Self diffusion coefficient simulations

Self-diffusion coefficients are dynamic properties that can be easily obtained by molecular dynamics simulation. The properties are obtained from mean-square displacement by the Einstein equation ... 

Figure 5.3.5 displays dynamic NMR microscopy of xenon gas phase Poiseuille flow with an average velocity of 25 mm s-1 and self-diffusion coefficient of 4.5 mm2 s-1 at 130 kPa xenon gas pressure with numerical simulation (A) and experimental flow profiles (B-D) of xenon gas. 

Results in Table I illustrate some of the strengths and weaknesses of the ST2, MCY and CF models. All models, except the MCY model, accurately predict the internal energy, -U. Constant volume heat capacity, Cv, is accurately predicted by each model for which data is available. The ST2 and MCY models overpredict the dipole moment, u, while the CF model prediction is identical with the value for bulk water. The ratio PV/NkT at a liquid density of unity is tremendously in error for the MCY model, while both the ST2 and CF models predictions are reasonable. This large error using the MCY model suggests that it will not, in general, simulate thermodynamic properties of water accurately (29). Values of the self-diffusion coefficient, D, for each of the water models except the CF model agree fairly well with the value for bulk water. 

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

Figure 12. Water self-diffusion coefficient of Nafion 117 (EW =1100 g/equiv), as a function of the water volume fraction Xy and the water diffusion coefficient obtained from a Monte Carlo (MC) simulation (see text). The proton conductivity diffusion coefficient (mobility) is given for comparison. The corresponding data points are displayed in Figure 14.

Although there has not been much theoretical work other than a quantitative study by Hynes et al [58], there are some computer simulation studies of the mass dependence of diffusion which provide valuable insight to this problem (see Refs. 96-105). Alder et al. [96, 97] have studied the mass dependence of a solute diffusion at an infinite solute dilution in binary isotopic hard-sphere mixtures. The mass effect and its influence on the concentration dependence of the self-diffusion coefficient in a binary isotopic Lennard-Jones mixture up to solute-solvent mass ratio 5 was studied by Ebbsjo et al. [98]. Later on, Bearman and Jolly [99, 100] studied the mass dependence of diffusion in binary mixtures by varying the solute-solvent mass ratio from 1 to 16, and recently Kerl and Willeke [101] have reported a study for binary and ternary isotopic mixtures. Also, by varying the size of the tagged molecule the mass dependence of diffusion for a binary Lennard-Jones mixture has been studied by Ould-Kaddour and Barrat by performing MD simulations [102]. There have also been some experimental studies of mass diffusion [106-109]. 

Figure 8. The ratio of the self-diffusion coefficient of the solute (Di) to that of the solvent molecules (D ) plotted as a function of the solvent-solute size ratio (

The study is performed at reduced temperature T = 0.75 and reduced density p = 0.844-0.92. This is precisely the system studied in computer simulations [102]. The variation of the self-diffusion coefficient with the solute size is shown in Fig. 8, where the size of the solute molecule has been varied from 1 to 1/20 times that of the solvent molecule. In the same figure the computer-simulated values [102] are also plotted for comparison with the calculated results. The calculated results are in good agreement with the computer simulations. Both the theoretical results and the computer simulation studies show an enhanced diffusion for size ratios TZ TZ = 01/02) between 1.5 and 15. This is due to the sharp decoupling of the solute dynamics from the solvent density mode. 

Even conclusions drawn from the different computer simulation studies also do not seem to be consistent. The classic computer simulation study of Alder and Wainwright was performed at low density [173]. In this study they have shown that the long-time diffusion coefficient diverges in 2-D systems due to the existence of persistent hydrodynamic flows. On the other hand, some recent molecular dynamics simulations of 2-D systems have reported estimates of the self-diffusion coefficient [174]. In particular, it appeared from these simulations that a diffusion coefficient might, after all, exist at higher densities due to the absence of the persistent hydrodynamic... 

