Model microscopic

In this section we consider electromagnetic dispersion forces between macroscopic objects. There are two approaches to this problem in the first, microscopic model, one assumes pairwise additivity of the dispersion attraction between molecules from Eq. VI-15. This is best for surfaces that are near one another. The macroscopic approach considers the objects as continuous media having a dielectric response to electromagnetic radiation that can be measured through spectroscopic evaluation of the material. In this analysis, the retardation of the electromagnetic response from surfaces that are not in close proximity can be addressed. A more detailed derivation of these expressions is given in references such as the treatise by Russel et al. [3] here we limit ourselves to a brief physical description of the phenomenon. 

The relation between the microscopic friction acting on a molecule during its motion in a solvent enviromnent and macroscopic bulk solvent viscosity is a key problem affecting the rates of many reactions in condensed phase. The sequence of steps leading from friction to diflfiision coefficient to viscosity is based on the general validity of the Stokes-Einstein relation and the concept of describing friction by hydrodynamic as opposed to microscopic models involving local solvent structure. In the hydrodynamic limit the effect of solvent friction on, for example, rotational relaxation times of a solute molecule is [ ]... 

Dardi P S and Dahler J S 1990 Microscopic models for iodine photodissociation quantum yields in dense fluids J. Chem. Phys. 93 242-56... 

Molecular dynamics tracks tire temporal evolution of a microscopic model system tlirough numerical integration of tire equations of motion for tire degrees of freedom considered. The main asset of molecular dynamics is tliat it provides directly a wealtli of detailed infonnation on dynamical processes. 

Monte Carlo simulations generate a large number of confonnations of tire microscopic model under study that confonn to tire probability distribution dictated by macroscopic constrains imposed on tire systems. For example, a Monte Carlo simulation of a melt at a given temperature T produces an ensemble of confonnations in which confonnation with energy E. occurs witli a probability proportional to exp (- Ej / kT). An advantage of tire Monte Carlo metliod is tliat, by judicious choice of tire elementary moves, one can circumvent tire limitations of molecular dynamics techniques and effect rapid equilibration of multiple chain systems [65]. Flowever, Monte Carlo... 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

The above phenomenological equations are assumed to hold in our system as well (after appropriate averaging). Below we derive formulas for P[Aq B, t), which start from a microscopic model and therefore makes it possible to compare the same quantity with the above phenomenological equa tioii. We also note that the formulas below are, in principle, exact. Therefore tests of the existence of a rate constant and the validity of the above model can be made. We rewrite the state conditional probability with the help of a step function - Hb(X). Hb X) is zero when X is in A and is one when X is ill B. 

Molecular dynamics simulation, which provides the methodology for detailed microscopical modeling on the atomic scale, is a powerful and widely used tool in chemistry, physics, and materials science. This technique is a scheme for the study of the natural time evolution of the system that allows prediction of the static and dynamic properties of substances directly from the underlying interactions between the molecules. 

The low-temperature chemistry evolved from the macroscopic description of a variety of chemical conversions in the condensed phase to microscopic models, merging with the general trend of present-day rate theory to include quantum effects and to work out a consistent quantal description of chemical reactions. Even though for unbound reactant and product states, i.e., for a gas-phase situation, the use of scattering theory allows one to introduce a formally exact concept of the rate constant as expressed via the flux-flux or related correlation functions, the applicability of this formulation to bound potential energy surfaces still remains an open question. 

M. Kroger, R. Makhloufi. Wormlike micelles under shear flow A microscopic model studied by nonequihbrium molecular dynamics computer simulations. Phys Rev E 55 2531-2536, 1996. 

The second important attribute of amphiphiles is their affinity to both water and oil. This aspect is retained in the microscopic models, which will be discussed in Sec. Ill and IV. Oil, water, and surfactant molecules are represented by simplified pseudoparticles. 

Models of a second type (Sec. IV) restrict themselves to a few very basic ingredients, e.g., the repulsion between oil and water and the orientation of the amphiphiles. They are less versatile than chain models and have to be specified in view of the particular problem one has in mind. On the other hand, they allow an efficient study of structures on intermediate length and time scales, while still establishing a connection with microscopic properties of the materials. Hence, they bridge between the microscopic approaches and the more phenomenological treatments which will be described below. Various microscopic models of this type have been constructed and used to study phase transitions in the bulk of amphiphihc systems, internal phase transitions in monolayers and bilayers, interfacial properties, and dynamical aspects such as the kinetics of phase separation between water and oil in the presence of amphiphiles. 

