Diffusion simulation studies

The theory of transport in microporous solids is complex and involves many aspects and steps. Although many aspects has been treated separately (e.g., sorption, diffusion, simulation studies, mechanisms, etc.) there are no coherent descriptions of permeation and separation in microporous membrcmes covering all the important aspects. In this chapter an attempt is made to introduce such a description. It is useful to give a qualitative picture first (Section 9.4.2.1). 

W. G. Tress, K. Leo and M. Riede, Optimum mobility, contact properties, and open-circuit voltage of OSCs A drift-diffusion simulation study, Phys. Rev. B, 2012, 85,155201. 

A crystallographic analysis of xenon binding to [NiFe] hydrogenase, together with a molecular dynamic simulation study of xenon and dihydrogen diffusion in the enzyme interior, suggests the existence of hydrophobic channels connecting the molecular surface with the active site 184). 

H. Dohle, R. Jung, N. Kimiaie, J. Mergel, and M. Muller. Interaction between the diffusion layer and the flow field of polymer electrolyte fuel cells—Experiments and simulation studies. Journal of Power Sources 124 (2003) 371-384. 

The complexity of modeling the adsorption of benzene in silicalite has already been discussed in the section concerned with diffusion. A TST study by Snurr et al. (106) led to the identification of 27 unique sorption minima in the asymmetric unit. Given this result, it is unsurprising that there have been relatively few simulation studies of this system. However,... 

This bimodal dynamics of hydration water is intriguing. A model based on dynamic equilibrium between quasi-bound and free water molecules on the surface of biomolecules (or self-assembly) predicts that the orientational relaxation at a macromolecular surface should indeed be biexponential, with a fast time component (few ps) nearly equal to that of the free water while the long time component is equal to the inverse of the rate of bound to free transition [4], In order to gain an in depth understanding of hydration dynamics, we have carried out detailed atomistic molecular dynamics (MD) simulation studies of water dynamics at the surface of an anionic micelle of cesium perfluorooctanoate (CsPFO), a cationic micelle of cetyl trimethy-lainmonium bromide (CTAB), and also at the surface of a small protein (enterotoxin) using classical, non-polarizable force fields. In particular we have studied the hydrogen bond lifetime dynamics, rotational and dielectric relaxation, translational diffusion and vibrational dynamics of the surface water molecules. In this article we discuss the water dynamics at the surface of CsPFO and of enterotoxin. 

In sharp contrast to the large number of experimental and computer simulation studies reported in literature, there have been relatively few analytical or model dependent studies on the dynamics of protein hydration layer. A simple phenomenological model, proposed earlier by Nandi and Bagchi [4] explains the observed slow relaxation in the hydration layer in terms of a dynamic equilibrium between the bound and the free states of water molecules within the layer. The slow time scale is the inverse of the rate of bound to free transition. In this model, the transition between the free and bound states occurs by rotation. Recently Mukherjee and Bagchi [14] have numerically solved the space dependent reaction-diffusion model to obtain the probability distribution and the time dependent mean-square displacement (MSD). The model predicts a transition from sub-diffusive to super-diffusive translational behaviour, before it attains a diffusive nature in the long time. However, a microscopic theory of hydration layer dynamics is yet to be fully developed. 

In the previous section we have discussed the relation between the time- and frequency-dependent friction and viscosity in the normal liquid regime. The study in this section is motivated by the recent experimental (see Refs. 80-87) and computer simulation studies [13,14, 88] of diffusion of a tagged particle in the supercooled liquid where the tagged particle has nearly the same size as the solvent molecules. These studies often find that although the fric-... 

Thus the existence of such inhomogeneity seems to be necessary to understand the decoupling of the self-diffusion from the viscosity for solutes having the same size as that of the solvent. The existence of such inhomogeneity has also been suggested by Taijus and Kivelson [89] from their computer simulation studies. 

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]. 

The most interesting result suggested by this theoretical investigation is the power law dependence of the solute diffusion on mass as has also been observed in computer simulation studies [99]. The power law dependence is clearly manifested in Fig. 7, where In (D1/D2) is plotted against In (M/m). Here D is the diffusion of the solvent and D2 is the diffusion of the solute. The slope of the line is found to be 0.099. This seems to suggest a weak mass... 

Although detailed microscopic calculations of the problem mentioned above are not available, there exist several computer simulation studies [102, 117], which also find the anomalous enhanced diffusion, even for simple model potentials such as the Lennard-Jones. The physical origin of the enhanced diffusion is not clear from the simulations. The enhancement can be as large as 50% over the hydrodynamic value. What is even more surprising is that the simulated diffusion constant becomes smaller than the hydro-dynamic prediction for very small solutes, with sizes less than one-fifteenth of the solvent molecules. These results have defied a microscopic explanation. 

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. 

Another surprising result obtained both in the computer simulation studies and in the theoretical analysis is the smaller value of the diffusion coefficient than that predicted by the SE relation for TZ > 15. The SE relation predicts that the friction is proportional to 1Z X whereas this present microscopic calculation shows a near constant value of the friction for 7Z > 15. This leads to a smaller value of the calculated diffusion coefficient than that predicted by the SE relation. 

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... 

In order to understand the above questions/paradoxes, a mode coupling theoretical (MCT) analysis of time-dependent diffusion for two-dimensional systems has been performed. The study is motivated by the success of the MCT in describing the diffusion in 3-D systems. The main concern in this study is to extend the MCT for 2-D systems and study the diffusion in a Lennard-Jones fluid. An attempt has also been made to answer the anomaly in the computer simulation studies. 

The numerical calculations are done both at low and high density at p = 0.6 and p = 0.7932 and at T = 0.7. The high-density calculations are performed to investigate if there is any existence of diffusion coefficient at high density as claimed by the simulation studies [174],... 

Figure 47. Diffusion coefficient D as obtained from a molecular dynamics simulation study of a binary Lennard-Jones system reaching temperatures below the crossover temperature of mode coupling theory (MCT). Solid line represents interpolation by MCT power law note the large temperature range covered by the power law. (From Ref. 371.)...

It has already been mentioned that zeolites are shape selective with respect to molecular adsorption. This property relates to their micropores stmcture. The zeolite framework shows a limited flexibility, which is essential. For instance, Yashonath et al. have shown in their classical dynamic simulations study of molecular diffusion within zeolite micropore that the zeolite framework flexibility affects significantly diffusion when the molecules have a size comparable with the micropore size. To get an idea of the order of magnitude of this flexibility, one can consider the hybrid semi-empirical DFT periodic study of chabazite zeolite of Ugliengo et al. V They introduced in the unit cell of chabazite Br0nsted acidic sites which are known to induce an increase of the volume of around 10 This increase of the volume relates with the difference of volume between a Si04 tetraheron and a... 

Paschek D, Geiger A. Simulation study on the diffusive motion in deeply supercooled water. J. Phys. Chem. B 1999 103 93. 

D. Huang, Y. Chen and K. A. Fichthom, A Molecular-Dynamics Simulation Study of the Adsorption and Diffusion Dynamics of Short n-alkanes on Pt (111),... 

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