Continuum interphase

The modulus recovery experiments allowed measuring the terminal relaxation time of reptation motion of bulk and surface immobilized chains, supporting the hypothesis that theie is no interphase per se when nano-scale is considered. In order to bridge the gap between the continuum interphase on the microscale and the discrete molecular structure of the matrix consisting of freely reptating chains in the bulk and retarded reptatiug chains in contact with the inclusions, higher order elasticity combined with a suitable molecular dynamics model could be utilized [151-155]. 

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

With this approach, even the dispersed phase is treated as a continuum. All phases share the domain and may interpenetrate as they move within it. This approach is more suitable for modeling dispersed multiphase systems with a significant volume fraction of dispersed phase (> 10%). Such situations may occur in many types of reactor, for example, in fluidized bed reactors, bubble column reactors and multiphase stirred reactors. It is possible to represent coupling between different phases by developing suitable interphase transport models. It is, however, difficult to handle complex phenomena at particle level (such as change in size due to reactions/evaporation etc.) with the Eulerian-Eulerian approach. 

Both radial and axial diffusion can be taken into account and the final equations to be solved are relatively simpler than those of the continuum model. Although, the equations of the model at steady state are algebraic equations, the dimensionality of the system increases considerably. McGuire and Lapidus (1965) used this model for the study of the stability of a packed bed reactor which included both interphase and intraparticle mass and heat transfer resistances. 

SBS Interphase. Since 20°C is below the 0-temperature for the polystyrene-cyclohexane systems, it was expected that the PBD phase would be permeable to cyclohexane, but the PS domains would be relatively impermeable. (It is known that PS swells almost fourfold in liquid cyclohexane and that SBS may be dissolved even in cyclohexane. However, the maximum uptake of cyclohexane vapor by SBS was approximately 40% of its original weight. Furthermore, a sample of pure PS did not absorb any vapor within the time scale of these experiments. It was concluded then that the pure PS domain was not penetrated by cyclohexane vapor in these experiments and that, except for the interface, the PS domains may be considered an impermeable phase dispersed within a permeable continuum.) Thus the diffusion coefficient would be expected to reflect the structure of the PBD phase and to be characteristic of diffusion in elastomers (i.e., Fickian diffusion). 

To account for the apparent restricted mobility of the PBD region attention must be paid to the interphase between pure PS domains and the pure PBD continuum. The presence of a significant interfacial region of mixed composition separating the PS domains from the PBD continuum has been demonstrated by analysis of dynamic viscoelastic behavior (25,26), TgS (27,28), and small angle x-ray scattering data (29,... 

In the present conribution, we develop a continuum-based model to describe experimentally observable interphases in thin adhesive films. The model is based on an extended contiuum theory, i.e. the mechanical behaviour in these interphases is captured by an additional field equation. The introduced scalar order parameter models the microscopical mechanical properties of the film phenomenologically. 

Due to the complexity of the formation of interphases, a completely satisfying microscopic interpretation of these effects cannot be given today, especially since the process of the interphase formation is not yet understood in detail. Therefore, a micromechanical model cannot be devised for calculating the global effective properties of a thin polymer film including the above-mentioned size effects governed by the interphases. On the other hand, a classical continuum-based model is not able to include any kind of size effect. An alternative to the above-mentioned classical continuum or the microscopical model is the formulation of an extended continuum mechanical model which, on the one hand, makes it possible to capture the size effect but, on the other hand, does not need all the complex details of the underlying microstmcture of the polymer network. 

In the present study an extended continuum mechanical model is derived which is able to predict either weak or stiff boundary layers in thin films. As a possible application, the formation of interphases in polymer films is investigated. In this case it was shown [7, 24, 37] that the local stiffness in the polymer depends on the combination of polymer and substrate. 

Lastly, in each section of the continuum, the compositions of the permeate and reject streams or phases may tend to offset one another. In other words, there is a window of opportunity where interphase transfer can be bidirectional for one or the other of the components. 

Nanoscopic particles, dispersed in a block copolymer, have dimensions that are appropriate for Brownian dynamics simulations (268). Clay composites have a range of length scales, but if the gallery spacing between the layers is not large, MD methods can be used (269) with periodicity in the directions parallel to the clay platelets. However, continuum mechanical models need to be invoked for the description of exfoliated clay systems (270). These materials have so much interfacial area that adhesion properties are very important (271). Traditional continuum bounds methods (130) usually ignore the interphases on the grounds that they comprise a very small volume fraction of the total material, and so are not expected to be very accurate for exfoliated clay systems. 

CONTINUUM AND STATISTICAL MECHANICS-BASED MODELS FOR SOLID-ELECTROLYTE INTERPHASES IN LITHIUM-ION BATTERIES... 

Chapter 6. Continuum and statistical mechanics-based models for solid-electrolyte interphases in lithium-ion batteries... 

Figure 7.4. (a) Visualizing the interphase considering only the micro-scale. Interphase is a continuum layer with a gradient of properties reflecting variations in its structure. The main role of the micro-scale interphase is to provide stable and effective ineans for stress transfer between inclusions and polymer matrix even under adverse conditions, (b) Visualizing the structure of a micro-composite considering also the nano-scale structural features when the discrete structure of the matrix and inclusions becomes evident [169]... 

Many computer modeling and simulation methods have been developed to study polymer nanocomposites with different nanofiller geometries. Resulting information on molecular simulation is very useful to understand the level of interaction at the interphase between polymer matrix and nanofiller. Molecular simulations results have been incorporated by several authors into continuum mechanics-based models in order to predict the mechanical behavior of polymer nanocomposites. 

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