Modified Rouse Model Research Articles

Inclusive Hyper- to Dilute-Concentrated Suspended Sediment Transport Study Using Modified Rouse Model: Parametrized Power-Linear Coupled Approach Using Machine Learning

The transfer of suspended sediment can range widely from being diluted to being hyper-concentrated, depending on the local flow and ground conditions. Using the Rouse model and the Kundu and Ghoshal (2017) model, it is possible to look at the sediment distribution for a range of hyper-concentrated and diluted flows. According to the Kundu and Ghoshal model, the sediment flow follows a linear profile for the hyper-concentrated flow regime and a power law applies for the dilute concentrated flow regime. This paper describes these models and how the Kundu and Ghoshal parameters (linear-law coefficients and power-law coefficients) are dependent on sediment flow parameters using machine-learning techniques. The machine-learning models used are XGboost Classifier, Linear Regressor (Ridge), Linear Regressor (Bayesian), K Nearest Neighbours, Decision Tree Regressor, and Support Vector Machines (Regressor). The models were implemented on Google Colab and the models have been applied to determine the relationship between every Kundu and Ghoshal parameter with each sediment flow parameter (mean concentration, Rouse number, and size parameter) for both a linear profile and a power-law profile. The models correctly calculated the suspended sediment profile for a range of flow conditions (0.268 mm≤d50≤2.29 mm, 0.00105gmm3≤particle density≤2.65gmm3, 0.197mms≤vs≤96mms, 7.16mms≤u*≤63.3mms, 0.00042≤cˉ≤0.54), including a range of Rouse numbers (0.0076≤P≤23.5). The models showed particularly good accuracy for testing at low and extremely high concentrations for type I to III profiles.

Diffusing Diffusivityin Dynamics of Unentangled Polymer Melts

AbstractThe exact results for Rouse chain dynamics in the crowded environment modeled bydiffusing diffusivity(N. Ahamad and P. Debnath,Phys. A2020,549, 124335) in explaining polymer melt dynamics is explored. Theory for mode accounts for anomalous and non‐Gaussian diffusion in the center of mass (c.m.) and monomer dynamics of united atom molecular dynamics simulation trajectories of unentangled polyethylene melts, with excellent agreement. Accurate estimates of scaling relations for c.m. and monomer mean square displacement with time interval are obtained. The corresponding c.m. and monomer diffusion coefficients as a function of confirmdiffusing diffusivityin polymer melts. It is shown shown that the dependent parameters of the modified Rouse model expressed in terms of all modes are reducible to an identical set of per mode parameter values on application to real data. Analysis of velocity autocorrelation functions indicates that the theory is tested on simulation trajectories of polymer melts with friction. The theory estimates for additional power‐law exponents in the exponential of displacement distributions agree with the data. The results show fairly convincingly that the Rouse model withdiffusing diffusivitymay find scope as an alternate theoretical formalism for polymeric liquids.

Viscoelasticity of a Stretched Semiflexible Polymer Chain with Fixed Ends

Regularities of the decay of time correlation functions of fluctuations of the microscopic stress tensor characterizing viscoelastic properties for statically stretched chains composed of elastic units with bending stiffness are studied by molecular dynamics and analytically. Chains with the persistent length smaller than the contour length at degrees of extension up to break are considered. In the region of high extensions, where distortions of the local structure of a chain appear, the decay of correlation functions can be described by the modified Rouse model consisting of subchains with anisotropic rigidities depending on the equilibrium chain extension. This makes it possible to determine the relaxation times and moduli of strongly stretched chains, which is necessary in studying the strength of polymer networks. For correlation functions, the analytical relation is formulated that takes into account the contribution of transverse bending fluctuations and agrees with the results of computer simulation at low extensions. A change in the correlation functions with the extension of long semiflexible chains is qualitatively similar to that for flexible chains, but quantitatively is appreciably different.

