The Microscopic Approach

The total interaction between two slabs of infinite extent and depth can be obtained by a summation over all atom-atom interactions if pairwise additivity of forces can be assumed. While definitely not exact for a condensed phase, this conventional approach is quite useful for many purposes [1,3]. This summation, expressed as an integral, has been done by de Boer [8] using the simple dispersion formula, Eq. VI-15, and following the nomenclature in Eq. VI-19  

The two volume integrals change the dependence on x and introduce the number density of atoms n (number/cm ) such that the energy now varies as 

A more detailed description of the interaction accounts for the variation of the polarizability of the material with frequency. Then, the Hamaker constant across a vacuum becomes 

The integration in Eq. VI-21 may be carried out for various macroscopic shapes. An important situation in colloid science, two spheres of radius a yields 

If the cylinders are of different radii, then a = (0102). In these cases, U(x) has a simple inverse dependence on x, so that the attraction is, indeed, long-range. 

---

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. 

From the above discussion, we can see that the purpose of this paper is to present a microscopic model that can analyze the absorption spectra, describe internal conversion, photoinduced ET, and energy transfer in the ps and sub-ps range, and construct the fs time-resolved profiles or spectra, as well as other fs time-resolved experiments. We shall show that in the sub-ps range, the system is best described by the Hamiltonian with various electronic interactions, because when the timescale is ultrashort, all the rate constants lose their meaning. Needless to say, the microscopic approach presented in this paper can be used for other ultrafast phenomena of complicated systems. In particular, we will show how one can prepare a vibronic model based on the adiabatic approximation and show how the spectroscopic properties are mapped onto the resulting model Hamiltonian. We will also show how the resulting model Hamiltonian can be used, with time-resolved spectroscopic data, to obtain internal... 

The microscopic approach looks at heterogeneous properties of the tissue and has been developed for plant material on the basis of plant physiology studies on the effect of osmosis on water balance and transport in growing plants. 

B. Formulation of the Microscopic Approach a Simple Model for Relaxation... 

The basic remark is that linearity of the macroscopic law is not at all the same as linearity of the microscopic equations of motion. In most substances Ohm s law is valid up to a fairly strong field but if one visualizes the motion of an individual electron and the effect of an external field E on it, it becomes clear that microscopic linearity is restricted to only extremely small field strengths.23 Macroscopic linearity, therefore, is not due to microscopic linearity, but to a cancellation of nonlinear terms when averaging over all particles. It follows that the nonlinear terms proportional to E2, E3,... in the macroscopic equation do not correspond respectively to the terms proportional to E2, E3,... in the microscopic equations, but rather constitute a net effect after averaging all terms in the microscopic motion. This is exactly what the Master Equation approach purports to do. For this reason, I have more faith in the results obtained by means of the Master Equation than in the paradoxical result of the microscopic approach. 

We understand very well that any book inavoidably reflects authors interests and scientific taste this fact is, first of all, usually seen in the selection of material which in our case is very plentiful and diverse. For instance, Chapter 2 gives examples of different general approaches used in chemical kinetics (macroscopic, mesoscopic and microscopic) and numerous methods for solving particular problems. We focus here on the microscopic approach based on the concept of active particles (structure elements, reactants, defects) whose spatial redistribution arises due to their diffusion affected by... 

More information about these approaches and their advantages readers could find in [14, 15, 64, 65] in this Section 7.2 we focus on the further improvement of the microscopic approach to the defect aggregation via taking into account elastic attraction between point defects. 

Equations (3.86) and (3.88) give some examples of the nonretarded van der Waals forces for ideal contact geometries. For retarded interactions, the exponent for the distance of separation increases by 1 with the change of the corresponding numerical coefficients. The preceding theory, assuming complete additivity of forces between individual atoms, is known as the microscopic approach to the van der Waals forces. 

Michel Mareschal, The Microscopic Approach to Complexity in Non-Equilibrium Molecular Simulations. Proceedings of the Euroconference held in Lyon, France, 15-19 July 1996, in Physica A (Amsterdam), 240 (1-2), Elsevier, Amsterdam, The Netherlands, 1997. 

