Determining polymer molecular weight distributions from rheological properties using the dual-constraint model

Although multiple models now exist for predicting the linear viscoelasticity of a polydisperse linear polymer from its molecular weight distribution (MWD) and for inverting this process by predicting the MWD from the linear rheology, such inverse predictions do not yet exist for long-chain branched polymers. Here, we develop and test a method of inverting the dual-constraint model (Pattamaprom et al., Rheol Acta 39:517–531, 2000; Pattamaprom and Larson, Macromolecules 34:5229–5237, 2001), a model that is able to predict the linear rheology of polydisperse linear and star-branched polymers. As a first step, we apply this method only to polydisperse linear polymers, by comparing the inverse predictions of the dual-constraint model to experimental GPC traces. We show that these predictions are usually at least as good, or better than, the inverse predictions obtained from the Doi–Edwards double-reptation model (Tsenoglou, ACS Polym Prepr 28:185–186, 1987; des Cloizeaux, J Europhys Lett 5:437–442, 1988; Mead, J Rheol 38:1797–1827, 1994), which we take as a “benchmark”—an acceptable invertible model for polydisperse linear polymers. By changing the predefined type of molecular weight distribution from log normal, which has two fitting parameters, to GEX, which has three parameters, the predictions of the dual-constraint model are slightly improved. These results suggest that models that are complex enough to predict branched polymer rheology can be inverted, at least for linear polymers, to obtain molecular weight distribution. Further work will be required to invert such models to allow prediction of the molecular weight distribution of branched polymers.     

Linear rheology of comb polymers with star-like backbones: melts and solutions

We investigate the linear rheology of model star-comb homopolymers consisting of star-like backbone chains with grafted branches. We show that the tube-based theory in the framework of full dynamic dilution, appropriately modified to account for the effects of the fluctuations of the free segments (the segments of the star arms between the outer branching points and the arms’ free end-monomers) of the backbone star arms, and the polydispersity accurately describes the linear viscoelastic spectrum. For these branched polymers, the relaxation is found to proceed hierarchically, similar to combs with linear backbones. However, in contrast to the latter, here, there is no reptation. The higher functionality star combs with sparse branching are particularly interesting because they resemble a Cayley tree structure and are treated in the context of asymmetric star polymers. We also employ solutions of star combs and test our model in static dilution conditions. In the region where despite the presence of both static (solvent) and dynamic dilution the polymers remain entangled, our model is particularly successful without adjusting any parameters.

Peripheral explosion in a self-gravitating gas ball and dynamic explosion of star equilibrium

The problem of peripheral explosion in a star initially at equilibrium is solved for an exponential density distribution. Qualitatively new flow modes, such as recurrent ejection of the star shell and partial scatter of its matter in interstellar space, are obtained. The critical energies corresponding to various flow modes are determined. Calculations conducted over a wide range of the determining parameters allow certain conclusions to be drawn concerning the possibility of explaining the phenomena occurring in the interior of pulsing and variable stars. The problem of dynamic explosion of star equilibrium, followed by the formation of a detonation wave travelling through a gravitating gas at rest, is also considered. It is shown that various solutions involving detonation may be constructed by choosing the adiabatic exponent and the exponent of the power density distribution. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 157–163, May–June, 1998.     

Existence and Non-linear Stability of Rotating Star Solutions of the Compressible Euler–Poisson Equations

We prove the existence of rotating star solutions which are steady-state solutions of the compressible isentropic Euler–Poisson (Euler–Poisson) equations in three spatial dimensions with prescribed angular momentum and total mass. This problem can be formulated as a variational problem of finding a minimizer of an energy functional in a broader class of functions having less symmetry than those functions considered in the classical Auchmuty–Beals paper. We prove the non-linear dynamical stability of these solutions with perturbations having the same total mass and symmetry as the rotating star solution. We also prove finite time stability of solutions where the perturbations are entropy-weak solutions of the Euler–Poisson equations. Finally, we give a uniform (in time) a priori estimate for entropy-weak solutions of the Euler–Poisson equations.     

