Construction of approximate entropic forces for finitely extensible nonlinear elastic (FENE) polymers

When the stress applied to a Rouse-like polymer chain is large enough, one must use anharmonic entropic spring forces in order to keep the chain contour length from increasing to unphysical values. Although one can derive “exact” equations relating the spring extension to the entropic force produced by a finitely extensible non-linear elastic (FENE) random-walk polymer, such expressions are usually of little interest because their complexity would entail large evaluation times in numerical studies by computer. Moreover, these expressions can rarely be used directly in analytical studies. In this article, we describe a systematic method to construct analytically simple yet numerically accurate expressions to relate the entropic force to the extension of an entropic spring for a random-walk polymer chain in arbitrary dimension d ≥ 2. These expressions are modified Pade approximants which yield the correct asymptotic behaviours in both the small and large extension limits. It is shown that the well-known Warner empirical approximation is but a limiting case (for infinite dimensions).     

A simple lennard-jones model for computer simulation of conformational disordering in polymers

A simple generalized model is proposed to describe the phenomenon of conformational disordering in polymers. The model represents a two-dimensional system of flexible particles interacting through a modified Lennard-Jones potential. Monte Carlo simulation of the phase behavior of the model system shows that the model is quite adequate to describe all essential features inherent in the transition to the conformationally disordered state.     

A simple model for multicomponent etching

We consider the situation where a multicomponent solid is etched using one or more acids. Of fundamental interest is the rate of surface etching but when this involves multicomponent surface reactions, it becomes unclear how the overall rate can be estimated. In this paper, we sketch a simple model designed to determine the effective etching rate by means of an atomic scale model of the etching process.     

Statistical mechanical model for diffusion of simple penetrants in polymers. II. Applications—nonvinyl polymers

The theory developed in Part I of this series is applied to a number of nonvinyl “smooth” chained homopolymers. The agreement between predicted and observed activation energies of diffusion for simple penetrants is generally good, particularly for polyethylene. Discrepancies observed for the smallest penetrants, He and H₂, in some polymers may be rationalized in terms of atomic scale irregularities on the polymer chain surface. It is shown that in favorable cases the theory may permit diffusion data to be used as an additional check on the accuracy of conformational energy maps for polymers.     

A simple n-state model for water

A simple n-state configurational excitation model which takes into account the presence of weakly connected pentamer units in liquid water is proposed. The model has features of both the “continuum” and “mixture” models. Calculations based on this model satisfactorily account for the important, diagnostic thermodynamic properties of water such as the density maximum, fraction of monomers and so on.     

Statistical mechanical model for diffusion of simple penetrants in polymers. I. Theory

Following Di Benedetto it is proposed that noncrystalline polymer regions possess an approximate semicrystalline order with chain bundles that are locally parallel along distances of several nanometers. Packing with on-average four nearest neighbors is assumed. A spherical molecule may move through such a substrate in two distinct ways: (a) along the axis of a “tube” formed by locally parallel chains or (b) perpendicular to this axis by two polymer chains separating sufficiently to permit passage of the molecule. The first process is relatively fast, generally requires little activation energy, and determines the effective jump length in diffusion. The second is responsible for the activation energy of diffusion, which is taken as the minimum energy necessary to produce a symmetrical chain separation which allows transfer of a molecule. This is calculated as a function of the penetrant diameter d and parameters Γ and β which characterize the interchain cohesion and chain stiffness, respectively. Γ is estimated from the polymer density and cohesive energy density by suitably linearizing a relation given by Di Benedetto for the potential between two polymer chains approximated as infinite strings of Lennard-Jones force centers. β is shown to be approximately obtainable from the polymer chain backbone geometry and bond rotation potentials. An expression for the diffusion coefficient D is developed which contains only one disposable parameter, the effective jump length.     

A simple model for polysoaps' thread-like aggregates

The equilibrium polymerization, a model for one-dimensional reversible aggregates is used under conditions of theta and bad solvent to describe thread-like aggregates of polysoaps. In two dimensions, the aggregate size distribution decreases always more slowly than an exponential distribution and the dependence of the mean aggregate size L on the density φ and end-cap energy E of the polysoap cylindrical micelle is of the form L[φexp(E/KT)]^δ with δ<1/2. On the other hand, in three dimensions in the bad solvent regime, the dependence of L on φ becomes exponential explaining the high φ dependence of the viscosity in experimental results.     

