Transient Network Model Research Articles

A molecular network model to describe the nonlinear viscoelastic behavior of polymers

A transient molecular network model is built to describe the nonlinear viscoelasticity of polymers by considering the effect of entanglement loss and regeneration on the relaxation of molecular strands. It is an extension of previous network theories. The experimental data on three thermoplastic polymers (ABS, PVC and PA6) obtained under various loading conditions are used to test the model. Agreement between the theoretical and experimental curves shows that the suggested model can describe successfully the relaxation behavior of the thermoplastic polymers under different loading rates by using relatively few relaxation modes. Thus the micromechanism responsible for strain-rate dependence of relaxation process and the origin of nonlinear viscoelasticity may be disclosed.

Transient network model for a multiphase polymeric fluid

Rheological behaviors of entangled polymer melts filled with rigid solid particles are phenomenologically described based on a Yamamoto type transient network model. Entanglement points are described by one kind of temporary junctions with a lifetime τp. Presence of fillers introduces another kind of network junctions with a lifetime τs, characteristic of adsorption of polymer chains at the filler surface. The degree of compatibility of filler surfaces with the polymer medium is characterized in terms of the adsorption time τs relative to the disentanglement time τp. By incorporating the effects of interfacial slippage into the deformation mechanism of the transient network, one can explain the variation of such rheological functions as shear viscosity with surface treatment of fillers and with molecular weight of the polymer matrix at a given filler volume fraction.

Quantitative rheological predictions of a transient network model of Lodge–Yamamoto type: Simple and multiaxial elongational flow

A transient network model is presented which is based on the ideas of Lodge and Yamamoto. The network thus consists of different types of strands and the creation and loss rates are configuration dependent. As the model cannot be solved analytically one has to rely upon numerical tools. Using a continuous time algorithm this network can be simulated very efficiently. The material functions can be calculated as time‐dependent ensemble averages. Concentration is put onto a quantitative comparison between the rheological predictions of the model and experimental data for polyisobutylene subjected to simple and multiaxial elongational flows. The predictions of the model are found to be in very good agreement with the experimental data.

Viscoelastic properties of physically crosslinked networks: Part 3. Time-dependent phenomena

The initial value problem is solved for the transient network model of physically crosslinked gels. Stress relaxation following a sudden macrodeformation is calculated for several realistic models of the chain breakage rate β( r). It is shown that, on large time scales, stress decay obeys a power law when β( r) has significant r-dependence, the precise value of the power being dependent on the high stretching behaviour of β( r). For instance, the shear stress decays as ≈ t −5 n if β( r) is proportional to r n at high stretching. Although the Lodge-Meissner relation still holds, time-strain separability loses its physical background. Overshoot phenomena in shear and normal stress, which appear after steady flows are started at the initial equilibrium state, are also analyzed. It is found that the shear stress first shows a transient maximum, and then maxima of the first and second normal stress differences follow. Larger overshoot is expected for larger values of the shear rate γ, but the time at which the stress reaches its maximum is almost independent of γ The elongational stress is also obtained as a function of time.

Viscoelastic properties of physically crosslinked networks: Part 1. Non-linear stationary viscoelasticity

The stationary solution for the transient network model of reversibly crosslinked gels is found under arbitrary macrodeformation. For shear flow with constant shear rate γ, the number of active chains, stationary viscosity, first and second normal stress differences are calculated as functions of γ. Elongational flow with constant elongational rate ϵ is also studied. It is found that these stationary properties depend rather sensitively on the chain breakage function β( r) and the recombination probability p of the sticky dangling ends. On the basis of the polymer statistics, a specific form of β( r) = β o exp kr is proposed, where β o and k are functions of the temperature T and the molecular weight M of the polymer chain. Stationary viscoelastic properties are shown to exhibit an exponential dependence on T due to the activation process for the junction dissociation, differing markedly from an uncrosslinked polymer melt whose viscosity varies as a power of the temperature. Thickening and thinning conditions for both types of flow are examined. It is shown that limiting behaviour of the shear viscosity under high shear rate obeys the scaling law η(γ) ≈ γ −2 n ( n+1) if β( r) is proportional to r n at high stretching.

Dynamic light scattering, mutual diffusion, and linear viscoelasticity of polymer solutions

A theoretical study of the dynamic light scattering spectrum from a binary solution consisting of a polymer and a small molecular weight solvent is carried out. The study is based upon a modified hydrodynamic equation previously derived from statistical mechanics by Bearman and Kirkwood for a multicomponent solution. The result shows that mutual diffusion arising from the osmotic pressure fluctuation and the viscoelasticity affecting mechanical properties of the polymer solution are, in general, mixed. The extent of mixing depends on the frequency and a coupling parameter β, which is proportional to the difference between the partial specific volume of the polymer and that of the small molecular solvent. At low frequency the coupling is negligible, and one obtains a diffusion equation with the mutual diffusion coefficient dependent only on the osmotic modulus. At the frequency in the plateau stress modulus region, the result predicts a cooperative diffusion coefficient which consists of both the osmotic modulus and the longitudinal elastic modulus of the polymer solution, provided that β=1. The result is equivalent to that previously obtained by de Gennes, assuming a transient gel network model for the semidilute polymer solution. The present calculation also provides new results for the polymer solution in which the transient gel network model is not applicable. The overall light scattering spectral density is calculated in terms of the autocorrelation functions of the local pressure and the local concentration, as well as cross correlation functions between them. The result shows that as the polymer concentration is gradually increased toward the bulk state, the concentration fluctuation spectrum losses its importance, and the pressure (or density) fluctuation spectrum dominates the overall light scattering spectral density.