Quantitative predictions of linear viscoelastic rheological properties of entangled polymers

The “dual constraint” model developed by Mead, Van Dyke et al. is here extended by inclusion of “early-time” contour-length fluctuations and constraint-release Rouse relaxation, and then evaluated by comparing its predictions with literature data for over 50 different linear and star polymers. By combining the reptation model of Doi and Edwards with contour-length fluctuations and constraint release, the model provides a systematic approach to prediction of the rheological properties of polymers. The parameters are taken from the literature and used consistently for linear polymers, star polymers, and their mixtures having the same chemical compositions. In most cases, the predictions of the model appears to agree well with data for monodisperse, bidisperse, and polydisperse linear and star polymers, except at low molecular weights. Received: 23 December 1999 Accepted: 28 March 2000     

Evaluation of drag correction factor for spheres settling in associative polymers

Drag correction factors are calculated for the creeping motion of spheres descending in various associative polymers of different concentration with various sphere-container ratios and Weissenberg numbers. The simple-shear rheology and linear viscoelasticity of these polymeric fluids have been previously presented and modeled with the BMP (Bautista–Manero–Puig) equation of state (Mendoza-Fuentes et al., Phys Fluids 21:033104, 2009). The drag on the sphere is initially kept nearly constant for small Weissenberg numbers, We < 0.1. As the Weissenberg number increases, We < 0.1, a reduction in drag is found. Experimental results show the presence of a critical Weissenberg number at which a drag reduction occurs. The reduction in the drag correction factor is associated to the onset of extension-thinning, which coincides with the formation of a negative wake. No increase in the drag correction factor was observed, due to the simultaneous opposing effects of extension-thickening and shear-thinning viscosity. The shape of the drag correction factor curve may be predicted considering the extensional properties of the solutions, as suggested elsewhere (Chen and Rothstein, J Non-Newton Fluid Mech 116:205–215, 2004).     

Hybrid approach for dynamic model identification of an electro-hydraulic parallel platform

In this paper, a hybrid optimization algorithm is proposed to identify the dynamic parameters of a 6-DOF electro-hydraulic parallel platform. The dynamic model of a parallel platform with arbitrary geometry, inertia distribution and frictions is obtained based on a structured Boltzmann–Hamel–d’Alembert formulation, and then the estimation equations are explicitly expressed in terms of a linear form with respect to the identified inertial and the friction coefficients in accordance with a linear friction model. However, when nonlinear friction models are considered, the parameter identification of the electro-hydraulic parallel platform is considered as an optimization process with an objective function minimizing the errors between the measurement and identification, and then an effective combination of the particle swarm optimization (PSO) method and the local quasi-Newton method is proposed to solve the identification problem. Experimental identification processes are carried out for the identified parameters, and the identified models are compared by the predicted forces between the LS method and the optimization technique as well as between the linear and nonlinear friction models.     

Determination of the ultimate states of elastoplastic bodies

The equations of quasistatic deformation of elastoplastic bodies are considered in a geometrical linear formulation. After discretization of the equations with respect to spatial variables by the finite-element method, the problem of determining equilibrium onfigurations reduces to integration of a system of nonlinear ordinary differential equations. In the ultimate state of a body of an ideal elastoplastic material, the matrix of the system degenerates and the problem becomes singular. A regularization algorithm for determining solutions of the problems for the ultimate states of bodies is proposed. Numerical solutions of test problems of determining the ultimate loads and equilibrium configurations for ideal elastoplastic bodies confirm the reliability of the regularization algorithm proposed. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 196–204, September–October, 2000.     

Local Well-Posedness of Dynamics of Viscous Gaseous Stars

We establish the local-in-time well-posedness of strong solutions to the vacuum free boundary problem of the compressible Navier–Stokes–Poisson system in the spherically symmetric and isentropic motion. Our result captures the physical vacuum boundary behavior of the Lane–Emden star configurations for all adiabatic exponents g < \frac65{\gamma < \frac{6}{5}} .     

