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.

The theory of a vitreous state inevitably includes the nucleation phenomena [

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 [

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 [

The classical theory of nucleation [

The most evident way to get the necessary properties is to govern the law of creation of the ideal metastability

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:

At first the

To construct the description of the multistage process which is the nucleation with the presence of the active heterogeneous impurities at first the

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

Under the influence of heterogeneously formed embryos the

To get the necessary optical characteristics it is necessary to formulate the

The final section of the paper is Conclusions where the results are summarized and the limitations and restrictions of the theory are described.

The rate of nucleation

The general formula for the rate of nucleation is the following:

It is reasonable to extract in

In the traditional approach mentioned above it is supposed that the intensity of absorption of new molecules by the embryo

The last formula (

Really, formula (

Instead of the previous formula (

In the liquid-vapor nucleation it is reasonable to suppose that the explicit dependence of

If

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

Certainly

One has to stress that

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 that

One can prove that the moment

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

The rate of nucleation

One can see that according to (

To see how

Later the mentioned estimate leads to the estimate

Also one can get the estimate analogous to (

To see that the exponential approximation really works one can present the characteristic behavior of the Zeldovich factor

Behavior of the exponential and preexponential factors for the nucleation rate.

Really, the relative change of the preexponential factor here has the order

To calculate

One can prove that in calculation of

Fortunately

On the base of (

Then balance equation (

Having inserted the mentioned linearizations and exponential approximation (

One can prove that

The last equation (

The first approximation calculated on the base of (

One can prove that already the first approximation is rather good and the size spectrum

The rate of nucleation in the first (see (

The fist and the second iterations for the nucleation rate as a function of the time shift. Here the nucleation rate is drawn in special renormalized units.

One can see that the first iteration is rather close to the second one. One can see the similarity of the forms of the first and the second iteration. If it is necessary to increase the accuracy one can make a simple shift.

One can choose

All what has been done is necessary to show that

Now the further evolution will be considered. One can see there two periods. The first period is the rapid consumption of the surplus substance

Later the supersaturation is fallen and there will be an asymptotic period where

Here one has to make one important notation. In the liquid-vapor transition the volume

In the opposite liquid-vapor transition it seems that the situation will be opposite and the whole volume begins to be the vapor phase. But here the main effect of the bubble formation is to take away the surplus stretching. It is necessary to keep in mind.

In the situation of the liquid-crystal transition the volume

Later without

Now the process of the heterogeneous condensation will be studied. It is supposed that the centers are relatively active. This means that all of them are exhausted in the process of nucleation. This property simplifies the theoretical description of kinetics of the phase formation.

One can note that the attribution of the centers (sites) to the group of the active ones depends on the intensity of the change of the external conditions. So this property is not absolute.

It is supposed also that the embryos on the heterogeneous centers can be described thermodynamically. It means that the number of molecules

The behavior of the heterogeneous embryo free energy together with the behavior of the homogeneous free energy is drawn in Figure

Heterogeneous and homogeneous free energies.

Since one can prove that

For heterogeneous nucleation the nucleation rate is given by

Instead of the general theory (see [

To solve (

One can easily estimate

Then (

The detailed justification of the presented method is rather long but it exists at the level of precise analytical derivation.

Now the formation of the pseudo-homogeneous embryos will be studied. The difference from the consideration of the homogeneous nucleation presented above will be the change of conditions for the new process because there are the heterogeneous consumers of metastability.

Certainly, the new forming embryos consume metastability and this leads to the perturbation of the rate of growth of the supercritical embryos formed on heterogeneous centers. But one can prove that this influence is very small and can be neglected. So one can see two separate problems here. The first problem is to determine the supersaturation which appears after the heterogeneous embryos have been formed. The second problem is to determine the characteristics of the embryos size spectrum when the homogeneous process takes place.

The balance equation with account of only heterogeneous centers is written as

So then (

Unfortunately it is impossible to inverse the dependence

Solution

One has to note that here it is impossible to use an approximation

Now the behavior of

The first possibility is to see the nucleation in conditions of the growing

Alternative possibility is the following: the supersaturation

One can use the exponential approximation based on

The second possibility is not interesting also from the practical point of view because it requires essential number of heterogeneous centers (sites).