Continuous time simulation of transient polymer network models

A continuous time simulation algorithm for polymer melts is presented. The method is introduced via an explicit application to transient polymer network models, but it may be applied to a much larger class of models. The central quantity of the simulation is the lifetime of a strand. This can be calculated according to a proper distribution, which may depend on the orientation and length of the generated new strand. As the equation of motion of the strand can be integrated analytically during the lifetime of the strand, the procedure works much more efficiently than a previous simulation algorithm for transient network models. The material functions can be calculated as ensemble averages. The new method is shown to work for time-dependent flows. A simplified algorithm for steady flows is also given. Some predictions are made for a specific configuration-dependent transient network model.

The rheological behaviour of suspensions of fat particles in oil interpreted in terms of a transient-network model

The rheological behaviour of suspensions of fat particles in oil interpreted in terms of a transient-network model

The Viscoelastic Behavior of Heat‐Set Ovalbumin Gels Explained in Terms of a Transient‐Network Model

The transient network model for concentrated dispersions, presented in a previous paper, is used for the interpretation of the viscoelastic behavior of heat‐set ovalbumin gels. The gels consist of a water phase and a network‐shaped phase of denatured ovalbumin molecules, which are crosslinked and entangled by sulphur bridges, hydrogen bridges, and hydrophobic bonds. The local stresses inside the dispersed phase are calculated from the local deformation and a modified Maxwell‐type constitutive equation that is used for the description of the rheological behavior of this phase. Subsequently, the macroscopic stress tensor is calculated making use of the concepts of the transient network model. Some remarkable model stress predictions are presented and interpreted. Linear viscoelastic measurements on the ovalbumin gels are used to quantify model parameters. Subsequently, the sample's behavior in simple shear and uniaxial compression experiments, including stress relaxation, are compared with model predictions. The model is thus found to satisfactorily describe the system's rheological behavior.

The rheological behaviour of suspensions of fat particles in oil interpreted in terms of a transient-network model

The transient-network model for concentrated dispersions, described in a previous paper, is used to describe the rheological behaviour of dispersions of glyceryl tristearate crystals in paraffin oil. The model prediction of the storage modulus of this system is compared with corresponding expressions given in literature. Model calculations are carried out to fit the linear viscoelastic behaviour of the system as well as its stress response in large amplitude shear experiments. Information is thus obtained about the stiffness and strength of the interparticle bonds, and the chance of them breaking in a state of rest or as the result of flow. It is concluded that the probability of interparticle bond fracture strongly depends on the measure of bond stretching. The general findings link up with the Lennard-Jones potential which is assumed to describe the potential energy of the bonds between the particles. Accurate measurements of the temperature dependence of the dynamic moduli by making use of a torsion resonator lead to the conclusion that the energy dissipation at a high frequency originates mainly from the flow of liquid around the particles.

Comment on the Berkeley kinetic network model

AbstractA kinetic model for the rheological behavior of polymeric systems, i.e. the Berkeley kinetic network model, is compared with a generalized transient‐network model. It turns out that the Berkeley kinetic network model fits quite well in the framework of the transient‐network model. From the point of view of the latter model, however, a number of parameters introduced in the kinetic‐network model are not independent constants as was supposed. Since the generalized transientnetwork model is totally based on microscopic assumptions, the parameters in this theory reveal information about the structure, whereas the physical interpretation of the parameters in the kinetic model is vague. Stress predictions of the generalized transient‐network model agree with experimental data from a 4 percent solution of polystyrene in arochlor, obtained from literature.

Analysis of the flow behavior of irradiated polyethylene through transient network models

AbstractThe chemical and physical properties of polymers can be dramatically affected by exposure to high‐energy radiation. Two polyethylene samples (HDPE and LDPE) were irradiated with a 60Co source. Radiated samples with doses ranging between 0.5 and 5 Mrad were melted and their viscoelastic properties characterized. Results show the normal drastic changes in their flow behavior: higher viscosity and interesting changes induced in their elasticity, particularly in the case of LDPE. An explanation of the change in flow behavior is forwarded in terms of transient network models using the concept of creation and destruction of entanglements.

