Diffusion First principles

Principles of Rigorous Absorber Design Danckwerts and Alper [Trans. Tn.st. Chem. Eng., 53, 34 (1975)] have shown that when adequate data are available for the Idnetic-reaciion-rate coefficients, the mass-transfer coefficients fcc and /c , the effective interfacial area per unit volume a, the physical solubility or Henry s-law constants, and the effective diffusivities of the various reactants, then the design of a packed tower can be calculated from first principles with considerable precision. 

In a packed column, operating at approximately atmospheric pressure and 295 K, a 10% ammonia-air mixture is scrubbed with water and the concentration of ammonia is reduced to 0.1%. If the whole of the resistance to mass transfer may be regarded as lying within a thin laminar film on the gas side of the gas-liquid interface, derive from first principles an expression for the rate of absorption at any position in the column. At some intermediate point where the ammonia concentration in the gas phase has been reduced to 5%. the partial pressure of ammonia in equilibrium with the aqueous solution is 660 N/nr and the transfer rate is ]0 3 kmol/m2s. What is the thickness of the hypothetical gas film if the diffusivity of ammonia in air is 0.24 cm2/s ... 

Box models and box-diffusion models have few degrees of freedom and they must describe physical, chemical, and biological processes very crudely. They are based on empirical relations rather than on first principles. Nevertheless, the simple models have been useful for obtaining some general features of the carbon cycle and retain some important roles in carbon cycle research (Craig and Holmen, 1995 Craig et al, 1997 Siegenthaler and joos, 1992). 

Computer simulations therefore have several inter-related objectives. In the long term one would hope that molecular level simulations of structure and bonding in liquid crystal systems would become sufficiently predictive so as to remove the need for costly and time-consuming synthesis of many compounds in order to optimise certain properties. In this way, predictive simulations would become a routine tool in the design of new materials. Predictive, in this sense, refers to calculations without reference to experimental results. Such calculations are said to be from first principles or ab initio. As a step toward this goal, simulations of properties at the molecular level can be used to parametrise interaction potentials for use in the study of phase behaviour and condensed phase properties such as elastic constants, viscosities, molecular diffusion and reorientational motion with maximum specificity to real systems. Another role of ab initio computer simulation lies in its interaction... 

It depends only on J sJkj A, which is a dimensionless group known as the Thiele modulus. The Thiele modulus can be measured experimentally by comparing actual rates to intrinsic rates. It can also be predicted from first principles given an estimate of the pore length =2 . Note that the pore radius does not enter the calculations (although the effective diffusivity will be affected by the pore radius when dpore is less than about 100 run). 

In addition to enhancing surface reactions, water can also facilitate surface transport processes. First-principles ab initio molecular dynamics simulations of the aqueous/ metal interface for Rh(l 11) [Vassilev et al., 2002] and PtRu(OOOl) alloy [Desai et al., 2003b] surfaces showed that the aqueous interface enhanced the apparent transport or diffusion of OH intermediates across the metal surface. Adsorbed OH and H2O molecules engage in fast proton transfer, such that OH appears to diffuse across the surface. The oxygen atoms, however, remained fixed at the same positions, and it is only the proton that transfers. Transport occurs via the symmetric reaction... 

As has been described in Ref. 70, this approach can reasonably account for membrane electroporation, reversible and irreversible. On the other hand, a theory of the processes leading to formation of the initial (hydrophobic) pores has not yet been developed. Existing approaches to the description of the probability of pore formation, in addition to the barrier parameters F, y, and some others (accounting, e.g., for the possible dependence of r on r), also involve parameters such as the diffusion constant in r-space, Dp, or the attempt rate density, Vq. These parameters are hard to establish from first principles. For instance, the rate of critical pore appearance, v, is described in Ref. 75 through an Arrhenius equation ... 

It only remains to specify the time constant, r0 [Eqs. (5.14) and (5.15)], related inversely to the attempt frequency with which the monomers attempt to cross barriers in the torsional potential (Fig. 1.2b). We have not attempted to calculate this time constant from first principles, but rather fixed it by comparison to experiment on chain self-diffusion at T = 450K [178]. This yields r0 1/50 picoseconds. This small number can be understood from the fact that because of the potentials, Eqs. (5.12) and (5.13), at T = 450 K only a few percent of the attempted hops of the effective monomers are successful the time constant for successful hops is of the order of 1 ps. These considerations... 

