Introduction

1. Discovery of strontium aluminate cements and current state of knowledge

The experimental study of the system CaO-Al₂O₃-H₂O have considerable practical value not only for its direct contribution to the setting of high-alumina cements (which are composed of mainly anhydrous calcium aluminates) and Portland cements, but also for possible applications in the fields of geology, soil equilibrium, water purification, and the extraction of purified hydrated alumina from crude bauxite [1-3,12]. Unfortunately, so far the study of the system CaO-Al₂O₃-H₂O has many difficulties, including [1]:

• The existence of numerous highly hydrated compounds, the determination of the degree of hydration of which is often a problem.

• Relatively low solubility of these compounds.

• The fact that most of the compounds are metastable, yet, once formed; persist over long periods of time.

• Close similarity in crystal habit and optical properties of compounds occurring as thin hexagonal (or pseudohexagonal) plates.

• The necessity for strict exclusion of carbon dioxide from air.

According to the results from numerous investigations of chemical reactions related to the hardening of hydraulic cements [4,12], lime and alumina in solution may combine to form isometric tri-calcium aluminate hexahydrate (3CaO∙Al₂O₃∙6H₂O, C₃AH₆). This compound is isomorphic to C₃FH₆ garnet as well as grossularite garnet (3CaO∙Al₂O₃∙3SiO₂, C₃AS₃) and andradite garnet (3CaO∙Fe₂O₃∙3SiO₂). Complete solid solutions exist between each of these compounds and the three others. The hydrous members of the series may be termed hydrogarnets [4,5,198]. The general formula may be written as 3CaO∙Al₂O₃∙mSiO₂∙(6-2m)H₂O, where m may vary from 0 to 3 [6]. Hydrogarnets crystallize in various cubic forms, of which icositetrahedra are probably the most usual at ordinary temperatures. In recent mineralogical nomenclature, the phases in the C₃AH₆ – C₃AS₃ series are collectively called hydrogrossular [7].

Tri-calcium aluminate hexahydrate decomposes under hydrothermal conditions at about equilibrium temperature 220 – 226°C, yielding calcium hydroxide and less basic aluminate. The equation may be written as [4,6]:

$$3(3CaO\cdot A{l}_2{O}_3\cdot 6H_{2}O)\leftrightarrow 4CaO\cdot 3{\mathrm{Al}}_2{O}_3\cdot 3H_{2}O+5\mathrm{Ca}{(OH)}_2+10H_{2}O$$E1

Although strontium is rather minor constituent of commonly applied cements such as Portland cement (PC) or alumina cement (AC), the strontium compounds may be formed via the reaction with silica, alumina and water. Despite the fact that the ionic radius of Sr²⁺(1.26 and 1.44 Å for the coordination number 8 and 12, respectively [92]) is larger than that of Ca²⁺(1.06 Å), strontium can partially replace calcium in numerous metal oxides and numerous studies concerned to Ca/Sr mixed oxides can be found in literature [8]. The existence of tri-strontium aluminate hexahydrate (3SrO∙Al2O3∙6SiO2) has been reported by Brandenger [9] and Maekawa [10].

Carlson [4] recognizes tri-strontium aluminate hexahydrate as the only hydrated phase formed in the system strontium – alumina – water. This phase is isomorphous with corresponding calcium compound of tri-calcium aluminate hexahydrate and the complete solid solution exists between those two phases. This is proved by the X-ray diffraction patterns, which indicate the same reflecting planes with such changes in relative intensities as might be expected for isomorphous substitutions. There is also a shift corresponding to somewhat larger unit cell of strontium compounds. The refraction index of tri-strontium aluminate hexahydrate is 1.588 [4,10].

3SrO∙Al₂O₃∙6H₂O has higher temperature stability than 3CaO∙Al₂O₃∙6H₂O, which decomposes under hydrothermal conditions according to Eq.1.1. Tri-strontium aluminate hexahydrate decomposes under hydrothermal conditions at temperatures as low as 305 °C (2 atm.) and 454 °C (420 atm.). The product of the decomposition is dependent on both the temperature and the pressure. In most cases the decomposition is accompanied by the liberation of strontium hydroxide. The lower hydrate with 1.5 molecules of water (3CaO∙Al₂O₃∙1/2H₂O, C₃AH_0.5) is probably formed. The hexagonal prism of crystalline phase 5SrO∙4Al₂O₃ was also identified as a product. It decomposes under further heating with the formation of SrO∙Al₂O₃ and 3SrO∙Al₂O₃ [4].

Therefore, with SrO formally replacing CaO in calcium aluminate cement, the strontium aluminate binder belongs to the family of aluminous cements based on the system of CaO – SrO – BaO – Al₂O₃ – ZrO₂ – HfO₂ [11-14]. The differences and the similarities between the strontium – alumina and lime – alumina systems are briefly mentioned below [14]:

1. There are no compounds like Sr₅A and Sr₄A in the lime – alumina system.

2. No analog of C₁₂A₇ in the strontium – alumina system exists

   Phase has not own field in the equilibrium diagram of SrO-Al₂O₃ system, but the synthesis of Sr₁₂A₇ is described by Yamaguchi at al [41].

   .

3. The similarities exist to some extent between Sr₃A and C₃A, SrA and CA, SrA₂ and CA₂ as well as between SrA₆ and CA₆, although the melting behaviour and the temperature differ significantly.

The discovery of calcium aluminates, the study of their crystal chemistry and phase equilibrium, subsequently developing the family of alumina cements and their ultimate field applications covered a span of one and half century, even though Bied received the French and the British patents for the method to produce calcium aluminate cement from bauxite and lime in a cupola furnace in 1908–1909, just about a century ago. The use of AC in basic refractories is rather limited as it leads to the formation of low-melting compounds like monticellite. In certain non-refractory applications AC shows comparatively poorer resistance to aggressive environments as well as to X-rays and gamma radiation [14].

Strontium aluminate cement can be prepared by solid-state sintering of approximately equimolar blend of SrCO₃ and Al₂O₃ at the temperature of 1500 °C. The main constituent of strontium aluminate cement is mono-strontium aluminate (SrO∙Al₂O₃), but strontium hexaluminate may also be present. Strontium aluminate cement hydrates relatively slowly, yielding to strontium aluminate hexahydrate phase of the approximate composition SrO∙Al₂O₃∙7-10H₂O as the main hydration product. In later stages of hydration this phase gradually converts to 6SrO∙Al₂O₃∙6H₂O and hydrous alumina. This reaction is associated with the loss of strength. The cement has the potential to be used in the production of refractory concretes [12].

Calcium and strontium aluminate cement contains Ca and Sr as the chemical elements from the second group of the periodic table (alkaline earth metals): Be (beryllium), Mg (magnesium), Ca (calcium), Sr (strontium), Ba (barium) and Ra (radium). Therefore, well-founded assumption exists to prepare similar binding system with barium, i.e. barium aluminate cement based on hydraulic properties of barium aluminate or monobarium aluminate (BaO∙Al₂O₃, BA).

Barium aluminate cement can be produced by solid-state sintering of equimolar blend of BaCO₃ (witherite) and Al₂O₃ at the temperature of 1400 – 1500°C. The sintering temperature can be reduced by the addition of suitable mineralizer such as MgO. The main constituent of such cement is barium aluminate, but small amount of tri-barium aluminate (3BaO∙Al₂O₃, B₃A) and barium hexaaluminate (BaO∙6Al₂O₃, BA₆) is formed. At low water to cement ratios (w/c < 0.35), the monobarium aluminate heptahydrate (BaO∙Al₂O₃∙7H₂O, BAH₇) is formed as the main hydration product. In the case that w/c ratio is higher; the tribarium aluminate hexahydrate (3BaO∙Al₂O₃∙6H₂O, B₃AH₆) is formed together with hydrous alumina. The reaction of barium aluminate cement with water is fast. Whole BA hydrates completely within three days at sufficiently high water to cement ratio. The hardened cement pastes show only a small strength loss when heated to high temperatures. Barium aluminate cements can be used for the production of refractory materials and they absorb radiation effectively [12,15,19].

The initiation of hydration reactions occurs with leaching of Ba²⁺ ions and with formation of hydroxides following the reaction [14]:

$$\mathrm{BA}+2H_{2}O\to {\mathrm{Ba}}^{2+}+2Al{O}^{2-}+H_3{O}^{+}+O{H}^{-}$$E2

Under low w/c ratio the following reactions take place:

$$\mathrm{BA}+6H\to {\mathrm{BAH}}_6$$E3

$$2BA{H}_6\to B_{2}A{H}_9+2A{H}_3$$E4

$$B_{2}A{H}_9+\mathrm{BA}\to B_{3}A{H}_6+2A{H}_3$$E5

Chaterje [14] described the pilot production process for the preparation of both BAC and SrAC cements, where raw materials in required stoichiometric proportions were cogrinded and next milled in a ball mill to the fineness of 5% residue on 212 μm sieve. The ground raw mix was treated with water and the pellets sized about 15 mm were prepared. Dried pellets were fed into a pilot rotary kiln. The sintering temperature was maintained at about 1620 °C and 1640 °C for SrAC and BAC respectively. The clinker was air quenched and ground to 300 ± 10 m²∙kg^-1 and 280 ± 10 m²∙kg^-1 for SrAC and BAC, respectively.

Several types of stoichiometric as well as non-stoichiometric aluminates are known. The most studied and also used aluminates of barium are the stoichiometric aluminates, such as: barium monoaluminate (BaO∙Al₂O₃, BA), tribarium monoaluminate (3BaO∙Al₂O₃, B₃A), barium tetraaluminate (BaO∙4Al₂O₃, BA₄) and barium hexa-aluminate (BaO∙6Al₂O₃, BA₆) [16,17,24].

