Discrete Relaxation Time Spectrum Research Articles

Rheology-determined critical conditions for shear-induced crystallization of biosynthesized polyhydroxyalkanoates

Shear-induced crystallization plays a crucial role in the manufacturing process of polymers. In this work, crystallization kinetics of biosynthesized polyhydroxyalkanoates (PHA) under different shear conditions were systematically investigated by rheometers. First, rheological properties of PHA melts were performed at different temperature to obtain mastercurves via the time-temperature superposition principle at 170 °C as a reference temperature. Then the stretch relaxation time and corresponding critical shear rate at different temperatures for the flow regime transition were calculated via the discrete Maxwell relaxation time spectrum and Arrhenius equation. Finally, the influence of shear temperature (Ts), shear time (ts) and shear rate (γ̇) on the crystallization process of PHA were discussed. The results showed the crystallization rate of PHA was improved significantly under high shear rate and long shear time. Interestingly, the t1/2 reached the minimum value when the γ̇ or the ts was large enough, which reached around 450 s at isothermal crystallization condition of 100 °C. Moreover, the nucleation density for PHA increased by appx.100 times than that under quiescent conditions. Therefore, this work may provide a useful theoretical guidance on the shear-induced crystallization of PHA.

Cooperativity in Molecular Dynamics Structural Models and the Dielectric Spectra of 1,2-Ethanediol

Linear relationships are established between the experimental equilibrium correlation factor and the molecular dynamics (MD) mean 〈cos ϕ〉 value of the О–Н···О bond angle and the longitudinal component of the unit vector of the mean statistical dipole moment of the cluster in liquid 1,2-ethanediol (12ED). The achievements of modern MD models in describing the experimental dispersion of the permittivity of 12ED by both continuous and discrete relaxation time spectra are analyzed. The advantage computer MD experiments have over dielectric spectroscopy for calculating relaxation time and determining the molecular diffusion mechanisms of the rearrangement of the network 12ED structure, which is more complex than water, is demonstrated.

Regularization methods for finding the relaxation time spectra of linear polydisperse polymer melts

The calculation of discrete or continuous relaxation time spectra from rheometric measurables of polydisperse polymers is an ill-posed problem. In this paper, a curve fitting method for solving this problem is presented and compared to selected models from the literature. It is shown that the new method is capable of correctly predicting the molecular mass distributions of linear polydisperse polymer melts as well as their relaxation time spectra.

Simulations and experimental investigation of paint film leveling

The surface structure of a paint film is the result of the interplay of a variety of physical influences, e.g., the superposition of droplets during spray application, the surface tension-driven leveling, and the viscosity increase in the leveling phase. A numerical simulation program is presented that incorporates all the relevant mechanisms of paint film structure formation during and after spray application. The simulation program was validated by comparing simulations and leveling experiments. The influence of the initial film geometry and viscosity on the leveling behavior is demonstrated. For the investigations, model liquids and commercial paints with an increasing complexity of the physical properties were chosen: Newtonian flow behavior without solvent evaporation, Newtonian flow behavior with solvent evaporation, viscoelastic paints with non-Newtonian flow behavior. Four variants are proposed regarding how thixotropy can be measured and how a mathematical model can be created. The advantages and disadvantages of the variants with regard to the implementation of thixotropy in the simulations are listed. A method to predict the leveling behavior of thixotropic paints with simultaneous recovery of the viscous and elastic properties from rheological measurements using discrete relaxation time spectra is presented.

