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A recently suggested relaxation mechanism which is expected to be dominant in fast flows of polymer melts is combined here with the tube model of Doi and Edwards, and the corresponding constitutive equation is derived. The mechanism consists of the renewal of topology by convective displacement of entanglements. Detailed predictions for the flow curve in steady shear are obtained in order to test the agreement with the Cox-Merz rule. With respect to the basic tube model, significant progress is achieved.

The classical dumbbell and Rouse models for dilute polymer solutions are here modified by introducing an anisotropic diffusivity which mimics the effect of topological interactions arising in polymeric solutions in the vicinity of the overlap concentration c ∗ . Closed-form constitutive equations are derived by making use of typical pre-averaging approximations. Predictions in steady and transient shear flows for tangential and normal stresses are compared to data recently obtained by Osaki et al. (J. Polym. Sci.: Polym. Phys. 38 (2000) 1917), on a polystyrene solution close to c ∗ . As regards the second normal stress difference, more rarely measured, we compare the model predictions with data by Magda et al. (Macromolecules 26 (1993) 1696).

The way of obtaining the Schrodinger equation for a quantum particle in an electromagnetic field is revisited, showing that very few physical assumptions are required. In fact, after having introduced the general formalism of non-relativistic quantum mechanics, it is shown that the structure of the Schrodinger equation for a spinless particle is obtained merely by requiring continuity of space and time, and covariance with respect to Galilean transformations. Both the Correspondence and Uncertainty principles then become «theorems».

In this paper the scalar quasi-static magnetic hysteresis is described by means of a new model based on a sequential algorithm, which appears to be a possible alternative to the classical Preisach model. A procedure leading to the identification of the model parameters is described. The hysteretic behavior obtained by using the new model is in good agreement with experimental measurements as well as with the predictions of the Preisach model. Finally, both models have been applied to study the dynamics of a circuit with an hysteretic inductor, driven by a sinusoidal voltage source. Good agreement between the two simulations was found, but our model appears computationally less cumbersome than that of Preisach.

Polymeric liquids often exhibit non-linear responses under flow conditions in spite of the fact that inertia effects are usually negligible. Non-linearities arise when, as a consequence of flow, the distribution of molecular conformations departs significantly from equilibrium. In turn, this will occur whenever the strength of the velocity gradient becomes larger than the inverse of the longest relaxation time of the polymer molecule. Based on this general concept, various detailed interpretations are given to the observed non-linearities (the non-Newtonian viscosity, for example) depending on the specific relaxation mechanism which prevails under different conditions (e.g. dilute solutions or concentrated systems, and branched or linear chains). Here, cases where new results were recently obtained are briefly discussed, including a very curious non-linear effect which is encountered in liquid crystalline polymers.

Shear-thickening effects are often observed in physically cross-linked networks formed by polymeric chains having a few localized, energetically favored interactions. The authors find that a possible explanation for these effects is the non-Gaussian behavior of the chains stretched by the shear flow. Two network models for unentangled telechelic chains are proposed, which differ in the way unattached chains relax after deformation. One of the models correctly portrays the qualitative shear-thickening features commonly observed.

A phenomenological model which represents a possible alternative to the well-known Preisach's model of magnetic hysteresis is here presented. The main feature of the proposed model, which differentiates it from Preisach's one, is that the individual elementary «particles» are not themselves hysteretic; it is the mechanism by which the state of these particles is modified that introduces hysteretic behaviour. The model appears to be computationally less cumbersome than Preisach's one, while having the same application range.

One of the (few) unrealistic features of the Doi-Edwards model for entangled polymers is that it predicts a steady-state shear stress which decreases indefinitely as the shear rate is increased. As shown by Marrucci and Grizzuti (Gazz. Chim. Ital., 118 (1988) 179) this feature remains, even when one takes into account chain stretching due to elongation of the tube by the flow. Here we suggest a further mechanism for chain stretching which arises because the tube, even when fully aligned with the flow, has a finite lateral dimension which exposes different parts of the chain to different flow velocities. Using a simplified model, we predict the asymptotic behaviour of the shear stress and first-normal-stress difference, and show that stable flow is recovered at high enough shear rates.