All three alkali-earth monoaluminates, namely CA, SrA and BA being the primary cement-forming phases, properties are listed in Table 1. As to melting temperatures of these phases, their high-temperature stabilities sequence is as follows [14]:

$$CA>SrA>BA.$$

Phase	Composition [%]			Molar mass	Density	Crystal system	Melting temperature/ behaviour
Me	Al	O	[g∙mol^-1]	[g∙cm^-1]
CA	25.4	34.1	40.5	158.04	2.98	Monoclinic	1605°C/ incongruent
SrA	42.6	26.3	31.1	205.58	3.82	Pseudo-hexagonal	1790°C/ congruent
BA	53.8	21.1	25.1	255.29	3.99	Cubic	1815°C/ congruent

Table 1.

Properties of MeAl₂O₄ phases in calcium, strontium and barium aluminate cements.

Further, the traces of the liquidus curves around SrA and BA are much flatter than those around the CA phase. Thus the compositional deviations in the CA line cements are more precarious for the CA cements than the compositional shifts in BA and SrA cements [14].

2. Phase relationship in SrO – Al₂O₃ system

In this part, the phase relationships in SrO – Al₂O₃ system (Fig.1) will be discussed. The equilibrium phase diagram is useful tool for the determination of phase composition of the system at given temperature and composition. The phase diagrams also enable to read the temperature of formation of eutectic or peritectic melts and equilibrium phases. Using the “lever rule” we can estimate the ratio of formed phases or the amount of melt. Moreover there is the possibility to calculate some thermodynamic parameters as will be described in Chapter 1.2.3.

Complete phase diagram of strontium–alumina binary system was constructed as early as 1979 [18,19]. The field of stability of five binary phases, i.e. tetra-strontium aluminate (Sr₄Al₂O₇, Sr₄A), cubic tristrontium aluminate (Sr₃Al₂O₆, Sr₃A), monoclinic strontium aluminate (SrAl₂O₄, SrA), strontium dialuminate (SrAl₄O₇, SrA₂) and strontium hexaaluminate phase (SrAl₁₂O₁₉, SrA₆) is given below.

High temperature modification of α-Sr₄A is stable in the temperature range from 1320 to 1690 °C. The transformation of tetrastrontium aluminate into low temperature β-Sr₄A phase takes place below 1320 °C. Under the temperature of 1125 °C α-Sr₄A phase is decomposed to Sr₃A and SrO. The phase diagram shows that the low temperature Sr₄A phase forms a solid solution with Sr₃A [18,19].

Figure 1.

Isobaric equilibrium phase diagram of the system of SrO – Al₂O₃ showing the composition and the temperature region of stability of binary compounds and the position of invariant points.

Liquidus temperature drop rapidly upon the addition of Al₂O₃ to SrO. Thus SrAl₁₂O₁₉ melts incongruently to Al₂O₃ at 1690°C. The compositions with the minimum melting temperatures are the eutectics between [18]:

• SrO∙2Al₂O₃ and SrO∙6Al₂O₃ located at 1780°C;

• SrO∙Al₂O₃ and SrO∙2Al₂O₃ located at 1760°C;

• 3 SrO∙Al₂O₃ and SrO∙Al₂O₃ located at 1505°C;

• 4 SrO∙Al₂O₃ and 3SrO∙Al₂O₃ located at 1630°C.

The following phases occur in the SrO – Al₂O₃ system:

Strontium oxide (strontia, strontium monoxide, SrO, S): is grayish-white colored, has elemental composition of 84.56 % Sr and 15.44 % O and cubic crystalline structure. The molar weight and the density of SrO are 103.63 g∙mol^-1 and 4.70 g∙cm^-3, respectively. It melts at the temperature of 2430 °C and vaporizes above 3000 °C. It reacts with water to strontium hydroxide (Sr(OH)₂) evolving heat. Strontium oxide can be prepared by thermal decomposition of strontium carbonate, hydroxide, or nitrate [91]:

$${\mathrm{SrCO}}_3(s)\to \mathrm{SrO}(s)+C{O}_2(g)$$E6

$$\mathrm{Sr}{(OH)}_2(s)\to \mathrm{SrO}(s)+H_{2}O(g)$$E7

$$\mathrm{Sr}{(NO)}_3(s)\to \mathrm{SrO}(s)+N_2{O}_5(g)$$E8

SrO is miscible with fused caustic potash, is slightly soluble in alcohol and insoluble in acetone and ether [91]. The methods used for industrial production of strontium carbonate as the intermediate for the preparation of other compounds of strontium is described in Chapter 2.1.1.

Tetrastrontium aluminate (Sr₄Al₂O₇, 4SrO∙Al₂O₃, Sr₄A): this phase is stable only within the temperature range from 1125 to 1630 °C. It decomposes into mixture of tri-strontium aluminate and free strontium oxide if the temperature is lower than 1125 °C, peritectic melting takes place at the temperature of 1630 °C. In the temperature of 1320 °C the high temperature modification of α-phase is transformed into low temperature β-phase. As can be read from the phase diagram in Fig.1, the low temperature modification of tetra-strontium aluminate forms solid solution with tri-strontium aluminate [14].

Tristrontium aluminate (Sr₃Al₂O₆, 3SrO∙Al₂O₃, Sr₃A): has cubic “stuffed-tridimite” structure with the space group Pa3 based on (AlO₃)₆ rings. The noteworthy feature of this structure is the puckered six-membered tetrahedral AlO₄ rings. There is a front layer and a back layer in this structure, each running in parallel to the other. This structure is typical for strontium aluminates where either tetrahedral AlO₄ or octahedral AlO₆ groups are present (or sometimes both). The lattice parameters are a=15.8556 Å, V=3986.3 ų, Z=24 [19]. The structure of strontium aluminate is shown in Fig.2(a).

Strontium aluminate (SrAl₂O₄, SrO∙Al₂O₃, SrA): white color, melts congruently at the temperature of 1790°C. The spinel type (AB₂O₄) compound has monoclinic structure that belengs to the P2₁ or P2₁/m space group with unit cell parameters a=5.1497 Å, b=8.836 Å, c=8.442 Å, β=93.43° and contains four formula per cell (Z=4). SrAl₂O₄ structure is derived from “stuffed tridimite structure”, where all Si⁴⁺cations are replaced by Al³⁺and the charge compensating cations Sr²⁺occupy the large open channels in the framework

Analogically to the structure of CaAl₂O₄ and BaAl₂O₄ that belong to the family of tridymite stuffed derivates of MAl₂O₄, where M = Ca, Sr, Ba. This structure is also typical for compounds such as BaNSiO₄ (N = Co, Mg and Zn) and (Na,K)ZXGeO₄ (Z = Aland Ga and X = Ge and Si [22]. The structure is similar to zeolites, but differs in the orientation of corner-connected tetrahedral units in the way that some of 6-membered rings convert to 4- and 8-mebered rings [23].

. Strontium aluminate is an indirect-band-gap oxide and its band gap was estimated to ~6.9 eV [19-23]. The structure of strontium aluminate is shown in Fig.2(b). The high temperature hexagonal polymorph is of P6₃ space group with the cell parameters: a=8.926 Å, c=8.4985 Å, γ=120° and Z=6 [24].

Strontium dialuminate (SrAl₄O₇, SrO∙2Al₂O₃, SrA₂): has two polymorphs, α-SrA₂ is stable at normal pressures, while high pressure form of β-SrA₂ can be prepared only at high pressure and temperature. The crystals of β-SrA₂ are orthorhombic with the space group Cmma, a=8.085 Å, b=11.845 Å, c=4.407, Z=4 and calculated density of 4.84 g cm^-3. The lattice consists of three-dimensional (Al₄O₇) network of AlO₆ octahedra (length bond from 1.795 to 1.968 Å) and AlO₄ tetrahedra (length bond from 1.449 to 1.537 Å). The structure has certain similarities to the structure of tetra-strontium aluminate. Each atom of Sr is surrounded by 10 O atoms with the distances ranging from 2.503 to 2.664 Å [19,25]. Monoclinic α-SrA₂ has the space group C12/c1 with the lattice parameters a=13.0389 Å, b=9.0113 Å, c=5.5358 Å, β=106.12° and Z=4 [26].

Strontium hexaaluminate (SrAl₁₂O₁₉, SrO∙6Al₂O₃, SrA₆): this phase has hexagonal magnetoplumbite structure

MP-type alkaline earth (AE) aluminum oxide (AEAl₁₂O₁₉). These compounds belong to the same space group as β-alumina (Chapter 4.2).

with space group P6/mmc. These compounds have a layer structure composed of spinel blocks and conduction layers which are stacked alternately (Fig.2(c)). There are five distinct Al³⁺sites in the lattice: one tetrahedral (Al(IV), AlO₄ with C_3v site symmetry) one trigonal bipiramidal (Al(V), AlO₅ with C_3v site symmetry) and three octahedral (Al(VI), AlO₆). The octahedral sites are of different symmetry: regular octahedron (D_3d, Al(VI)-1), antiprism (C_3v, Al(VI)-2) and distorted octahedron (C_s, Al(VI)-3). The Al(V) sites are present only in so called “central atom model” (residual factor 0.0340) while half occupied site of distorted tetrahedron (Al(IV)-d of C_3v site symmetry) results from later “split atom model” (residual factor 0.0331). The unit cell is composed of spinel-structured slabs containing Al³⁺cations separated by mirror planes (2b

Wyckoff position denotes the point belonging to a set of points for which the site symmetry groups are conjugate subgroups of the space group.

site in the central atom model and 4e with half occupancy in the split atom model) which contain Sr²⁺and three oxygen ions per unit cell. The band gap of strontium aluminate is about 7.6 eV [19,27,28,29,30].