The dramatic increase of water density at a charged surface was observed by Toney et al. in their in situ X-ray scattering experiments, which has not yet been confirmed by simulation results.58,70 In another MD simulation work, Kiselev et al. found that selfdiffusion coefficient strongly decreases with increasing electric field.27 However, no difference between the self-diffusion coefficients for motion parallel and perpendicular to the external field was observed. 

MD simulation is advantageous for obtaining dynamic properties directly, since the MD technique provides not only particle positions but also particle velocities that enable us to utilize the response theory (e.g., the Kubo formula [175,176]) to calculate the transport coefficients from time-dependent correlation functions. For example, we will examine the self-diffusion process of a tagged PFPE molecular center of mass (Fig. 1.49) from the simulation to gain insight into the excitation of translational motion, specifically, spreading and replenishment. The squared displacement of the center mass of a molecule or a bead is used as a measure of translational movement. The self-diffusion coefficient D can be represented as a velocity autocorrelation function... 

SIMULATIONS OF LIQUID HMX 3.1. Viscosity and self-diffusion coefficient... 

The practical advantage of these relations is that, in MD simulations, single molecule properties like the self-diffusion coefficient and rotational relaxation times converge much faster than system properties due to additional averaging over the number of molecules in the ensemble. We applied eqs. 10 and 11 to our MD results using data at 800 K as a reference point in order to predict the viscosity over the entire temperature interval. In Fig. 7 we compare the predicted values with those obtained from simulation. It appears that in the temperature interval 600 K to 800 K predictions of Eq. (10) are more consistent with MD results than are the predictions of Eq. (11). This leads us to conclude that the viscosity temperature dependence in liquid HMX is more correlated... 

MD and MC simulations have provided data on layer spacings, thermodynamic properties, as well as interlayer water configurations, interlayer-species self-diffusion coefficients, and total radial distribution functions that are consistent with experimental data. Most of the clay surface is relatively... 

Self-Diffusion Coefficients of Ions and Solvent Water (Dj in a 2.2 molal Ltl Solution Obtained from MD Simulation and Experiments at 305... 

The classical MD simulations performed in task I provide self-diffusion coefficients for water and also for hydronium ions, which is strictly the vehicular component of the proton diffusivity. These diffusion coefficients are calculated from the mean square displacement of H2O and HsO using the Einstein relation. The numerical values for Nation and SSC membranes at the four hydration levels are hsted in Table 5 along with the experimental values. ... 

Properties of the membrane studied included the pressure on all six faces of the simulated volume, displacements of lipid molecules and atomic displacements, deuterium order parameter (S p) as a function of chain position, fluctuations of chain and head group dihedral angles, penetration of water, surface area per lipid molecule, and the lipid self-diffusion coefficient. Interesting re-... 

Figure 10. Mass dependence of self-diffusion coefficient of hard-sphere test particle of mass mg in hard-sphere fluid of particles of mass m for Rg = R at density pR, = 0.9 for Rodger-Sceats friction (-----) and Smoluchowski friction (--------). 0—simulation results of Herman and Alder.

To close this Section we comment on two papers that do not fit under any neat heading. The first of these is by Xiao et al,261 who study the final stages of the collapse of an unstable bubble or cavity using MD simulations of an equilibrated Lennard-Jones fluid from which a sphere of molecules has been removed. They find that the temperature inside this bubble can reach up to an equivalent of 6000 K for water. It is at these temperatures that sonolumines-cence is observed experimentally. The mechanism of bubble collapse is found to be oscillatory in time, in agreement with classical hydrodynamics predictions and experimental observation. The second paper, by Lue,262 studies the collision statistics of hard hypersphere fluids by MD in 3, 4 and 5 dimensions. Equations of state, self-diffusion coefficients, shear viscosities and thermal conductivities are determined as functions of density. Exact expressions for the mean-free path in terms of the average collision time and the compressibility factor in terms of collision rate are also derived. Work such as this, abstract as it may appear, may be valuable in the development of microscopic theories of fluid transport as well as provide insight into transport processes in general. 

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