Flexible polymeric chains (Kuhn chains) made up of nematogenic moieties are considered here. The simplest approach to the dynamics of such a system, in both the isotropic and the nematic phases, is that of modeling the chain as an elastic dumbbell where the ‘‘spring’’ law also accounts for the nematic interaction. The constitutive equation of this ‘‘nematic dumbbell model’’ is thereby derived. The rheological predictions so obtained are qualitatively compared with the most prominent features of LCP rheology as well as with previous theoretical results.

The viscosity of liquid crystalline polymers (LCPs) in the nematic phase does not approach a Newtonian plateau in the limit of vanishing shear rates. This non-linear effect is probably linked to large distortions of the LCP polydomain, i.e. to a dominant role of the defect structure. To account for spatial distortions inclusive of defects a molecular theory is used, based on a generalized Maier-Saupe potential. The resulting mathematical problem, too complex to be solved exactly, is here suitably simplified. A set of two differential equations is thus obtained, and solutions are found numerically for a representative case. The results indicate that defects can indeed play a leading role in LCP behaviour at low shears by acting as pseudo-walls for director anchoring.

In view of the limitations inherent to phenomenological approaches, the problem of the internal structure of defects in nematics is here afforded from a molecular viewpoint. By using an extension of the Maier-Saupe mean-field potential which accounts for distorted states of nematics, the general equation applicable inside a defect is obtained by self-consistency. The specific example of the hedgehog point defect is developed in detail. It is shown that the size of the defect core depends on the order parameter characterizing the undistorted nematic phase: the core size decreases when the order parameter increases.

The classical Maier-Saupe theory of spatially uniform nernatics is here extended to the case of distorted configurations. Under the assumption that the molecules are rodlike, the mean field potential is here defined as an average over a suitable neighborhood of the test molecule. The energy density thereby calculated automatically accounts for the distortion energy, and properly reduces to the classical expres- sion for the homogeneous situation. In the linear limit, contrary to existing predictions based on Maier- Saupe potential, the three constants of Frank elasticity are found to be different from one another, and ordered as K 1

The classical equations of thermodynamics are here rewritten in the form of a field equation in then-dimensional space of the state variables. This result is obtained by proving the validity of a theorem for Pfaffian forms inn-dimensional spaces. By using the field expression thus obtained, it becomes possible to reformulate classical thermodynamics in a very compact way.

The tumbling behavior of LCPs at low shear rates is investigated with the help of the Leslie-Ericksen theory restricted to the planar case and to the one-constant approximation. It is shown that the director distortions induced by the tumbling nature of the nematic may easily reach down to the molecular level, where they get saturated. A rotational motion of the director becomes unavoidable in such a case. The time-dependent structure of the director field is then investigated, thus obtaining definite predictions about the shear dependence of the viscosity in the low shear rate range

To understand the flow behaviour of polymeric liquids, it is important to analyse the microscopic origin of the stress. The classical molecular theories relate the stress to the tensile force of entropic origin acting along the chains. The momentum transfer due to collisions between different molecules is not explicitly considered. Though certainly adequate for dilute solutions, that assumption becomes questionable in concentrated solutions and melts

Two-dimensional simulations of the Brownian dynamics of polymers in a grid of topological obstacles were carried out in this study. A first model made use of the exact expression for the free energy of a chain with fixed ends interacting with an obstacle and can be used to derive the elastic force acting on the subchains of a Rouse-like model. With this model we calculated the diffusion coefficient of chains of different lengths constrained by the fixed obstacles. A second model, less rigorous but more efficient computationally, was used to simulate the chain dynamics in fast flows. In such a case, the obstacles were convected according to the shear gradient. During the simulation, the configurational changes of the chain constrained by the obstacles could be monitored and the stresses which developed were calculated. These simulations reveal important effects of the conformational fluctuations at high shear rates

We here report on some modifications of the Primitive Chain Network (PCN) model, originally proposed in [Y. Masubuchi et al., J. Chem. Phys. 115, 4387 (2001)], which both refine the model and make it suitable for predicting nonlinear rheological response in fast flows. Simulation results are compared with some existing viscoelastic data on monodisperse polystyrene melts [Schweizer et al., J. Rheol. 48, 1345 (2004); Bach et al., Macromolecules 36, 5174 (2003)], by using as single adjustable parameter (to be assigned once and for all) a basic relaxation time which relates to the coarse graining of the PCN model. Essentially quantitative prediction of linear viscoelasticity is achieved. Without further parameters, the nonlinear behavior of the polystyrene melts in start‐up of shear and uniaxial elongational flow is also reproduced by the simulations. Specifically, the viscosity growth functions are quantitatively predicted, whereas transient elongational viscosities are reasonably reproduced only up to extensional rates of the order of the reciprocal Rouse time. At higher elongation rates, although the strain hardening effect is correctly captured by the simulations, steady state values are not. Notwithstanding this latter limitation, the simulation model appears adequate to portray the rheological behavior of entangled melts, both in the linear and the nonlinear range.

