Three clearly isolated avenues differing by the initial raw material emerged since the beginning of the second half of the 20th century in the problem of the synthesis of sitalls. The proponents of the first avenue based on mixtures of pure oxides are developing technical sitalls of the pyroceram and photositall kind, for example, spodumenic, cordieritic, and so on. The second avenue leans toward petrositalls from basic rocks with pyroxene composition [1,2,3,4]. The third avenue, which is driven by an awareness of environmental problems, studies as the initial raw material the possibility of waste-free complex and environmentally clean use of large-tonnage industrial wastes [5,6,7] (Table 1).

Table 1. Chemical Composition of Initial Melts
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The information published in the scientific literature on the physical and chemical processes involved in obtaining cast stone are experimental and reflect particular cases of the development of a technology according to definite types of raw material for obtaining one or another type of product [8, 9].

Glassy crystalline materials are obtained by molding blanks from melt followed by heat-treatment (cast stone) or crystallization of glass for the formation of a fine-crystalline structure for sitalls.

Crystallization processes proceed so that a glassy phase (the matrix) always remains in the glassy crystalline material and crystals of one size or another are immersed in it. For this reason their properties, especially mechanical, will be determined by the form and amount of the phases, the boundaries of interaction between these phases in the surface layer, and the size and number of the precipitated crystals.

The strength properties of sitalls are determined by their fine-crystalline structure, which is determined by the high rate of nucleation and low growth rate of the crystals. A thermodynamic model of the crystallization of sitalls with a metasilicate composition is proposed in [10]. The mechanisms of nucleation are examined in the present work.

Models of Nucleation in Melts

More than 100 years ago G. Tamman proposed a theory of homogeneous crystallization of supercooled melts [11]. In this theory two types of molecules exist for each liquid — isotropic and anisotropic. The rate of transition of anisotropic molecules into isotropic molecules increases with increasing temperature, so that the crystallization properties of the silicate melt decrease with increasing heating time of the melt. The fallaciousness of such arguments is indicated in [12]. But the significance of G. Tamman’s work lies in the derivation of experimental dependences of the growth and nucleation rates on the degree of supercooling, which possess maxima at certain temperatures, for glassy systems.

The driving force of the crystallization process is associated with the difference of the free energies of the liquid and crystal. On cooling of melt below Tcr the driving force increases, but kinetic effects start to play a role in the form of an increase of viscosity and decrease of the diffusion rate [13]. A barrier to nucleation arises; this barrier is associated with the appearance of particles with excess surface energy. These particles arise as a result of density fluctuations, and the excess energy Φ is determined by the formula [14]

$$ \Phi =4\pi {r}^2\upsigma, $$
(1)

where r and σ are, respectively, the radius and surface energy of a spherical nucleus.

Only nuclei with size exceeding a certain critical radius r* (r > r* ) grow. For this reason, for high rates of cooling a metastable state of a supercooled liquid arises, which is proved experimentally for the example of obtaining glasses with different composition [15, 16]. For homogeneous crystallization the rate I of formation of centers is given by the relation [14]

$$ I=p\exp \left(-\frac{E_d+{\Phi}^{\ast }}{RT}\right), $$
(2)

where Ed is the activation energy of diffusion or other transport processes; Φ* is the difference of the free energies of the glass and crystal phases; and, p is a constant. In addition the authors of [9, 10] note that there exists a stationary rate of nucleation of crystals and a certain induction period of non-stationary nucleation. Relations for these processes have been derived and are presented in [11]. The rate Ist of stationary nucleation is equal to

$$ {I}_{\mathrm{st}}=2N{\left(\frac{\sigma {l}^2}{kT}\right)}^{1/2}\frac{1}{\uptau_0}\exp \left(-\frac{\Phi_{\uptau}+{\Phi}^{\ast }}{kT}\right), $$
(3)

where N is the number of structural units of the crystallized melt per unit volume; σ is the surface energy at the glass-crystal boundary; l is the linear size of a structural unit transitioning from glass into a nucleus; τ0 is a period equal to the vibrational time of the atoms during the switching of a structural unit; Φτis the free energy of activation of a transition of structural units; Φ* is the difference of the free energies of the glass and crystal phases:

$$ {\Phi}^{\ast }=\frac{16{\uppi \upsigma}^3}{3\Delta \upvarphi}; $$
(4)

Δφ is the difference of the free energies of the glass and crystal per unit volume of a nucleus.

The authors of [17 – 19] do not discuss what such a structural group is. They believe that the rate of the induction period is also determined by the thermodynamic parameters of the melt, first and foremost, the quantity Φ*.

If the melt is cooled sufficiently slowly, then such an approximation correctly reflects the quasi-static process. But when the melt is poured into a mold and is clearly in a non-equilibrium state, the notion of cybotaxic groupings, which store a portion of the energy in internal degrees of freedom or boundaries between phases, must be introduced. The derivatograms of glass with pyroxene composition, which exhibit energy absorption peaks in the nucleation range, attest this.

