Morphology and Dynamics of Phase‐Separating Fluids with Viscosity Asymmetry

Summary: The effects of viscosity asymmetry of the components on morphology and dynamics of phase‐separating AB fluids are investigated numerically based on a modified Model H. For critical mixtures, in the early stage of phase separation the co‐continuous morphology with droplets of A in B and B in A is observed. In the late stage of phase separation, the viscosity asymmetry leads to morphological change from co‐continuous structure to completely dispersed structure where the less viscous component forms droplet. The pathway of this transformation is accompanied by the breakdown of balance of volume fraction between droplets with different viscosity. Domain growth is characterized by a crossover from a faster growth at intermediate time under the influence of hydrodynamics to Lifshitz–Slyozov behavior at late times. For off‐critical mixture, viscosity asymmetry only plays an important role for domain growth in the intermediate stage of phase separation and the domain growth depends on whether the more viscous phase is dispersed or continuous, and the late stage of domain growth follows Lifshitz–Slyozov power law independent of which phase is dispersed.

Result for the evolution of phase‐separating domains for critical fluid mixtures = 0.5 for t = 1 500 with viscosity asymmetry: η_A = 0.8, η_B = 0.2. A‐rich regions and B‐rich regions are represented by white and black, respectively.     

Long-time growth kinetics of first order phase transitions in the presence of a boundary layer

The late stage growth mechanism for a first order phase transition, either through nucleation growth or spinodal decomposition, is well understood to be an Ostwald ripening or coarsening process, in which larger domains grow at the expense of smaller ones. The growth kinetics in this regime was shown by Lifshitz and Slyozov to follow at(1/3) law. However, the kinetics is altered if there exists a barrier ahead of the growth front, irrespective of the physical origin of the boundary layer. We present an analytic calculation for the growth kinetics in the presence of a boundary layer, showing that in the limit of barrier-dominated growth, the domains grow with at(1/2) law. This result holds true in the dilute regime independent of whether the growing nuclei are spherical or cylindrical.     

Pre-Averaged Sampling On the Entanglement Kinetics for Polymer Dynamics

A new model for entangled polymer dynamics based on pre-averaged sampling of the entanglement structure is proposed. Although it has been reported that sliplink simulations are powerful and promising to predict entangled polymer dynamics, it is still unpractical to calculate polymers with many entanglements. In the present study, a possible approach to achieve fast calculation is proposed by pre-averaged sampling of entanglement structure with skipping detail kinetics of entanglements dominated by chain ends in conventional sliplink models. To achieve time development of the chain conformation and entanglement structure, i) number of entanglement per chain and number of monomers for each segment are randomly obtained from the equilibrium distribution proposed by Schieber [J. Chem. Phys. 2003 , 118, 5162] and ii) the renewed entanglement structure is mechanically equilibrated. The established power-laws on molecular weight dependence of chain dimension, the longest relaxation time and self-diffusion coefficient were reasonably reproduced. Comparison on linear viscoelastic response is also discussed.     

Stochastic entangled chain dynamics of dense polymer solutions

We propose an adjustable-parameter-free, entangled chain dynamics model of dense polymer solutions. The model includes the self-consistent dynamics of molecular chains and solvent by describing the former via coarse-grained polymer dynamics that incorporate hydrodynamic interaction effects, and the latter via the forced Stokes equation. Real chain elasticity is modeled via the inclusion of a Pincus regime in the polymer's force-extension curve. Excluded volume effects are taken into account via the combined action of coarse-grained intermolecular potentials and explicit geometric tracking of chain entanglements. We demonstrate that entanglements are responsible for a new (compared to phantom chain dynamics), slow relaxation mode whose characteristic time scale agrees very well with experiment. Similarly good agreement between theory and experiment is also obtained for the equilibrium chain size. We develop methods for the solution of the model in periodic flow domains and apply them to the computation of entangled polymer solutions in equilibrium. We show that the number of entanglements Π agrees well with the number of entanglements expected on the basis of tube theory, satisfactorily reproducing the latter's scaling of Π with the polymer volume fraction φ. Our model predicts diminishing chain size with concentration, thus vindicating Flory's suggestion of excluded volume effects screening in dense solutions. The predicted scaling of chain size with φ is consistent with the heuristic, Flory theory based value.     

Models for adhesion at weak polymer interfaces

The weak interfaces between immiscible polymer pairs typically fail through chain scission. The critical facture toughness for such interfaces is closely related to the density of intermolecular entanglements at the interface. From scaling analysis, a simple correlation between facture toughness and chain entanglement was developed. It predicts well the interfacial adhesion for many immiscible polymer pairs found in the literature. For an interface with block copolymer reinforcement, its critical fracture toughness comes from both intermolecular entanglements of homopolymers and copolymer bridges. In the chain scission regime (low copolymer coverage), the block copolymer contribution is found proportional to copolymer interfacial coverage, with the coefficient being the energy to stretch and break a copolymer chain. The chain‐breaking energy for different copolymers was evaluated and compared to literature data. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 47: 2313–2319, 2009     