The theory of rodlike polymers in the nematic phase is now sufficiently well developed as to allow predictions of the rheological behaviour that qualitatively compare well with the experimental observations. One of the main results of the theory is the prediction that rodlike polymers are nematics of the tumbling type at low shear rates, whereas they become shear oriented at high rates: a nonlinear effect, which is absent in low molecular weight nematics. This aspect is here reviewed in an effort to highlight the intuitive aspects of the matter. First, the low shear-rate situation is discussed in order to investigate the conditions that determine the existence of a stationary solution as opposed to a periodic one (tumbling). Then, the high shear-rate range is considered, where the shear-oriented situation prevails under all conditions. The intermediate range of shear rates is the most interesting one for its peculiar rheological behavior.

The two-dimensional model introduced by Marrucci and Maffettone is explored in the limit of small shear rate. A tumbling motion for the director n of the nematic phase has been found. Explicit expressions for the shear and normal stresses have been derived for both the elastic and viscous contributions. The present results indicate that the model is able to describe the experimentally observed positive to negative to positive normal stresses transitions with increasing shear rate

In this paper we derive a few analytical results for the case of polymeric solutions flowing past a spherical obstacle. The calculations are meant to ascertain the possible importance of flow-induced migration (i.e., of changes in concentration), making use of the, recent two-fluid theory of Doi and Milner. In order to obtain solutions in closed form, the problem is suitably linearised; in particular, we have considered both the case of Newtonian behaviour of the polymeric liquid flowing past the sphere, and that of a linearly elastic polymeric gel perturbed by a localised force. The results, though confirming migration effects, indicate that, in these linear limits at least, the concentration change decays quadratically with the reciprocal distance from the centre of the perturbation, i.e., no long ''wake'' is predicted. Another outcome of these calculations refers specifically to the viscous case. The classical Stokes result is recovered only as a singular solution of the two-fluid problem

The flow of polymer melts and solutions differs substantially from that of other liquids, because the long polymer chains can become entangled and cannot cross each other freely in their motion. In his Perspective, Marrucci provides an overview of theoretical attempts to explain polymer flow. He highlights the report by Bent et al., who have performed sophisticated experiments on linear polymers that prove key elements of the latest theory. Furthermore, the approach used by the authors will become an important experimental tool for investigating more complex polymer systems.

The flow of polymer melts and solutions differs substantially from that of other liquids, because the long polymer chains can become entangled and cannot cross each other freely in their motion. In his Perspective, [Marrucci][1] provides an overview of theoretical attempts to explain polymer flow. He highlights the report by [ Bent et al .][2], who have performed sophisticated experiments on linear polymers that prove key elements of the latest theory. Furthermore, the approach used by the authors will become an important experimental tool for investigating more complex polymer systems. [1]: http://www.sciencemag.org/cgi/content/full/301/5640/1681 [2]: http://www.sciencemag.org/cgi/content/short/301/5640/1691

The viscosity of polymeric liquids crucially depends on polymer concentration c and molar mass M. These dependencies can generally be predicted from relatively simple theories, because what really matters is the chain-like structure of the polymer molecule rather than its detailed chemistry. In this chapter we summarize the main concepts leading to these predictions throughout the concentration range, i.e., from dilute solutions up to polymer melts. However, only the so-called zero-shear viscosity will be considered because the nonlinear effects arising from the coupling between flow and molecular “structure” are more complex and fall outside the scope of this chapter. In dilute solutions the important concept is that of “intrinsic” viscosity which, through dependence on M, reveals structural features of the polymer (flexible chain vs. rigid rodlike, for example) as well as the solvent quality. In semidilute solutions of long polymers, the overall structure of the system becomes that of an impermanent network of entangled chains. The viscosity is then related to the elasticity of the network and to the kinetics of chain disengagement. Typically, the viscosity scaling takes the form of power laws in both M and c. The M dependencies remain the same in the particularly relevant case of polymer melts. The chapter ends with a brief description of systems with localized interactions (or sticky points) in which the viscosity is particularly sensitive to the strength of such interactions.