New insights into the use of multi-mode phenomenological constitutive equations to model extrusion film casting process

This article is concerned with the effect of the individual viscoelastic relaxation modes of a polymer melt on its behavior in polymer melt extrusion film casting process. We compare the predicted versus experimentally obtained film necking or neck-in profile as a function of draw ratio. The predicted necking profile was obtained using well-established one-dimensional isothermal flow kinematics and consisted of using two different phenomenological constitutive equations, upper convected Maxwell and Phan-Thien–Tanner, with a discrete spectrum of relaxation times. The numerical simulations, containing the two different phenomenological constitutive equations, provided an insight into the effect of the slow and the fast relaxing modes on the stresses, strains, and strain/extensional rates that develop in the molten polymer film as it is stretched from the die exit to the chill-roll. The slow relaxing modes follow trends that are directly proportional to strain (similar to Hookean solids), whereas the fast relaxing modes follow trends that are directly proportional to the stretch rate (in accordance with Newton’s law of viscosity). Comparing the numerical predictions with the experiments showed that predictions using the upper convected Maxwell constitutive equation best described the long-chain branched polymers (like low-density polyethylene, which shows extensional strain hardening) in the extrusion film casting process. On the other hand, predictions using the Phan-Thien–Tanner constitutive equation best described the linear polymers (like linear low-density polyethylene, which does not show noticeable extensional strain hardening) in the extrusion film casting process.

A systematic approximation of discrete relaxation time spectrum from the continuous spectrum

Most of viscoelastic models contain linear and nonlinear viscoelastic parameters and the linear viscoelastic parameters correspond to relaxation time spectra of materials. The relaxation spectra can be classified into continuous and discrete ones. Discrete relaxation time spectrum is more convenient to simulate multi-mode models than continuous one because it demands shorter calculation time. It has been demonstrated that the continuous spectrum is uniquely determined in views of theoretical (Fuoss and Kirkwood, 1941; Davies and Anderssen, 1997) [1,23] as well as empirical approaches (McDougall et al., 2014) [5]. Whereas, it is reported that different algorithms for discrete spectrum infer the different results from the same data (Malkin and Masalova, 2001) [14]. This is the study on a systematic method for discrete spectrum on the basis that a discrete spectrum must be consistent with the continuous one. We suggest a simple method to extract discrete relaxation spectrum as a systematically approximated continuous spectrum by means of the Levenberg–Marquardt method. The new algorithm is tested and compared with previous algorithms using synthesized model spectra and experimental data.

Study on discrete relaxation time spectrum for PBO/PPA liquid crystal polymer solution

Based on KBKZ integral type rheological equation, discrete relaxation spectra were attained by use of the least square linear and non-linear regressions. The results show that the number of relaxation mode ranging from 9 to 11 for linear regression method has little effect on the non-linear regression method. Apart from that the relaxation modulus coefficient related with the different time range was gained, the relaxation time range had little effect on the time spectrum. The trend for relaxation time spectrum of PBO solution is similar to common polymer solution except that there is not a platform modulus region for high elastic state. The relaxation time spectrum at 160°C was obviously different from that at 170°C. However, the difference diminished when the temperature exceeded 170°C, and no significant distinction was found at glass transition region. While at 170°C, the influence on the movement of the chain segment was trivial as the molecular weight increases, mainly resulting in the movement restriction of polymer chains.

Rheologically Determined Critical Shear Rates for Shear-Induced Nucleation Rate Enhancements of Poly(lactic acid)

Both the nonisothermal and isothermal crystallization kinetics under the influences of different shear conditions for poly(lactic acid) (PLA) were investigated by rheometry. The nucleation and growth of PLA spherulites during isothermal crystallization with different shear conditions were observed by polarized optical microscopy (POM). Shear-induced nucleation rate enhancements of PLA were studied on the basis of the prerequisite determination of the critical shear rates, for which the stretch of the longest chains (high molecular mass tails) of PLA would be expected. The transitions between different shear flow regimes for shear-induced crystallization of PLA at the temperature of 135 °C were determined by two characteristic Weissenberg numbers on the basis of reptation time and Rouse time for the high molecular mass tails, which were determined through combination of the discrete Maxwell relaxation time spectra of PLA at the reference temperature of 190 °C and the Arrhenius type of temperature dependence for the horizontal shift factor, aT. It was then found that the crystallization process of PLA was greatly enhanced by shear compared to the quiescent condition, and the crystallization kinetics could be accelerated by the increased shear rate and/or shear time. It was more interesting to find that there existed a critical shear time under a certain shear rate, and a further increase in the shear time did not lead to further acceleration of the crystallization kinetics. POM observation indicated that the acceleration of crystallization kinetics was obviously brought about by the enhanced nucleus density under the application of shear and the subsequent spherulitic growth rates kept about constant. Thus, a kinetic model based on directly relating the extra number of activated nuclei promoted by shear to the shear rate was further applied to well predict the effects of shear time on the shear-induced isothermal crystallization kinetics of PLA.