The unit cell parameters of SrAl₁₂O₁₉ hexagonal cell are: a=5.562, c=21.972 Å and Z=2 [31,32]. Small divalent cations Mg²⁺can be incorporated in the spinel block with replacing trivalent Al³⁺ions and the difference of their ionic valences causes the defects in the structure or sometimes the modification of the structure [33]:

• Mg replaces Al in the Al-spinel block, inducing a valence deficiency. A charge compensation with positively charged conduction layer is required.

• Spinel unit (Mg₂Al₄O₈) will be inserted in the Al-spinel block.

• Al³⁺ions in octahedral sites should be replaced by Fe³⁺[34,35], Cr³⁺[36], Co³⁺[36,37], Mn²⁺[38] and rare earth elements ions such as Y³⁺[39] or Ga³⁺[40].

Figure 2.

Structure of tristrontium aluminate (a) and strontium aluminate (b), schematic drawing of the structure of magnetoplumbite compounds (c) according to [24], where large shaded circles represent large cations (Na⁺, K⁺, Sr²⁺…), white and black circles represent O^2-and Al³⁺, respectively.

Aluminium oxide (Al₂O₃, A): belongs to the R $$\overline{3}$$ c space group and forms hexagonal crystal (in the only thermodynamically stable crystallographic modification of corundum, α-Al₂O₃) with the lattice cell parmeters (a=4.754 Å and c=12.99 Å). O^2-ions are arranged in close hexagonal arrangement, with cations Al³⁺occupying two-thirds of the octahedral interstitial positions. Corundum has the density of 3.97 g∙cm^-3 and the elemental composition of 52.91% Al and 47.08% O.

The molar weigh of aluminium oxide is 101.96 g∙mol^-1. Al₂O₃ is available or prepared in several forms for various commercial applications. Some of these are α-alumina (corundum), activated aluminas (such as, γ-alumina, η-alumina and ρ-alumina), hydrated aluminas including aluminum oxide monohydrate, Al₂O₃∙H₂O and aluminum oxide trihydrate, Al₂O₃∙3H₂O (natural gibbsite) and acidic, neutral and basic aluminas (no definite chemical compositions, they are prepared by the addition of various amounts of water to activated aluminas). Alumina exhibits amphoteric behaviour [91,424]. The properties and the methods of industrial production of Al₂O₃ are described in Chapter 2.1.2.

Other phases found and described within the SrO – Al₂O₃ oxide system, which aren’t plotted in the equilibrium phase diagram (Fig.1) are:

• Sr₁₂Al₁₄O₃₃ (12SrO⋅7Al₂O₃): has cubic structure of space group I43d and following cell parameters: a=12.325 Å and Z=2 [41].

• Sr₁₀Al₆O₁₉ (10SrO⋅3Al₂O₃): has monoclinic structure (space group C12/c1, a=35.5823 Å, b=7.8460 Å, c=15.7485 Å, β=103.68° and Z=8, Fig.3), which belongs to the group of oligoaluminates, where the prominent structural features are [Al₆O₁₉] – groups. The arrangement of tetrahedra within hexamers is not linear but highly puckered. In detail, the geometry can be characterized as open five-membered tetrahedral rings with one additionally attached sixth tetrahedron. The Al-O bond distances in Sr₆Al₁₀O₁₉ range between 1.73 and 1.81 Å [42]. Sr₆Al₁₀O₁₉ is isotypic with strontium gallate α-Sr₁₀Ga₆O₁₉ [43].

Figure 3.

Single [Al₆O₁₉] hexamer in the structure of Sr₁₀Al₆O₁₉ with two linking tetrahedra (1) and the arrangement of cations, anions and vacancies in the corresponding idealized unit of perovskyte (2) containing atoms of Sr (big dark spheres) and Al (light gray spheres), oxygen ligands (small gray spheres) and vacancies (small dark spheres). Al-O bonds within [Al₆O₁₉] hexamer are represented by dark rod (a). The structure of Sr₁₀Al₆O₁₉, projection is parallel with [010] (b). Big gray spheres corresponding to Sr cations and tetrahedra (T) belonging to one of single [Al₆O₁₉] hexamers are marked [42].

• Sr₇Al₁₂O₂₅ (7SrO⋅6Al₂O₃): is trigonal phase with the space group P3 and following cell parameters a=11.91 Å, γ=120° and Z=3 [44].

• Sr₄Al₁₄O₂₅ (4SrO⋅7Al₂O₃): is orthorhombic phase of Pnna space group with following cell parameters: a=24.7451 Å, b=8.4735 Å, c=4.8808 Å and Z=2 [45].

• Sr₂Al₆O₁₁ (2SrO⋅3Al₂O₃): is orthorhombic phase of Pnnm space group with following cell parameters: a=21.914 Å, b=4.884 Å, c=8.4039 Å and Z=4 [46].

2.1. Application of Gibbs phase law on the phase diagram

The Gibbs phase rule [47] enables to calculate the number of degrees of freedom of the system. The most general and well known formulation is:

$$v=s-f+2-C$$E9

where v, s and f denote the number of degrees of freedom, species (constituents) and phases, respectively. Number “two” represents the temperature and pressure. The value of parameter C is derived from other restriction conditions in given thermodynamic system, for example from the number of independent chemical reactions (Gibbs stoichiometric law, Chapter 1.4), from the Dalton law, from the principle of electroneutrality, etc.).

The Gibbs phase law can be derived from the following consideration which includes the system formed from s species and f phases. There are certain minimal numbers of parameters which are necessary to describe the state of this system (f (s – 1)+2):

$$\begin{array}{l}{x}_s^1=1-{\displaystyle \sum _{j=1}^{s-1}{x}_j^1}\\ ⋮\\ x_s^f=1-{\displaystyle \sum _{j=1}^{s-1}{x}_j^f}\end{array}$$E10

There is also certain minimal number of parameters which are necessary to describe the equilibrium in the system. The equilibrium is defined as the equivalence of chemical potential of s components across f – 1 phases, i.e. s (f – 1) variables.

$$\begin{array}{l}{\mu }_1^1={\mu }_1^2=\dots ={\mu }_1^{f-1}\\ {\mu }_2^1={\mu }_2^2=\cdots ={\mu }_2^{f-1}\\ ⋮\\ {\mu }_s^1={\mu }_s^2=\cdots ={\mu }_s^{f-1}\end{array}$$E11

For example, if we consider a system that consists of two components and three phases (e.g. some of eutectic points or the formation of peritectic melt in Fig.1), the condition can be explained as follows: if μ₁₁=μ₁₂ and μ₁₂=μ₁₃ then μ₁₁=μ₁₃. Therefore, only 2×(3-1)=4 terms are necessary for the description of equilibrium.

The Gibbs phase law is a simple difference between the number of parameters defining the state and the equilibrium in the system:

$$v=f(s-1)+2-s\left(f-1\right)=fs-f+2-fs+s$$E12

$$v=s-f+2$$E13

In the case of isobaric type [p] of phase diagram in Fig.1 without other restricting condition (C=0) it is possible to write:

$$v=s-f+1$$E14

2.2. Phase diagram and lever law

The equilibrium phase diagram or isobaric equilibrium phase diagram is a useful toll, which enables to predict the system behaviour during the thermal treatment. It is possible to estimate:

• Final phase composition of material;

• Field of stability (temperature and composition) of formed phases;

• Formation and decomposition of solid solutions;

• Melting temperature;

• Composition and temperature of invariant points (eutectic and peritectic points);

• Behaviour during melting (congruent or incongruent);

• Immiscibility gap in melt;

• Behaviour during crystallization;

• Ratio of formed phases.

Moreover is possible to predict the structure of cooled product (for example grain of precipitate dispersed in matrix of solidified eutectic melt) or to calculate melting enthalpy from the liquids line of diagram (described in Chapter 2.3).

It should be pointed out, that the equilibrium phase composition is not often established during industrial production of materials, especially during fast firing process of ceramics. Moreover metastable phases exhibit higher reactivity, therefore are often prepared purposely. For example, fast cooling of clinker of ordinary Portland cement (OPC) is applied in order to prevent it from the decomposition of tricalcium silicate (C₃S, 3CaO⋅SiO₂) into dicalcium silicate and free lime (C, CaO), as well as to keep the formed dicalcium silicate (C₂S, 2CaO∙SiO₂) in the hydraulic reactive polymorph of β-C₂S. Stable modification of α-C₂S is not capable of hydration and contribution to hardening of concretes and mortars.

Five binary compounds are present in the phase diagram of SrO – Al₂O₃ system (Fig.1). Since 4SrO Al₂O₃ and SrO 6Al₂O₃ melt incongruently, there are two peritectic points. Other known strontium aluminate phases melt congruently. Five eutectic points are present in the phase diagram. Both, the eutectic and the peritectic points are invariant points without any degree of freedom (Gibbs Phase Rule, Chapter 2.1), but the system may contain thermodynamically stable melt at the temperature lower than the temperature of peritectic point. There is no stable melt present at the temperature lower than the temperature of eutectic point in the binary system.