Modeling the effect of compatibilizers in the coarsening of ternary polymer blends with core‐shell morphology

Numerical simulations of the phase separation and coarsening of particulate, ternary polymer blends have been performed using a ternary form of the modified Cahn–Hilliard equation. The third component was chosen to be a compatibilizer, typically being a random copolymer of the major components. The results show that compatibilized blends follow the same Lifshitz–Slyozov coarsening law as binary systems. Slower coarsening rates, indicating system stabilization, were observed for blends containing ∼10% compatibilizer and exhibiting a core‐shell morphology. Larger amounts of compatibilizer resulted in significantly higher coarsening rates. This appears to be a result of the greater affinity of the compatibilizer for the major component and warrants further experimental investigation. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 1301–1306, 2000     

Entanglement effects in model polymer networks

Many fundamental questions for the understanding of polymer networks are more suitably addressed by current computer simulations than by experiments. Details of the microscopic topology, such as the elastically active cluster or loop entanglements, can be identified as well as controlled. In particular, it is possible to isolate and quantify their effects on macroscopic observables such as the elastic modulus. The constraints due to connectivity and conserved topology are more clearly present for networks than for melts. Already for strand lengths between crosslinks which are relatively short, the effect of the conserved topology is important. The mode relaxation in a network is significantly different from that of a melt. For weakly crosslinked systems the melt entanglement length is the relevant scaling parameter. The elastic modulus of a long chain network under ideal conditions reaches an asymptotic value which is about 2.2 times smaller than the prediction of an affine model for a network made of strands of the melt entanglement length. An analysis of the stress reveals that in the linear regime the contribution from the excluded volume is dominant compared to that from the connectivity along the strands. For larger elongations, however, the non-linear elastic response is dominated by the chemically and topologically shortest paths through the system, where chemical crosslinks and topological entanglements between meshes of the network play a similarly important role.     

Theory of nanoparticle diffusion in unentangled and entangled polymer melts

We propose a statistical dynamical theory for the violation of the hydrodynamic Stokes-Einstein (SE) diffusion law for a spherical nanoparticle in entangled and unentangled polymer melts based on a combination of mode coupling, Brownian motion, and polymer physics ideas. The non-hydrodynamic friction coefficient is related to microscopic equilibrium structure and the length-scale-dependent polymer melt collective density fluctuation relaxation time. When local packing correlations are neglected, analytic scaling laws (with numerical prefactors) in various regimes are derived for the non-hydrodynamic diffusivity as a function of particle size, polymer radius-of-gyration, tube diameter, degree of entanglement, melt density, and temperature. Entanglement effects are the origin of large SE violations (orders of magnitude mobility enhancement) which smoothly increase as the ratio of particle radius to tube diameter decreases. Various crossover conditions for the recovery of the SE law are derived, which are qualitatively distinct for unentangled and entangled melts. The dynamical influence of packing correlations due to both repulsive and interfacial attractive forces is investigated. A central finding is that melt packing fraction, temperature, and interfacial attraction strength all influence the SE violation in qualitatively different directions depending on whether the polymers are entangled or not. Entangled systems exhibit seemingly anomalous trends as a function of these variables as a consequence of the non-diffusive nature of collective density fluctuation relaxation and the different response of polymer-particle structural correlations to adsorption on the mesoscopic entanglement length scale. The theory is in surprisingly good agreement with recent melt experiments, and new parametric studies are suggested.     

Network architecture and modulus of miscible heteropolymer blends

The architecture and entanglement density of a multicomponent polymer network composed of miscible, flexible chains is related to the corresponding properties and concentrations of the pure components. It is assumed that the coupling frequency between different chain species is proportional to their fractional participation in the blend and that the entanglement ability of polymers can be affected by the presence of heteropolymer neighbors. The theory, equally applicable to temporary and permanent networks, estimates the molecular weight between entanglements and their total number along a chain in the blend. This information is used in establishing a mixing law for the rubbery plateau modulus of a fluid polymer blend and in deriving a relationship for the equilibrium modulus of an interpenetrating polymer network containing trapped entanglements and dangling segments. The theoretical predictions are compared with experimental results from the literature on several miscible polymer blends.     

Phase behavior and viscoelastic properties of entangled block copolymer gels

Triblock copolymers in midblock‐selective solvents can form physical gels. However, at low triblock contents (near the percolation threshold), the bridging of chains between micelles can lead to macrophase separation. Adding a styrene–isoprene diblock to a styrene–isoprene–styrene triblock copolymer in squalane can eliminate macrophase separation, yielding a wide range of stable, single‐phase gels with a disordered arrangement of micelles. The plateau modulus of these triblock gels scales with the 2.2 power of polymer content, indicating the importance of entanglements in dictating the modulus. Comparing gels made from the midblock‐saturated derivative of the same polymer [styrene‐(ethylene‐alt‐propylene)‐styrene] in squalane reveals that the modulus differences in the gels are a direct consequence of the difference in the entanglement molecular weight of the midblock homopolymer in bulk. Finally, the broad relaxation spectrum of these triblocks is well‐described by a recent theory for the dynamics of entangled star polymers, with the breadth of the relaxation spectrum dictated by the number of entanglements per midblock in the gel. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2183–2197, 2001     

