Mesoscopic/macroscopic models

Figure 25.4 gives an overview of the various levels of mesoscopic/macroscopic models that are typically employed in the literature. The modeling and simulation of any physical problem involves several steps. The first step is to make a set of approximations to constmct a mathematical representation of the physical problem that can be later solved using the computational resources available. The second step is to discretize the set of equations, as the problem typically cannot be solved in continuous space and time. The third step is to use the appropriate set of numerical methods (robust, accurate, etc.) to solve the discretized equations and simulate the physical problem. The fourth step is to visualize/analyze the results to... 

Raimondeau, S. Aghalayam, P. Vlachos, D.G. Katsoulakis, M. Bridging the gap of multiple scales from microscopic to mesoscopic to macroscopic models. In Foundation of molecular Modeling and Simulation AIChE S5miposium Series No. 325 2001 Vol. 97, 155-158. 

In the future, new spectroscopic techniques such as time-resolved X-ray scattering should provide a first direct look into the microscopic scale dynamics of cations in clays. From the modeling point of view, the understanding of multivalent ions and their description in continuous theories remains a challenge. Adopting a multiscale strategy, whereby each level of description is calibrated on a more fundamental one, from ab-initio and molecular simulations to macroscopic models via mesoscopic descriptions, will be essential to achieve this goal. 

The models for chemically reacting media discussed above described the evolution of the system on macroscopic scales. In some instances, especially when one considers applications of nonlinear chemical dynamics to biological systems or materials on nanoscales, a mesoscopic description will be more appropriate or even essential. In this section, we show how one can construct mesoscopic models for reaction-diffusion systems and how these more fundamental descriptions relate to the macroscopic models considered previously. 

Generally, the computational effort increases when going from the macroscale over the mesoscale up to the microscale. If real sensor or actuator apphcations are of interest, in most cases mesoscopic or macroscopic modeling is predestined. In order to understand the physicochemical mechanisms underlying the conversion, i.e., the mechanical deformation, the theoretical and experimental facts behind the swelling process have to be investigated. 

In Sect. 5, the statistical theory, a macroscopic model, is explained. In Sect 6 two different mesoscopic models are presented first, die chemo-electro-mechanical transport model (Sect. 6.1), then a continuum model based on porous media (Sect. 6.2) are described in detail. 

Usually, the mesoscopic, kinetic models are considered to be well suited for predicting dynamic properties of polymer solutions on macroscopic scales. Details of the fast solvent dynamics are in most cases irrelevant for macroscopic properties. Exceptions are polyelectrolytes, where the motion of counterions in the solvent can have a major influence on polymer conformation. Therefore, more microscopic models of polyelectrolytes with explicit counterions are sometimes employed [34] (see also the contribution by M. Muthukumar in this volume). Another exception is the dynamics of individual biopolymers, for example, protein folding, which is modeled with an all atomistic model including an explicit treatment of the (water) solvent molecules [35]. 

These theories are examples of mesoscopic or macroscopic models that lead to closed-form constitutive equations. Furthermore, they can all be described in the context of the single-generator bracket [175] or the GENERIC [167] formalisms of nonequilibrium thermodynamics. 

The mesoscopic and macroscopic models come with a number of parameters, for example, mean field potentials, friction coefficients, effective relaxation times, and so on, whose connection with molecular terms is not straightforward. It is the purpose of thermodynamically guided, systematic coarse-graining methods to address this... 

It is almost impossible to cover the entire range of models in Figure 25.1, and in this chapter we will limit ourselves to the different modeling approaches at the continuum level (micro-macroscopic and system-level simulations). In summary, there are computational models that are developed primarily for the lower-length scales (atomistic and mesoscopic) which do not scale to the system-level. The existing models at the macroscopic or system-level are primarily based on electrical circuit models or simple lD/pseudo-2D models [17-24]. The ID models are limited in their ability to capture spatial variations in permeability or conductivity or to handle the multidimensional structure of recent electrode and solid electrolyte materials. There have been some recent extensions to 2D [29-31], and this is still an active area of development As mentioned in a recent Materials Research Society (MRS) bulletin [6], errors arising from over-simplified macroscopic models are corrected for when the parameters in the model are fitted to real experimental data, and these models have to be improved if they are to be integrated with atomistic... 

Another important class of materials which can be successfiilly described by mesoscopic and contimiiim models are amphiphilic systems. Amphiphilic molecules consist of two distinct entities that like different enviromnents. Lipid molecules, for instance, comprise a polar head that likes an aqueous enviromnent and one or two hydrocarbon tails that are strongly hydrophobic. Since the two entities are chemically joined together they cannot separate into macroscopically large phases. If these amphiphiles are added to a binary mixture (say, water and oil) they greatly promote the dispersion of one component into the other. At low amphiphile... 