This chapter reviews all aspects of the 2D NMR of relaxation and diffusion. Firstly, numerous pulse sequences for the 2D NMR and the associated spin dynamics will be discussed. One of the key aspects is the FLI algorithm and its fundamental principle will be described. Applications of the technique will then be... 

The substance-specific kinetic constants, kx and k2, and partition coefficient Ksw (see Equations 3.1 and 3.2) can be determined in two ways. In theory, kinetic parameters characterizing the uptake of analytes can be estimated using semiempirical correlations employing mass transfer coefficients, physicochemical properties (mainly diffusivities and permeabilities in various media), and hydro-dynamic parameters.38 39 However, because of the complexity of the flow of water around passive sampling devices (usually nonstreamlined objects) during field exposures, it is difficult to estimate uptake parameters from first principles. In most cases, laboratory experiments are needed for the calibration of both equilibrium and kinetic samplers. 

In this chapter, I have provided a brief overview of the QMC method for electronic structure with emphasis on the more accurate diffusion Monte Carlo (DMC) variant of the method. The high accuracy of the approach for the computation of energies is emphasized, as well as the adaptability to large multiprocessor computers. Recent developments are presented that shed light on the capability of the method for the computation of systems larger than those accessible by other first principles quantum chemical methods. 

Since about 15 years, with the advent of more and more powerfull computers and appropriate softwares, it is possible to develop also atomistic models for the diffusion of small penetrants in polymeric matrices. In principle the development of this computational approach starts from very elementary physico-chemical data - called also first-principles - on the penetrant polymer system. The dimensions of the atoms, the interatomic distances and molecular chain angles, the potential fields acting on the atoms and molecules and other local parameters are used to generate a polymer structure, to insert the penetrant molecules in its free-volumes and then to simulate the motion of these penetrant molecules in the polymer matrix. Determining the size and rate of these motions makes it possible to calculate the diffusion coefficient and characterize the diffusional mechanism. 

In order to develop a consistent free-volume diffusion model, there are some issues which must be addressed, namely i) how the currently available free-volume for the diffusion process is defined, ii) how this free-volume is distributed among the polymer segments and the penetrant molecules and iii) how much energy is required for the redistribution of the free-volume. Any valid free-volume diffusion model addresses these issues both from the phenomenologic and quantitative points of view such that the diffusion process is described adequately down to the microscopic level. Vrentas and Duda stated that their free-volume model addresses these three issues in a more detailed form than previous diffusion models of the same type. Moreover, it was stated that the model allows the calculation of the absolute value of the diffusion coefficient and the activation energy of diffusion mainly from parameters which have physical significance, i.e. so-called first principles . In the framework of this model the derivation of the relation for the calculation of the self-diffusion coefficient of the sol-... 

From the point of view of earlier discussions, namely the true prediction of diffusion coefficients for volatile and nonvolatile organic penetrants in glassy polymers, the diffusion equations derived in the framework of the DST have only a limited usefulness. That means that, because the parameters of the DST models are not directly related to first principles , the equations can be used with success to correlate experimental results, but not to truly predict diffusion coefficients. 

Somewhat closer to the designation of a microscopic model are those diffusion theories which model the transport processes by stochastic rate equations. In the most simple of these models an unique transition rate of penetrant molecules between smaller cells of the same energy is determined as function of gross thermodynamic properties and molecular structure characteristics of the penetrant polymer system. Unfortunately, until now the diffusion models developed on this basis also require a number of adjustable parameters without precise physical meaning. Moreover, the problem of these later models is that in order to predict the absolute value of the diffusion coefficient at least a most probable average length of the elementary diffusion jump must be known. But in the framework of this type of microscopic model, it is not possible to determine this parameter from first principles . 

It was shown in the above section that as a rule, at the base of the classical or microscopic diffusion models, there are ad hoc (heuristic) assumptions on a certain molecular behaviour of the polymer penetrant system. The fact that the mathematical formulae developed on such bases often lead to excellent correlations and even semipredictions of diffusion coefficients must be aknowledged. It is true that the classical models are not capable to predict diffusion coefficients only from first principles but this is often not an obstacle to hinder their use in certain types of investigations. Therefore we are quiet sure that this type of diffusion models will certainly be used in the future too for the interpretation of diffusion experiments. 

---