A simple method for determination of discrete relaxation time spectrum

A simple method for determining the discrete relaxation time spectrum from loss modulus data was developed. The method is based on the wavelet transform, which is a kind of peak-sharpening processes. Although several methods have been developed, most cannot determine the number of relaxation times in a single step without an adjustable parameter. However, the method suggested here provides a way of determining the number of relaxation times as well as the arrangement of the relaxation times automatically in a single numerical operation.

Glacial-Isostatic Adjustment-II. The Inverse Problem

Summary The principle of correspondence is invoked to determine Laplace transform domain solutions to the surface mass loading problem for radially stratified visco-elastic (Maxwell) spheroids. These Laplace transform domain solutions are expressed in terms of visco-elastic analogues of the conventional surface load Love numbers of elasticity. These visco-elastic Love numbers may be approximately transformed to the time domain using an extremal technique. Application of this technique shows that the Love number time histories may be well approximated by the linear superposition of a discrete set of purely exponential relaxations plus a constant term. Alternatively the discrete spectrum of relaxation times involved in the synthesis of each Love number time history may be found exactly by solving the associated homogeneous problem. Such solutions determine the set of eigen-decay times associated with the normal modes of viscous gravitational relaxation of the visco-elastic planetary model. The solution of the inhomogeneous problem may be expressed in the form of a normal mode expansion. This normal mode expansion is employed as the basis for the construction of a rigorous first-order perturbation theory for the inference of the viscosity of the deep interior of the planet. A variational principle is derived which determines to first order that shift in position of a free decay pole in the relaxation spectrum which is forced by the addition of a radially-distributed perturbation of viscosity to the starting model. This determines the differential kernels required for the solution of the inverse problem. The uniqueness of the state of isostatic equilibrium for the viscously incompressible Maxwell models employed in this analysis is demonstrated and this uniqueness allows us to calculate the shift in the residue which is associated with the shift in position of a free decay pole for the inhomogeneous problem. The inhomogeneous problem is thus closed to first order. A formula is derived for the differential kernels appropriate to the inhomogeneous problem. The time domain form of these kernels may be calculated analytically. The structure of the full inverse theory is sufficiently simple that it may be employed to rigorously test the compatibility of the simple Maxwell model with the observed relaxation of the Earth's shape which accompanied deglaciation at the end of the last ice age.

Viscoelastic Shear Response and Network Structure in Polycyanurates

The shear response of polycyanurate networks with different cross-link densities, varied by changing the ratio of difunctional to monofunctional cyanate ester, is measured from shear stress relaxation and dynamic experiments. Master curves are constructed following the time−temperature superposition principle, and the temperature dependence of the shift factors is examined. The discrete relaxation time spectra are calculated from the viscoelastic responses and are found to be independent of cross-link density over the time/frequency range measured. The cross-link density, determined from the rubbery modulus, and the sol content, measured from sol extraction experiments, are modeled for the fully cured polycyanurate networks using the recursive method; a monomer cyclization reaction is assumed in the modeling based upon the chemical composition of the sol which was determined by mass spectroscopy. The effect of monomer cyclization on the conversion at gelation of dicyanate esters is discussed.

