Model for the viscoelastic and viscoplastic responses of semicrystalline polymers

Constitutive equations are derived for the time‐dependent behavior of semicrystalline polymers at isothermal loading with small strains. A semicrystalline polymer at temperatures above the glass‐transition point for its amorphous phase is thought of as a network of macromolecules bridged by junctions (physical crosslinks, entanglements, and crystalline lamellae) that can slide with respect to their reference positions in the bulk material under straining. The network is assumed to be highly inhomogeneous, and it is modeled as an ensemble of mesoregions (MRs) with various strengths of interchain interaction. Two types of MRs are distinguished: passive, where these interactions prevent detachment of strands from junctions; and active, where active strands separate from junctions and dangling strands merge with the network at random times as they are thermally agitated. The viscoelastic response of a semicrystalline polymer reflects reformation of strands in active MRs, whereas its viscoplastic behavior is associated with sliding of junctions. Stress–strain relations for uniaxial deformation are developed by using the laws of thermodynamics. Adjustable parameters in the constitutive equations are found by fitting experimental data for isotactic polypropylene in a tensile test with a constant strain rate and in tensile relaxation tests at various strains. Fair agreement is demonstrated between the observations and the results of numerical simulation. It is revealed that the viscoplastic flow of junctions strongly affects the rearrangement process in active MRs, whose rate reaches a threshold value in the vicinity of the apparent yield point. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 88: 1438–1450, 2003     

Conduction and dissipation in the shearing flow of granular materials modeled as non-Newtonian fluids

After providing a brief review of the constitutive modeling of the stress tensor for granular materials using non-Newtonian fluid models, we study the flow between two horizontal flat plates. It is assumed that the granular media behaves as a non-Newtonian fluid (of the Reiner-Rivlin type); we use the constitutive relation derived by Rajagopal and Massoudi [Rajagopal, K. R. and M. Massoudi, “A Method for measuring material moduli of granular materials: flow in an orthogonal rheometer,” Topical Report, DOE/PETC/TR-90/3, 1990] which can predict the normal stress differences. The lower plate is fixed and heated, and the upper plate (which is at a lower temperature than the lower plate) is set into motion with a constant velocity. The steady fully developed flow and the heat transfer equations are made dimensionless and are solved numerically; the effects of different dimensionless numbers and viscous dissipation are discussed.     

Modeling the rheology of gelatin gels for finite deformations. Part 1. Elastic rheological model

The rheological behavior of gelatin gels is studied through a constitutive equation derived in the literature from the classical hyperelasticity theory by using a nonlinear strain energy density. Gelatin gels are assumed to be an elastic solid here, which at a given maturation time has a degree of physical cross-links composing a permanent elastic network. It is also studied in particular: (a) the validation of the elastic rheological model proposed here in relation to the prediction of two data sets pertaining to different experimental tests (shear and simple compression) by using fixed values of rheological parameters, (b) the increase of the stress-strain nonlinearity with increasing maturation time, at a given protein concentration, (c) the evolution of the stress-strain nonlinearity with increasing gelatin concentration. It is found that gelatin gels exhibit a strong nonlinear strain energy density, which is expressed through two shear modules related to nucleation and growing of junction zones.     

The effect of strain rate on the viscoplastic behavior of isotactic polypropylene at finite strains

Two series of uniaxial tensile tests are performed on isotactic polypropylene with the strain rates ranging from 5 to 200 mm/min. In the first series, injection-molded specimens are used without thermal pre-treatment. In the other series of experiments, the samples are annealed for 51 h at 160 °C prior to testing.A constitutive model is developed for the viscoplastic behavior of isotactic polypropylene at finite strains. A semicrystalline polymer is treated as equivalent heterogeneous network of chains bridged by permanent junctions (physical cross-links and entanglements). The network is thought of as an ensemble of meso-regions connected with each other by links (lamellar blocks). In the sub-yield region of deformations, junctions between chains in meso-domains slide with respect to their reference positions (which reflects sliding of nodes in the amorphous phase and fine slip of lamellar blocks). Above the yield point, the sliding process is accompanied by displacements of meso-domains in the ensemble with respect to each other (which reflects coarse slip and fragmentation of lamellar blocks). To account for alignment of disintegrated lamellar blocks along the direction of maximal stresses (which is observed as strain-hardening of specimens in the post-yield regions of deformations) elastic moduli are assumed to depend on the principal invariants of the right Cauchy-Green tensor for the viscoplastic flow.Stress-strain relations for a semicrystalline polymer are derived by using the laws of thermodynamics. The constitutive equations are determined by five adjustable parameters that are found by matching observations. Fair agreement is demonstrated between the experimental data and the results of numerical simulation. A noticeable difference is revealed between the mechanical responses of non-annealed and annealed specimens: (i) necking of samples not subjected to thermal treatment precedes coarse slip and fragmentation of lamellar blocks, whereas cold-drawing of annealed specimens up to a longitudinal strain of 80% does not induce spatial heterogeneity of their deformation; (ii) the elastic modulus increases with the strain rate for non-annealed specimens and decreases for annealed samples.     