Endothermal peaks at 680 and 710°C can be seen in Fig. 1. And an additional amount of heat is lost to ordering of ‘frozen’ cybotaxic groups so that a certain induction period is required for such a transition in order for a structural rearrangement into glass to occur. The area of the endothermal effect (710°C) corresponds to the absorption of heat on nucleation, associated with the start of melting and structural rearrangement of the initial glass. Therefore the stationary weight of nucleation is an effective quantity dependent on the manifestation of cybotaxic groupings in the form of nuclei.

Fig. 1.
figure 1

DTA curves of glass with the compositions 1 and 2 (see Table 1): ΔT ) temperature difference between an inert standard and the sample; ΔT is proportional to the endothermal effect of nucleation a(Q – Qst), where Q and Qst are, respectively, the heat absorbed by the sample and the standard; 1, 2 ) glasses with metasilicate composition with high and low rate of crystallization, respectively.

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This show in the non-equilibrium conditions occurs spontaneously, and the probability dWs of a transition of cybotaxis into a structural grouping can be expressed as [20]

$$ {dW}_{\mathrm{s}}= AN\hbox{'} dt, $$
(5)

where N' is the number of cybotaxic groups per energy level; dt is the transition time; a is a coefficient.

But not every grouping can form a nucleus. There exists a certain probability that on reaching a certain size l2 a cybotaxis will transition into a nucleus, (Eq. (3)). And this occurs on cooling of the melt and enlargement of cybotaxic groups in a polymerization process.

The number of groups will depend on the temperature of the melt at which they were formed. And the lower the temperature Tm , the more groups are formed, so that an inversely proportional relation appears, and in the general case the coefficient A can be represented as a ratio

$$ A\approx {T}_g/{T}_m, $$
(6)

where Tg is the vitrification temperature and Tm is the production temperature of the melt.

The temperature Tm depends on the viscous properties of the melt, which are presented in Table 2. Tm is equal to 1250 – 1300°C in the melts 1 – 3 (see Table 2) and 1100 – 1200°C in the melts 5 – 7 (see Table 2). The vitrification temperature is related with the viscosity because of the polymerization of the melt and is not a function of state, but rather it depends on the cooling rate of the melt [21].

Table 2. Technological Properties of Melts and the Phase Composition of Glassy Crystalline Materials
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At Tm = Tg the melt loses its elastic properties, and a cybotaxic grouping becomes a structural grouping. During cooling of the melt a certain latent period is needed for this process to occur, so that from the formulas (5) and (6) the rate of spontaneous transitions under non-equilibrium conditions can be written in the form

$$ {J}_{\mathrm{sp}}={A}^{\prime }{\left({T}_g/{T}_m\right)}^{\updelta}, $$
(7)

where δ is a coefficient characterizing the degree of polymerization as the melt cools.

The total nucleation rate IΣ taking into account the difference Δφ of the free energies of the glass [6]

$$ \Delta \upvarphi =\uplambda \left({T}_{\mathrm{melt}}-{T}_{\mathrm{f}}\right)/{T}_{\mathrm{melt}}, $$
(8)

can be expressed as a product of the stationery nucleation rate Ist (3) and the spontaneous transformation rate Isp (7):

$$ {I}_{\Sigma}=\frac{M}{T_{\mathrm{f}}^{1/2}{T}_{\mathrm{m}}^{\updelta}}\exp \left(-\frac{F}{T_{\mathrm{f}}}-\frac{\lambda {T}_{\mathrm{m}\mathrm{elt}}^2}{\Delta {TT}_{\mathrm{f}}}\right), $$
(9)

where M, E, λ, and δ are experimentally determined coefficients.The melt temperature Tm, the formation temperature Tf, supercooling temperature difference ΔT, and the melting temperature Tmelt are technological parameters, forming the structure of the melt as it cools.

EXPERIMENTAL STUDY OF THE NUCLEATION PROCESS FOR METASILICATE MELTS

The objects of the investigation were metasilicate compositions based on native raw material from deposits in Siberia (see Table 1). The crystal-chemical formulas of pyroxenes and technological parameters are presented in Table 2.

The melting temperature of the batch was determined from the phase diagram of the system SiO2–Al2O3–CaO– Na2O and chosen by means of the spreading of a drop on an aluminum substrate. The temperature of homogenization of the melt was in the range 1340 – 1370°C. The batch was melted in aluminum crucibles in Silit furnaces with heating rates 2.5 K/min and melt dwell time for homogenization 1 – 1.5 h.

The temperature Tf of the molten glass, °C, was chosen from the DTA data according to the endothermal peak. After casting, the blank was held in a muffle furnace in order to form centers of crystallization. The number of nuclei was determined with the aid of a MBS-2 microscope after the surface was polished with diamond pastes. The number of grains was determined per 1 mm2 of the surface. Next, knowing the dwell time, the volumetric nucleation rate I was calculated; the values are presented in Fig. 2.

Fig. 2.
figure 2

Nucleation rate IΣ of metasilicate melt versus the temperature of the molten glass, °C: ×) experimental; Δ) values calculated from the relation (9).

