Atomistic considerations

The Stranski - Kaischew theory ofthe mean separation works [1.21-1.23] provides the first consistent atomistic description of the elementary processes taking place on the crystal surface. Considering the nucleation 

The present Chapter comments upon some specific properties of the very small clusters - those consisting of several atoms [1.106,1.107]. The size n of such nuclei cannot be considered as a continuous variable and one cannot derive a simple analytical relation between ric and Afi using the classical approach described in Chapter 1.22.4. 

Certainly, it would be trivial to point out here the reasons for which small clusters have always attracted considerable attention. Their unusual behavior has proved significant for a number of physical phenomena such as catalysis, adsorption, photography, electrochenrical deposition of metals, semiconductors and alloys etc. Several authors already in the seventies obtained valuable information on the stmcture, the energetics and the thermodynamic properties of the microclusters [1.108-1.113]. This Chapter, however, is not going to discuss the basic achievements in this field. Here we shall provide information on the small clusters behavior obtained by means of illustrative model considerations. For the purpose we shall firstly calculate the nucleation work of 1- to 19-atomic clusters formed on a stmctureless foreign substrate [1.107] (see also [1.67] and [1.114]). 

For small clusters the size n is a discrete variable and we shall calculate the nucleation work using the general formula (1.32) rewritten in the form  

Since the values of the quantities A, E are not known we shall calculate the AG(n) vs. n relationship according to equation (1.145). This means to neglect the potential dependence of the free energy excess ( ), which is a frequently used approximation. Besides, we assume that y/ = 0.25 y/ and that the bulk new phase is a crystal with a face centered cubic lattice for which p n,u The inequality y/ V l is a necessary condition for clusters with a five-fold symmetry axis formed on a stmctureless foreign substrate [1.107]. For the sake of convenience the supersaturation Ap will be measured in ip units. 

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Atomistic theory of nucleation — The theory applies to very small clusters, the size n of which is a discrete variable and the process of nucleus formation must be described by means of atomistic considerations. Thus, the thermodynamic barrier AG ( ) that has to be overcome in order to form an n-atomic nucleus of the new phase is given by the general formula [i-v]... 

In practice, since we are only interested in an estimate of the effect, we resort to an approximate analysis in which the relevant chemical potentials and mass flux, and attendant strain rate are all evaluated heuristically. The argument begins with reference to fig. 11.6 with the claim that the vacancy formation energy for the faces subjected to tensile stresses differs from that on the faces subjected to compressive stresses. Again, a rigorous analysis of this effect would require a detailed calculation either of the elastic state of the crystal or an appeal to atomistic considerations. We circumvent such an analysis by asserting that the vacancy concentrations are given by... 

Thermodynamic properties of a system can also be obtained from the atomistic considerations. Molecular dynamics or Monte Carlo methods have been successfully used to smdy polymers. The success stems from the fact that many properties can be projected from dynamics of relatively simple, oligomeric models. Unfortunately, miscibility strongly depends on the molecular weight and so far it cannot be examined by these methods. 

Science is in incessant evolution it grows with more precise theories and better instrumentation. The thermodynamic theories of polymers and polymeric systems move toward atomistic considerations for isomeric species modeled mathematically by molecular dynamics or Monte Carlo methods. At the same time good mean-field theories remain valid and useful—they must be remembered not only for the historical evolution of human knowledge, but also for the very practical reason of applicability, usefulness, and as tools for the understanding of material behavior. 

Return to Macroscopic View The remarkable thing here is that this entire process has been developed using atomistic considerations, but the execution of it makes no use of them. Indeed, only macroscopic bodies are moved, brought into contact, separated, compressed, and expanded. Finally, ice cubes are counted. These are aU manipulations that can be carried out when nothing is known about atoms. In order to have a well-directed approach, it is enough to remember the concept mentioned in Sect. 3.2 that all things contain a movable, producible, but indestmctible... 

The infortnation provided in this chapter can be divided into four parts 1. introduction, 2. thermodynamic theories of polymer blends, 3. characteristic thermodynamic parameters for polymer blends, and 4. experimental methods. The introduction presents the basic principles of the classical equilibrium thermodynamics, describes behavior of the single-component materials, and then focuses on the two-component systems solutions and polymer blends. The main focus of the second part is on the theories (and experimental parameters related to them) for the thermodynamic behavior of polymer blends. Several theoretical approaches are presented, starting with the classical Flory-Huggins lattice theory and, those evolving from it, solubility parameter and analog calorimetry approaches. Also, equation of state (EoS) types of theories were summarized. Finally, descriptions based on the atomistic considerations, in particular the polymer reference interaction site model (PRISM), were briefly outlined. 

Another chapter deals with the physical mechanisms of deformation on a microscopic scale and the development of micromechanical theories to describe the continuum response of shocked materials. These methods have been an important part of the theoretical tools of shock compression for the past 25 years. Although it is extremely difficult to correlate atomistic behaviors to continuum response, considerable progress has been made in this area. The chapter on micromechanical deformation lays out the basic approaches of micromechanical theories and provides examples for several important problems. 