Effect of non-Gaussian chains on fluctuations of junctions in bimodal networks

A simple tetrahedron model is used to study the effect of non-Gaussian chains on fluctuations of junctions in bimodal networks. The four chains are assumed to meet at a junction with their other ends being fixed at the vertices of the tetrahedron. It is assumed that the angles between mean end-to-end vectors of all four chains connected at the junction are tetrahedral, but the lengths of edges of the tetrahedron may differ due to the difference in the lengths of the chains. The central junction is free to fluctuate, subject to the constraints imposed by the pendant chains. The long chains are chosen to be Gaussian. The short chains are assumed to be non-Gaussian. Calculations show that the non-Gaussian nature of the short chains imposes severe restrictions on the fluctuations of the central junction. The strength of these restrictions directs attention to the importance of anharmonic modes in networks.     

泡盖式蒸馏塔的效率

<正>蒸馏塔的分离效率是评价这个塔好坏的指标之一。到目前为止已经有很多作者来探讨了影响蒸馏塔分离效率的若干因素并从而建立了估计分离效率的经验或半经验公式。所谓分离效率就是来衡量一个蒸馏塔实际的分离情况与理想的分离情况（每块塔板上达到平衡）间相差的程度。在一般文献上所载的分离效率通常有下列三种定义:     

From discretization to regularization of composite discontinuous functions

Discontinuities between distinct regions, described by different equation sets, cause difficulties for PDE/ODE solvers. We present a new algorithm that eliminates integrator discontinuities through regularizing discontinuities. First, the algorithm determines the optimum switch point between two functions spanning adjacent or overlapping domains. The optimum switch point is determined by searching for a “jump point” that minimizes a discontinuity between adjacent/overlapping functions. Then, discontinuity is resolved using an interpolating polynomial that joins the two discontinuous functions.This approach eliminates the need for conventional integrators to either discretize and then link discontinuities through generating interpolating polynomials based on state variables or to reinitialize state variables when discontinuities are detected in an ODE/DAE system. In contrast to conventional approaches that handle discontinuities at the state variable level only, the new approach tackles discontinuity at both state variable and the constitutive equations level. Thus, this approach eliminates errors associated with interpolating polynomials generated at a state variable level for discontinuities occurring in the constitutive equations.Computer memory space requirements for this approach exponentially increase with the dimension of the discontinuous function hence there will be limitations for functions with relatively high dimensions. Memory availability continues to increase with price decreasing so this is not expected to be a major limitation.     

The effects of temperature and molecular weight on the mechanical response and strength of elastomers

Summary Constitutive equations are derived for the mechanical behavior of rubbery polymers at finite strains. The model is based on the concept of rigid-rod networks, where breakage of chains is treated as bond scission. Adjustable parameters in the stress-strain relations are found by fitting observations in tensile tests for ethylene-octene copolymers. It is revealed that the constitutive equations correctly describe stress-strain curves up to the break points. Young's modulus and the critical strength per bond monotonically decrease with temperature and increase with molecular weight. Received: 17 January 2001/Accepted: 27 February 2001     

POSSIBILITY OF CONCENTRATION POLARIZATION OCCURRING INSIDE THE MEMBRANE DURING STEADY STATE PERVAPORATION

Mathematical equations are derived for describing the pervaporation transport of pure penetrant through polymeric membranes by assuming the chemical potential gradient as the driving force for the flow of penetrant. An imaginary phase (liquid or vapor) is assumed to be present and is in thermodynamic equilibrium with the membrane phase for deriving these equations. The mathematical equations obtained are similar to those derived by Okada and Matsuura (1991) using the pore-flow model. The possibility of concentration polarization occurring inside the membrane is predicted based on the analysis of binary mixture pervaporation. Experimental profiles of binary penetrant concentration in the membrane are established and these profiles substantiate the prediction of concentration polarization occurring inside the membrane. Pervaporation and sorption data from liquid and vapor phase at 25° C are reported for the acetic acid-water-polyamide system.     