Rouse model in crowded environment modeled by “diffusing diffusivity”

We derive exact results for the dynamics of Rouse model in crowded environment modeled by stochastically varying diffusivity, the concept of “diffusing diffusivity”. We specifically generalize the initial analytical formulation of this concept of diffusivity in single particle diffusion to many particle systems with localized interactions, particularly to chains of connected beads, the Rouse model. Independent Rouse modes are allowed to diffuse with stochastically varying time dependent diffusivities to model diffusion in the crowded rearranging environment, where zeroth mode represents the chain center of mass coordinate. We develop an analytical formalism to calculate the probability distribution of these individual mode displacements and subsequently average bead displacement distribution of the Rouse chain and its moments in terms of these modes. The resulting diffusion behavior of thus modified Rouse model is a function of mode resolved parameters which model local interactions and crowded environment collectively. Our analytically tractable implementation of the “diffusing diffusivity” concept to Rouse model is one of its kind that yields non-Gaussian diffusion with the scope of being anomalous and plausibly applied to dynamics in polymeric liquids.

Globally Suppressed Dynamics in Ion-Doped Polymers.

We investigate how ion-polymer complexation suppresses molecular motion in conventional polymer electrolytes using molecular dynamics (MD) simulations of lithium hexafluorophosphate in poly(ethylene oxide) and a modified Rouse model. The employed model utilizes an inhomogeneous friction distribution to describe ion-polymer interactions and provides an effective way to examine how ion-polymer interactions affect polymer motion. By characterizing the subdiffusive Li+ transport and polymer relaxation times at several salt concentrations, we observe that increases in local friction due to ion-polymer complexation are significantly smaller than previously assumed. We find that a Rouse-based model that only considers local increases in friction cannot simultaneously capture the magnitude of increased polymer relaxation times and the apparent power-law exponent for Li+ subdiffusion observed in MD simulations. This incompatibility is reconciled by augmenting the modified Rouse model with a term that increases the global friction with the square of the salt concentration; this significantly improves the agreement between the model and MD, indicating the importance of ion-ion interactions and distributions on ion/polymer mobility.

Description of poly(ethylenepropylene) confined in nanopores by a modified Rouse model.

A recent model for unentangled polymer chains in confinement [M. Dolgushev and M. Krutyeva, Macromol. Theory Simul. 21, 565 (2012)] is scrutinized by small-angle neutron scattering (SANS) with respect to its static prediction, the single-chain structure factor. We find a remarkable agreement although the model simplifies the effect of the confinement to a harmonic potential. The effective confinement size from fits of SANS data with the model agrees well with the actual pore size. Starting from this result we discuss the possibility of an experiment on the dynamic structure factor predicted by the model. It turns out that such an experiment would need a large ratio polymer dimension/pore size which is difficult but not impossible to achieve.

Shear Dynamic Moduli of Stretched Polymer Chains and Networks: Modified Rouse Model

AbstractThe dynamic mechanical properties of polymer chains and networks under large static strains are studied. Polymer chains are modeled as sequences of Gaussian subchains whose elasticity constants are different for motions parallel ($K_{||} $) and perpendicular ($K_ \bot $) to the stretching force, f, and are functions of the elongation ratio (a modified Rouse model). The frequency‐dependent storage and loss shear moduli are shown to be different for different geometries under small oscillating shear deformation. Three principal geometries under shear deformation denoted by D‐, V‐ and G‐geometry (see the image) are considered. Static elongation leads to the broadening of the frequency dependencies of the loss moduli for all considered geometries. The predicted storage and loss moduli are in a good agreement with experimental data. magnified image

