Relaxation spectra in polymers

Methods of determining the continuous and discrete relaxation spectra are analyzed with the object of choosing a characteristic describing the actual relaxation processes. It is shown that the discrete spectrum is a physically sounder characteristic and makes it possible to distinguish the most probable elementary relaxation processes. Problems of predicting the relaxation processes in real polymers can be simplified by using the discrete relaxation spectrum.Leningrad Branch, Scientific-Research Institute of the Rubber Industry. Translated from Mekhanika Polimerov, No. 1, pp. 151–154, January–February, 1973.     

Estimation of the molecular characteristics of polymers by the SPRT method and study of their influence on the properties of compositions

Pressure relaxation was examined in the cylinder of an MPT Monsanto processability tester after stopping the piston. The experimental function of the pressure drop F(t) was smoothed over and approximated by cubic splines. The spectra of pressure relaxation times (SPRT) were obtained according to the method of Schwarzl-Staverman. The SPRT method served well for estimating the spectra of the molecular-mass distribution (MMD) of polymers close in their physical sense to the SPRT. The correlation of the characteristic relaxation times and average molecular mass of ethylene-propylene rubbers and polyethylenes obtained by gel permeation chromatography was approximated by optimum models used for calculating the the molecular mass of rubbers according to the measurement results of the relaxation pressure of melts. The SPRT and characteristic relaxation times were used to analyze the significant technical properties of compositions based on polyethylene and rubber. The SPRT method was used to examine the failure of the cure network of butyl rubber and the dependence of the mechanical properties of thermoplastic elastomers on the molecular features of the decomposite.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Kazan State Technological University, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 5, pp. 691–698, September–October, 1998.     

Rheology and molecular structure of polyethylene melts. 2. Effect of molecular structure on the viscoelastic characteristics

The viscoelasticity of low-density polyethylene is investigated as a function of molecular structure. The viscoelasticity is characterized by three numbers — the zero-shear viscosity, the characteristic relaxation time, and the index of the rate of fall of dynamic viscosity with deformation frequency. The molecular structure is also characterized by three numbers — , , g_w. The zero-shear viscosity is shown to depend on the polydispersity of the molecules and a corresponding analytic expression is obtained. The method of determining the characteristic relaxation time from the maximum of the frequency relaxation spectrum is justified and an expression giving the characteristic relaxation time as a function of the principal characteristics of the molecular structure is derived. It is shown that long-chain branching of the molecules has an incommensurably greater (several orders) effect on the rheological characteristics of the branched polymer than the polydispersity of the molecular weight. The effect of the polydispersity of the molecules of commercial low-density polyethylenes on the viscoelastic characteristics cannot be established without taking the branching of the molecules into consideration.For communication 1 see [1].Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 1, pp. 120–127, January–February, 1976.     

Rheology and molecular structure of polyethylene melts 1. Melt identification from mechanical properties

It is proposed to use periodic deformation for the identification of the molecular structure of polyethylenes. By comparing the dependence of the complex and apparent viscosities on frequency and shear rate, respectively, it is shown that these functions are not equivalent for low-density polyethylene melts. The need for a numerical characteristic of the rheological functions, which should be related to the numerical characteristics of the molecular structure of the polymer, is established. The possibilities of using generalized relations for investigating the molecular structure are discussed. It is shown that the dynamic viscosity of polyethylene melts, described by a three-parameter equation, quite accurately reflects the viscoelastic properties of polyethylene melts and makes it possible, with the aid of the calculated function of the relaxation spectrum, to construct a frequency dependence of the modulus corresponding to the experimental data. Three numbers characterizing the viscoelasticity of polyethylene, which must be related with three molecular characteristics, are established.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 523–532, May–June, 1975.     

Graphoanalytical determination of rheological coefficients

A graphoanalytical method is proposed for determining the most probable discrete values of the relaxation times. This method consists in the quantile partition of the relaxation spectrum distribution function.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 4, No. 4, pp. 754–757, July–August, 1968.     

Determination of the relaxation and retardation spectra – a view from the up-to-date signal processing perspective

The problem of determination of relaxation and retardation spectra (RRS) is considered from the viewpoint of up-to-date signal processing. It is shown that the recovery of RRS represents the Mellin deconvolution problem, which transforms into the Fourier deconvolution problem for data on a logarithmic time or frequency scale, where it can also be treated as the inverse filtering problem. On this basis, discrete deconvolution (inverse) filters operating with geometrically sampled data are proposed to use as RRS estimators. Appropriate frequency responses and algorithms are derived for estimating RRS from eight different material functions. The noise amplification coefficient is suggested to use as a measure for quantifying the degree of ill-posedness and illconditioness of the RRS recovery problem and algorithms. A methodology is developed for designing RRS estimators with a desired noise amplification, producing maximum accurate spectra for available limited input data. Practical algorithms for determining RRS are proposed, and their performance is studied. The algorithms suggested are compared with the so-called moving-average formulae. It is demonstrated that the minimum frequency range for recovering the point estimate of a relaxation spectrum depends on the allowable noise amplification (the degree of ill-conditioness) and is in no way limited by 1.36 decades, as it is stated by the sampling localization theorem.     