Melt or liquid phase can be formed via congruent or incongruent melting process. Incongruent melting means the formation of peritectic melt and other solid phase. Example for strontium hexaaluminate it can be written as:

$$\mathrm{SrO}\cdot 6A{l}_2{O}_3(s)\stackrel{T_P}{\to }{\mathrm{Al}}_2{O}_3(s)+peritecticmelt$$E15

It means that the peritectic point is shifted to the left from the stoichiometric composition of strontium hexaaluminate. The peritectic liquid then contains 1 % less alumina than strontium hexaaluminate. The entire surplus of alumina stays in solid phase, therefore the temperature cannot rise if SrO⋅6Al₂O₃ solid is present (invariant point). Once SrO⋅6Al₂O₃ melts the system gets one degree of freedom and the composition of melt can change according to the line of liquidus until the rest of alumina is dissolved. The temperature as well as final composition of melt phase can be determined by the extrapolation of SrO⋅6Al₂O₃ composition to the line of liquidus of Al₂O₃ (1920 °C, x_SrO=14 %). The liquidus line can be defined as the borderline separating heterogeneous system, which consists of precipitate and melt from the field of stability of homogeneous melt.

The ratio of formed phases can be derived by the “lever rule” as is described below (Al₂O₃ : peritectic melt=1 : 14). On the contrary, congruently melting strontium dialuminate forms the liquid phase of the same composition as original solid:

$$\mathrm{SrO}\cdot 2{\mathrm{Al}}_2{O}_3(s)\stackrel{T_T}{\to }\mathrm{melt}$$E16

The melting process turns the complex structures into the melt, negative charge of [AlO₄]^5-anions is compensated by cations of Sr²⁺. Basic oxides also provide O^2-anion, that is needed for the modification of melted compounds into the simpler structures [48,49] and affects the equilibria of redox reactions (e.g. Fe³⁺/Fe²⁺equilibrium) [50,51,52].

The system composition can be expressed by the molar or weight ratio of constituents. The rule must be fulfilled that the sum of molar (x_j) as well as weight (w_j) ratio is equal to one (or 100 %) (Eq.17), therefore x₂=1 – x₁ or w₂=1 – w₁ in the binary system.

$${\displaystyle \sum _{i=1}^j{x}_j}=1$$E17

$${\displaystyle \sum _{i=1}^j{w}_j}=1$$E18

These ratios can be recalculated one to another as follow:

$$w_i=\frac{M_i{x}_i}{{\displaystyle \sum _i{M}_i{x}_i}}$$E19

$$x_i=\frac{\frac{w_i}{M_i}}{{\displaystyle \sum _i\frac{w_i}{M_i}}}$$E20

where M is the molecular mass of i-th species.

The behaviour of the system during cooling of melt phase will be explained on the examples in Fig.4 with the composition given by line (1) and (2). Starting with composition (1) when melted phase contains 90 % SrO, the first SrO solid appears at the temperature corresponding to the point A (2200 °C). The infinitesimal amount of SrO(s) (A´) is in the equilibrium with the melt of composition (A). This equilibrium is described by Eq.39

the line of liquidus in the phase diagram given in Fig.4 is a function of this equation, but increasing non-ideal behaviour of melt leads to the differences between experimental and calculated data as the content of the second constituent increases.

. Using general formula of Gibbs phase law, the system has one degree of freedom. Therefore, the composition of the system may change with decreasing temperature according to the line of liquidus. Strontium oxide precipitating from the melt leads to the liquid phase enriched with Al₂O₃ (dark red arrows).

Figure 4.

Crystallization pathway for the system A and B (a) with graphical illustration of “lever rule” and cooling curve.

At the temperature of 1900 °C that corresponds to the point B, the system consists of SrO(s) and melt in the ratio given by the lever rule:

$$\frac{\mathrm{SrO}(s)}{\mathrm{melt}}=\frac{\left|B{B}^{\prime }\right|}{\left|{B^{\prime }}^{\prime }{B}^{\prime }\right|}\Rightarrow \mathrm{SrO}(s):\mathrm{melt}=1:1.\overline{3}$$E21

Therefore pure SrO is formed (B´´), the system consists of SrO precipitate dispersed in the equilibrium melt, the composition of which is given by the line of liquidus. Three phases are present if the system reaches the composition and the temperature corresponding to the peritectic point: peritectic melt (P), tetra-strontium aluminate (Sr₄A) and strontium oxide (Sr). There is zero degree of freedom and the temperature cannot be changed until the solidification of whole melt proceeds.

Within the temperature interval from 1690 to 1320 °C, the solid system consists of strontium oxide and tetra-strontium aluminate in the ratio |CC| : |C´´C´|=1:1. The ratio stays unchanged to the temperature 1125°C. Below the temperature of 1125 °C, Sr₄A is decomposed to SrA and Sr₃A in the ratio 1 : 1.5.

The crystallization of the system with the composition given by line (2) begins at the temperature of point (a) where the first infinitesimal amount of strontium oxide appears. The composition of melt changes according to the line of liquidus as the temperature decreases. On the contrary in the system (1) the free strontium oxide disappears at the temperature of peritectic point (P). The Sr₄A solid and melted phase, the composition of which changes according to the line of liquidus, is the equilibrium as the temperature decreases from P to E. At the temperature of eutectic point, the system becomes invariant again and the temperature stays constant until whole eutectic melt is solidified. Sr₄A crystals appear in the fine crystallized matrix of eutectic melt, but both phases are pure crystalline phases.

Below the temperature of 1575 °C (b) the Sr₃A is dissolved in S₄A phase. Because there are three equilibrium phases (Sr₄A, Sr₄A(ss) and Sr₃A) in the two component system at constant pressure, the system becomes invariant until entire Sr₃A is dissolved. Formed solid solution is being enriched by dissolved Sr₄A as the temperature decreases from point (b) to (c). That can be easily proved by the lever rule. At the temperature of point (c) Sr₄A dissolves in the solid solution and line (2) heads to the homogeneous field of solid solution, but at the temperature of 1320 °C (d) the Sr₄A phase is formed again.

Tri-strontium aluminate starts precipitating from the solid solution at the temperature of point (e) and this process continues to the temperature of point (f) where the solid solution does not exist any longer. Below this temperature, the decomposition of Sr₄A to free SrO and Sr₃A takes place.

Tri-strontium aluminate and strontium aluminate are neighboring congruently melting compounds. Therefore, they form relatively simple binary subsystem of the SrO – Al₂O₃ system that can be made as separate phase diagram (Fig.5). From this point of view, the phase diagram in Fig.1 consists of four simple systems, which are formed between oxide species and congruently melting phase or two congruently melting phases: SrO – Sr₃A, Sr₃A – SrA, SrA – SrA₂ and SrA₂ – SrA₆.

Figure 5.

Binary system formed between tri-strontium aluminate and strontium aluminate.

Fig.5 shows the lever rule applied to the system containing slightly higher amount of SrO with regard to strontium aluminate stoichiometry heated to the temperature of 1600 °C. The solid to melt ratio has the value:

$$\frac{solid}{melt}=\frac{\left|ab\right|}{\left|bc\right|}=\frac{47.5-35}{50-47.5}=\frac{5}{1}$$E22

The original system containing 45.7 % Al₂O₃ and 100 – 45.7=54.3 % SrO consists of 1 piece of melt and 5 pieces of solid. That can be verified as follows:

$$\begin{array}{ccccccc}\mathrm{Melt}& {\mathrm{Al}}_2{O}_3[\%]& 35& & 2.5\to & 1& \\ & & & 45.7& & & \\ \mathrm{Solid}& {\mathrm{Al}}_2{O}_3[\%]& 50& & 12.5\to & 5& \end{array}$$E23

The same rule can be used to calculate the ratio for two systems mixed in order to reach required composition. In the case, that x_Al2O3 is recalculated to w_Al2O3 (Eq.19), the following relation can also be used for these purposes:

$$m_1{w}_1+m_2{w}_2+\dots m_i{w}_i=W(m_1+m_2+\cdots m_1)$$E24

For example, if the system containing 50 % Al₂O₃ should be prepared by mixing 100 g of system with 30 % Al₂O₃ with pure Al₂O₃ (100 %), the solution is:

$$\begin{array}{l}100\cdot 30+m_{A{l}_2{O}_3}\cdot 100=50(100+m_{A{l}_2{O}_3})\\ m_{A{l}_2{O}_3}=40g\end{array}$$E25

2.3. Calculation of melting enthalpy from equilibrium phase diagram

The equilibrium phase diagram constructed from the experimental data on the behaviour of system during heating can be used for the calculation of melting enthalpy (Δh_m) of the constituents. The solution is named as “Schröder and Le Chatelier's law” and it will be demonstrated on the SrO – Al₂O₃ diagram (Fig.1). It must be pointed out, that the following solution is derived for the equilibrium of pure crystalline solids with melt where no solid-solution is formed. That means that presented solution is applicable for all congruently melting solids in the binary system of SrO – Al₂O₃.

The liquidus line in the phase diagram represents the equilibrium state between solids and melt phase at given temperature. In the equilibrium state the chemical potential of both phases must fulfill the term:

$${\mu }_{A,s}^{\ast }={\mu }_{A,\ell }$$E26

where μ*_A,s and μ_Aℓ denote the chemical potential of species A as the pure (*) solid phase (s) and in the melt (ℓ), respectively. The chemical potential of pure solids is given as follows:

$${\mu }_{A,s}^{\ast }=f(T,p)$$E27

$$d{\mu }_{A,s}^{\ast }={\left(\frac{\partial {\mu }_{A,s}^{\ast }}{\partial T}\right)}_{p}dT+{\left(\frac{\partial {\mu }_{A,s}^{\ast }}{\partial p}\right)}_{T}dp$$E28

$$d{\mu }_{A,s}^{\ast }=-s_A^{*}dT+v_A^{*}dp$$E29

where s^* and v^* are the molar entropy and volume.

The chemical potential of species A in the melt with ideal behaviour can be expressed by the equation:

$${\mu }_{A,\ell }=f(T,p,x_A)$$E30

$$d{\mu }_{A,\ell }=-{\overline{s}}_{A,\ell }dT+{\overline{v}}_{A,\ell }dp+RTd\ln x_A$$E31

where $$\overline{s}$$ and $$\overline{v}$$ are partial molar entropy and volume.

