Advances in Optical Technologies

Volume 2016, Article ID 7561952, 13 pages

http://dx.doi.org/10.1155/2016/7561952

## Theoretical Description of the Glass Preparation with the Necessary Optical Properties

Saint Petersburg State University, Universitetskaya Embankment 7/9, Saint Petersburg 199034, Russia

Received 27 July 2015; Revised 27 December 2015; Accepted 29 December 2015

Academic Editor: José Luís Santos

Copyright © 2016 Victor Kurasov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The method to get the glass media with necessary optical characteristics is proposed. This method is based on inserting a necessary number of heterogeneous centers of given activity into the system. The theoretical description of the nucleation process in such situation is given and the resulting formulas allow choosing the characteristics of heterogeneous centers necessary to get the required optical characteristics of the media.

#### 1. Introduction

The theory of a vitreous state inevitably includes the nucleation phenomena [1]. Every vistreous medium contains the embryos of a crystal phase which are growing extremely slowly. So one can say that they are practically stable objects. Such embryos are the centers of scattering and they affect the optical properties of glass. The start of systematic scattering theory was given by Lord Rayleigh [2]; the scattering on fluctuations was considered by Smoluchowsky and Einstein [3, 4]. Essential contribution in the consideration of the scattering with account of magnetic properties was made by Cabannes [5, 6]. A bright example of the theory of scattering on particles of essential sizes is given by the famous Mie scattering theory [7]. This approach was essentially generalized by Jobst [8] and Debye [9].

In every scattering theory the distribution of the embryos sizes is reflected in the optical coefficients of material. Ordinary it is necessary to know only several first momentums of the distribution of the embryos over sizes. Sometimes it is sufficient to know only the mean size of the embryos.

Impurities are inevitably presented in the media and they are like some heterogeneous centers. The nucleation theory on heterogeneous centers was investigated already by Frenkel in [10] and this paper completes the classical theory of nucleation created by Becker and Doering [11], Volmer [12], Kramers [13], and Zeldovich [14]. The full analysis of the stationary nucleation rate in the case of heterogeneous condensation was given by Kuni [15]. Even without the solution of kinetic equation one can evidently see that the main factor in the rate of nucleation is the Boltzmann distribution , where is the free energy of the critical embryo counted in the thermal units. This value is the central object of the theory and the main efforts were paid to find a correct expression for . Unfortunately it is rather difficult to do and even in the simplest case of the charged nuclei it is necessary to build a rather complex theory with decompositions [16–18]. These decompositions were constructed in frames of the general Gibbs dividing surfaces theory [19]. In the formalism of these decompositions there appear several coefficients with very specific physical meaning.

During the process of the glass preparation it is necessary to have the glass with the necessary optical properties. So it is necessary to have impurities in the necessary quantity and of the given size.

Around the embryos there appear specific profiles of density of the substance. The role of the density profiles in the light propagation has been studied in [20]. The analogous property in frames of the embryos in nucleation theory was investigated in [21].

The classical theory of nucleation [22, 23] is like an elegant necklace which has to be around the free energy of critical embryo. The problem here is not only in the necessity to know what the surface tension has to be put in these constructions [24] and to describe the surface layer [25], but also to know the influence of the heterogeneous center on these objects. Even in the case without the heterogeneous center the question of the free energy profile is one of the most actual questions [26]. Considering the possibility of the direct measurement of the free energy of embryos one has to state that except special situations (see [27]) there is no clear way to do it. The direct application of the statistical mechanical approach [28] also causes certain difficulties. The same can be said about the comparison with the 2D Ising model as it is proposed in [29]. The density functional approach [30–33] also can not give the absolutely precise result. But the accuracy of the last approach is rather satisfactory. One has to mention also the possibility of using the approximate scaling formulas [34–36] based on some characteristic basic points for decompositions as parameters. One can also insert here the generalizations associated with the dependence of the surface tension on the size [37, 38], the effects in constrained systems [39], and the deformation of heterogeneous centers [40]. Here the possibility of fluctuations [41] and the Tolman-like corrections [42] is not considered. The refinements of [43, 44] can be introduced here directly.

The most evident way to get the necessary properties is to govern the law of creation of the ideal metastability , that is, the imaginary metastability which would be in the system when the process of the new phase formation is forbidden. The value of the ideal metastability is fully governed externally and it seems that this function can be regulated to have the necessary properties. Unfortunately, the preparation of the glass requires the absolutely prescribed conditions to have the glass with optimal properties and it is impossible to change the conditions of the glass preparation. So as a function of time is supposed to be fixed.

