Microscopic theory nucleation

Microscopic theory of nucleation in metastable alloy states... 

Martensitic traasfonnation Master ec[uations Mean field crossover to Ising Mechanical properties Metallic alloys Metallic glasses Metastable alloys Microhardness test Microscopic theory of nucleation... 

It should also be emphasised that an initial period of interaction of elementary substances when there is still no compound layer and consequently there is only one common interface at which substances A and B react directly, is outside the scope of the proposed macroscopic consideration. The stage of nucleation of a chemical compound between initial phases is to be the subject of examination within the framework of a microscopic theory which would have to provide, amongst other parameters of the process, a minimal thickness sufficient to specify the interaction product formed at the A-B interface as a layer of the chemical compound ApBq possessing its typical physical and chemical properties. However, it can already now be said with confidence that this value is small in comparison with really measured thicknesses of compound layers and therefore can hardly have any noticeable effect on the shape of the layer thickness-time kinetic dependences observed in practice. 

As E is decreased one observes a change from the unimodal distribution for subcritical clusters to a bimodal form indicating growth of supercritical clusters. Because the system is adiabatic, the biomodal distributions also represent stationary states in which there are maximum supercritical cluster sizes, which, if exceeded, result in destruction of that supercritical cluster size new bonds formed in the system increase the cluster kinetic energy and decrease the pressure of the monomer gas. In the future it would be desirable to extract from the molecular-dynamics calculation accurate values for the free energy of formation of clusters. Such calculations would resolve the differences between the B - D theory and the Lothe-Pound theory. In the future, molecular-dynamics calculations should make possible development of correct mesoscopic and microscopic theories of homogeneous and even heterogeneous nucleation. 

The microscopic theory of the physically consistent cluster due to Reiss and co-workers [25, 63-68] addresses the rigorous calculation of the energetics of embryo formation from statistical mechanics. This approach is only applicable to nucleation in supercooled vapors. The key result of the theory is an expression for the free energy of embryo formation. 

In Chapter 4, different modifications of nucleation theory in a concentration gradient field are described. Using the thermodynamic approach, we have introduced the notion of a critical concentration gradient above which nucleation becomes thermodynamically prohibited. Different microscopic schemes (nucleation modes) have been applied to the description of the nucleation mechanism. 

It is our aim to examine the details of the elementary steps of the early stages of homogeneous nucleation. It appears that a microscopic theory of the process is appropriate since the cluster sizes involved are sufficiently small that macroscopic, continuum concepts are often of questionable validity. Furthermore, it appears that such a microscopic treatment is feasible since the steps involved are similar to well studied chemical processes (the main difference being in the bond strengths), and since the cluster sizes involved are sufficiently small that current numerical methods are applicable. The direct calculation of the rate parameters for the elementary nucleation steps would enable us to compute the steady state nucleation rate without making reference to questionable macroscopic assumptions. 

In the past half-decade, extensive studies have focused on aerosol nucleation in aircraft exhaust plumes [79]. This interest has brought attention to the formation of volatile aerosols that might eventually evolve into cloud condensation nuclei [80], Measurements of ultrafine particles reveal remarkably high abundances in jet wakes at very early times (within 1 second of emission) (e.g., [81]). As in the background atmosphere, the classical homogeneous nucleation theory has been applied to explain the number and size distribution of these volatile microscopic particles [82,83], However, while achieving some initial success, the theory has not been able to explain more recent, detailed observations. 

In order to verify which of the above nucleation mechanisms accurately represents hydrate nucleation, it is clear that experimental validation is required. This can then lead to such qualitative models being quantified. However, to date, there is very limited experimental verification of the above hypotheses (labile cluster or local structuring model, or some combination of both models), due to both their stochastic and microscopic nature, and the timescale resolution of most experimental techniques. Without experimental validation, these hypotheses should be considered as only conceptual aids. While the resolution of a nucleation theory is uncertain, the next step of hydrate growth has proved more tenable for experimental evidence, as discussed in Section 3.2. 