We close these introductory remarks with a few comments on the methods which are actually used to study these models. They will for the most part be mentioned only very briefly. In the rest of this chapter, we shall focus mainly on computer simulations. Even those will not be explained in detail, for the simple reason that the models are too different and the simulation methods too many. Rather, we refer the reader to the available textbooks on simulation methods, e.g.. Ref. 32-35, and discuss only a few technical aspects here. In the case of atomistically realistic models, simulations are indeed the only possible way to approach these systems. Idealized microscopic models have usually been explored extensively by mean field methods. Even those can become quite involved for complex models, especially for chain models. One particularly popular and successful method to deal with chain molecules has been the self-consistent field theory. In a nutshell, it treats chains as random walks in a position-dependent chemical potential, which depends in turn on the conformational distributions of the chains in... 

In coarse-grained microscopic models, the amphiphiles, oil, and water molecules are still treated as individual particles, but their structure is very much... 

Whereas microscopic models for bulk systems incorporate the amphiphihc character and often the orientational properties of the surfactants as basic ingredients, models for bilayers and monolayers are constructed to reproduce internal transitions, such as the gel-fluid transition, and therefore concentrate on rather different aspects of the surfactant structure. 

M. Schick, W. H. Shih. Simple microscopic model of a microemulsion. Phys Rev Lett 59 1205-1208, 1987. 

A. Ciach, J. S. Hoye, G. Stell. Microscopic model for microemulsion. I. Ground state properties. J Chem Phys 90 1214-1221, 1989. 

A. Ciach, J. S. Hoye, G. Stell. Microscopic model for microemulsion. II. Behavior at low temperatures and critical point. J Chem Phys 90 1222-1228, 1989. A. Ciach. Phase diagram and structure of the bicontinuous phase in a three dimensional lattice model for oil-water-surfactant mixtures. J Chem Phys 95 1399-1408, 1992. 

M. Seesselberg, G. J. Schmitz, B. Nestler, I. Steinbach. Macroscopic and microscopic modeling of the growth of YBaCuO bulk material. IEEE Trans Appl Supercond 7 11>9, 1997. 

More detailed theoretical approaches which have merit are the configurational entropy model of Gibbs et al. [65, 66] and dynamic bond percolation (DBP) theory [67], a microscopic model specifically adapted by Ratner and co-workers to describe long-range ion transport in polymer electrolytes. 

Their theory can also be regarded as the begiiming of micro-macro thinking in the written history of science in a philosophical manner, macroscopic properties are projected, but not transferred on a pttre hypothetical microscopic model (Weillbach, 1971). 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

In most of the gas lubrication problems in nano-gaps, gas flow usually locates in the slip flow or the transient flow regime, depending on working conditions and local geometry. Therefore, both of the macroscopic and microscopic models are introduced to analyze the gas lubrication problems. 

In spite of the absence of periodicity, glasses exhibit, among other things, a specific volume, interatomic distances, coordination number, and local elastic modulus comparable to those of crystals. Therefore it has been considered natural to consider amorphous lattices as nearly periodic with the disorder treated as a perturbation, oftentimes in the form of defects, so such a study is not futile. This is indeed a sensible approach, as even the crystals themselves are rarely perfect, and many of their useful mechanical and other properties are determined by the existence and mobility of some sort of defects as well as by interaction between those defects. Nevertheless, a number of low-temperamre phenomena in glasses have persistently evaded a microscopic model-free description along those lines. A more radical revision of the concept of an elementary excitation on top of a unique ground state is necessary [3-5]. 

The structure of the interface between two immiscible electrolyte solutions (ITIES) has been the matter of considerable interest since the beginning of the last century [1], Typically, such a system consists of water (w) and an organic solvent (o) immiscible with it, each containing an electrolyte. Much information about the ITIES has been gained by application of techniques that involve measurements of the macroscopic properties, such as surface tension or differential capacity. The analysis of these properties in terms of various microscopic models has allowed us to draw some conclusions about the distribution and orientation of ions and neutral molecules at the ITIES. The purpose of the present chapter is to summarize the key results in this field. 

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