Simulation of Transport Properties

Equilibrated structures from CGMD simulations could be used as input for simulations of transport properties in PEMs. The self-diffusion coefficient of water is obtained by taking the slope of the mean-square displacement (MSD) at long time 

At short timescales (t 500 ps) the MSD exhibits a quadratic time dependence. This is known as the ballistic regime, where particle collisions are infrequent. In nanoporous materials, an intermediate regime starts when particles are colliding but only with a subset of the other particles due to the confinement. When particles are able to escape the local environment and explore the full macroscopic network, the diffusive regime is reached. In the diffusive regime, the MSD becomes linear with time, exhibiting a slope of one on a log-log scale. 

For the calculation of self-diffusivity in hydrated Nafion, an atomistic structure of the water channel can be generated, based on the mesoscopic structure obtained from DPD or CGMD calculations. This remapped atomistic pore model can be optimized further with atomistic MD calculations. This procedure was employed in Malek et al. (2008) at different X. 

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In the next section we describe the basic models that have been used in simulations so far and summarize the Monte Carlo and molecular dynamics techniques that are used. Some principal results from the scaling analysis of EP are given in Sec. 3, and in Sec. 4 we focus on simulational results concerning various aspects of static properties the MWD of EP, the conformational properties of the chain molecules, and their behavior in constrained geometries. The fifth section concentrates on the specific properties of relaxation towards equilibrium in GM and LP as well as on the first numerical simulations of transport properties in such systems. The final section then concludes with summary and outlook on open problems. 

Computer simulations of transport properties using Green-Kubo relations... 

In this first task, each excess proton is permanently attached to a hydronium ion. This assumption prohibits stractural diffusion of the proton. However, for the purposes of the first task, namely the generation of molecular-level stmcture of the hydrated membrane and its interfaces, this approximation is adequate. For the second task, namely the generation of transport properties, this limitation is removed. Although, the classical MD simulations in task I cannot quantitatively characterize the stmctural diffusion mechanism, from the analysis of the hydration structure of the hydronium ions in these simulations the characteristics of Zundel and Eigen ion (which are necessary for structural diffusion) can be studied. 

The topic of this article is the study of transport properties of liquid crystal model systems by various molecular dynamics simulations techniques. It will be shown how GK relations and NEMD algorithms for isotropic liquids can be generalised to liquid crystals. It is intended as a complement to the texts on transport theory such as the monograph "Statistical Mechanics of Nonequilibrium liquids [8] by Evans and Morriss and "Recent Developments in Non-Newtonian Molecular Dynamics [9] by Sarman, Evans and Cummings and textbooks on liquid crystals such as "The physics of liquid crystals" [2] by de-Gennes and Frost and "Liquid Crystals" [3] by Chandrasehkar. 

We have presented EMD and NEMD simulation algorithms for the study of transport properties of liquid crystals. Their transport properties are richer than those of isotropic fluids. For example, in a uniaxially symmetric nematic liquid crystal the thermal conductivity has two independent components and the viscosity has seven. So far the different algorithms have been applied to various variants of the Gay-Beme fluid. This is a very simple model but the qualitative features resembles those of real liquid crystals and it is useful for the development of molecular dynamics algorithms for transport coefficients. These algorithms are completely general and can be applied to more realistic model systems. If the speed of electronic computers continues to increase at the present rate it will become possible to study such systems and to obtain agreement with experimental measurements in the near future. 

As discussed in the Introduction, considerable effort has been devoted to the study of transport properties. Specifically, viscosity estimates are of potential importance in the rational design of lubricants. In these simulations, two main approaches have been used. The earlier of the two focused on a linear response theory based Green-Kubo formula ... 

Alternatively, one could use SLLOD equations to do direct simulations, such as shear a system under planar Couette flow and measure the shear stress. As we have already discussed, this approach has been used successfully to calculate a host of transport properties. It is important to remember, however, that direct simulation is often unable to simulate realistic materials at experimentally accessible shear rates. At low shear rates, the nonequilibrium response becomes small compared to the magnitude of the equilibrium fluctuations that naturally arise. The extremely small signal-to-noise ratio would demand prohibitively long simulations before any meaningful answers could be obtained. 

The computation of the equilibrium properties of quantum systems is a challenging problem. The simulation of dynamical properties, such as transport coefficients, presents additional problems since the solution of the quantum equations of motion for many-body systems is even more difficult. This fact has prompted the development of approximate methods for dealing with such problems. 

Before comparing the glass transition observed in simulations with that observed in the laboratory, it is necessary to review briefly the temperature and density dependence of transport properties. In some of the model systems studied (specifically, hard and soft spheres) there is only one system variable, and temperature- or density-dependent representations of the properties are a matter of choice only. With other systems and of course with all laboratory systems, the two types of plot display independent aspects of the system s behavior. 

