Diffusion coefficient from simulations

Estimation of Diffusion Coefficients from Simulation of Polymer Microstructure... 

Table 2 Ratio of diffusion coefficients from simulations with (Dnuc) and without (Dbuik) a nucleation site with different potentials...

To obtain atomistic insight into orthoborate-based ionic liquids Wang et al. have first developed a force field for this new class of halogen-free chelated orthoborate-phosphonium ionic liquids. NMR data is used to partially calibrate the FF. The diffusion coefficients from simulations show dynamically three different regimes telling about the high complexity of these systems. 

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

Diffusion Coefficients from Computer Simulation and Experiment. .. 131... 

The diffusion coefficients calculated from a simulation employing a flexible framework were all between 5 and 10 times larger than those calculated from fixed lattice simulations. A comparison between flexible framework results and NMR measurements (57) illustrated the influence of the cations in the experimental sample calculated diffusion coefficients from the cation-free (flexible) framework were approximately 5 times higher than the experimental results. The increase in diffusion coefficient as a function of loading found in experimental studies was reproduced by the simulations. 

In the third simulation example, we carried out an analysis of some of the aspects that characterize the case of the mass transfer of species through a membrane which is composed of two layers (the separative and the support layers) with the same thickness but with different diffusion coefficients of each entity or species. To answer this new problem the early model has been modified as follows (i) the term corresponding to the source has been eliminated (u) different conditions for bottom and top surfaces have been used for example, at the bottom surface, the dimensionless concentration of species is considered to present a unitary value while it is zero at the top surface (iii) a new initial condition is used in accordance with this case of mass transport through a two-layer membrane (iv) the values of the four thermal diffusion coefficients from the original model are replaced by the mass diffusion coefficients of each entity for both membrane layers (v) the model is extended in order to respond correctly to the high value of the geometric parameter 1/L. 

Figure 3.2 shows the mean square displacement (MSD) for water in the glassy and supercooled glucose solutions in a log-log scale. The most remarkable result is that water diffuses at 220 K, in the glass, as observed in the experiments. The diffusion at 220 K, however, occurs in a scale comparable with the /rs of the simulation and cannot be quantified from the data in Figure 3.2. For the five supercooled solutions, T = 250 to 365 K, we computed the diffusion coefficient from the long time dependence of...

A stochastic dynamics simulation requires a value to be assigned to the collision frequency friction coefficient 7. For simple particles such as spheres this can be related to the diffusion constant in the fluid. For the simulation of a rigid molecule it may be possible to derive 7 via the diffusion coefficient from a standard molecular dynamics situation. In the more general case we require the friction coefficient of each atom. For simple molecules such as butane the friction coefficient can be considered to be the same for all atoms. The optimal value for 7 can be determined by trial and error, performing a stochastic dynamics simulation for different values of 7 and comparing the results with those from experiment (where available) or from standard molecular dynamics simulations. For large molecules the atomic friction coefficient is considered to depend upon the degree to which each atom is in contact with the solvent and is usually taken to be proportional to the accessible surface area of the atom (as defined in Section 1.5). 

Table 1. Diffusion coefficients from Molecular Dynamics simulations and from experiment, at 300 K unless indicated otherwise...

Figure 14.9 Estimation errors of diffusion coefficients from the two models (ej and 2) and the proposed model ( 3) as a function of a. Simulated data (Mp/Mp, versus time) are generated from equation (14.27) assuming Dq = 10 " cm /s and Lp = 10 pm. (Reproduced with permission from Chung and co-workers. Food Additives and Contaminants, 2002, 19, 6, 611 [22]. 2002, Taylor Francis)...

Molecular dynamics calculations are more time-consuming than Monte Carlo calculations. This is because energy derivatives must be computed and used to solve the equations of motion. Molecular dynamics simulations are capable of yielding all the same properties as are obtained from Monte Carlo calculations. The advantage of molecular dynamics is that it is capable of modeling time-dependent properties, which can not be computed with Monte Carlo simulations. This is how diffusion coefficients must be computed. It is also possible to use shearing boundaries in order to obtain a viscosity. Molec-... 

Typical dynamic properties like the scaling of relaxation times, e.g., ri, or diffusion coefficient with N are found in the simulation to change systematically from typical Rouse-like behavior Dj< ocN, t oc to... 

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