Thermodynamically consistent incorporation of entanglement spatial fluctuations in the slip-link model.

We evaluate the thermodynamic consistency of the anisotropic mobile slip-link model for entangled flexible polymers. The level of description is that of a single chain, whose interactions with other chains are coarse grained to discrete entanglements. The dynamics of the model consist of the motion of entanglements through space and of the chain through the entanglements, as well as the creation and destruction of entanglements, which are implemented in a mean-field way. Entanglements are modeled as discrete slip links, whose spatial positions are confined by quadratic potentials. The confinement potentials move with the macroscopic velocity field, hence the entanglements fluctuate around purely affine motion. We allow for anisotropy of these fluctuations, described by a set of shape tensors. By casting the model in the form of the general equation for the nonequilibrium reversible-irreversible coupling from nonequilibrium thermodynamics, we show that (i) since the confinement potentials contribute to the chain free energy, they must also contribute to the stress tensor, (ii) these stress contributions are of two kinds: one related to the virtual springs connecting the slip links to the centers of the confinement potentials and the other related to the shape tensors, and (iii) these two kinds of stress contributions cancel each other if the confinement potentials become anisotropic in flow, according to a lower-convected evolution of the confinement strength or, equivalently, an upper-convected evolution of the shape tensors of the entanglement spatial fluctuations. In previous publications, we have shown that this cancellation is necessary for the model to obey the stress-optical rule and the Green-Kubo relation, and simultaneously to agree with plateau modulus predictions of multichain models and simulations.

Derivation of free energy expressions for tube models from coarse-grained slip-link models.

We present the free energy of a single-chain mean-field model for polymer melt dynamics, which uses a continuous (tube-like) approximation to the discrete entanglements with surrounding chains, but, in contrast to previous tube models, includes fluctuations in the number density of Kuhn steps along the primitive path and in the degree of entanglement. The free energy is obtained from that of the slip-link model with fluctuating entanglement positions [J. D. Schieber and K. Horio, J. Chem. Phys. 132, 074905 (2010)] by taking the continuous limit of (functions of) the discrete Kuhn-step numbers and end-to-end vectors of the strands between entanglements. This coarse-graining from a more-detailed level of description has the advantage that no ad hoc arguments need to be introduced. Moreover, the thermodynamic consistency of the slip-link model [J. D. Schieber, J. Non-Equilib. Thermodyn. 28, 179 (2003)] can be preserved. Fluctuations in the positions of entanglements lead to a harmonic bending term in the free energy of the continuous chain, similar to that derived by Read et al. [Macromolecules 41, 6843 (2008)] starting from a modified GLaMM model [R. S. Graham, A. E. Likhtman, T. C. B. McLeish, and S. T. Milner, J. Rheol. 47, 1171 (2003)]. If these fluctuations are set to zero, the free energy becomes purely Gaussian and corresponds to the continuous limit of the original slip-link model, with affinely moving entanglements [J. D. Schieber, J. Chem. Phys. 118, 5162 (2003)]. The free energy reduces to that of Read et al. under their assumptions of a homogeneous Kuhn-step number density and a constant degree of entanglement. Finally, we show how a transformation of the primitive-path coordinate can be applied to make the degree of entanglement an outcome of the model instead of a variable. In summary, this paper constitutes a first step towards a unified mathematical formulation of tube models. The next step will be to formulate the dynamics of the primitive-path conformation and the entanglement density along the primitive path. Now that the free energy is known, statistical mechanics can be employed for this purpose.

Laplace's law and the interfacial momentum source in two-phase models.

A two-phase flow model with liquid-solid transformation [M. Hütter, Phys. Rev. E 64, 011209 (2001)] is discussed, focusing on two elements: (1) the driving force for nucleation and growth and (2) the contribution of phase interfaces to the momentum balance. According to the model, nucleation and growth are partly driven by deviations from the equilibrium pressure difference between the phases, obtained as the surface tension times the ratio of the rates of change of two structural variables: the interfacial area per unit volume and the solid volume fraction. This is shown to be the proper extension of Laplace's law to nondilute conditions. Contrary to the classical result, the equilibrium pressure difference changes sign at a volume fraction around 50% because the amount of interfacial area lost due to impingement starts to outweigh the amount gained by growth. Hütter did not notice this and consequently misinterpreted a source term in his evolution equation for the momentum density. This term involves the surface tension times the interfacial area per unit volume, which is always nonnegative and hence not related to Laplace's law, as assumed in earlier two-phase models [M. Ishii, (Eyrolles, Paris, 1975); J. Ni and C. Beckermann, Metall. Trans. B 22, 349 (1991)]. An alternative derivation of the interfacial momentum source is presented here, which shows that Hütter's result correctly expresses the balance of forces on a representative volume element and should have been presented as a correction, rather than a corroboration, of the previous works mentioned.