With respect to Eq.26 the equilibrium state requires:

$$d{\mu }_{A,s}^{\ast }=d{\mu }_{A,\ell }$$E32

$$-s_A^{*}dT+v_A^{*}dp=-{\overline{s}}_{A,\ell }dT+{\overline{v}}_{A,\ell }dp+RTd\ln x_A$$E33

$$\left({\overline{s}}_{A,\ell }dT-s_A^{*}\right)dT=\left({\overline{v}}_{A,\ell }+v_A^{*}\right)dp+RTd\ln x_A$$E34

The term in the brackets corresponds to the change of entropy and volume during melting. Therefore, the applied equilibrium phase diagram is constructed for the isobaric conditions [p] which can be written:

$$\Delta s_{A,m}dT=RTd\ln x_A$$E35

where Δs_m is the melting entropy of the process, which can be expressed using the definition law of Gibbs energy for the equilibrium state (Eq.36).

$$\Delta g_m=\Delta h_m-T\Delta s_m=0\Rightarrow \Delta h_m=T\Delta s_m\Rightarrow \Delta s_m=\frac{\Delta h_m}{T}$$E36

From the combination of Eqs.35 and 36 the following can be derived:

$$\frac{d\ln x_A}{dT}=\frac{\Delta h_{A,m}}{R{T}^2}$$E37

$${\displaystyle \underset{1}{\overset{x_A}{\int }}d\ln x_A}=\frac{\Delta h_{A,m}}{R}{\displaystyle \underset{T_A}{\overset{T_{x_A}}{\int }}\frac{dT}{T^2}}$$E38

$$\ln x_A=-\frac{\Delta h_{A,m}}{R}\left(\frac{1}{T_{x_A}}-\frac{1}{T_A}\right)\approx -\frac{\Delta h_{A,m}}{R}\frac{T_A-T_{x_A}}{T_A^2}\left[x_A\to 1\Rightarrow T_A\approx T_{X_A}\right]$$E39

where T_A and T_xA are the melting temperature of pure species A and of the system of composition x_A, respectively. The melting (fusion) enthalpy can be assessed then from the slope (term-Δh_m/R) of plot ln x_A to be the difference (T_xA^-1-T_A^-1).

Figure 6.

Estimation of melting enthalpy from phase diagram.

The value T_xA should be the closest possible value to T_A (x_A is close to one) due to the presumption of ideal behaviour of melted phase. In the case that the value of melting enthalpy is known, the liquidus line can be calculated from Eq.39. The example of the estimation of melting enthalpy of SrO from the equilibrium phase diagram is shown in Fig.6.

2.4. Ternary system SrO-CaO-Al₂O₃

The phase relationships of the SrO – CaO – Al₂O₃ system (Fig.7) were described by Massazza at al. [53,54]. The ternary diagram shows the formation of solid solution between corresponding EA_xA_y phases, where EA denotes alkali earth element (Ca or Sr): CaAl₁₂O₁₉ (CA₆)-SrAl₁₂O₁₉ (SrA₆), CaAl₄O₇ (CA₂)-SrAl₄O₇ (SrA₂), CaAl₂O₄ (CA) – SrAl₂O₄ (SrA) and Ca₃Al₂O₆ (C₃A)-Sr₃Al₂O₆ (Sr₃A) [55].

Figure 7.

Phase relationship in the ternary system of CaO – SrO – Al₂O₃ [53].

The interesting fact is the stability of β-Sr₄A solid solution up to the melting temperature in the ternary system, while in the binary system SrO – Al₂O₃ the β-Sr₄A is stable only up to the temperature of 1575 °C (Fig.1).

2.5. Quaternary system CaO – SrO – Al₂O₃ – ZrO₂

The compounds formed in the system CaO – SrO – Al₂O₃ – ZrO₂ (Fig.8) are of great interest for the production of refractories due to high melting temperatures. This system contains 14 binary and ternary compounds, but the SrO – ZrO₂ system is very complicated and the literature contains contradictionary information on it. The connection lines in Fig.8 divide the quaternary system into the 21 elementary tetrahedra. There are two series of continuous solid solutions CaAl₂O₄ – SrAl₂O₄ and CaZrO₃ – ZrSrO₃ [55].

Figure 8.

Elementary tetrahedra of the quaternary system CaO-SrO-Al₂O₃-ZrO₂ (a) [55] with highlighted elementary tetrahedron including CA-SrA-SrZ region (b) favourable for the production of refractory binders [56].

The region CA – SrA – SrZ is considered as favourable for the production of special cements for refectory purposes. While strontium and calcium aluminates possess the bonding properties, the calcium and strontium zirconates have high melting temperatures. These cements are characteristic of water to cement ratio from 0.20 to 0.31 and rapid hardening [56].

3. Thermodynamics of SrO-Al₂O₃ system

3.1. Estimation of molar thermal capacity

The thermal capacity of solid inorganic compounds, which are formed from elements or simple solid compounds (Eq.40), can be estimated by the Neumann-Koop rule according to the relation 41. The method is applicable for intermetallic compounds and mixed oxides.

$$mA(s)+nB(s)\to A_m{B}_n(s)$$E40

$$c_{pm}^{\circ }(A_m{B}_n)=m{c}_{pm}^{\circ }(A)+n{c}_{pm}^{\circ }(B)$$E41

For example, the thermodynamic data of binary compounds from SrO – Al₂O₃ system (Fig.1) can be calculated as follows:

$$\begin{array}{l}{c}_{pm}^{\circ }({\mathrm{Sr}}_{3}A{l}_2{O}_6,298K)=3c_{pm}^{\circ }(\mathrm{SrO},298K)+c_{pm}^{\circ }({\mathrm{Al}}_2{O}_3,298K)\\ c_{pm}^{\circ }({\mathrm{SrAl}}_4{O}_7,298K)=3\cdot 45.241+79.029=214.752J\cdot mo{l}^{-1}{K}^{-1}\end{array}$$E42

The overview of results of other binary compounds and the comparison with experimental value is shown in Table 2. More methods can be found in literature. Kellogg [57] originally suggested a method for the estimation of heat capacity of predominantly ionic, solid compounds at 298 K. It is analogous to the Latimer's method for the estimation of standard entropies [58,59]. The method is based on the summation of contributions from the cationic and anionic groups in the compound

For example, the cationic contribution of Sr to heat capacity at 298 K is 25.52 J⋅K^-1. The anionic contribution for Al₂O₄ and Al₂O₆ is 98.52 and 135.46 J⋅K^-1, respectively. For strontium aluminate the heat capacities can be then calculated to be 25.52 + 98.52 = 124.04 J⋅mol^-1⋅K^-1. For tri-strontium aluminate the value 3 25.52 + 135.46 = 212.02 J⋅mol^-1⋅K^-1 at 298 K. can be found. The results show good agreement with data in Table 2.

.

Binary compounds	c pm° (25 °C); [J∙mol^-1K^-1]		A	∙10^-3 B	∙10^5 C	∙10^-6 D	Difference; [%]
Estimated ^(a)	Determined	c pm° = A + BT + C/T^2 + DT^2; [J.mol^-1K^-1]
4SrO∙Al2O3	259.686	253.659	308.958	36.507	-58.833	0.008	2.38
3SrO∙Al2O3	214.445	209.152	257.294	31.873	-51.243	-0.012	2.56
SrO∙Al2O3	123.963	119.030	177.192^(1)	4.937	-53.11	---	4.14
146.105^(2)	29.288	---	---
SrO∙2Al2O3	202.685	?	269.527^(e)	48.85	-71.815	-9.84	---
SrO∙6Al2O3	517.572	?	707.381^(e)	138.6	-203.60	-30	---

Table 2.

Estimation of molar thermal capacity of binary compounds from SrO-Al₂O₃ system from their oxides.

(1) Temperature range 298.15 – 932 K.

(2) Temperature range 932 – 2063 K.

(a) HSC software.

(e) Results based on experimental data are not available, constants A, B, C and D were determined from estimated data.

The temperature dependence of c_pm° of pure oxides in the SrO – Al₂O₃ system:

$$\begin{array}{l}{c}_{pm}^{\circ }(SrO,T)=46.047+14.163\cdot {10}^{-3}T-\frac{3.885\cdot {10}^5}{T^2}-7.4\cdot {10}^{-6}{T}^2\\ \left[298.15-900K\right]\end{array}$$E43

$$\begin{array}{l}{c}_{pm}^{\circ }(SrO,T)=83.372-36.439\cdot {10}^{-3}T-\frac{65.775\cdot {10}^5}{T^2}+12.175\cdot {10}^{-6}{T}^2\\ \left[900-1700K\right]\end{array}$$E44

$$c_{pm}^{\circ }(A{l}_2{O}_3,T)=109.474+22.414\cdot {10}^{-3}T-\frac{32.949\cdot {10}^5}{T^2}-4.155\cdot {10}^{-6}{T}^2$$E45

the c_pm° of binary compounds at required temperature can be calculated (Fig.9).

Figure 9.

Temperature dependence of thermal capacity of binary compounds from SrO – Al₂O₃ system (a-e) and dependence of c_pm° on the system phase composition at temperature of 298.15 K (f).

3.2. Estimation of enthalpy and entropy

Using the Hess´ Law, the concept of formation reaction is employed for the numerical determination of enthalpy changes. In general, the reaction of the formation of pure substance B from the elements (E_i) is:

$$B={\displaystyle \sum {\nu }_i{E}_i}$$E46

where ν_i is the stoichiometric coefficient. The standard enthalpy of formation if defined by the formula:

$$\Delta H_{f,B}^{\circ }(T)=H_{f,B}^{\circ }(T)-{\displaystyle \sum {\nu }_i{H}_{E,i}^{\circ }}$$E47

Standard enthalpies of elements in their reference phases at ambient standard conditions

Standard ambient conditions are defined in Chapter 3.3.

and their reference phases are set to zero (ΔH_f,i°=0) [303].