Inference of polymer structure by simultaneous analysis of chromatographic and rheological measurements

A Bayesian framework that integrates chromatographic and rheological measurements to infer the structure of an unknown binary linear blend, with a high molar mass tail, is proposed and explored. A visualization method based on clustering of multidimensional data is introduced to facilitate the comprehension of the joint probability distribution that results from the Bayesian analysis. The relationship between the rheological and chromatographic data is found to be synergetic: the simultaneous analysis of both sets of data circumvents problems of degeneracy and insensitivity associated with rheology and chromatography, respectively.     

Complex-potential method in the nonlinear theory of elasticity

For materials characterized by a linear relation between Almansi strains and Cauchy stresses, relations between stresses and complex potentials are obtained and the plane static problem of the theory of elasticity is thus reduced to a boundary-value problem for the potentials. The resulting relations are nonlinear in the potentials; they generalize well-known Kolosov's formulas of linear elasticity. A condition under which the results of the linear theory of elasticity follow from the nonlinear theory considered is established. An approximate solution of the nonlinear problem for the potentials is obtained by the small-parameter method, which reduces the problem to a sequence of linear problems of the same type, in which the zeroth approximation corresponds to the problem of linear elasticity. The method is used to obtain both exact and approximate solutions for the problem of the extension of a plate with an elliptic hole. In these solutions, the behavior of stresses on the hole contour is illustrated by graphs. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 133–143, January–February, 2000.     

Brownian dynamics simulations of star-branched polymers in dilute solutions in extensional flow

Using Brownian dynamics (BD) simulations of FENE bead-spring models, the dynamics of star-branched polymers in dilute solutions under extensional flow have been investigated. Studies on star polymers in transient extensional flow reveal that the initial transient stress response at low strains is governed by both the number of arms and the shortest arm. On the other hand, the steady-state behavior of star polymers in elongational flow is limited by the maximum effective “contour” length of the molecules. The distribution of arm extension and birefringence of the star-branched molecule are broader and the mean is shifted to lower values, when compared to equivalent linear systems. As a result, the degree of arm extension at steady-state decreases as the number of arms in the star increases. Both an analysis of individual ensembles in Brownian dynamics simulations and a study of a simple force balance indicate that the constraint imposed on the star arms by the central branch point and contributions from “asymmetric” arm arrangements give rise to overall less extended and oriented star-branched molecules with broader arm extension and birefringence distributions. The results obtained from stress-conformation hysteresis simulation indicate that less-stretched arms exhibit more retarded relaxation, as the number of arms increases in star-branched molecules. The effect of excluded volume (EV) interactions, incorporated through the Lennard–Jones potential, on the dynamics of star polymers in extensional flow appears unimportant.     

One-dimensional motion of an emulsion with solidification

A mathematical model is proposed for the process of solidification of an emulsion with a small disperse-phase concentration moving under the action of thermocapillary forces and microgravity. The first-approximation problem that arises when solutions are represented as asymptotic series in a small parameter is examined. Conditions for the partial and complete displacement of the impurity from the solidified part and conditions for the accumulation of the impurity in the solidified mixture are obtained. The problem of producing a composite with a specified disperse-phase distribution is considered. Exact solutions that adequately reflect various features of the qualitative behavior of the general solution under different input data are obtained and examined. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 128–136, May–June, 1999.     

Permeability identification of a weakly permeable partially saturated porous rock

The present paper deals with the determination of permeability in partially saturated conditions for weakly permeable porous continua such as argillites or deep clayey formations. The permeability can be deduced from measurements of transient weight loss of a sample submitted to a laboratory drying test: a decrease of relative humidity is imposed by saline solution in an hermetic chamber. Assumptions of constant gas pressure equal to atmospheric pressure and of negligible Fickean diffusive transport of vapour are adopted. The only transport phenomenon taken into account inside the sample is the Darcean advective transport of the water liquid. The forward problem is solved by following two modelling approaches: a linear one and a nonlinear one. The parameter identification procedure is based upon the solution of corresponding inverse problems. In the two cases, the Levenberg–Marquardt algorithm has been used for the minimization problem. In the linear approach, the solution of the forward problem is explicit. In the non linear approach, finite volume method for the spatial discretization combined with a Newton–Raphson algorithm has been used to solve the non linear forward problem. The identification method enables variations of permeability and capillary capacity to be estimated. Comparisons between linear and non linear approaches show that the first one is useful to give mean values and order of magnitude of permeability and capacity. A more complete information is deduced from the non linear approach as variations of equivalent capacity and permeability during a test are significant in most cases. The analysis of the obtained results shows that the basic modelling assumption of constant gas pressure inside the sample would not be relevant for lower range of relative humidities and liquid permeability than those investigated.     