A transient-network model describing the rheological behaviour of concentrated dispersions

Attractive forces acting between particles in dispersions may cause a three-dimensional structure to be built up. A temporary-network model is postulated that describes the rheological behaviour of such systems. Chains of particles are assumed to be created and broken by thermal actions and by applied deformation. The relation between the network structure and the macroscopic stress tensor is deduced. One of the main model features is that no use is made of the common assumption of affinity of the motion of the chain vectors with the gradient of the macroscopic velocity field. Instead, the chain deformations are assumed to depend on the forces acting on them, i.e. their deformations depend on their stiffness and on the applied deformation, whereas fracture of chains may cause stress relaxation in the rest of the network. The chains may behave as highly non-linear springs, whereas the probability that the chains will break in some time interval may be an explicit function of the chain length itself. Integral equations are derived, from which the stress-tensor components can be calculated in any flow experiment, that obeys creeping-flow conditions. Analytical expressions are obtained for the relaxation spectrum of such systems in terms of the microscopic parameters.

Transient Hydrodynamic and Salinity Simulations in the Chesapeake Bay Network

A transient network model is applied to the Chesapeake Bay and its tributary estuaries. Calibration of the model is based on only three external parameters: a friction factor that is spatially described, and two global constants required to calibrate a dynamic dispersion relationship that depends on both the local salinity gradient and hydraulic conditions. The transient hydrodynamics and the transient salinity distribution of the Bay and its tributary estuaries are simulated for the period of one month and comparisons made between calculated and observed salinities.

A generalized network model for polymeric liquids

A kinetic model was developed for relating the molecular structure and the rheological behaviour of polymer-like materials in which bonds are being created and broken. In particular, the stress contribution of molecules that are not a part of the network was taken account of. In two limiting cases the model introduced here is identical to the transient-network model and the bead-spring model, respectively. Model predictions were interpreted and compared with measurements on an L.D.-polyethylene melt, available from literature, and an S.I.S.-blockcopolymer solution. In both cases information was obtained about the network and the kinetic processes occurring in a state of rest and during steady-shear flow, assuming the model to be valid.

Viscoelastic properties of polydisperse homopolymers

Abstract The transient network model, developed in this laboratory to predict linear viscoelastic properties of entangled monodisperse polymers and their binary and ternary blends, has been extended to apply to polydisperse polymers. This model incorporates both the viscous and the elastic interactions among the entanglements. The viscous effect is taken into account by assigning enhanced frictional coefficients to the entangled beads. The elastic effect is modeled by connecting these beads with additional coupling springs. It was recognized that, in a multiple blend or a polydisperse sample, entanglements form not only among chains of the same length (intracomponent entanglements), but also among chains of different lengths (intercomponent entanglements). The multitudinous interactions among the various chains in such a system are taken into account by a mixing rule generalized from the assumptions previously applied to binary/ternary blends. A polydisperse polymer with a broad molecular weight distribut...

Viscoelastic properties of entangled polymers: Ternary blends of monodisperse homopolymers

In a previous publication from this laboratory, the Rouse-Bueche-Zimm molecular theory of viscoelasticity has been extended by using a transient network model to apply to binary blends of monodisperse polymers with chain entanglements. The dynamics of the entanglements were modeled both by the enhanced frictional coefficients and by the additional elastic couplings. It was recognized that entanglements not only may form between chains of the same lengths (intracomponent entanglements) but also between those of different lengths (intercomponent entanglements). At a given intercomponent entanglement, the longer chain was assumed to have the frictional coefficient of the shorter chain. Similarly, for blends consisting of several monodisperse components with different molecular weights, such modifications are also required to predict their linear viscoelastic behavior. The frequency of these interactions is assumed to be proportional to the weight ratio of the respective component chains in the blend. Equations of motion are formulated for each component and solved numerically for the relaxation time spectra. Linear viscoelastic properties such as the dynamic mechanical moduli, stress relaxation moduli, and zero-shear viscosity can then be computed for these systems by linear summation of those of the components. The reduced steady-state shear compliance of the blends can also be computed from the component relaxation times. Results are found to be in good agreement with experimental data on polystyrene ternary blends.

Viscoelastic Properties of Entangled Polymer. IV. Binary Blends of Monodisperse Homopolymers

In a previous publication from this laboratory, the Rouse‐Bueche‐Zimm molecular theory of viscoelasticity has been extended by using a transient network model to apply to monodisperse polymers with chain entanglements. Effects of the entanglements were modeled both by the enhanced frictional coefficients and by the additional elastic couplings resulting from the transient entanglement network. For binary blends consisting of two monodisperse polymers with different molecular weights, additional modifications are now required to predict their linear viscoelastic behavior. It is recognized that entanglements not only may form between chains of the same length, but also between those of different lengths. For the latter case, the longer chain will in fact have the frictional coefficient of the shorter chain at the point of entanglement. The frequency of such interactions is assumed to be proportional to the weight ratio of the respective component chains in the blend. Equations of motion are formulated for each component and solved numerically for the relaxation spectra. Linear viscoelastic parameters such as the dynamic mechanical moduli, stress relaxation moduli, and zero shear viscosity can then be computed for the blends by linear summation of those of the components. The reduced steady state shear compliance of the blends can also be computed from knowledge of the component relaxation times. Results are found to be in excellent agreement with literature data on polystyrene, poly(dimethyl siloxane), and poly(methyl methacrylate).