But one can propose another rather simple and effective method. The main idea is to inject a given number of heterogeneous centers of the given activity to change the effective conditions of the nucleation of the main quantity of impurities which occurs pseudo-homogeneously (i.e., without exhaustion of the potential sites). The theoretical description of this method will be the matter of the current paper. This paper is organized as follows: (i)At first the* properties of the nucleation rate* will be recalled. The exponential approximation for the nucleation rate as a function of a supersaturation will be introduced. (ii)To construct the description of the multistage process which is the nucleation with the presence of the active heterogeneous impurities at first the* nucleation without heterogeneous impurities* will be studied. This process is more simple but the extraction of the characteristic features will help in construction of the more complex process. (iii)Having started the description of the nucleation process with the presence of the active heterogeneous impurities one can consider the separate process of exhaustion of these impurities or the* formation of the embryos on the active heterogeneous impurities*. This part of consideration will give the parameters of the influence of the embryos on heterogeneous centers on the process of formation of the pseudo-homogeneous embryos of a crystal phase. (iv)Under the influence of heterogeneously formed embryos the* formation of the pseudo-homogeneous embryos* will be studied. Namely, this part of consideration will give the necessary optical characteristics of the media. (v)To get the necessary optical characteristics it is necessary to formulate the* conditions on parameters of the active heterogeneous centers*. This is the matter of the next section. Here the ways to solve the system of algebraic equations of parameters are also discussed. (vi)The final section of the paper is Conclusions where the results are summarized and the limitations and restrictions of the theory are described.

#### 2. Properties of the Nucleation Rate

The rate of nucleation is proportional to exponent of the free energy of the critical embryo. This factor is the main factor and that is why the free energy of the critical embryo is the main object of interest in the investigations devoted to the nucleation.

The general formula for the rate of nucleation is the following:In (1) is the number of sites (or the molecules) which can be the starting point for the nucleation (crystallization) formation of the embryo and the factor is the Zeldovich factor. The value of is given by the numerous considerations but the approach appearing in the classical theory of nucleation is not yet radically reconsidered. Although there are some attempts to refine the Zeldovich factor these refinements do not lead to the change of this value in the order of the magnitude. Some interesting ideas for appear in the multidimensional consideration [45, 46] but this is not the subject of the current paper.

It is reasonable to extract in the kinetic factor (the number of adsorbed molecules in the time unit) and to writespeaking about the truncated Zeldovich factor because contrary to the liquid-vapor nucleation or the solid-liquid nucleation in the situation considered here (i.e., in the process of vitrification) the kinetic factor has specific behavior: at some moment of time it becomes so small that even the supercritical embryos practically stop growing. Otherwise it is impossible to explain why the crystal embryos do not gradually transform the whole volume of the glass into the set of crystals.

In the traditional approach mentioned above it is supposed that the intensity of absorption of new molecules by the embryo iswhere is the density of the molecules in the vicinity of the embryo, is the mean thermal velocity of the molecule, is the condensation coefficient, and is the surface area of the embryo.

The last formula (3) is supposed to be valid both for the critical and for the supercritical embryos. The evolution of undercritical embryos is not too interesting because in the region of undercritical embryos there is the quasi-equilibrium state.

Really, formula (3) has a rather general range of application. Then it follows that the peculiarities of are caused by the peculiarities of . This explains the factorization (2). The condensation coefficient contains the main dependence specific for the vitrification process. Ordinary it is presented asimplying that in the process of adsorption there is a necessity to overcome some barrier. The height of barrier is denoted by . This barrier can have a very specific origin including the barriers for displacement of the molecules in the environment of the embryo in order to give enough space for the appropriate installation of the new molecule in the embryo (this seems to be the main reason for annulation of ). So only some general remarks can be made concerning (4).

Instead of the previous formula (4) it is better to writeimplying that even when the thermodynamic conditions correspond to the absence of imaginary barrier there remain other factors (like thermal relaxation, noncorrespondence of the position of the molecule and the profile of the adsorbing region of the embryo, etc.) which affects the probability of adsorption. Note that barrier in (5) is an abstract effective barrier.