Shear Dynamic Modulus of Nematic Elastomers: Modified Rouse Model

We develop a microscopic theory of dynamic mechanical properties of nematic elastomers taking the chain structure of network strands explicitly into account. We use an approach in which network strands are modeled as a sequence of Gaussian subchains, whose elasticity constants and friction coefficients are different for motions parallel and perpendicular to the LC-director: K∥ ≠ K⊥ and ζ∥ ≠ ζ⊥ (a modified Rouse model). We show that the dynamic modulus of an ordered nematic elastomer, G∗ = G′ + iG′′, should demonstrate a frequency behavior very similar to that of usual (nonordered) rubbers; especially, it should display a frequency domain with a Rouse-like behavior, G′ ≅ G′′ ∼ ω1/2, a feature which is confirmed by experiments. In contrast to the usual rubbers, nematic elastomers are characterized by the anisotropy of the dynamic mechanical behavior with respect to the LC director, n. In agreement with experiment we show that for prolate systems in the D-geometry (when n is parallel to the shear velocity) G′D greatly decreases around the nematic isotropic phase transition, whereas in the V-geometry (when n is perpendicular to the shear plane) GV′ does not demonstrate such a singularity. We discuss the predictions of our theory for other geometries under shear deformation.

Rheology of Sodium Hyaluronate Saline Solutions for Ophthalmic Use

The aim of this work was to investigate the rheological properties of different saline solutions of sodium hyaluronate (NaHA) with special interest for medical applications. The experimental results were compared with literature data for commercial ophthalmic viscosurgical devices (OVDs) used in cataract surgery. We offer some tools to tailor the rheological behavior of OVDs for different purposes. We have investigated to which extent surgical requirements can be fulfilled by adjusting either the molecular weight of NaHA or its concentration, parameters that are in some respects equivalent but not in others. Furthermore, we demonstrate that moduli and complex viscosities of NaHA saline solutions are adequately falling on master curves, using either empirical or calculated shift factors, the latest ones being based on a modified Rouse model.

Melt Rheology of Tin Phosphate Glasses

Abstract The melt rheology of a low T g tin phosphate glass [Pglass] has been studied with oscillatory shear flow experiments to accelerate efforts to melt process the glass with different organic polymers. The w dependence of the complex viscosity η* of the Pglass is easily predicted by a modified Rouse model with two relaxation times. The complex viscosity of the glass at different temperatures and frequencies can be superposed and described by the Arrhenius equation. At higher temperatures, the melt viscosity of the Pglass increased monotonically with time. This viscosity rise is thought to be due to sample crystallization. The Pglass was melt-mixed with two different thermoplastic polymers (low-density polyethylene and polystyrene) to produce unique hybrid materials with interesting microstructures.

On the Validity of the Maxwell Model and Related Constitutive Equations; a Study Based on the First Normal Stress Coefficient

In a previous publication ( J. Phys. I France 4 (1994) 1791) the semi-microscopical Rouse model was projected onto macroscopic Langevin equations for concentration and stress tensor fluctuations. Adding afterwards appropriate convective terms due to a simple shear flow the resulting equations were shown to be a generalization of the upper convected Maxwell model. Based on the projected equations the first normal stress coefficient ψ 1,0 is now calculated in this publication: ψ 1,0 = 2η 0 2 /κ B Tc 0 where η 0 and c 0 denote the zero-shear viscosity and the monomer concentration respectively. The resulting steady state compliance J s is found to be J s = 1/κ B Tc 0 . This turns out to be in disagreement with experimental findings. Based on the derivation of the Lodge equation one can conclude the following: In order to get the right first normal stress coefficient it is necessary to incorporate the shear flow at first into the Rouse model and to project afterwards the modified Rouse model.

Viscoelastic properties of polyelectrolyte solutions

AbstractViscoelastic properties of polyelectrolyte solutions were studied in the absence and presence of added salt over a wide range of polymer concentration in comparison with those of non‐ionic polymers in good solvents.In the semidilute region where the polymer concentration C is higher than the overlap concentration for viscosity C*v, but lower than C** (ca. 0.3 kg/dm3), the polymer concentration dependence of specific viscosity at zero‐shear rate η increases with increasing added‐salt concentration Cs and at high Cs it almost agrees with that of non‐ionic polymers in good solvents. In the concentrated and entangled region where C>C and C>C**, η more steeply increases with C even in salt‐free solutions, implying the screening of electrostatic interaction and the increase of local frictional coefficients as considered in concentrated non‐ionic polymer solutions.In the dilute region where C is lower than the overlap concentration for steady‐state compliance C, the steady‐state compliance Je more strongly depends on molecular weight and C than in non‐ionic polymer solutions. In the semidilute region where C>C, but C<C**, the polymer concentration dependence of Je is lower than that of non‐ionic polymer solutions. In the concentrated and entangled region where C>C and C>C**, the polymer concentration dependence becomes similar to that of non‐ionic polymer solutions. In the concentrated and non‐entangled region where C<C and C>C**, the frequency dependences of storage and loss moduli are well explained by the modified Rouse model like non‐ionic polymer systems in the same region.The experimental results in the semidilute region are well explained by our theory of viscoelastic properties of polyelectrolyte solutions based on the reptation model with electrostatic interactions.