Determination of regular discrete relaxation spectra from the viscoelastic functions

A procedure for determining regular discrete relaxation spectra from the viscoelastic functions is presented as a generalization of a previously proposed procedure for determining the conditional rheological characteristics in a piecewise-exponential approximation of the experimental creep, relaxation, and – curves. The calculated data are in good agreement with the experimental results.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 801–807, September–October, 1971.     

Calculation of relaxation spectra from dynamic test data

A procedure is proposed for calculating relaxation spectra from experimental data on uniaxial tension over a broad range of strain rates. The relaxation spectra are calculated for low-density polyethylene and for a fabric-reinforced glass laminate in three directions relative to the reinforcement.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 349–353, March–April, 1971.     

Relaxation spectra as a means of estimating rubber inhomogeneity

The results of an investigation of the effect of rubber inhomogeneity on the nature of the relaxation spectra are presented. There is a correlation between the spectrum parameters and the inhomogeneity characteristics of the rubbers.Yaroslavl' Technological Institute. Translated from Mekhanika Polimerov, No. 5, pp. 916–919, September–October, 1969.     

Determination of the parameters of the reticulate structure of amorphous linear polymers subjected to orientational stretching

As a result of interlacing the macromolecules in linear block polymers form a sort of molecular network. The structure of this network affects the strength characteristics of oriented amorphous linear polymers in the brittle state. An important characteristic of the reticulate structure is the number of network chains per unit volume. The literature describes a method of determining this quantity based on the degree of stretching and the stress developed during stretching of the softened polymer. This method is not sufficiently accurate, since it disregards the presence in linear pclymers of network nodes of different stability. The proposed method of calculating this quantity is based on an analysis of the "equilibrium contraction" curves and takes into account the relaxation processes in the polymers. It is more accurate than that previously employed.Mekhanika Polimerov, Vol. 2, No. 2, pp. 163–169, 1966     

Investigation of the change in the composition of a methyl methacrylate — Styrene copolymer through a spectrometric measurement of the dielectric characteristics

Conclusions 1. The most probable relaxation time and the increment of dielectric permeability are the two most informative characteristics for determining the percentage of the components in methyl methacrylate-styrene copolymers in the range of dipole-radical interaction. These values are derived by approximating the experimental data by the Havriliak-Negami dispersion equation.2. Analysis of the experimental data of dipole-radical processes has shown that the most probable relaxation time as a function of the methyl methacrylate concentration decreases exponentially. The spectrum of the dielectric relaxation times does not vary up to a methyl methacrylate content of 60%, beyond which it narrows down. With respect to the change of percentage of the components, the copolymer investigated is a rheological complex material.3. The principle of concentration-frequency analogy may be applied to the complex dielectric permeability of the copolymer under investigation up to a methyl-methacrylate content of 60%. The results obtained by the approximation may be used in order to establish a criterion for the applicability of the analogy principle and to determine the reduction coefficients.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 903–908, September–October, 1978.     

Resin cure monitoring system based on nondestructive dielectric spectrometry

An on-line indirect cure monitoring prototype system is described based on dielectric spectrometric measurements (frequency dependences of the dielectric modulus and/or dielectric permittivity and/or dielectric loss factor and/or relaxation spectrum). A key block of the system is the Dielectric Spectrometer. Model DS 7056-C interfaced with a standard PC. A variation of the dielectric frequency spectra during the cure process for two epoxy systems based on araldite resins LY556 and LY561-1 are considered as an illustration of capabilities of the described hardware and software.Institute of Polymer Mechanics, Riga, Latvia LV-1006. Published in Mekhanika Kompozitnykh Materialov, Vol. 32. No. 3, pp. 401–409, May–June, 1996.     

Thermal-fluctuation model of a thixotropic rubbery liquid

The isothermal deformation of a viscoelastic liquid, whose structural elements are simulated by the Maxwell elements of the relaxation spectrum, is considered. It is assumed that in the presence of temperature fluctuations the elements of the spectrum can be destroyed and restored. A comparison is made with experiment. A procedure for calculating the relaxation spectrum from the results of shear and normal stress measurements in the steady-state shear flow regime is proposed.Presented at the Symposium on Polymer Rheology, Institute of Petrochemical Synthesis, Academy of Sciences of the USSR, Moscow, May, 1970.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 307–317, March–April, 1971.     