Properties of clays and clayey rocks, and also the processes in them depend on a number of factors. Then the mathematical simulation of the properties and processes, as one of the methods of their examination, is a rather difficult problem. Physically it is clear that the speciflc properties of clay rocks (low permeability, plasticity in moist condition) are caused by the existence of clay minerals in their composition, and these properties are a manifestation of surface capacities, which exist between particles of the clay minerals, which are included in the composition of clays. The most useful conception of the activity of surface capacities is the conception of disjoining pressure between colloid particles, Mitchell (1976). In this work we provide a description of the physical and mechanical clay properties and transport processes in them. The description is based on methods of theory of filtration consolidation. Nikolaevskiy (1996), and also on the theory of stability of lyophobic colloids (theory of Deijaguin-Landau-Verwey-Overbeeck, or DLVO theory), which uses the conception of disjoining pressure. 

Guevara-Carrion, G., Nieto-Draghi, C., Vrabec, J., Hasse, H. Prediction of transport properties by molecular simulation methanol and ethanol and their mixture. J. Phys. Chem. B 112, 16664-16674 (2008)... 

Computer simulations such as molecular dynamics (MD) and Brownian dynamics (BD) permit the study of transport properties. Self-diffusion coefficients can easily be obtained by differentiation of mean-square displacements or by integration of the velocity self-correlation functions of the ion. In contrast, the evaluation of conductivity by means of cross-correlation functions is cumbersome and computer-time-consuming and can only scarcely be executed. 

Kataoka Y (1987) Studies of liquid water by computer simulations. V. Equation of state of fluid water with Carravetta-Clementi potential. J Chem Phys 87 589-598 Kataoka Y (1989) Studies of hqttid water by computer simulations. VI. Transport properties of Carravetta-Clementi water. Bull Chem Soc Jpn 62 1421-1431... 

Measurements of transport properties would provide another means to explore the metastable regime. In particular, studies based on simulation focused on the supercooled regime [104] correlate the breakdown of the Stokes-Einstein relation Dv = constant) with the Widom line w P), locus of the correlation length maxima emanating down from the proposed liquid-liquid critical point toward lower pressures (Fig. 3b). Measurements of diffusivity could be performed at negative pressure by NMR on static samples (e.g., via MVLE or inclusions) and viscosity could be measured by capillary rheometry with the MVLE method. 

In this chapter we will review recent developments in modelling proton transport in different media. We will thereby narrow the topic to atomistic modelling of transport properties and processes only. The majority of studies in this area employ molecular dynamics (MD) to get insight into the mechanisms. For large systems classical force fields are used, small systems are often studied with ab-initio molecular dynamics, especially with Car-Parinello MD simulations. These methods are well known and documented, including their drawbacks, as e.g. finite-size effects in periodic simulations." Therefore, we will abandon explicit comments on the computational details, and refer the interested reader to the cited references or ordinary textbooks. 

The classical equations of motion are deterministic therefore once the initial coordinates and velocities are known, the coordinates and velocities can be determined later. The coordinates and velocities for a complete dynamics run are called the trajectory. Thus solving the classical equations of motion as a function of time (typically over a period limited to tens of nanoseconds) generates the microscopic states of the system. The systan may thus relax to equilibrium (provided the time for the relaxation falls within the time accessible to MD simulation), leading to the extraction of transport properties, which at the macroscopic scale describe the relaxation of the system in response to inhomogeneities. 

Although experimental transport properties are measured at different temperatures and pressures, it is the density, or molar volume, which is the theoretically important variable. So, for the prediction of transport properties, it is necessary to convert data at a given temperature and pressure to the corresponding temperature and density, or vice versa, by use of a reliable equation of state. Accordingly, an account is given in this volume of the most useful equations of state to express these relationships for gases and liquids. For dense fluids, it is possible to calculate transport properties directly by molecular simulation techniques under specified conditions when the molecular interactions can be adequately represented. A description is included in this book of these methods, which are significant also for the results which have aided the development of transport theory. 

The motivation for establishing accurate values of the various transport properties has been discussed in previous chapters. The current stams of fundamental kinetic and statistical mechanical theory, computer simulation and experimental technique and data acquisition impose severe limits on the accuracies achievable in any description of a transport property surface for a real fluid. For instance, it will not be possible to determine viscosities to one part in 10 for the near future. Thus, the accuracies associated with primary standards for transport properties fall an order of magnitude below those associated with primary measurement standards for equilibrium thermodynamic properties. Fortunately, the technological applications of transport property information do not require extreme accuracies. 

The transport properties of the supercritical fluids fall somewhat in between the gas and the liquid and also depend on how removed one is from the critical point. Dense gasses have the solubilizing power of liquids and the mobility of gasses as depicted in Table 20.1.3. There are quite a few empirical correlations and theoretical models, which are primarily extensions of corresponding low-pressure liquid and gas counter parts. Similarly, some of the classical experimental methods can be used for measurement of transport properties of supercritical fluids. A rather brief overview of the methods applicable for supercritical fluids will be presented since specialized reviews in the area give a good account of the state of the art. " " For engineering purposes, one can use applicable property estimation methods available in flowsheet simulators such as ASPEN PLUS, PROll, G-PROMS, and CHEMCAD. These methods are discussed in a text classical in the field." ... 

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