete this second-generation RISC floating-point unit design. >

Glass transition, compatibility and viscosity of blends of polymethylmethacrylate with its copolymers

The glass transition temperatures and viscosities of binary PMMA blends with the statistical copolymers of methylmethacrylate with the ethyl-, butyl- or octyl-methacrylates and also PMMA blends with polyethyl-, polybutyl- and polyoctyl-methacryates have been studied. From the glass transition temperatures, the authors characterize the regions of the mono- and diphasic states of the blends studied. Blends of PMMA and methylmethacrylate copolymers with ethylmethacrylate may form monophasic systems over a wide range of compositions of the blends and the copolymers. Transition to methylmethacrylate copolymers with butyl- and especially octylmethacrylates sharply narrows the ranges of the compositions of these copolymers and their blends with PMMA in which they form monophasic systems. The dependences of the boundaries of compatibility of the polymers on their solubility parameters have been established. The viscosity of the monophasic systems studied changes with the concentration of the blend in line with the logarithmic additivity rule. Passage to diphasic blends in most cases leads to a negative deviation of the logarithm of viscosity from its value predicted by the logarithmic additivity rule. In the region of the transition from the monophasic to the diphasic state viscosity may sharply rise.

Steady-state viscosity of the liquid two-phase disperse system water-casein-sodium alginate

The steady-state viscosity ( η ∗ ) of the liquid two-phase disperse system water-casein-sodium alginate of varying composition, with a disperse particle diameter of about 30 μm, has been investigated in the shear rate range γ ̇ = 10 −2−10 7 s −1 . The flow curves obtained are similar in shape. They are invariant in composition and can be fitted to the flow curve equation ( η ∗ η ∗ 0 ) = f(k γ ̇ ) where η ∗ 0 is the Newtonian viscosity obtained from data at low shear stresses and k is an empirical parameter. Both η ∗ 0 and k depend on the composition of the two-phase system. The experimental dependence of the viscosity on the composition follows a logarithmic additivity law at high shear stresses, but at low shear stresses the observed viscosities are lower than would be predicted by this law. The experimental dependence of the ratio k η ∗ 0 (the mean activation volume) on the composition is also not in agreement with the additivity law. These special features of the steady-state flow of the investigated disperse system are explained by the presence of a low-concentration interphase layer having a lower viscosity and less pronounced non-Newtonian behaviour than the disperse and continuous phases. This interphase layer exists in other two-phase systems which consist of a solvent and two incompatible polymers. For this reason, the rheological behaviour of the system investigated may be expected to be common for two-phase systems of this type, particularly protein-polysaccharide mixtures in water.

Stochastic modeling of community noise levels

A modeling technique is proposed for combining two or more noise histograms (i. e., noise level versus duration histories) into a new histogram. By considering the time varying nature of independent noise sources, much more accurate community noise measures may be computed. Logarithmic addition of statistical measures (e. g., Leq or L50) invariably overestimates the true result. The method is particularly useful in modeling a multisource environment such as a construction site where the duty cycles of independently operating equipment are accurately known. The analytical procedure is presented with a construction noise example.

Two-component monolayers. III. Evaporation resistances of the octadecanol—docosyl sulphate system

The evaporation resistances of monolayers of the octadecanol-docosyl sulphate system have been measured by the Langmuir-Schaefer technique. Eleven different compositions spanning the entire composition range were studied. At surface pressures above 15 mN m −1 evaporation resistances of the mixed monolayers were greater than resistances calculated from the Barnes-La Mer logarithmic addition rule. At lower surface pressures the pattern was more complex, but of special note was a sharp drop in resistance from the high value of a pure octadecanol monolayer to the low values of the various mixtures. These results are discussed in terms of the energy barrier theory and excess free energies of activation ( ΔG ≠E) are reported. A relationship between these values and the excess free energies of mixing ( ΔG E): ΔG ≠E = − KΔG E, is established, provided allowance is made for the (unmeasured) contributions of the high area regions of the surface pressure-area isotherms to AGE. When the evaporation resistance data are plotted against molecular area, as suggested by the accessible area theory, all of the points fall roughly on the same curve except for the points for pure octadecanol monolayers. The high resistances of octadecanol are attributed to a low value of the evaporation coefficient.

All n–p–n 4-quadrant analogue multiplier

A technique is described that utilises the addition of logarithms in an all n–p–n bipolar configuration, to derive the algebraic product of two variables. The circuit is ideal for integrated-circuit implementation, and uses a simple method of distortion cancellation that is shown to be independent of device current gain β and base-spreading resistance.

Viscous flow and structure of linear polymers

AbstractThe viscosity of linear polymers obeys the logarithmic additivity rule. This is associated with the fact that the activation energy of the viscous flow within certain limits of parameters changes does not depend on temperature, stress, molecular weight, and content of active filler. The mechanism of viscous flow of linear polymers is associated with the reversible destruction of an unstable supramolecular structure typical of the polymers in the state of viscous flow. The effect of shear stress on the viscosity of linear polymers does not obey the Eyring equation. It is supposed that the effect of shear stress on viscosity is associated with the peculiar mechanism of the destruction of the supramolecular structure. At higher rates of viscous flow of the polymer, the activation energy can decrease as a result of the deep destruction of the supramolecular structure. This is a reversible process, as after removal of stress the supramolecular structure slowly reverts to its original state.