Influence of molecular weight distribution on viscoelastic constants of polymer melts in the terminal zone. New blending law and comparison with experimental data

A new blending law for the viscoelastic constants of polymer melts in the terminal zone is proposed on the basis of the well known power law for viscosity ( η 0 ∼ M a w) and a simple assumption of the dependence of the weight-average relaxation time on molecular weight distribution. This blending law reduces to the modified Rouse model if the exponent a takes the value of 3. The recoverable compliance of binary blends can be calculated as a function of zero-shear viscosities and recoverable compliances of their components and of parameter a. Measurements have been carried out on binary blends of nearly monodisperse PS melts. For these results as well as for various experimental data from the literature, satisfactory agreement between calculated and experimental values of the recoverable compliance has been observed. The blending law is extended to polymer melts with continuous molecular weight distribution.

On the dynamics of coiled macromolecules

A modified Rouse model of dilute polymer solution in flow was considered and applied to describe the rheological behaviour of this system. The modification of the model was carried out in the following direction. The restoring force acting between two neighbouring beads was calculated accounting the fact, that the deformation of the real chain was accompanied by change not only of entropy, but also of internal energy, since during the deformation a certain number of links can shift from one energetical state to another. The calculations were carried out in the rotational isomeric state approximation.

Effect of Internal Viscosity on the Deformation of a Linear Macromolecule in a Sheared Solution

The deformation of macromolecules in a solution subjected to time-dependent shear flow is described by means of a modified Rouse model. Into every segment of this model an internal viscosity force is introduced which is proportional to the rate at which the end-to-end distance of the segment is changed, and acts in the direction of the tie line between the end points. No a priori assumptions have been made on the motion of the segments themselves. An analytical solution for the distribution function of the segment lengths can now no longer be derived. At low values of the internal viscosity, an approximate solution can be obtained by means of a perturbation calculus. The effect of internal viscosity on the moments of the distribution function and, hence, on the average dimensions of the macromolecules, can be calculated. It appears that with constant shear flow at high rates of shear, as well as with oscillatory shear at high frequencies, an increase of the internal viscosity results in a decrease of the average deformation of the molecules. The flow birefringence at high shear rates decreases as the internal viscosity increases, whereas the optical extinction angle is hardly influenced by it. In the literature there is no unanimity on the relation between the deformation of the molecules and the shear stress; in the present study we use the formula of Kirkwood and Riseman. In the case of constant shear flow, the apparent viscosity of the solution decreases with increasing rates of shear. With oscillatory shear flow the internal viscosity leads to a finite limiting value of the dynamic viscosity at high frequencies, and this value should increase when a constant shear rate is superimposed on the oscillatory shear rate. This theoretical prediction conflicts with experimental data.

Influence of rate of shear on the apparent viscosity of dilute polymer solutions

AbstractThe measured viscosity of polymer solutions varies with the magnitude of the applied shearing stress. In this paper a modified Rouse model of a polymer molecule (which has some stiffness) is used, and the diffusion equation is solved with respect to the co‐ordinates system moving with the polymer molecule. Results indicate: (1) The above phenomena is a natural consequence of the fact that the molecules are caused to rotate by the shearing action. (2) This is the same mechanism with the dynamic viscosity caused by the applied periodic shearing stress.