The use of a continuous relaxation spectrum for describing the viscoelastic properties of polymers

For describing the viscoelastic properties of polymeric materials, the continuous relaxation spectrum is proposed instead of the discrete spectrum of relaxation time distribution, which is usually employed in practical calculations. For polymer melts and solutions, such a continuous spectrum is assumed to be represented by a power function and to have a certain maximum relaxation time as its upper bound. The spectrum is determined by three constants: the upper bound (maximum relaxation time), slope in the logarithmic scale (exponent), and height (form factor). The necessity to find a small number of parameters is an advantage in comparison with the need to determine a large number of constants within the concept of a discrete relaxation time spectrum and corresponding weights. The central idea is that the parameters of the continuous spectrum can be unambiguously determined from the experimentally found integral characteristics (moments) of the viscoelastic behavior of the material: the viscosity, normal stress coefficient, and area under the relaxation curve after the cessation of steady flow. The proposed choice of the relaxation spectrum shape is shown to provide an adequate description of the specific features of viscoelastic behavior for melts of polymers with a varying MM.

Indirect detection of structural changes in micellar solutions by rheological measurements

At higher concentration levels the inner structure of micellar solutions cannot be detected directly by optical means. Nevertheless the flow behavior of the micellar solutions reflect their micellar structures. Hence, in this study the material behavior of micellar surfactant solutions was investigated by rheometrical means in steady and oscillatory shear flows with and without thermal preload. The rheological behavior of the solutions was found to be strongly dependent on the concentration. The complex material structure is modeled according to the flow behavior by discrete and continuous relaxation time spectra depending on the concentration. It is found that the material possesses besides the memory for mechanical preload also a strong memory for thermal preload which depends on the strength of preload.

An experimental study on the flow behavior of micellar solutions

At higher concentration levels, the inner structure of micellar solutions cannot be detected directly by optical means. Nevertheless, the flow behavior of the micellar solutions reflects their micellar structures. Hence, in this study the material behavior of micellar surfactant solutions was investigated by rheometrical means in steady and oscillatory shear flows. The flow behavior of the solutions was found to be strongly dependent on the concentration of the surfactants. At very low concentrations, the surfactant solution shows Newtonian behavior. With increasing concentration, a transition to shear thinning behavior and increasing viscoelasticity was found. The complex material structure is modeled according to the flow behavior by discrete and continuous relaxation time spectra, depending on the concentration.

Torsional flow stability of highly dilute polymer solutions

We examine the torsional flow stability, to axisymmetric disturbances, of a variety of multimode and non-linear constitutive models in a bounded parallel plate geometry. The analysis is facilitated by the construction of a regular perturbation scheme in the ratio of polymer to total viscosity. As a model for Boger fluids this corresponds to the assumption that the Boger fluid is highly dilute. The consequent simplification provided by the perturbation scheme allows us to examine the effects of a discrete spectrum of relaxation times, shear thinning, second normal stress difference, and finite extensional viscosity on the torsional instability.

Influence of molecular structure on the rheological and processing behavior of polyethylene resins

AbstractThe rheological and processing behavior (melt fracture performance) of linear lowdensity polyethylenes (LLDPEs) is studied as a function of both the weight average molecular weight (Mw) and its distribution (MWD). A number of LLDPE resins having different molecular characteristics were tested, with essentially one characteristic (Mw or MWD) changing at a time. The first series of resins consisted of nine samples having a wide range of polydispersities (3.3–12.7) and nearly constant Mw and short chain branching. The second series had six resins with varying Mw (51,000–110,000) but fixed MWD (about 4). The influence of Mw and MWD on the viscosity profiles, linear viscoelastic moduli as expressed by means of a discrete spectrum of relaxation times, extrudate swell, and melt fracture behavior for these resins is reported. Correlations between the molecular characteristics of the resins and their rheological and processing behavior are also reported. It is found that for a given molecular weight, the optimum melt fracture performance is obtained at a specific polydispersity value, and it is characterized by a minimum relaxation time for the resin defined in terms of recoverable shear.

