Nucleation space function

As we have just observed, in all the cases, two phenomena intervening during nucleation, that is, the creation of the precursors and their condensation, occur in zones that keep their dimensions constantly equal to each other, so we can define the reactivity (surface or volume according to whether nucleation occurs on the surface or the bulk) and the space function Consequently, for nucleation on the surface, if... 

In all the pseudo-steady state modes of nucleation, as the space function always has the same value for the whole of the steps, the theorem of the equahty of the speeds (see section 7.4.2) can be applied to the reactivities and thus... 

To find the form of the solution of a pure mode, we assume that the rate constants k and k are infinite except if p = i. As in nucleation, the various steps proceed all in the same zone or zones having same sizes (same space function), the pseudo-steady state conditions, and as the multiplying coefficients all are identical, we can thus have the equality of the reactivities... 

If y t) and S t) are the specific frequency of nucleation and its space function, respectively (free area or free volume following the case), nucleation always has the form ... 

This space function includes the space function of nucleation S (t). ... 

Moreover, nucleation is in pseudo-steady state mode then, with T and P constants, ydoes not depend any more on r and the space function becomes ... 

We assume that the reactivity of growth and the specific frequency of nucleation are independent of time (pseudo-steady state modes at constant tenperature and partial pressures). We will thus refer to relations [10.16] and [10.18], but in this case, a nucleus corresponds to a grain we can thus reveal in these expressions the space function of growth of a grain. 

We will not reconsider the modeling of the space function for growth already abimdantly covered in Chapter 10 but will examine some chemical models for growth and nucleation. 

For the reaction between massive solids, from the experimental knowledge of the kinetic law, it is theoretically possible to deduce the specific nucleation frequency and the growth reactivity, with the condition of having modeled the space function. 

Perrin [PER 02] has shown, starting from kinetic curves and the morphological modeling of the space function, that the preceding reduction proceeds according to a one-process model with instantaneous nucleation with inward development of the formed layer. The rate determining step occins at the external interface. These data are supplemented by the pseudo-steady state and separable rate tests, which are both satisfied. 

We carry out pressure switches for a fractional extent a = 0.25, under an initial pressure Pq toward various pressures P. Calculate the value of the space function with a = 0.25 at the Ho pressure. Deduce the value of the reactivity of growth and the specific frequency of nucleation at each pressure Hi. 

Since the space functions of all the steps of nucleation are identical, in pseudosteady state mode, the reactivity will be given according to the reactivity of any of the steps by ... 

Figure 18.25 shows that the only knowledge of the fractional extent is not enough to determine the rate. This latter one depends on the past of the sample. For the same pressure and the same temperature (thus same reactivity), the difference between the end of the two curves can only be devoted to the space function and to the fact that the fractional extent does not determine this functioa From this, we deduce that we will not be able to be satisfied with a one-process model of slow nucleation and fast growth or that of slow growth and fast nucleatioa It is necessary to consider the simultaneous run of the growth and nueleation and use a two-process model. 

In addition, we have a one-process model with instantaneous nucleation and slow growth, thus the space function is completely determined by the fractional extent, that is,... 

To apply the preceding relations, it is necessary to be able to clarify the nucleation-specific frequency and the space function of growth. 

In the first case, this is the model of nucleation, whieh we considered in section 8.5.1.2.1, We know whereas that if we work at constant temperature (which is the only run of creation of these precursors), we will be able to consider quasi-steady state modes (the space function is independent of time) for which nucleation frequency will not be time dependent (see sections 8.5.4.2 and 8.5.4.3). We will also be able to consider no quasi-steady state modes, leading to nucleation-specific frequency function of time, such as the models of section 8.5.4.4 and equation [8.55], which we again give in [A.9.9] ... 

To obtain the rate, it is enough to use equation [A.9.6] and to substitute into it the nucleation-specific frequency and the space function of selected growth. In the same way, we use [A.9.7] for kinetic law. We will consider three examples. 

The second part (Chapters 7 to 11) presents the modeling of the reactions of solids by the introduction of the general concepts with the installation of the mechanisms and their resolutions in a single process (Chapter 7), the study of the nucleation process of a new solid phase (Chapter 8), the growth of the nucleus (Chapter 9), and the superposition of the two processes of nucleation and growth (Chapter 10). This part finishes with Chapter 11 which makes it possible to connect the concepts introduced by modeling to the experimental data. This part is largely devoted to space function. 

The growth space function is often complex because it incorporates the process of nucleation. 

In section 14.6.1, we saw that the nucleation process involves two stages, each of them including elementary steps occurring on the whole surface of the sohd. The respective rate will therefore have the same space function. We will examine the type of elementary steps that could describe these two stages. 

The present work aims to derive fully microscopic expressions for the nucleation rate J and to apply the results to realistic estimates of nucleation rates in alloys. We suppose that the state with a critical embryo obeys the local stationarity conditions (9) dFjdci — p, but is unstable, i.e. corresponds to the saddle point cf of the function ft c, = F c, — lN in the ci-space. At small 8a = c — cf we have... 

An interesting class of exact self-similar solutions (H2) can be deduced for the case where the newly formed phase density is a function of temperature only. The method involves a transformation to Lagrangian coordinates, based upon the principle of conservation of mass within the new phase. A similarity variable akin to that employed by Zener (Z2) is then introduced which immobilizes the moving boundary in the transformed space. A particular case which has been studied in detail is that of a column of liquid, initially at the saturation temperature T , in contact with a flat, horizontal plate whose temperature is suddenly increased to a large value, Tw T . Suppose that the density of nucleation sites is so great that individual bubbles coalesce immediately upon formation into a continuous vapor film of uniform thickness, which increases with time. Eventually the liquid-vapor interface becomes severely distorted, in part due to Taylor instability but the vapor film growth, before such effects become important, can be treated as a one-dimensional problem. This problem is closely related to reactor safety problems associated with fast power transients. The assumptions made are ... 

Chemical behaviour depends on chemical potential and electromagnetic interaction. Both of these factors depend on the local curvature of space-time, commonly identified with the vacuum. Any chemical or phase transformation is caused by an interaction that changes the symmetry of the gauge field. It is convenient to describe such events in terms of a Lagrangian density which is invariant under gauge transformation and reveals the details of the interaction as a function of the symmetry. The chemically important examples of crystal nucleation and the generation of entropy by time flow will be discussed next. The important conclusion is that in all cases, the gauge field arises from a symmetry of space-time and the nature of chemical matter and interaction reduces to a function of space-time structure. 

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