A simple adsorption model for ionic surfactants

In the past, few theoretical attempts have been made to describe quantitatively the adsorption of ionic surfactants at liquid interfaces. Well-known adsorption isotherms due to Frumkin or Hill–de Boer cannot respond to the specific electrostatic and geometric properties of the surfactant molecules. Our approach is based on a combination of the Gouy–Chapman theory with a modified Frumkin isotherm. The modification implies that the system is free to choose an optimal head group area and an optimal arrangement of the surfactant molecules in the interface as a function of bulk concentration. Interaction energies between neighbouring adsorbed surfactant molecules and between surfactant and water molecules are taken into consideration. The minimum of the Gibbs free energy of the system is equivalent to a minimal interfacial tension. Thus, the thermodynamically stable isotherm can be obtained as the lower envelope of the family of σ versus ln c isotherms resulting from different choices of the model parameters, including the area per molecule. According to the Gibbs equation, the Γ versus ln c adsorption isotherm is obtained as the derivative of this envelope. By variation of the model parameters, the envelope of the calculated adsorption isotherms can be fitted to experimental data of the interfacial tension versus bulk concentration. A computer program is used to calculate the σ versus c and the Γ versus ln c curves as well as to fit the parameters. Received: 28 October 1999/Accepted: 8 February 2000     

A simple model for a reaction surface

An analytical expression, which has some claim to be the simplest possible, is proposed for the potential governing a collinear reaction. It shows the desired qualitative features but, with only one available parameter, cannot fit a given surface accurately everywhere. The quality of fitting attainable is shown using the surface for the O + H₂ reaction.Because of the simple form of this expression, it is possible to make broad generalizations about such reactions. From a plausible assumption about the parameter value the energy barrier and the transition state geometry can be predicted. These barriers agree well with those suggested by Johnston and Parr for hydrogen transfer reactions.     

A simple theoretical model for dual phosphorescence

A simple theoretical model is presented to explain the observed anomalous dual phosphorescences of certain aromatic carbonyl compounds in some rigid media. The phenomenon of dual phosphorescence for large molecules violates the well-known Kasha rule stating that the emission can occur only from the lowest excited electronic state of a given multiplicity. For a small energy gap between the second triplet state (T₂) and the first triplet state (T₁), the sparse density of T₁ vibronic levels, isoenergetic with the T₂ vibrationless level, leads to a rather slow T₂ → T₁ radiationless process which is unable to quench the T₂ emission completely. Two cases of T₁ = ³nπ^*, T₂ = ³ππ^* and T₁ = ³ππ^*, T₂ = ³nπ^* are discussed at both the low-temperature and the high-temperature limits.     

A mathematical model for stress-driven diffusion in polymers

A model for case II diffusion into polymers is presented. The addition of stress terms to the Fickian flux is used to produce the characteristics progressive front. The stress in turn obeys a concentration-dependent evolution equation. The model equations are analyzed in the limit of small diffusivity for the problem of penetration into a semiinfinite medium. Provided that the coefficient functions obey two monotonicity conditions, the solvent concentration profile is shown to have a steep front that progresses into the medium. The formulas governing the progression of the front are developed. After the front decays away, the long time behavior of the solution is shown to be a similarity solution as in Fickian diffusion. Two techniques for approximating the solvent concentration and the front position are presented. The first approximation method is a series expansion; formulas are given for the initial speed and deceleration of the front. The second approximation method uses a portion of the long time similarity solution to represent the short time solution behind the front.     

Statistical mechanical model for diffusion of simple penetrants in polymers. III. Applications—vinyl and related polymers

The theory developed in Part I of this series is modified to accommodate polymers that possess closely spaced, bulky side groups on the chains. The side groups give rise to free space between the chain “cores,” which reduces the chain separation required for penetrant motion transverse to the local chain axis. The theory is then identical to that of Part I, except that penetrant diameters minus a constant factor are employed in place of the normal diameters. In most of the cases studied the reduction factor for a given polymer may be estimated with reasonable precision from chain geometry data. This diameter-reduction effect is the likely explanation of the apparent proportionality between the activation energy of diffusion and the square of the penetrant diameter reported earlier for vinyl polymers. The data quoted here and in Part II are analyzed to give a semitheoretical correlation between the effective jump length L? and ΔE, the activation energy of diffusion. This correlation appears to be equally valid for glassy and rubbery noncrystalline polymers.     

