Rheology and molecular structure of polyethylene melts

Conclusions 1. The molecular structure indices of low-density polyethylene (LDPE), i.e., the weighted-average (wt.-av.) molecular weight, the number of branch points of long-chain branchings per molecule, and the polydispersity index, were calculated from the data of gel-permeation chromatography, assuming that the Mendelson—Drott hypothesis holds. They show a fairly close correlational relationship to the index of the mechanical characteristics of this melt.2. The correlation between the highest Newtonian viscosity and the molecular characteristics, taking into account the polydispersity index, is improved if the low-molecular-weight part of the molecular weight distribution MWD (M < 800) is neglected.3. The index of the exponential equation, which determines the dependence of the highest Newtonian viscosity of LDPE on the wt.-av. molecular weight, increases when the influence of the polydispersity is taken into account.4. Commercial samples of LDPE synthesized under similar technological conditions show a fairly good correlation between the polydispersity and the degree of branching of the molecule.5. From the highest Newtonian viscosity, the specific relaxation time of the LDPE, and the dependence of the degree of branching of the molecules on the polydispersity of the molecular weights, common for the given technology of the synthesis, it is possible to calculate the molecular structure indices of LDPE with satisfactory accuracy.For communication 3, see [3].The Walter Ulbricht Enterprise "Leunawerke," German Democratic Republic. Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. The Okhtinsk Scientific and Production Union "Plastpolimer," Leningrad. Translated from Mekhanika Polimerov, No. 2, pp. 283–293, March–April, 1977.     

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.     

Rheology of filled polymers. Steady-state shear flow and periodic deformation. 1. Relaxation time spectra,viscosity

Conclusions 1. The validity of the temperature-frequency superposition in the finite relaxation region has been confirmed for a concentrated solution of PIB filled with solid dispersed particles;2. the apparent heat of activation of the viscoelastic relaxation times increases at and above a certain filler concentration (c=15%);3. in the case of fillers for which the apparent heat of activation of the viscoelastic relaxation times is independent of the filler concentration, the filler only shifts the frequency functions of the components of the complex shear modulus along the logarithm of the modulus axis and the relaxation time spectrum H() along the log H axis. In this case G_C/G_M=G_C/G_M=_OC/_OM=_rel;4. the effective filler concentration for ccrit, as determined from the concentration dependence of the initial viscosity, exceeds the nominal filler concentration by a factor of 2.4 to 2.8;5. at a filler concentration c>c_crit, the contour of the relaxation time spectrum changes on account of the formation of a secondary filler structure;6. when the filler concentration ccrit, the dependencies of the viscosity on the shear rate for the composites and the matrix can be reduced with respect to two parameters;7. possible methods for the reduction of flow curves by means of linear coordinate transformations have been systematized.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 6, pp. 1075–1083, November–December, 1977.     

Flow characteristics of a metasilicate-filled low-density polyethylene melt

Certain of the flow characteristics of a filled low-density polyethylene melt have been investigated. The results obtained are compared with the results of measurements in the single-frequency vibrational shear regime.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 693–697, July–August, 1969.     

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.     

Rheological characteristics of a low-molecular branched polyethylene melt near the melting point

Data are presented on the rheological characteristics of low-density polyethylene with near-critical molecular weight at 110°C. A comparison of the shear rate dependence of the viscosity in repeat measurements indicates that the material has a kinetic memory for isothermal, shear flow-initiated crystallization from the melt.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga; Plastpolimer Research and Production Association, Leningrad. Translated from Mekhanika Polimerov, No. 6, pp. 1129–1132, November–December, 1973.     

The characteristic of convexity of a Banach space and normal structure

We present some sufficient conditions for normal structure of Banach spaces and their dual spaces in terms of the characteristic of convexity, the James constant, and the coefficient of weak orthogonality. Many known results are improved and strengthened. We also show that some of our results are sharp.     

On the characteristic polynomial of a special class of graphs and spectra of balanced trees

Let H be a simple graph with n vertices and G be a sequence of n rooted graphs G₁,G₂,…,G_n. Godsil and McKay [C.D. Godsil, B.D. McKay, A new graph product and its spectrum, Bull. Austral. Math. Soc. 18 (1978) 21-28] defined the rooted product H(G), of H by G by identifying the root vertex of G_i with the ith vertex of H, and determined the characteristic polynomial of H(G). In this paper we prove a general result on the determinants of some special matrices and, as a corollary, determine the characteristic polynomials of adjacency and Laplacian matrices of H(G).Rojo and Soto [O. Rojo, R. Soto, The spectra of the adjacency matrix and Laplacian matrix for some balanced trees, Linear Algebra Appl. 403 (2005) 97-117] computed the characteristic polynomials and the spectrum of adjacency and Laplacian matrices of a class of balanced trees. As an application of our results, we obtain their conclusions by a simple method.     

The crystal and molecular structure of 3, 4-dihydroxy L-phenyl-alanine

The molecule crystallizes in spacegroup P2₁ with two molecules per unit cell. The unit cell dimensions area = 6.044 Å,b = 13.607 Å,c = 5.311 Å,γ = 97.55°. The density was calculated to be 1.512 g.ml.^?1 and found to be 1.51 g.ml.^?1 The major atoms were located by the reliable image method and the hydrogen atoms were located from a difference electron density map. Full-matrix least squares refinement of the parameters yielded an unweighted residual indexR of 0.088. The bond lengths and angles of the amino acid grouping are consistent with values in other amino acids. The structural parameters of the aromatic system are very similar to those of noradrenaline and dopamine hydrochlorides. The crystal structure is dominated by a three-dimensional intermolecular hydrogen bonding system. The molecular conformation is different from that displayed by any other aromatic amino acid or peptide whose crystal structure is known.     

The Laplacian spectra of graphs with a tree structure

Trees are very common in the theory and applications of combinatorics. In this article, we consider graphs whose underlying structure is a tree, except that its vertices are graphs in their own right and where adjacent graphs (vertices) are linked by taking their join. We study the spectral properties of the Laplacian matrices of such graphs. It turns out that in order to capture known spectral properties of the Laplacian matrices of trees, it is necessary to consider the Laplacians of vertex-weighted graphs. We focus on the second smallest eigenvalue of such Laplacians and on the properties of their corresponding eigenvector. We characterize the second smallest eigenvalue in terms of the Perron branches of a tree. Finally, we show that our results are applicable to advancing the solution to the problem of whether there exists a graph on n vertices whose Laplacian has the integer eigenvalues 0, 1, …, n ? 1.     

An electromagnetic free convection flow of a micropolar fluid with relaxation time

In the present investigation, we study the influence of a transverse magnetic field on the one-dimensional motion of an electrically conducting micropolar fluid through a porous medium. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the microrotation and the velocity distributions as well as for the induced magnetic and electric fields are carried out and represented graphically.