Shear and extensional rheology of solutions of mixtures of poly(ethylene oxide) and anionic surfactants in ionic environments

Interactions between a high molecular weight poly(ethylene oxide) (PEO) and the anionic surfactant sodium dodecyl benzene sulfonate (SDBS) in aqueous solutions were investigated by shear and extensional rheometry. Results for mixtures between PEO and sodium dodecyl sulfate (SDS) are also presented for comparison purposes. Addition of anionic surfactants to PEO solutions above the critical aggregation concentration (CAC), at which micellar aggregates attach to the polymer chain, results in an increase in shear viscosity due to PEO coil expansion, and a strengthening of interchain interactions. In extensional flows, these interactions result in a decrease of the critical shear rate for the onset of the characteristic extension thickening of the PEO solutions that is due to transient entanglements of polymer molecules. The relaxation times associated with these transient entanglements are not directly proportional to the shear viscosity of the solutions, but rather vary more rapidly with surfactant concentration. In the presence of an electrolyte, coil contraction results in lower shear viscosities and a decrease in the extension thickening effects at surfactant concentrations just beyond the CAC. The relaxation times associated with transient entanglement reach a minimum at the same surfactant concentration as the shear viscosity, which indicates that coil contraction is responsible for the observed effects in both types of flow. However, the increase in extensional-flow entanglement relaxation times is much more abrupt than the decrease in shear viscosity. All these results point to a greater sensitivity of extensional flows on the molecular conformation of PEO/surfactant complexes.     

Dynamic mechanical properties of cross-linked rubbers. V. An approximate analysis of the modulus contributions of trapped and untrapped entanglements

Using the criterion that a coupling entanglement in a polymer network is trapped if all four strands radiating from it terminate in chemical cross-links (an approximation to a more rigorous treatment of Langley), an equation is derived relating the equilibrium modulus to the magnitude of the compliance in the frequency region where all coupling entanglements, whether trapped or not, contribute to the elasticity. The latter value is estimated from the storage compliance at the frequency where the storage compliance of the uncross-linked polymer corresponds to the entanglement compliance derived from integration over the loss compliance. The theory agrees rather well with data on vulcanizates of natural rubber. For 1,4-polybutadiene and styrene–butadiene rubbers, the agreement is somewhat less satisfactory, but the results support the hypothesis that the low-frequency losses observed in lightly cross-linked rubbers are due to relaxation of untrapped entanglements.     

The effect of chain architecture on “sharkskin” of metallocene polyethylenes

Dynamic viscoelastic and extrusion capillary results of metallocene based polyethylenes are analyzed. Three samples show very high viscosities at low frequencies and large relaxation times, which is a symptom of the presence of small amounts of long chain branching (LCB). A linear correlation is found between the sharkskin dynamics (periodicity) and a characteristic entanglement‐disentanglement time. It is found that this correlation does not hold for samples suspected of LCB.     

A highly coarse-grained model to simulate entangled polymer melts

We introduce a highly coarse-grained model to simulate the entangled polymer melts. In this model, a polymer chain is taken as a single coarse-grained particle, and the creation and annihilation of entanglements are regarded as stochastic events in proper time intervals according to certain rules and possibilities. We build the relationship between the probability of appearance of an entanglement between any pair of neighboring chains at a given time interval and the rate of variation of entanglements which describes the concurrence of birth and death of entanglements. The probability of disappearance of entanglements is tuned to keep the total entanglement number around the target value. This useful model can reflect many characteristics of entanglements and macroscopic properties of polymer melts. As an illustration, we apply this model to simulate the polyethylene melt of C(1000)H(2002) at 450 K and further validate this model by comparing to experimental data and other simulation results.     

Effect of chain entanglements on plastic deformation behavior of ultra‐high molecular weight polyethylene

Samples of ultra‐high molecular weight polyethylene, in which the chain topology within the amorphous component was altered using two‐stage processing, including crystallization at high pressure in the first step, were produced and their deformation behavior in the plane‐strain compression was studied. Deformation and recovery experiments demonstrated that the state of the molecular network governed by entanglement density is one of the primary parameters controlling the response of the material on the imposed strain, especially at moderate and high strains. Any change in the concentration of entanglements markedly influences the shape of the true stress–true strain curve. The strain hardening modulus decreases while the onset of strain hardening increases with a decrease of the entanglement density within the amorphous component. Density of entanglements also influences the amount of rubber‐like recoverable deformation and permanent plastic flow. In material of the reduced concentration of entanglements permanent flow appears easier and sets in earlier than in the material with a higher entanglement density, becoming a favorable deformation mechanism at moderate strains. As a result, strong strain hardening is postponed to higher strain when compared with the samples of equilibrium entanglement density. In the samples of an increased entanglement density the molecular network becomes stiffer, with a reduced ability of strain induced disentangling of chains. Consequently, there is a less permanent flow and strain hardening begins earlier than in the reference material of an unaltered chain topology. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 276–285, 2010