Various empirical methods were developed to estimate the enthalpy of formation of mixed oxides from either the elements or the constituent binary oxides [60-63]. The Aronson’s method [60], which is based on the Pauling’s relation between the enthalpy of formation and the differences between the electronegativities of elements forming the compound, is the most general one:

$$\Delta H_f^{\circ }=-96.5n_O{(X´-X_O)}^2\left[kJ\cdot mo{l}^{-1}\right]$$E48

where n_O is the number of oxygen atoms in the formula unit, X_O is the Pauling´s electronegativity of oxygen and X´is the weighted geometrical mean of so-called “pseudoelectro-negativities” of the oxide forming elements. The pseudoelectronegativity values are derived from known values of enthalpy of formation of relevant binary oxides. The enthalpy of formation of oxide can be then calculated as [61]:

$$X^{\prime }=X_O+\sqrt{\frac{-{\Delta }_{f}H}{96.5n_O}}$$E49

Another method, also based on the concept of electronegativity, was proposed by Zhuang et al. [62]. The method calculates the enthalpy of formation of binary oxides according to the relation:

$${\Delta }_f{H}^{\circ }=(n_1+n_2)x_1{x}_2\lambda$$E50

Where n_i and x_i are the number of moles and the mole fraction of i-th constituent of mixed oxide and λ is the constant similar to the interaction parameter used in regular solution model. Assuming the formation of double oxide from the two simple oxide species:

$$n_1{A}_a{O}_x+n_2{B}_b{O}_y\to n_1{A}_{a}X\cdot n_2{B}_{b}Y$$E51

Once the formation enthalpy is known for any double oxide n₁(A_aO_x) n₂(B_bO_y), the value of λ can be calculated from the formation enthalpy of double oxide with given stoichiometry and used for oxides of another stoichiometry. Otherwise, λ can be estimated using the following formula [61,62]:

$$\lambda =-96.5z{(X_A-X_B)}^2[kJ\cdot mo{l}^{-1}]$$E52

where X_A and X_B are the Pauling’s electronegativities for A and B elements and z is the stoichiometric factor (double number of oxygen atoms or ions given by oxide formula (Eq.51)) is given by the relation:

$$z=2\left(\frac{x}{a}+\frac{y}{b}\right)$$E53

A new method of estimation of the enthalpy of Al₂O₃ – Ln₂O₃ mixed oxides formation was derived by Voňka and Leitner [61]. The method is based on the Pauling’s concept of electronegativity and, in particular, on the relation between the enthalpy of formation of binary oxide and the difference between the electronegativities of the oxide-forming element and oxygen. This relation can be extended also for the calculation of enthalpy of other types of mixed oxides.

When the Pauling´s electronegativity of the central cations is ≥ 1.9, the method suggested by Šesták at al [64] enables to calculate the formation enthalpy of mixed oxide as follows:

$${\Delta }_f{H}^{\circ }=-29.288M$$E54

where M is the number of oxygen atoms or ions in the double oxide. The Pauling´s electronegativities of Al and Sr are 1.61 and 0.95, respectively. Therefore the assumption for Šesták method is not fulfilled in SrO – Al₂O₃ system.

Le Van [65] described the method based on the assumption of additivity of bond energies for the estimation of Δ_fH°:

$${\Delta }_f{H}^{\circ }(298K)=n_{-}P+n_{+}Q+4.184(4n_{+}^2)+4.184n_{-}^2[kJ\cdot mo{l}^{-1}]$$E55

where n₊and n_-denote the number of cations and anions, respectively. The P and Q are characteristic parameters

The characteristic values of Q(Sr²⁺) = -862 and P(Al₂O₄^2-) = -1494 enable to estimate the formation enthalpy of strontium aluminate to be -862 -1494 + [4.184 42] + 4.184 = -2285 kJ⋅mol^-1.

for cations and anions.

As was mentioned in Chapter 1.3.1, the Latimer's method of estimation of standard entropies (predominantly ionic compounds) uses the empirically found values of anion and cation contributions

The anionic and cationic contributions of Sr²⁺ and Al₂O₄^2- ions are 48.7 and 57 J⋅K^-1 mol^-1, respectively. The entropy of strontium aluminate can be then estimated to 52.9 + 57 = 109.9 J⋅K^-1 mol^-1. Using the contribution of O^2- for divalent cation (2.5 J⋅K^-1⋅mol^-1), the value of (3⋅52.9) + (2⋅2.5) + 57 = 220.7.

[58,59].

The comparison of experimental and calculated standard enthalpy of formation and the standard entropy for known binary compounds (mixed oxides) in the SrO-Al₂O₃ system is given in Table 3.

Binary componds	ΔHf,B° (298 K) [kJ∙mol^-1]		Sm° (298 K) [kJ∙K^-1mol^-1]
Determined	Estimated^(a)	Determined	Estimated^(a)
SrO	-591.999	---	55.580	---
4SrO∙Al2O3	-4118.725	-4407.794	322.570	283.947
3SrO∙Al2O3	-3544.681	-3717.308	255.375	229.160
SrO∙Al2O3	-2338.897	-2336.337	108.800	120.219
SrO∙2Al2O3	?	-3994.452	?	180.945
SrO∙6Al2O3	?	-5424.412	?	423.852
Al2O3	-1675.690	---	50.949	---

Table 3.

Standard enthalpy of formation and standard entropy for binary compounds in the SrO-Al₂O₃ system.

^(a) HSC software

Fig.10 (a) shows that estimated value of Δ_fH°(298 K) is quite different from the extrapolation of fitted experimental data.

Figure 10.

Estimated enthalpy (a) and entropy (b) of binary compounds in the SrO-Al₂O₃ system.

Therefore, the values expected from fitted dependence were applied for the calculation of temperature phase stability (Chapter 1.3.3). The estimated entropy (Fig.10 (b)) shows good agreement with predicted results (dashed line). The enthalpy and the entropy reach the maximum at the composition that corresponds to strontium aluminate.

The enthalpy and the entropy of species can be recalculated to required temperature using the following relations:

$$H(T)=H(298.15)+{\displaystyle \underset{298.15}{\overset{T_{\alpha \to \beta }}{\int }}{c}_{pm}^{\circ ,\alpha }}dT+\Delta H_{\alpha \to \beta }+{\displaystyle \underset{T_{\alpha \to \beta }}{\overset{T}{\int }}{c}_{pm}^{\circ ,b}}dT$$E56

$$S(T)=S(298.15)+{\displaystyle \underset{298.15}{\overset{T_{\alpha \to \beta }}{\int }}\frac{c_{pm}^{\circ ,\alpha }}{T}}dT+\frac{\Delta H_{\alpha \to \beta }}{T}+{\displaystyle \underset{T_{\alpha \to \beta }}{\overset{T}{\int }}\frac{c_{pm}^{\circ ,b}}{T}}dT$$E57

The definition law can be then used for the calculation of Gibbs energy as follows:

$$G(T)=H(T)-TS(T)$$E58

3.3. Temperature phase stability

There are five binary compounds in the phase diagram in Fig.1: 4SrO∙Al₂O₃, 3SrO∙Al₂O₃, SrO∙Al₂O₃, SrO∙2Al₂O₃ and SrO∙6Al₂O₃. The thermodynamic considerations enable to evaluate the stability of these phases at given temperature. For example, all compounds mentioned above are stable at the temperature of 600 °C. This question can be solved by the calculation of reaction Gibbs energy (Δ_rG°) related to the formation of compounds from simple oxides (SrO and Al₂O₃):

$${\mathrm{Al}}_2{O}_3+4SrO\to {\mathrm{Sr}}_{4}A{l}_2{O}_7$$E59

$$A{l}_2{O}_3+3SrO\to S{r}_{3}A{l}_2{O}_6$$E60

$${\mathrm{Al}}_2{O}_3+SrO\to {\mathrm{SrAl}}_2{O}_4$$E61

$$2A{l}_2{O}_3+SrO\to {\mathrm{SrAl}}_4{O}_7$$E62

$$6A{l}_2{O}_3+SrO\to {\mathrm{SrAl}}_{12}{O}_{19}$$E63

While the reactions 59-61 are exothermic (Δ_rH°< 0), the formation of SrAl₁₂O₁₉ is endothermic process at the temperature of 600 °C. Since Δ_rG° < 0 (Table 4), all these reactions lead to thermodynamically stable products. The question is, if the products are stable than basic oxides SrO and Al₂O₃.

Eq.	ΔrH°		ΔSm	ΔrG°		ΔrG°^(bo)	xSrO
[kJ·mol^-1]		[J·K^-1mol^-1]	[kJ·mol^-1]		[kJ·mol^-1]	[%]
59	-75.664	EXO	47.509	-117.147	Δ rG°< 0	-23.429	80.0
60	-94.403	34.681	-124.685	-31.129	75.0
61	-70.513	3.230	-73.333	-36.667	50.0
62	-54.211	17.517	-69.506	-46.337^(e)	33.3
63	10.342	ENDO	46.971	-30.670	-26.289^(e)	14.3

Table 4.

Reaction thermodynamics of the formation of binary compounds in the SrO-Al₂O₃ system at 600 °C.

^(bo) The reaction Gibbs energy recalculated to one mol of basic oxides (SrO+Al₂O₃)

^(e) Calculated from estimated thermodynamic data (Chapter 1.3.1 and 1.3.2.).