In the liquid-vapor nucleation it is reasonable to suppose that the explicit dependence of on the time is rather smooth. In the vitrification process has to go to zero when grows. It is rather easy to explain why it is so. It is supposed that ordinary there are no thermal effects of nucleation and the creation of is attained by the cooling of the system. Then the vapor consumption leads to the decrease of the number of the molecules but the temperature is not affected by embryos. The decrease of ordinary leads to increase of the activation barriers. If it is supposed that , where is the activation barrier of accommodation, then we need the increase of with decrease of temperature. Really, it is quite natural because the activation barrier is measured in thermal units. Then it can be seen that goes to zero.

If is linearized as a function of then it is possible to come towith the linearization coefficient . The linearization (6) invokes a question of the dependence of on time. Here it is also supposed that the linearization can be made and one can come towith parameters and . Parameter can be put to zero by the appropriate shift of the time scale.

Certainly, it is necessary to have some concrete dependence in order to present some concrete formulas. Analogous theory can be constructed for some other concrete dependence of on .

Certainly can not be greater than . The exponential approximation does not correspond to this property. A more simple and effective approximation is the following: for and for . Namely, this approximation will be used. Fortunately it is possible to determine , at least to estimate in order to ensure the correct final size of embryos.

One has to stress that appears both in expression for and in expression for the rate of growth. So after there will be no appearance of the supercritical droplets nor growth of the already existing droplets. The process is therefore stopped.

Since it is required that the system is metastable it means that it is supposed that without active heterogeneous centers there will be some embryos of the new phase. This means thatwhere is the moment of intensive formation of the pseudo-homogeneous embryos. Certainly the moment in (8) of characteristic time of appearance of embryos on the active heterogeneous centers has to be smaller than . Then

One can prove that the moment of the pseudo-homogeneous formation in the presence of active heterogeneous centers satisfies inequalityThenInequalities (8), (9), (10), and (11) presented above are rather important because they allow stating that the process of nucleation can be considered at the practically constant value of .

A special question which has to be analyzed here is whether the rate of nucleation is really the stationary one. Justifications of the stationarity are quite similar to the liquid-vapor case.

The mechanism of the embryos growth can be chosen as the free molecular one which corresponds to the case of crystallization. This leads to , where is the mean concentration of the molecules in the noncrystalline phase.

The rate of nucleation is the function of the power of metastability which is called the supersaturation. Ordinary is presented aswhere is the concentration of molecules in the noncrystalline phase at the state of the phase coexistence. Certainly thenwhere is the number of the molecules in the crystalline phase calculated in units of . Because of the growth of the crystals the value grows also. At the value stops to grow but the process of the new droplets formation stops also.

One can see that according to (12), (13) the supersaturation is not governed purely externally but depends on the process of nucleation and on the process of the substance consumption and the media heating by the regular growth of supercritical embryos. So there appears the self-consistent problem which is the subject of investigation in the theories of the global nucleation.

To see how depends on it is reasonable to notice that the main dependence occurs through the factor . The free energy is the smooth regular function of thermodynamic parameters and of and it is reasonable to linearize it on the deviation in . It leads towhere is some base for decomposition and is a parameter of linearization. For in (14) one can getwhere is the chemical potential and is the number of the molecules in the critical embryo. The function is rather smooth and takes moderate values. For of the mother phase system like the ideal gas (i.e., for every mother phase where the mean field approximation works) one can easily come toSo according to (15), (16) one can estimate as and it is possible to see that gets the big values ( to ensure the thermodynamic description of the critical embryo). It means that the dependence of on is rather sharp.

Later the mentioned estimate leads to the estimateduring the period of intensive nucleation. Here is the value of at some characteristic moment which belongs to the nucleation period.

Also one can get the estimate analogous to (17) for and on the base of (18) one can see that alsoThe last estimate shows that the nucleation period is relatively short in time. Namely, estimate (19) allows speaking above about , , and as some moments of time for nucleation.

To see that the exponential approximation really works one can present the characteristic behavior of the Zeldovich factor for the pseudo-homogeneous case. It is shown in Figure 1. One can see here two lines: the practically constant line which is the reduced Zeldovich factor and the exponent which is the main Boltzmann factor. So one can see that the preexponential factor (Zeldovich factor) does not really change. Here the worst value for the parameter is and the worst value for the renormalized surface tension is (the biggest possible value in the normal conditions). Namely, these values are taken for this example. Certainly, one can not include here the explicit time dependence of the kinetic coefficient of adsorption.