Finally, we want to describe two examples of those isolated polymer chains in a sea of solvent molecules. Polymer chains relax considerably faster in a low-molecular-weight solvent than in melts or glasses. Yet it is still almost impossible to study the conformational relaxation of a polymer chain in solvent using atomistic simulations. However, in many cases it is not the polymer dynamics that is of interest but the structure and dynamics of the solvent around the chain. Often, the first and maybe second solvation shells dominate the solvation. Two recent examples of aqueous and non-aqueous polymer solutions should illustrate this poly(ethylene oxide) (PEO) [31]... 

The interatomic interaction is described by an EAM potential specifically developed for NiAl in the B2 structure [12]. Compared to the older potential [16], which was used in most of the previous atomistic studies, our new potential gives considerably higher antiphase boundary (APB) energies = 0.82 J/m, yj pg = 1.06 J/m in good agreement with the APB... 

Many practically important polymers have a chemical structure that is considerably more complicated than PE, and this fact further complicates the simulation of macromolecular materials. As a consequence of all these arguments, it is clear that a simulation of fully atomistic models of a sufficiently large system over time scales for which thermal equilibration could be reached at practically relevant temperatures, is absolutely impossible thus a different approach must be taken ... 

Point defects are only notionally zero dimensional. It is apparent that the atoms around a point defect must relax (move) in response to the defect, and as such the defect occupies a volume of crystal. Atomistic simulations have shown that such volumes of disturbed matrix can be considerable. Moreover, these calculations show that the clustering of point defects is of equal importance. These defect clusters can be small, amounting to a few defects only, or extended over many atoms in non-stoichiometric materials (Section 4.4). 

As a final point, we note that typical surfaces are usually not crystalline but instead are covered by amorphous layers. These layers are much rougher at the atomic scale than the model crystalline surfaces that one would typically use for computational convenience or for fundamental research. The additional roughness at the microscopic level from disorder increases the friction between surfaces considerably, even when they are separated by a boundary lubricant.15 Flowever, no systematic studies have been performed to explore the effect of roughness on boundary-lubricated systems, and only a few attempts have been made to investigate dissipation mechanisms in the amorphous layers under sliding conditions from an atomistic point of view. 

In contrast, the chemists Lespieau admired were Wurtz, whose La theorie atomique (1879) "converted" him to atomistic ideas, and Grimaux, whose Theories et notations (1883) left Lespieau with the impression "that a sensible man could not help but adopt atoms." 19 Grimaux merited admiration for his modern approach to chemistry after 1898, he also merited considerable sympathy when he was fired from the Ecole Polytechnique after signing the petition supporting Alfred Dreyfus.20... 

Periodic surface profiles on vicinal surfaces have received considerable attention in the past, both from a continuum as well as an atomistic point of view [8-18], Here we describe briefly some recent work for surfaces of miscut a (about 3-10°) based on continuum mechanics specifically designed to take the anisotropy of y(0) into account [18], The approach is based on eq. (1) and the excess chemical potential given by [2]... 

As a continuum approximation, this approach should break down by the atomistic level. For islands it is presumably inappropriate for the small clusters imaged with FIM. More importantly, in many cases the stiffness may not be nearly anisotropic, as we have assumed it to be in our analysis. Then, as perhaps for Ag( 100) islands, new mechanisms may play a role. For vacancy clusters, there can be trapping in corners in systems that might seem to be cases of PD from consideration of vicinal surfaces. 

Atomistic simulation of an atactic polypropylene/graphite interface has shown that the local structure of the polymer in the vicinity of the surface is different in many ways from that of the corresponding bulk. Near the solid surface the density profile of the polymer displays a local maximum, the backbone bonds of the polymer chains develop considerable parallel orientation to the surface [52]. This parallel orientation due to adsorption can be one of the reasons for the transcrystallinity observed in the case of many anisotropic filler particles. 

Single crystals of /S-A1203 are essentially two dimensional conductors. The conducting plane has hexagonal symmetry (honeycomb lattice). This characteristic feature made -alumina a useful model substance for testing atomistic transport theory, for example with the aid of computer simulations. Low dimensionality and high symmetry reduce the computing time of the simulations considerably (e.g., for the calculation of correlation factors of solid solutions). 

The structure of crystalline surfaces is described briefly in Sections 9.1 and 12.2.1 and in Appendix B. All surfaces have a tendency to undergo a roughening transition at elevated temperatures and so become general. Even though a considerable effort has been made, many aspects of the atomistic details of surface diffusion are still unknown.6... 

A method using the Atomistic approach (Masuda, 1996) was published recently and claims an improved performance from consideration of solute SASA as well as proximity effects of substituent groups. Measured values for 500 solutes were taken as a test set, but just how substituent proximity was taken into account was not explained. For a set of 20 pharmaceuticals not in the original test set, five methods other than the SASA-scaled atomistic method were compared. Table 5.1 shows the statistical results for the best three methods. 

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