The value of Δ_rG° was recalculated to one mol of basic oxides Δ_rG°^(bo) and plotted as the function of x_SrO in the compound (Fig.11(a)). The value calculated for SrO⋅6Al₂O₃ phase is placed over dashed line. That means that the compound is unstable at given conditions and should be transformed into neighboring stable phase:

$${\mathrm{SrAl}}_{12}{O}_{19}\to {\mathrm{SrAl}}_4{O}_7+4A{l}_2{O}_3$$E64

The negative value of Δ_rG° (-63.321 kJ) of the process provides further evidence about the thermodynamical instability of SrO⋅6Al₂O₃ phase

Thermal treatment of raw meal (Chapter 4) shows that formation SrA₆ is preferred from the formation of SrA₂ due to reasons discussed in Chapter 1.5.

.

Figure 11.

Thermal stability of binary compounds in the SrO – Al₂O₃ system at the temperature of 600 °C (a) and 1200 °C (b).

3.4. Equilibrium composition in ternary system

The calculation of equilibrium composition of ternary system will be demonstrated with the example for the temperature of 1000°C. The first step of this solution includes the construction of ternary plot with the points corresponding to the composition of all known phases (Fig.12). Now it is possible to draw all possible connecting lines between all points. The intersectional points present the chemical reaction of corresponding phases.

Figure 12.

Possible chemical reactions in the ternary system CaO-SrO-Al₂O₃.

For example, the intersectional point R1 in Fig.12, corresponds to the chemical reaction, which can be expressed by usual notation in cement chemistry as follows:

$${\mathrm{Sr}}_{4}A+3C\stackrel{?}{\leftrightarrow }{C}_{3}A+4Sr$$E65

The decision, which pair of compounds is stable under given conditions (are the species on the left side of Eq.65 or the products on the right side stable?). This decision is based on the standard Gibbs energy of reactions (Δ_rG°) at given temperature:

Δ_rH°(Eq.65, 1000°C)=67 580 J (Δ_rH° > 0 → endothermic reaction)

Δ_rS°(Eq.65, 1000°C)=-7.531 J K^-1

Δ_rG°(Eq.65, 1000°C)=67 580+(1273,13 7.531)=77 168 J

$${\mathrm{Sr}}_{4}A+3C\stackrel{1000°C}{\leftarrow }{C}_{3}A+4Sr$$E66

Therefore if Δ_rG° > 0, the products of reaction according to Eq.65 are thermodynamically unstable under given temperature (Eq.66) and the connection line between C₃A and Sr must be withdrawn from the diagram. These calculations should be repeated until all interconnection points are solved. The ternary diagram in Fig.13 shows the equilibrium phase composition of Al₂O₃ – SrO – CaO system calculated for the temperature of 1000°C.

Figure 13.

Phase stability in the ternary system Al₂O₃ – SrO – CaO at the temperature of 1000°C.

The equilibrium ternary diagram in Fig.13 can be a useful tool to predict the phase composition of the mixture of raw materials under the thermal treatment, e.g. by thermal treatment of mixture, the composition of which corresponds to the point A (50 % Al₂O₃, 25 % SrO and 25 % CaO). This composition provides the system consisting of SrA, C₂SrA and CA₆ in the ratio 1:1.5:1. It is also possible to assess the ratio in which the constituents should be mixed in order to obtain the mixture of required composition. For example, in order to get the system A from the systems B and C, they must be mixed in the ratio |AC| : |BA|=x_B : x_C=1:1.55.

Another example is the system D prepared in the corresponding ratio of systems X, Y and Z, which, after the thermal treatment to the temperature of 1000 °C provides compounds SrA. Sr₃A and C₂SrA in the ratio 2.3:1:1.

4. System with chemical reaction

The number of independent chemical reactions and their stoichiometry can be derived from the Gibbs Stoichiometric Law. The system is assumed, which contains N chemical compounds consisting of M elements (C₁, C₂... C_N) where the following reaction takes place:

$${\displaystyle \sum _{i=1}^N{\nu }_i{C}_i}=0$$E67

where ν_i are the stoichiometric coefficients. The reaction can be expressed in the form of matrix equation:

$${\nu }^T\text{[}1,N\text{]}C\left[N,1\right]=0$$E68

where ν^T is the stoichiometric coefficient matrix (1 row, N column) and C is the species matrix (N row, 1 column). The composition of chemical compounds can be expressed as:

$${\displaystyle \sum _{j=1}^M{A}_{ij}{B}_j}=C_i$$E69

Where A_ij is the constitution coefficient related to the amount of atoms of j-element in the molecules of i-compounds. Therefore, it can be written:

$$A^T\text{[}N,M\text{]}B=C$$E70

Where A^T is the transposed matrix of constitution coefficient (N row, M column). The combination of Eq.70 and Eq.68 gives the following formula:

$$A^T\text{[}N,M\text{]}B=C$$E71

$$B^{T}A\nu =0$$E72

Since B ≠ 0 and thus B^T ≠ 0, the following term must be fulfilled:

$$A[M,N]{\nu }_N=0$$E73

For example we may consider the following system consisting of seven compounds and five elements:

N=7: SrCO₃, SrO, Al₂O₃, Al(OH)₃, SrAl₂O₄, CO₂ and H₂O;

M=5: C, Sr, Al, O and H.

	ν1	ν2	ν3	ν4	ν5	ν6	ν7
SrCO3	SrO	Al2O3	Al(OH)3	SrAl2O4	CO2	H2O
C	1	0	0	0	0	1	0
Sr	1	1	0	0	1	0	0
Al	0	0	2	1	2	0	0
O	3	1	3	3	4	2	1
H	0	0	0	3	0	0	2

Table 5.

Constitution coefficients in reaction system.

Via the substitution from the table of constitution coefficients (Table 5) to Eq.73, the following expression should be derived:

$$\left[\begin{array}{ccccccc}1& 0& 0& 0& 0& 1& 0\\ 1& 1& 0& 0& 1& 0& 0\\ 0& 0& 2& 1& 2& 0& 0\\ 3& 1& 3& 3& 4& 2& 1\\ 0& 0& 0& 3& 0& 0& 2\end{array}\right]\left[\begin{array}{c}{\nu }_1\\ {\nu }_2\\ {\nu }_3\\ {\nu }_4\\ {\nu }_5\\ {\nu }_6\\ {\nu }_7\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\end{array}\right]$$E74

The order of matrix A [M,N] can be now determined

Except for the switch of the row with column (order of column must remain unchanged), all usual mathematical operation can be used. These calculation includes: The change of the order of row (1), The multiplication of row by a constant (2), The summation of row with other row or their linear combination (3), The exclusion of row which is a linear combination of other rows (e.g. last row in Eq.75).

:

$$\left[\begin{array}{ccccccc}1& 0& 0& 0& 0& 1& 0\\ 1& 1& 0& 0& 1& 0& 0\\ 0& 0& 2& 1& 2& 0& 0\\ 3& 1& 3& 3& 4& 2& 1\\ 0& 0& 0& 3& 0& 0& 2\end{array}\right]\Rightarrow \left[\begin{array}{rrrrrrr}\hfill 1& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill 1& \hfill -1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 0& \hfill 1& \hfill 0& \hfill 0& \hfill \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\\ \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0& \hfill 0\end{array}\right]$$E75

The last row of matrix A is a liner combination of other rows and the order of matrix of constitution coefficients (h) is equal to 4. From the Gibbs Stoichiometric Law the following can be now calculated:

$$R=N-h=7-4=3$$E76

Therefore, there are three independent chemical reactions in the system. Eq.74 and the solution 75 lead to the following results:

$${\nu }_1+{\nu }_6=0$$E77

$${\nu }_1+{\nu }_2-{\nu }_6=0$$E78

$${\nu }_3+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\nu }_4+{\nu }_5=0$$E79

$${\nu }_4+\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\nu }_7=0$$E80

The stoichiometry of these processes can be calculated choosing three parameters:

1. The choice ν₇=1, ν₆=0 and ν₅=0 leads to the solution:

$${\nu }_4=-\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$$E81

$${\nu }_3=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$$E82

$${\nu }_2=0$$E83

$${\nu }_1=0$$E84

$$\begin{array}{l}{H}_{2}O+\frac{1}{3}A{l}_2{O}_3=\frac{2}{3}\mathrm{Al}{(}_O\\ 3H_{2}O+{\mathrm{Al}}_2{O}_3=2\mathrm{Al}{(}_O\end{array}$$E85

From the calculation of the reaction Gibbs energy it can be estimated, that the reaction 85 proceeds to right side (product) if the temperature is lower than 113 °C (Δ_rG < 0) and further, the dehydration of gibbsite takes place if the temperature is higher (Δ_rG > 0).

1. The choice ν₇=0, ν₆=1 and ν₅=0 leads to the solution:

$${\nu }_4=0$$E86

$${\nu }_3=0$$E87

$${\nu }_2=1$$E88

$${\nu }_1=-1$$E89

$${\mathrm{SrCO}}_3=\mathrm{SrO}+C{O}_2$$E90

This reaction reaches the equilibrium state at the temperature of 1071°C (Δ_rG=0).

1. The choice ν₇=0, ν₆=0 and ν₅=1 enables to find the last independent reaction in the system where ν₄=0, ν₃=0, ν₂=1 and ν₁=-1:

$$\mathrm{SrO}+A{l}_2{O}_3={\mathrm{SrAl}}_2{O}_4$$E91

The formation of strontium aluminate from the oxide species (Eq.91), the thermal decomposition of gibbsite (Eq.85) and strontium carbonate (Eq.90) are three independent reactions in the above mentioned system. The solution represents the base vectors (ν₁,ν₂,ν_3,) of 3D reaction space (N-h=3) which come through the beginning of 7D (N) vector space:

Eq.85: (001);

Eq.90: (-110);

Eq.91: (0-1-1).