The predictions of tube models in fast flows have recently been improved by incorporating convective constraint release (CCR). In this paper, we revisit CCR showing that, whenever chains are not stretched, an additional effect named convective conformation renewal (CCR2 to distinguish it from CCR) should also be considered. CCR2 consists in the loss of orientation of the unstretched chain due to flow-induced lengthening of tube segments. A simple quantitative model including both CCR and CCR2 is then developed, which reduces to double reptation in the linear limit. Although the model ignores fluctuations, the predicted flow curve is monotonic, and compares favourably with data for nearly monodisperse entangled polymers. For the case of a single relaxation time, typical of polymer-like micelles, a maximum in the flow curve still remains.

The basic concepts of entangled polymer dynamics, like that of ‘tube’ of constraints and of ‘reptation’, have not yet exhausted their momentum. Further progress has taken place recently with detailed calculations of tube length fluctuations and constraint release, not only for branched polymers but also for linear ones. Progress in the non-linear range has occurred with the concept of convective constraint release and, for branched polymers, with the so-called ‘pom-pom’ model. Lack of complete agreement between theory and experiments stimulates work to proceed along various lines.

It is well known that the flow behaviour of polymeric liquids can be, and often is, much more complicated than that of ordinary fluids. The ultimate reason for the complication is the molecular anisotropy, obviously very large in a polymer chain. Indeed, although most materials are isotropic at rest (because the molecular segments are randomly orientated), the anisotropy reveals itself when the polymeric liquid is set in motion, as the molecules become oriented. The flow-induced molecular orientation, and the consequent material anisotropy, even persists for a while after the flow has been switched off, i.e. during relaxation.

Between 1976 and December 1992 a total of 80 patients aged 80 or over underwent surgery for colorectal cancer at the Surgical Clinic of Pisa University. Twelve patients died, 18 presented non-lethal complications and 48 had a normal postoperative recovery. The type of surgery did not appear to be correlated with mortality. Urgency, associated with the advanced stage of cancer (according to Duke's classification modified by Aster-Coller), was the factor which most severely influenced mortality. A marked prognostic improvement can be obtained by an early diagnosis and careful evaluation of the patient's overall conditions.

Between 1976 and December 1992 a total of 748 patients with colorectal cancer were observed at the Surgical Clinic of Pisa University; in 9 of these a second colorectal tumour appeared between 30 months and 18 years after the original tumor. Coloscopy is the treatment of choice for evaluating the colon operated due to cancer since it allows an early diagnosis of any metachronous tumours, reveals synchronous tumours which have been overlooked prior to the operation and performs a therapeutic and prophylactic function in terms of the endoscopic removal of adenomatous polyps which have appeared in the meantime.

All the complexity of the rheological behaviour of polymeric materials is ultimately rooted in their molecular structure or, in the case of suspensions, blends, block copolymers, or other complex materials, is due to the “structure” of the material at the scale of, say, 1 µm. Thus a valuable alternative to the classical approach to rheology, originally founded in continuum mechanics, is one which starts from a consideration of the micro-structure of the system, and attempts to model those structural features which are most likely to be important in determining the rheological behaviour which is observed macroscopically. This is, generally speaking, the approach known as micro-rheological modelling, though in the present context only the case of solutions and melts of ordinary polymers will be considered, for which the names of “molecular modelling” or “molecular theories” are more often employed.

The paper presents a thermodynamic derivation of the most general expressions for the force density, the electromagnetic surface traction, and the polarization work in an electrically or magnetically polarized solid, which is totally based on a continuum description. The notion of internal stress in polarized materials is discussed. © 1995 Società Italiana di Fisica.

Liquid crystalline polymers (LCPs) seem to be tumbling nematics in most cases, which perhaps explains why they are also defect laden. Their behaviour in a shear flow appears to be dominated by tumbling, and by the dynamics of the defects. After briefly reviewing our present understanding of this complex situation, some tentative way of modelling the defect dynamics is presented.

The theory of rodlike polymers in the nematic phase is now sufficiently well developed to allow predictions of rheological behaviour which compare favourably with many experimental observations. These include peculiar effects such as negative normal stresses, damped oscillations in start-up flows and oscillation frequencies inversely proportional to shear rate. The basic points of the theory are presented, together with a more direct interpretation of the above-mentioned phenomena.