Analysis of mathematical models of microstrip transmission lines

Solution methods for eigenvalue problems nonlinear in the spectral parameter are considered for model problems of the theory of microstrip transmission lines. The spectral properties of the Fredholm operator function F() in Sobolev weighting classes are considered. A numerical method is proposed for determining the approximate characteristic values of F(). We prove theorems on nonemptiness, discreteness, and factorization of the spectrum and on convergence of the approximate spectra and characteristic values to the exact quantities.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 175–198, 1986.     

Effect of relaxation phenomena on the strength of plastics exposed to harmonic loads and impacts

It is shown that if the frequency of cyclic loading is varied, the total heat-up is small, and relaxation processes do not play a role (proceed too fast or too slowly), the loading times are additive. It is shown that deviations from this additivity rule manifest themselves in a disproportionally slow increase or even in a decrease of the strength with rising frequency; these deviations may be caused by relaxation factors, i.e., by structural changes and local warming up. It is established that the degree of deviation depends on various factors determining the rate of relaxation, i.e., on the temperature, plastification, and orientation, which is in accordance with ideas on the role of relaxation phenomena in fatigue rupturing but conflicts with the view that the number of load cycles should be the decisive factor. The phenomenon of fatigue rupturing of plastics exposed to repeated impacts is discussed; specific features of impacts are the high loading rate and the rest between the impacts. It is shown that, in spite of the much shorter action time of the load, the strength under repeated impacts is lower than that under harmonic loading, owing to the stronger effect of local excess stresses and disorientation during unloading.Scientific-Research Institute of Plastics, Moscow. Translated from Mekhanika Polimerov, No. 1, pp. 165–168, January–February, 1972.     

Method of determining the volume and shear characteristics of elastico-viscous hereditary media from uniaxial-tension (compression) experiments

A method is proposed for determining the elastic constants — instantaneous modulus of elasticity, Poisson's ratio, shear modulus, bulk modulus, and the shear and volume influence functions — the shear creep kernel, the shear creep rate kernel, and the corresponding relaxation kernels from the data of creep or relaxation tests.Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 754–758, July–August, 1969.     

Thermocreep of polyethylene in the cyclic temperature regime

The shear creep and recovery of low-density polyethylene has been investigated in the constant and cyclic temperature regimes. It is shown that in the cyclic regime the relaxation processes are accelerated and the nonisothermal creep strains become progressively greater than the isothermal strains at the maximum temperature of the cycle. In recovery the relaxation spectra are shifted along the time scale relative to the creep spectra, the shifts for simple shear and uniaxial tension being nonidentical.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 227–235, March–April, 1969.     

Choice of kernels in solving problems involving creep and relaxation

For the memory theory analysis of creep and relaxation processes whose rates have a singularity at the beginning of the process it is proposed to employ a kernel and the corresponding resolvent in the form of a series whose convergence is proved. An estimate of the convergence and an asymptotic formula for large times are also presented. The kernel, the resolvent and their integrals have been tabulated. Theoretical curves are constructed for certain parameters, and it is shown how they can be used in analyzing creep problems when the object is to find relaxation (creep) curves from experimental creep (relaxation) curves. It is noted that, given a suitable choice of parameters, it is possible to use the proposed functions to describe both finite and unlimited creep.Mekhanika Polimerov, Vol. 2, No. 4, pp. 483–497, 1966     

Kinetics of formation of the actual contact area in rubbery materials

The kinetics of formation of the actual contact area in rubbery materials is investigated. A procedure for determining the relaxation spectrum of the segmental motion of the macromolecules in the surface layer of a polymer in contact with a solid surface is proposed. The effect of temperature and the applied specific load on H() is determined. The activation energy of segmental motion of the macromolecules in a thin surface layer of the specimen is calculated.Second Moscow Pirogov State Medical Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1047–1052, November–December, 1971.     

Nonlinear shear response of fluid polymers during periodic deformation. Basic harmonics

Conclusions 1. A model with a relaxation spectrum independent of accumulated elastic strains describes the combination of data on the amplitude and frequency dependences of the components of the shear modulus in terms of basic harmonics for low-molecular-weight polyisobutylene and a filled-polyethylene melt.2. The greater the amplitude of the shear strain, the broader the region of invariance of the relaxation functions relative to the amplitude of the rate of oscillatory shear.3. The kinetic (s) function affects the amplitude dependences of the modulus in terms of basic harmonics. Its effect is significantly greater on the real component of the modulus than on the imaginary one.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, pp. 1093–1100, November–December, 1977.