On the characterization of nonlinear viscoelastic materials

Starting with specific constitutive equations, methods of evaluating material properties from experimental data are outlined and then illustrated for some polymeric materials; these equations have been derived from thermodynamic principles, and are very similar to the Boltzmann superposition integral form of linear theory. The experimental basis for two equations under uniaxial loading and the influence of environmental factors on the properties are first examined. It is then shown that creep and recovery data can be conveiently used to evaluate properties in one equation, while two-step relaxation data serve the same purpose for the second equation. Methods of reducing data to accomplish this characterization and to determine the accuracy of the theory are illustrated using existing data on nitrocellulose film, fiber-reinforced phenolic resin, and polyisobutylene. Finally, a set of three-dimensional constitutive equations is proposed which is consistent with nonlinear behavior of some metals and plastics, and which enables all properties to be evaluated from uniaxial creep and recovery data.     

A model for estimation of electromechanical characteristics of piezoelectric polymer composites

The constitutive equations, of piezoelectric polymer composite materials used for their electromechanical characterisation are complicated and their analytical solution is difficult. Earlier models assumed either stress or strain components to be constant. In this paper we present a model for electromechanical characterisation of piezoceramic polymer composites. The assumptions are (i) the samples have complete stress free boundary conditions at the polymer piezoelectric interfaces, (ii) the materials are lossless, (iii) the co-ordinate axes are pure mode propagation directions and (iv) the electric potential and the displacements are assumed to be independent degrees of freedom. Based on these assumptions, expressions for impedance, phase velocities and electromechanical coupling constant are developed for piezoelectric polymer composites.     

CALCULATION METHODS FOR DISTILLATION SYSTEMS WITH REACTION

A calculation procedure is presented for the solution of the equations describing a distillation process in which reaction is taking place. A mathematical model based on matrix notation is used to formulate the system equations. It is assumed that the distillation process can be represented by a system of interconnected stages, each of which is well mixed with the vapor and liquid phases in equilibrium. Reaction may take place in the vapor phase, liquid phase, or both. The equations describing this system are first presented. Then a calculation sequence and a correction algorithm are derived for the case in which the liquid and vapor solutions are ideal—that is, the equilibrium ratios are functions only of temperature and pressure and not of composition. The calculation sequence uses as variables of iteration the vectors of vapor flow rate, temperature, and reaction extents. A derivative correction method is used, and equations for the calculation of the Jacobian matrix are derived. The application of the method is illustrated by solution of two sample problems. The special case in which the heat of reaction is small, and constant molal overflow can be assumed, and the case in which the reaction rate is very fast and chemical equilibrium exists in each stage are considered.     

Propagation of Signals of Finite Concentration in Gas Chromatography. I. The Quasi-Ideal Model

Abstract

A general system of partial differential equations describing the propagation of signals of finite concentrations in a chromatographic column is derived. These equations are related to the mass-balance equations for the solutes and the carrier gas. The model used assumes that there is no temperature or pressure variation at any point in the column when the signal is eluted, and that the equilibrium between stationary and mobile phases is instantaneous. It is shown that this model, which leads to a tractable set of equations, is generally valid. The solution of this system of equations gives new insight into the phenomena which are responsible for the peak deformations and broadening in preparative scale chromatography.     

Theoretical analysis of formation of defects in hafnia

Quasichemical equations describing the dependences of the number density of defects formed in HfO₂ by the Schottky, Frenkel, and anti-Frenkel mechanisms as a function of the partial pressure of oxygen and the temperature are derived and solved. Equations describing chemical disordering in the ion and electron subsystems and equations describing the processes of ionization of defects in oxides are derived for ZrO₂ and HfO₂ as an example. The theoretical data obtained can be used for deriving equations for calculating the number density of defects in HfO₂ in any range of temperature and pressure. Results of an experimental test of the derived disordering equations by means of a precision thermomassometric method are presented. An evaluation has shown that the value of the deviation in HfO₂ from index 2 amounts to about 0.0002 from the fraction O₂. It is assumed that three-dimensional diffusion of oxygen by Ginstling’s equation is the limiting stage of the decrease in the mass. Translated from Ogneupory i Tekhnicheskaya Keramika, No. 2, pp. 9–14, February, 2000.     