Dynamic shear rheology of high molecular weight polydimethylsiloxanes: comparison of rheometry and ultrasound

The viscoelastic properties of three linear polydimethylsiloxanes (PDMS) of high molecular weights are investigated using rheometrical as well as ultrasonic tests over a large range of temperatures. Classical shear rheometrical measurements are carried out in the low frequency range from 10−1 to 102 rad s−1 between −50 and +20°C. The frequency range is enlarged using the time–temperature superposition principle, allowing coverage of about 6–7 decades of pulsation. Ultrasonic tests use an inclined incidence wave reflection technique to measure the complex shear mechanical impedance from 1.5 to 25 MHz, between −10 and +50°C. Rheometrical and ultrasonic experiments are then combined for the three PDMSs at the same reference temperature. They give the reduced shear elastic and loss moduli for reduced frequencies covering 10 decades. A discrete relaxation time spectrum is first deduced from the master curve in each case. More accurate predictions may be obtained using a molecular weight distribution and BSW (Baumgaertel–Schausberger–Winter) model for polydisperse systems.

The experimental observation and numerical prediction of planar entry flow and die swell for molten polyethylenes

We report experimental observations and matching numerical simulations for the planar entry flow and die swell of two high-density polyethylenes (HDPEs) and one low-density polyethylene (LDPE). Experimental data for stress fields, centreline velocities and die swell are reported for each polymer. These results are compared with numerical simulation. The materials are characterized in simple shear using a Wagner integral constitutive equation with a discrete spectrum of relaxation times and a single parameter damping function. The numerical simulation has been carried out using a finite element software package, Polyflow. Self consistency in the stress and die swell data are found for one HDPE, but the other HDPE and the LDPE show an extensional strain hardening response which is not predicted using the simple shear rheology data. In the latter cases, the numerical predictions consistent with entry flow experimental observations can be achieved if extensional flow damping parameters, rather than simple shear damping parameters, are chosen. For the LDPE, an increase in the strain hardening parameter results in the numerical prediction of upstream recirculation vortices in the entry region, which qualitatively agrees with the experimental observations. Apparent inconsistencies in the absolute values of measured and simulated velocity profiles are explained in terms of the 2D nature of the simulation and a 3D component to the experimental flow.

Some Aspects of Discrete Relaxation Time Spectra as Obtained from Dynamic Moduli Data

The problem of finding a relaxation time spectrum best fitting dynamic moduli data in the least-squares sense is shown to be well-posed and to yield a discrete spectrum, provided the data cannot be fitted exactly, i.e., without any deviation of data and calculated values. Properties of the resulting spectrum are discussed. Examples of discrete spectra obtained from simulated literature data and experimental literature data on polymers are given. The problem of smoothing discrete spectra when continuous ones are expected is discussed. A detailed study of an integral transform inversion under the non-negativity constraint is given in Appendix.

Analytical and numerical analysis of Debye–Ramm’s equation for the polarization density distribution in polar materials

Due to dipole interaction, the molecular polarization brought about by an external field is significantly lower in condensed matter (liquids) than in a gas. In addition to this, the response of interacting dipoles to stepwise changes of the external field does not follow a simple time exponential. Instead, a spectrum of relaxation times is required to describe such a response. Debye and Ramm [Ann. Phys. 28, 28 (1937)] have attempted to describe the effects associated with rotational hindrances due to dipole-dipole interaction by the following differential equation: ∂f/∂t =(kT/ρ)Δf+(1/ρ)div(f grad u), where f denotes the distribution function specifying the number of dipoles pointing in a certain solid angle, t the time, ρ a friction coefficient, and u the potential of the forces acting on the dipoles. The latter quantity depends both on the external field and on the contribution from the dipole-dipole interaction (internal field). Although unable to solve the above equation explicitly, Debye and Ramm (DR) made some predictions about the solution, concluding, among other things, that the inclusion of an internal field E would yield a process with a discrete spectrum of relaxation times. Finding such prospect highly interesting, we subjected the DR equation to a close study using some advanced mathematical tools (Fourier integral operators etc.). Contrary to the conclusions of DR, we found that the above equation cannot be solved in the way originally described, and that the conjectured eigenfunctions and eigenvalues do not exist. Furthermore, we show that, in contrast to DR’s statements, the above equation is not solved by certain classical expressions relating to free-rotating dipoles (no internal field). The lack of physical content of this equation appears to be due to a number of not permissible simplifications.