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Several parameters were changed simultaneously in the nucleation experiment: molten glass production temperature Tm , stabilization temperature Tf, sample dwell time tmold in the mold, and the number of nuclei N in 1 mm3. The experimental data are presented in Table 3.

Table 3. Heat-treatment Parameters and Number of Nuclei of the Crystalline Phase of the Metasilicate Composition 2
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CALCULATION OF THE MODEL PARAMETERS

To calculate the coefficients of the function IΣ (9) the number of formed nuclei was divided by the dwell time of the casting in the mold. Next, taking its logarithm we obtain

$$ 1\mathrm{n}\kern0.5em {I}_{\Sigma i}=1\mathrm{n}\kern0.5em {x}_0+\frac{1}{2}1\mathrm{n}\kern0.5em {T}_{\mathrm{f}i}--{x}_21\mathrm{n}\kern0.5em \cdots {T}_{\mathrm{m}i}={x}_3/{T}_{\mathrm{f}i}+{x}_4\left(\frac{T_{\mathrm{m}\mathrm{elt}}}{\Delta {TT}_{\mathrm{f}}}\right), $$
(10)

where xi are unknown coefficients.

The problem of calculating the unknown coefficients xi was solved so as to minimize the deviations εi of the computed values from the experimental value [22]:

$$ {\sum}_j{\left(1\mathrm{n}\kern0.5em {I}_{ij\exp }-1\mathrm{n}\kern0.5em {I}_{ij\mathrm{comp}}\right)}^2={\upvarepsilon}_i, $$
(11)

where j is the number of experiments and i is the number of unknown parameters.

The solution of the system of equations (11) gave the numerical values of the coefficients which were subsequently refined on a computer by a random search method [23, 24]. The coefficients are presented in Table 4. The average \( {\overline{\upvarepsilon}}_i \) is equal to 0.28; M is a collective coefficient, including several values. The quantity δ shows that the cybotaxic groupings are formed in large numbers as the temperature of the melt decreases.

If the activation energy is calculated according to the coefficient x3 by multiplying by the gas constant, we obtain 16,300 × 8.317 = 135.55 J/mol = 32.35 kcal/mol, which is close to the viscous flow energy E = 33.55 kcal/mol for a specified composition. This confirms the physical meaning of the model parameters. And this temperature influences the nucleation rate (Fig. 3) so that IΣ decreases with increasing Tm.

Fig. 3.
figure 3

Effect of the production temperature of the composition on the nucleation rate of metasilicate melt.

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DISCUSSION

Comparing Figs. 1 and 2 we can see that the left-hand side of the curves is actually associated with the nucleation of sitallized glasses and the right-hand side with the formation of nuclei from supercooled melt. As the glass warms up, the nucleation rate increases, which is caused by the accumulation of energy in the internal degrees of freedom or liquation boundaries of glasses with this composition. If the energy is sufficient to overcome the kinetic barrier, nucleation and ordering of the glass occur.

During the formation of the casting the nucleation rate increases with decreasing temperature of the supercooled melt. This process is associated with not only an increase of the thermodynamic potential but also the component composition of the studied melts (see Table 1), which contain iron oxides. If a diopside motif is present in the melt, then in the presence of chromite they can interact and CaFeSi2O6 can be formed [10], though it is unstable below 1000°C and promotes composition fluctuations and the formation of nuclei in a supercooled melt.

The nucleation model (9) proposed in this work is applicable to melts in which predominantly monomineral pyroxene solutions are formed. However, if additional phases precipitate in the course of cooling (compositions 5 6, see Table 1), then the model requires that nucleation kinetics be taken into account. In this case the nucleation mechanism can change.

So, the structural elements of spinelide-pyroxene melt are: spherulite, nuclei of spherulite – spinelide, spherulite shell – pyroxene and a glassy interlayer. A different model of the structure of a nucleus spinelide–pyroxene of cast stone arises on the basis of the elements found. And the main element of the model – spherulite – possesses a bilayer structure. It consists of a nucleus formed by a spinel-like phase and a shell consisting of pyroxene with a 2 – 4-chain structure. Spherulites spread by diffusion in the glass phase in a manner such that the glass phase forms an interlayer between them.

Therefore, the new creation rate will depend not only on the degree of polymerization of the melt but also on the diffusion processes inside the spherulite nucleus. The higher the degree of polymerization, the lower the nucleation rate is, since the glassy phase becomes more stable. And these subtle mechanisms require additional study.

Conclusions

The results of an experimental study of melting and nucleation processes in metasilicate systems under non-equilibrium conditions were presented. The viscous properties of the melts at the melting temperature and the crystallographic formulas of the precipitated solid solutions of diopside were studied.

A model of the nucleation, taking into account the thermodynamic and kinetic factors of phase transformation in metasilicate melts with a diopside motif, was proposed on the basis of the notions of cybotaxic groupings. It was shown that the casting temperature of the melt and the heating temperature of the mold influence the nucleation rate. The model makes it possible to optimize the choice of technological parameters parameters to obtain glassy crystalline materials with a finecrystalline structure.