The hydrodynamics of a hydrocyclone based on a three-dimensional multi-continuum model

The concept and principles of applying a multi-continuum model for calculating a hydrocyclone performance is presented. In this model the carrying liquid is described as one continuum, and each particle fraction, with its characteristic size is described as a separate continuum. Particle–particle and particle–fluid interactions derived from a lubrication theory and a collision theory are discussed. A set of governing partial differential equations consisting of mass and momentum conservation equations together with constitutive expressions is discussed. These equations were discretized by applying an unstructured grid consisting of tetrahedral elements. A numerical solver based on a finite element method combined with a segregated approach is described. The numerical approach is subject to ongoing research.     

Preparative Fractionation by Frontal Countercurrent Distribution

Abstract

The frontal (multiple-input) approach is applied to countercurrent distribution. Equations are derived for calculating the number of tubes required for a given separation or for calculating the amount of solute emerging at the intersection of two solute profiles. In both cases ideal behavior is assumed. The applicability of these equations to preparative fractionations is evaluated and the utility of the frontal technique demonstrated.     

A time-averaged model for gas-solids flow in a one-dimensional vertical channel

In this study, we are interested in deriving time-smoothed governing and constitutive equations for gas-solids flow in moderately dense systems where particle-particle collision is the main energy dissipation mechanism. Results obtained from dynamic simulations of a gas-solids flow in a 1D channel are used to show that it is possible to obtain expressions for the time-averaged constitutive relations based on Taylor series expansion. We demonstrate, by comparing with time-averaged transient results, that the 1st term (or laminar) in the series expressions of most non-linear constitutive relations can yield inaccurate quantitative and qualitative results. This means that steady-state models derived by simply removing the partial time derivative from the governing equations are not suitable for gas-solids flows. This study shows that it was necessary to include many terms of the Taylor series expression of non-linear constitutive relations (such as the granular energy dissipation term) due to large-scale oscillations that were computed for all flow variables at all locations in the 1D domain. In some cases, the Taylor series expansion diverged and the Euler transformation was used to improve the convergence of these series. In this moderately dense flow system, turbulence in the gas-phase was found to be just a reaction to turbulence in the solids phase that resulted from the large-scale motion of solids clusters. This resulted in a negative turbulent gas viscosity computed due to the fact that gas (in the horizontal direction) flows only to occupy regions vacated by clusters of solids. The steady-state results obtained using the time-smoothed gas-solids flow model compared well with the time-averaged results obtained using the transient model for all flow variables.     

Correlations for wall and particle shape effects on fixed bed bulk voidage

Correlations have been derived, by both geometrical arguments and empirical treatment of data, for the bulk void fraction in a fixed bed. The void fraction has been correlated as a function of particle-to-tube diameter ratio for packings of spheres and equilateral solid cylinders. Prediction of void fraction for equilateral hollow cylinders can be made from the solid cylinder correlation, provided a correction factor is included that allows for internal voidage and interpenetration of packings.     

Statistical moments of elution bands and their application in gel permeation chromatography of polydisperse macromolecules

Equations have been derived that relate the statistical moments of uncorrected and spreading-corrected chromatograms for a general form of the spreading function in gel permeation chromatography of polydisperse macromolecules. The first moment (centroid) of the chromatogram is shown to be directly given by the centroid, In M^*, of a suitably defined molecular weight distribution function of the polydisperse sample, regardless of the position of the calibration dependence, provided it is linear. Both the molecular weight M^* associated with the centroid of the chromatogram and its second central moment (variance) are but little sensitive to the shape of sample molecular weight distribution and can be easily calculated from the polydispersity index M_w/M_n, at least for polymers of a not excessively broad distribution. The derived relations are shown to find application in the calibration of GPC columns by means of characterized, polydisperse standards and in the separation of independent contributions to peak width which originate in sample polydispersity and in band broadening processes in the column. Improved column- and packing performance criteria are also proposed.