Multiparticle collision dynamics describes the interactions in a many-body system in terms of effective collisions that occur at discrete time intervals. Although the dynamics is a simplified representation of real dynamics, it conserves mass, momentum, and energy and preserves phase space volumes. Consequently, it retains many of the basic characteristics of classical Newtonian dynamics. The statistical mechanical basis of multiparticle collision dynamics is well established. Starting with the specification of the dynamics and the collision model, one may verify its dynamical properties, derive macroscopic laws, and, perhaps most importantly, obtain expressions for the transport coefficients. These features distinguish MPC dynamics from a number of other mesoscopic schemes. In order to describe solute motion in solution, MPC dynamics may be combined with molecular dynamics to construct hybrid schemes that can be used to explore a variety of phenomena. The fact that hydrodynamic interactions are properly accounted for in hybrid MPC-MD dynamics makes it a useful tool for the investigation of polymer and colloid dynamics. Since it is a particle-based scheme it incorporates fluctuations so that the reactive and nonreactive dynamics in small systems where such effects are important can be studied. 

Fig. 6 compares the nuclearity effect on the redox potentials [19,31,63] of hydrated Ag+ clusters E°(Ag /Ag )aq together with the effect on ionization potentials IPg (Ag ) of bare silver clusters in the gas phase [67,68]. The asymptotic value of the redox potential is reached at the nuclearity around n = 500 (diameter == 2 nm), which thus represents, for the system, the transition between the mesoscopic and the macroscopic phase of the bulk metal. The density of values available so far is not sufficient to prove the existence of odd-even oscillations as for IPg. However, it is obvious from this figure that the variation of E° and IPg do exhibit opposite trends vs. n, for the solution (Table 5) and the gas phase, respectively. The difference between ionization potentials of bare and solvated clusters decreases with increasing n as which corresponds fairly well to the solvation free energy of the cation deduced from the Born solvation model [45] (for the single atom, the difference of 5 eV represents the solvation energy of the silver cation) [31]. 

Note here that the relation between mesoscopic and microscopic approaches is not trivial. In fact, the former is closer to the macroscopic treatment (Section 2.1.1) which neglects the structural characteristics of a system. Passing from the micro- to meso- and, finally, to macroscopic level we loose also the initial statement of a stochastic model of the Markov process. Indeed, the disadvantages of deterministic equations used for rather simplified treatment of bimolecular kinetics (Section 2.1) lead to the macro- and mesoscopic models (Section 2.2) where the stochasticity is kept either by adding the stochastic external forces (Section 2.2.1) or by postulating the master equation itself for the relevant Markov process (Section 2.2.2). In the former case the fluctuation source is assumed to be external, whereas in the latter kinetics of bimolecular reaction and fluctuations are coupled and mutually related. Section 2.3.1.2 is aimed to consider the relation between these three levels as well as to discuss problem of how determinicity and stochasticity can coexist. 

Since around the mid-1990s, there has been a proliferation of hydrate time-dependent studies. These include macroscopic, mesoscopic, and molecular-level measurements and modeling efforts. A proliferation of kinetic measurements marks the maturing of hydrates as a field of research. Typically, research efforts begin with the consideration of time-independent thermodynamic equilibrium properties due to relative ease of measurement. As an area matures and phase equilibrium thermodynamics becomes better defined, research generally turns to time-dependent measurements such as kinetics and transport properties. 

Finally, a key highlight of this investigation is that the systematic estimation of the effective transport parameters for the porous CL and GDL from the mesoscopic modeling can quantitatively predict the fuel cell performance from the macroscopic fuel cell models. 

During the past few decades, various theoretical models have been developed to explain the physical properties and to find key parameters for the prediction of the system behaviors. Recent technological trends focus toward integration of subsystem models in various scales, which entails examining the nanophysical properties, subsystem size, and scale-specified numerical analysis methods on system level performance. Multi-scale modeling components including quantum mechanical (i.e., density functional theory (DFT) and ab initio simulation), atom-istic/molecular (i.e., Monte Carlo (MC) and molecular dynamics (MD)), mesoscopic (i.e., dissipative particle dynamics (DPD) and lattice Boltzmann method (LBM)), and macroscopic (i.e., LBM, computational... 

Yet another promising line of research lies in creating mesoscopic representations whose fundamental scale is somewhere within the 1010 range of distance scales and for which one defines closed (or fully consistent) equations of motion. At the macroscopic limit, hydrodynamic models are a very successful and standard example. More recent approaches include the Cahn-Hillard coarse-grained models and phase-field models. In some cases, one aims to ascertain the degree to which the systems exhibit self-similarity at various length scales hence the lack of a specific parameterization—which would be necessary using reduceddimensional models—is not of much importance. 

One can think that this situation, described by equations (3.1), can be visualised as a picture of interacting (and connected in chains) Brownian particles suspended in anisotropic viscoelastic segment liquid . Introduction of macroscopic concepts is unavoidable consequence of transition from microscopic to mesoscopic approach, or better to say, from the microscopic model of interacting Kuhn-Kramers chains to mesoscopic model of interacting chains of Brownian particles. 

---