Macroscopic description of the NLO phenomena is very similar to the microscopic approach presented previously. The macroscopic quantities of interest are the susceptibilities of various orders defined by... 

The generalization of the microscopic approaches for description of real many-chain polymer systems such as networks, concentrated solutions and melts... 

This equation can be used in conjunction with (6.9) for the estimation of Fa. The microscopic approach points out clearly that the key parameters controlling drug absorption are three dimensionless numbers, namely, absorption number An, dissolution number Dn, and 9. The first two numbers are the determinants of membrane permeation and drug dissolution, respectively, while 9 reflects the ratio of the dose administered to the solubility of drug. 

The formulations of the population balance equation based on the continuum mechanical approach can be split into two categories, the macroscopic- and the microscopic population balance equation formulations. The macroscopic approach consists in describing the evolution in time and space of several groups or classes of the dispersed phase properties. The microscopic approach considers a continuum representation of a particle density function. 

H.C. Hamaker, in 1937, was the first to treat London dispersion interactions between macroscopic objects. He started with the most basic case, to determine the interaction of a single molecule with a planar solid surface. He considered a molecular pair potential and its relation with the molecules present within the solid surface, to derive the total interaction potential by summing the attractive interaction energies between all pairs of molecules, ignoring multibody perturbations. In this way, he built up the whole from the parts. Thus, Hamaker s method is called the microscopic approach. 

The microscopic approach has been particularly successful in the treatment of the Hall effect in electrolytes, summarized in an earlier overview [5]. As in the case of Hall conductivity, the magnitude of the magnetic field effect on diffusion is very small [6,7] but not negligible in a rigorous sense. The Llelmezs-Musbally formula [6] based on the theory of irreversible thermodynamics for bi-lonic systems ... 

Much ofthe phase behavior of binary and ternary amphiphilic systems has been reproduced by the models that we have described. Distinguishing which behavior is characteristic of generic multicomponent systems and which is particular to amphiphilic systems has been one of the successes of the microscopic approach, which is able to describe both kinds of systems. 

The aforementioned calculations have been performed with this formula. In addition, the values for and hsted in table 7 have been obtained in this way. Finally, the fits given in figs. 49 and 50 using the microscopic approach correspond to expressions such as eq. (135b) (Thalmeier 1988). 

In the original treatment, also called the microscopic approach, the Hamaker constant was calculated from the polarizabilities and number densities of the atoms in the two interacting bodies. Lifshitz presented an alternative, more rigorous approach where each body is treated as a continuum with certain dielectric properties. This approach automatically incorporates many-body effects, which are neglected in the microscopic approach. The Hamaker constants for a number of ceramic materials have been calculated from the Lifshitz theory using optical data of both the material and the media (Table 9.1) (9). Clearly, all ceramic materials are characterized by large unretarded Hamaker constants in air. When the materials interact across a liquid, their Hamaker constants are reduced, but still remain rather high, except for silica. 

This section opened with an example of the macroscopic theory which is based, of course, on the conservation laws. The "mesoscopic" description (a term due to VAN KAMPEN [2.93) permits knowledge not only of the average behavior of an aerosol but also of its stochastic behavior through so-called master equations. However, this mesoscopic level of description may require (in complex systems) some physical assumptions as to the transition probabilities between states describing the system. Finally, the microscopic approach attempts to develop the theory of an aerosol from "first principles"—that is, through study of the dynamics of molecular motion in a suitable phase space. Master equations and macroscopic theory appear from the microscopic theory by the reduction of the complete dynamical description of the system in a suitable phase space to small subsets of chosen variables. 

The aim of the microscopic approach has been to obtain exact theories of Brownian motion from "first principles" which permit generalization to the study of many diverse statistical phenomena. As an example, one can begin the microscopic theory by writing down the classical Hamiltonian H for a system containing Np Brownian particles and (monatomic) gas molecules ... 

The microscopic approach to nucleation problems has apparently not yet been carried out. There have been a number of mesoscopic developments for homogeneous nucleation [2.19,36-38]. The mesoscopic approach is successful in giving information on fluctuations, which are, of course, central to the process of nucleation. In this, the mesoscopic approach improves on the macroscopic approach. However, the transition probability is not known from "first principles" and, therefore, must retain some phenomenological elements. 

---