The only necessary fact which remains to be clarified is the existence of the time of the end of the rapid growth

How can one calculate

The time

It is also necessary to determine the mean size

Now the first possibility will be considered. Here consideration is similar to the case of the pure homogeneous nucleation but it is necessary to consider

The function

The balance equation is

The last equation (

The first approximation on the base of (

One can choose

Now the further evolution will be considered. Since

The final values will be

The value

The value

The known value of

Changing

The second possibility when the supersaturation is determined by the heterogeneously formed embryos is not too attractive because here the number of heterogeneous centers is greater than in the first variant and can be even greater than the number of homogeneously formed embryos. The only positive property is that in the majority of situations

In the first variant the number

The most serious limitation of the theory is the free molecular regime of growth. It leads to the collective consumption of metastability. But when the quantity of the heterogeneous centers is small the region of influence of every heterogeneous center is large enough and it is impossible to keep the free molecular regime here. Fortunately there is a certain analogy between the situation of the collective consumption and the situation with the zones of depletion. This analogy was established in [

Surface area of the embryo

Parameter in the equation for the evolution after the heterogeneous nucleation stops

Surplus chemical potential

Parameter in the equation for the evolution after the heterogeneous nucleation stops

Parameter in the model for the free energy of the embryo on heterogeneous center

Index to mark the critical embryo

Index to mark to equilibrium embryo

Free energy of the embryo measured in the thermal units

Size spectrum of the embryos

Number of the molecules in the crystalline phase calculated in units of

Rate of nucleation

Condensation coefficient

Basis for the approximation for

Coefficient in linearization of

Number of molecules in a new phase at the end of intensive growth

Mean size for the homogeneously formed embryos

Number of heterogeneously formed droplets calculated in approximation of the total number of the free heterogeneous centers

Mean concentration of the molecules in the noncrystalline phase

Number of sites (or the molecules) which can be the starting point for the nucleation (crystallization) formation of the embryo

Equilibrium distribution

Number of the free heterogeneous centers (unoccupied by the supercritical embryos)

Density of the molecules in the vicinity of the embryo

Total number of heterogeneous centers

Concentration of molecules in the noncrystalline phase at the state of the phase coexistence

Linear size of the embryo

Absolute temperature

Time

Time of the “freezing” of the system, which is a parameter in approximation for kinetic coefficient

Moment of intensive formation of the pseudo-homogeneous embryos

Time of appearance of embryos on the active heterogeneous centers

Moment of the pseudo-homogeneous formation in the presence of active heterogeneous centers

Time of the end of the rapid growth of the embryos sizes

Mean thermal velocity of the molecule

Kinetic factor (intensity of adsorption of the molecules by the embryo)

Zeldovich factor

Truncated Zeldovich factor

Coordinate of the spectrum of the heterogeneously formed embryos

Height of the activation barrier of nucleation

Width of the equilibrium zone

Parameter of decompositions for the free energy, which ordinary equals the total number of the molecules inside the homogeneous critical embryo or the difference between the number of molecules in the critical embryo and the number of molecules in the equilibrium embryo

Parameter

Chemical potential

Number of molecules in the embryo

Number of molecules in the critical embryo

Number of molecules in the equilibrium embryo

Activation barrier of accommodation

Imaginary metastability which would be in the system when the process of the new phase formation is forbidden

Basis for linearization of

Coefficient in linearization of

Supersaturation that would be in a system when the heterogeneous formation is allowed and the homogeneous mechanism is forbidden

Parameter of linearization of

A cubic root of

Renormalized surface tension

Characteristic time, parameter in the law of growth, which ordinary equals the mean time between collisions of some given molecule of the saturated disordered phase with other molecules in this phase

Power of metastability or the supersaturation

Basis for decompositions for

Index denoting the values at the period of nucleation.

The author declares that there is no conflict of interests regarding the publication of this paper.