Brownian dynamics simulations with stiff finitely extensible nonlinear elastic-Fraenkel springs as approximations to rods in bead-rod models

A very stiff finitely extensible nonlinear elastic (FENE)-Fraenkel spring is proposed to replace the rigid rod in the bead-rod model. This allows the adoption of a fast predictor-corrector method so that large time steps can be taken in Brownian dynamics (BD) simulations without over- or understretching the stiff springs. In contrast to the simple bead-rod model, BD simulations with beads and FENE-Fraenkel (FF) springs yield a random-walk configuration at equilibrium. We compare the simulation results of the free-draining bead-FF-spring model with those for the bead-rod model in relaxation, start-up of uniaxial extensional, and simple shear flows, and find that both methods generate nearly identical results. The computational cost per time step for a free-draining BD simulation with the proposed bead-FF-spring model is about twice as high as the traditional bead-rod model with the midpoint algorithm of Liu [J. Chem. Phys. 90, 5826 (1989)]. Nevertheless, computations with the bead-FF-spring model are as efficient as those with the bead-rod model in extensional flow because the former allows larger time steps. Moreover, the Brownian contribution to the stress for the bead-FF-spring model is isotropic and therefore simplifies the calculation of the polymer stresses. In addition, hydrodynamic interaction can more easily be incorporated into the bead-FF-spring model than into the bead-rod model since the metric force arising from the non-Cartesian coordinates used in bead-rod simulations is absent from bead-spring simulations. Finally, with our newly developed bead-FF-spring model, existing computer codes for the bead-spring models can trivially be converted to ones for effective bead-rod simulations merely by replacing the usual FENE or Cohen spring law with a FENE-Fraenkel law, and this convertibility provides a very convenient way to perform multiscale BD simulations.     

A simple low-temperature DTA model for teaching purposes

A simple model for low-temperature DTA for teaching purposes is described, and its utility is demonstrated by measurements on mercury, pentane, cyclohexane and ammonium nitrate.     

Interacting stereo-irregular chains: A model for conducting polymers

Interacting stereo-irregular chains of hydrogen atoms, which simulate the topological structure of many conducting polymers, are generated by a computer and solved numerically with the unrestricted Hartree–Fock method with a modified spin polarized potential. The electron localization is investigated, and a mechanism for the interchain tunneling is discovered. Local antiferromagnetic ordering is derived which may explain the AF behavior observed in some conducting polymers.     

A molecular model for flow induced crystallization of polymers

We present in this paper a thermodynamic model for flow induced crystallization of a thermoplastic. The thermomechanical framework (generalized standard materials) allows us to couple in a very natural way the kinetics of crystallization with the mechanical history experienced by the thermoplastic^[1]. In describing the viscoelastic properties of the polymer with a molecular theory, we obtain a model for flow-induced crystallization that couples the chain conformation to the kinetics of crystallization. This model intends to be valid both for shearing and elongation. We present the equations for two cases: Maxwell and Pom-Pom constitutive equations. We finally illustrate our model with injection molding simulations achieved with a dedicated Finite Element code.     

A simple model for calorimeters with temperature programming

We present a theoretical study of an RC-model constituted only by one heat capacity and one coupling with the thermostat. It is assumed that the thermostat temperature varies as a function of time, and the heat capacity variation is due to its dependence either on temperature or on the mass exchange with the exterior.The results are parallel to the corresponding RC-models where the thermostat temperature is constant. The variations of sensibility are shown, as well as a criterion for the applicability of inverse filtering as a deconvolution technique in calorimeters with temperature programming.     

A simple model for an amperometric channel biosensor

A simple model is presented for the channel biosensor where an oxidase enzyme layer is located upstream of a detector electrode. In this model the enzyme kinetics are restricted to the linear region. The model enables the elucidation of the parameters important in tuning the response of the biosensor. Two extreme regimes of operation are identified; the kinetically limited and mass transport limited regimes both provide features which overcome reproducibility problems associated with the classical enzyme electrode geometry.     

A simple strategy for detecting synthetic polymers on solid surfaces using liquid crystal

In this study, we developed a liquid crystal (LC)-based detection method for polymer films synthesized on solid surfaces. A dark to bright transition in the optical appearance of nematic 4-cyano-4′-pentylbiphenyl (5CB) was observed after transferring a poly(methyl methacrylate) (PMMA) film onto a glass substrate functionalized with n-octyltrichlorosilane (OTS). This phenomenon indicates an orientational transition of 5CB from a homeotropic to a planar-random state. The optical response of 5CB was then evaluated directly through polymerization reactions on the OTS-functionalized glass substrate. Polymer films of PMMA, poly(glycidyl methacrylate) (PGMA), and poly(dimethylsiloxane) (PDMS) were synthesized on OTS surfaces covered with their reaction mixtures. All polymer films displayed bright signals of 5CB, which corresponded to the planar-random orientation of LCs. However, no change in orientation was observed for the control experiments. We confirmed the formation of polymer films on the OTS surface using atomic force microscopy. Overall, our results suggest that LCs can be used to construct optical monitoring systems for the product of polymerization reactions.