The conclusion is, therefore, that both spontaneous and forced rupture of foam bilayer by a-particles are mediated by microscopic holes of surfactant vacancies and can be described from a unified point of view with the aid of the nucleation theory of bilayer rupture [399,402,403]. However, studying the effect of a-particle irradiation of the bilayer lifetime is an independent way of proving the applicability of the hole mechanism of bilayer rupture. 

Rupture of emulsion bilayers. Experimental verification of the theory [399,402,403] of hole nucleation rupture of bilayer has also been conducted with emulsion bilayers [421]. A comparative investigation of the rupture of microscopic foam and emulsion bilayers obtained from solutions of the same Do(EO)22 nonionic surfactant has been carried out. The experiments were done with a measuring cell, variant B, Fig. 2.3, a large enough reservoir situated in the studied film proximity was necessary to ensure the establishment of the film/solution equilibrium. The emulsion bilayer was formed between two oil phases of nonane at electrolyte concentration higher than Cei,cr-... 

It is well known that water dispersions of amphiphile molecules may undergo different phase transitions when the temperature or composition are varied [e.g. 430,431]. These phase transitions have been studied systematically for some of the systems [e.g. 432,433]. Occurrence of phase transitions in monolayers of amphiphile molecules at the air/water interface [434] and in bilayer lipid membranes [435] has also been reported. The chainmelting phase transition [430,431,434,436] found both for water dispersions and insoluble monolayers of amphiphile molecules is of special interest for biology and medicine. It was shown that foam bilayers (NBF) consist of two mutually adsorbed densely packed monolayers of amphiphile molecules which are in contact with a gas phase. Balmbra et. al. [437J and Sidorova et. al. [438] were among the first to notice the structural correspondence between foam bilayers and lamellar mesomorphic phases. In this respect it is of interest to establsih the thermal transition in amphiphile bilayers. Exerowa et. al. [384] have been the first to report such transitions in foam bilayers from phospholipids and studied them in various aspects [386,387,439-442]. This was made possible by combining the microscopic foam film with the hole-nucleation theory of stability of bilayer of Kashchiev-Exerowa [300,402,403]. Thus, the most suitable dependence for phase transitions in bilayers were established. 

Let us summarise the conditions of formation of a microscopic foam film in order to serve the in vivo situation. These are film radius r from 100 to 400 pm capillary pressure pa = 0.3 - 2.5-102 Pa electrolyte (NaCl) concentration Ce 0.1 mol dm 3, ensuring formation of black films (see Section 3.4) and close to the physiological electrolyte concentration sufficient time for surfactant adsorption at both film surfaces. Under such conditions it is possible also to study the suitable dependences for foam films and to use parameters related to formation and stability of black foam films, including bilayer films (see Section 3.4.4). For example, the threshold concentration C, is a very important parameter to characterise stability and is based on the hole-nucleation theory of bilayer stability of Kashchiev-Exerowa. As discussed in Section 3.4.4, the main reason for the stability of amphiphile bilayers are the short-range interactions between the first neighbour molecules in lateral and normal direction with respect to the film plane. The binding energy Q of a lipid molecule in the foam bilayer has been estimated in Section 11.2. 

Classical nucleation theory assumes that the surface energy density, a, is independent of the size of the nucleus. This is probably not true when the nucleus is very small and consists of just a few molecules. Also, the theory assumes that the interface between the nucleus and the amorphous phase is sharp. On a microscopic scale, the interface is probably diftuse with a width that could be comparable with the nucleus size at high supercooling. 

This type of equation is also encountered in other areas, such as nonlinear waves, nucleation theory, and phase field models of phase transitions, where it is known as the damped nonlinear Klein-Gordon equation, see for example [165, 355, 366]. In the (singular) limit r 0, (2.15) goes to the reaction-diffusion equation (2.3). Front propagation in HRDEs has been studied analytically and numerically in [149, 150, 152, 151, 374]. The use of HRDEs in applications is problematic. Such equations are obtained indeed very much in an ad hoc manner for reacting and dispersing particle systems, and they can be derived neither from phenomenological thermodynamic equations nor from more microscopic equations, see below. 

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