Approximation of periodic solutions of linear differential-difference equations of neutral type by spline-collocation method

We construct a method for the approximation of periodic solutions of linear differential-difference equations of the neutral type by using cubic splines and investigate conditions for the convergence of the proposed approximation scheme. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 147–154, April–June, 2006.     

Structure and rheology of heat-set gels of globular proteins

The structure and the rheology of systems resulting from heating at 80°C isoelectric solutions of bovine serum albumin (BSA) in the concentration range 10–200mg/ml were studied.  Small-angle neutron scattering measurements view the systems as being formed of large aggregates of micrometric size with a close packed arrangement of denatured protein molecules. No indication of a fractal structure stands out.  The viscoelastic behaviour is linear up to about 5% strain, except in the BSA concentration range 30–90mg/ml where the linearity limit is below 1% strain. The viscoelastic response was analysed in the linear domain, or as close as possible to it, by combining the results of dynamic and creep recovery measurements. The dependence on concentration of the steady state viscosity, of the steady state compliance, and of the average retardation time shows a marked change around a concentration C ₀∼50mg/ml, corresponding probably to a percolation threshold. Received: 10 March 1998 Accepted: 1 May 1998     

Solving initial–boundary-value creep problems

The Galerkin–Bubnov method with global approximations is used to find approximate solutions to initial–boundary-value creep problems. It is shown that this approach allows obtaining solutions available in the literature. The features of how the solutions of initial–boundary-value problems for oneand three-dimensional models are found are analyzed. The approximate solutions found by the Galerkin–Bubnov method with global approximations is shown to be invariant to the form of the equations of the initial–boundary-value problem. It is established that solutions of initial–boundary-value creep problems can be classified according to the form of operators in the mathematical problem formulation     

Interaction of an external supersonic flow with the subsonic layer near a wavy wall

A solution of the problem of supersonic flow past a wavy wall with an adjacent subsonic layer is obtained. The solution is a generalization of the well-known solutions [1] of the linear problem of purely subsonic and purely supersonic flow past a wavy wall and goes over into these solutions in the limiting cases of infinite and zero wall-layer thickness, respectively. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 97–103, January–February, 1997.     

Numerical simulation of the interaction of gravity waves with elastic ice floes

The self-consistent motion of a fluid and elastically oscillating plates partially covering the fluid is simulated numerically in the linear approximation. The problem is reduced to the simultaneous solution of the Laplace equation for the fluid and the equation of elastic plate oscillations for the ice. The numerical and analytical solutions, the latter obtained from an integral equation containing the Green’s function, are compared. To solve the problem numerically, the boundary element method for the Laplace equation and the finite element method for the equation describing the elastic plate are proposed. The coefficients of transmission and reflection of surface gravity waves from the floating plates are calculated. It is shown that the solution may be quasi-periodic with characteristics determined by the initial values of the wave and ice-floe parameters. The ice floes may exert a filtering effect on the surface wave spectrum, essentially reducing its most reflectable components. Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 123–131, May–June, 2000.     

Physically and geometrically nonlinear deformation of conical shells with an elliptic hole

The elastoplastic state of conical shells weakened by an elliptic hole and subjected to finite deflections is studied. The material of the shells is assumed to be isotropic and homogeneous; the load is constant internal pressure. The problem is formulated and a technique for numerical solution with allowance for physical and geometrical nonlinearities is proposed. The distribution of stresses, strains, and displacements along the hole boundary and in the zones of their concentration is studied. The solution obtained is compared with the solutions of the physically and geometrically nonlinear problems and a numerical solution of the linear elastic problem. The stress-strain state around an elliptic hole in a conical shell is analyzed considering both nonlinearities __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 69–77, February 2008.