All other solutions are their linear combinations. All these reactions also increase the value of parameter C in Eq.1.9.

The system base consists of first (N-h) species in Table 5. Therefore, we can choose different order of species and solve different three independent reactions. The number of possible system bases in this system can be calculated from the equation:

$$\left(\frac{N}{h}\right)=\frac{N!}{h!(N-h)!}=\frac{7!}{4!3!}=35$$E92

For example, if we rearrange the system as follows (system base is marked bold):

SrAl₂O₄, SrO, H₂O, CO₂, Al(OH)₃, SrCO₃, Al₂O₃,

the following independent reaction is determined:

1. The choice ν₇=1, ν₆=0 and ν₅=0 leads to the solution:

$${\mathrm{SrAl}}_2{O}_4=\mathrm{SrO}+A{l}_{2}O{}_3$$E93

1. The choice ν₇=0, ν₆=1 and ν₅=0 leads to the solution:

$$\mathrm{SrO}+C{O}_2={\mathrm{SrCO}}_3$$E94

1. The choice ν₇=0, ν₆=0 and ν₅=1 leads to the solution:

$$\begin{array}{l}\frac{1}{2}SrA{l}_2{O}_4+\frac{3}{2}{H}_{2}O=\frac{1}{2}\mathrm{SrO}+Al{(}_O\\ {\mathrm{SrAl}}_2{O}_4+3H_{2}O=\mathrm{SrO}+2Al{(}_O\end{array}$$E95

But not all possible system bases are valid; the stoichiometry of independent reaction can be calculated only from the system base that contains all elements occurring in the system. For example the choice:

SrAl₂O₄, SrO, Al₂O₃, CO₂, Al(OH)₃, SrCO₃, H₂O;

is not acceptable, because the compounds in the system base do not contain hydrogen.

In order to better explain the geometric meaning of the Gibbs Stoichiometric Law, the following simple reaction system is applied:

• N=3: SrO, Al₂O₃ and SrAl₂O₄.

• M=3: Sr, Al and O.

$$\begin{array}{cc}& \begin{array}{ccc}{\nu }_1& {\nu }_2& {\nu }_3\end{array}\\ \begin{array}{c}Sr\\ Al\\ O\end{array}& \left[\begin{array}{ccc}1& 0& 1\\ 0& 2& 2\\ 1& 3& 4\end{array}\right]\end{array}\begin{array}{c}\\ \left[\begin{array}{c}{\nu }_1\\ {\nu }_2\\ {\nu }_3\end{array}\right]\end{array}=\begin{array}{c}\\ \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]\end{array}$$E96

$$\left[\begin{array}{ccc}1& 0& 1\\ 0& 2& 2\\ 1& 3& 4\end{array}\right]\Rightarrow \dots \Rightarrow \left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 0& 0& 0\end{array}\right]\Rightarrow h=2$$E97

$$R=N-h=3-2=1$$E98

Therefore we search for only one independent reaction, where ν₃=1:

$${\nu }_2=-{\nu }_3=-1$$E99

$${\nu }_1=-{\nu }_3=-1$$E100

$$\mathrm{SrO}+A{l}_2{O}_3=SrA{l}_2{O}_4$$E101

The results represent the coordinates vector (-1-11) related to the line passing through the beginning P[0,0,0] of 3D reaction space (Fig.14). As results from Eq.92, there are three options for the system base.

Figure 14.

Geometric drawing of the solution resulting from the Gibbs Stoichiometric Law.

5. Density of cement phases

The densities of cement phases are fundamental parameter with many uses, such as calculating the space filling in pastes by solid cement substances as well as the assessment of the potential for changes of size and porosity within hardened pastes undergoing the phase changes. The substances containing cement can be divided into two groups: crystalline and amorphous. The principal amorphous substance of Portland cement is C–S–H. It poses certain problems in determining its density as it is insufficiently crystalline for the application of conventional crystallographic methods: only experimentally-determined measurements are useful. Crystalline solids, on the other hand, are amenable for the calculation of density from crystallographic constants using the relationship [66]:

$$\rho =\frac{ZM}{N_{A}V}[kg\cdot m^{-3}]$$E102

where ρ is the calculated density, Z is the number of formula units per unite cell, M is the formula weight, V is the unite cell volume (V=M/ρ) and N_A=6.02214 10²³ mol^-1 is the Avogadro´s number.

The volumes of the unit cell can be calculated as follows:

• Isometric /cubic crystal system: V=a³;

• Tetragonal crystal system: V=a²c;

• Orthorhombic crystal system: V=abc;

• Hexagonal /trigonal crystal system: V=a²c sin (60°);

• Monoclinic crystal system: V=abc sin (β);

• Triclinic crystal system: V=abc sin ((1-cos²α – cos²β – cos²γ)+2(cos α cos β cos γ))^1/2

where a, b, c are the unit cell axial dimensions and α,β, γ are the relevant angles. The parameters of basic seven lattice systems are shown in Fig.15.

Phase in SrO-Al2O3 system	Sr3A		SrA		α-SrA2
3SrO·Al2O3	SrO·Al2O3	SrO·2Al2O3
Crystal system	Cubic		Monoclinic		Monoclinic
M [kg·mol^-1]	0.41282		0.20558		0.30754
Z	24		4		4
a [m]	1.586·10^-9		5.150·10^-10		1.304·10^-9
b [m]	---		8.836·10^-10		9.011·10^-10
c [m]	---		8.442·10^-10		5.536·10^-10
β [°]	---		93.43		106.12
V [m^3]	3.986·10^-27		3.384·10^-28		6.249·10^-28
ρ [kg.m^-3]	4127		3561		3269
Phase in SrO-Al2O3 system	β-SrA2		SrA6		Sr10A3
SrO·2Al2O3	SrO·6Al2O3	10SrO·3Al2O3
Crystal system	Orthorhombic		Hexagonal		Monoclinic
M [kg·mol^-1]	0.30754		0.71539		1.34208
Z	4		2		8
a [m]	8.085·10^-10		5.562·10^-10		3.558·10^-9
b [m]	1.185·10^-9		---		7.846·10^-10
c [m]	4.407·10^-10		2.197·10^-9		1.575·10^-9
β [°]	---		60		103.68
V [m^3]	4.220·10^-28		5.886·10^-28		4.272·10^-27
ρ [kg.m^-3]	4840		4036		4174

Table 6.

Calculation of the density of strontium aluminate cement phases.

Some examples including calculated volume and the density of compounds from SrO – Al₂O₃ system are given in Table 6.

Figure 15.

Parameters of basic lattice system.

Calculated data from Table 6 enable to explain how the density of compounds formed in the binary system of SrO-Al₂O₃ depends on the SrO/ Al₂O₃ ratio. Fig.16(a) shows that there are two exponential functions, which are cross each other in the composition corresponding to the SrA₂ phase (please consult with Fig.11).

Figure 16.

The influence of Sr/A ratio on the density (a) and its linearized dependence (b).

These exponential functions should be linearized as is shown in Fig.16(b). It is also a possible reason, why the formation of SrA₆ phase is preferred from the formation of SrA₂; lesser energy is required for the deformation of surroundings by newly formed crystal. Therefore the phase should be prepared via precipitation form melt, but the solid state synthesis is difficult [67,68].

6. Kinetics of heterogeneous processes

Since the thermal decomposition of solids is the most significant field of heterogeneous kinetics, the common analytical techniques include the methods such TG, DTA, DSC, TDA, etc., but other techniques could be also used by kinetic methods. These measurements are usually applied for the investigation of kinetics and mechanism of heterogeneous processes using the data measured as the function of time or temperature [69,70].

6.1. Mechanism dependent methods

Mechanism dependent or reaction model fitting method should be divided into two large groups [63,67]:

• Isothermal method;

• Non-isothermal method.

Both use the reaction models, which describe the dependence of degree of conversion (fractional conversion or the extent of conversion, α) on time (isothermal method) or temperature (non-isothermal method). The reaction model usually refers to the rate limiting step of process and the reaction system geometry (Table 7)

The fractional conversion is usually defined as the ratio of some system additional properties (mass, length, area...), the change of volume or mass of the system are mostly used:

$$\alpha =\frac{N_0-N_t}{N_0-N_{\infty }}$$E103

where N₀, N_t and N_∞ are the initial value, the value reached in time t and the final value, respectively. The value α is normalized in the range from 0 to 1 (or 0 to 100 %).

The following differential kinetic equation was used for the description of kinetics of investigated process via the isothermal method [71-73,456].

$$\frac{d\alpha }{dt}=k(T)\cdot f(\alpha )$$E104

The integration of Eq.104 leads to the formula:

$${\displaystyle \underset{0}{\overset{\alpha }{\int }}\frac{d\alpha }{f(\alpha )}}=k\cdot {\displaystyle \underset{t=0}{\overset{t}{\int }}dt}\Rightarrow g(\alpha )=kt$$E105

where k is the constant of reaction rate. In the case that correct kinetic function was applied, the plot of g(α) versus t provides straight line. The rate constant k can be then calculated as the slope of this plot.

The value of k for several temperatures must be determined for the calculation of activation energy (E_a) and frequency factor (A) via the logarithmic form (Eq.107) of Arrhenius law (Eq.106):

$$k=A\exp \left(-\frac{E_a}{RT}\right)$$E106

$$\ln k=\ln A-\frac{E_a}{R}\frac{1}{T}$$E107

where R denotes the universal gas constant (8.314 J∙K^-1∙mol^-1) and other symbols have their usual meaning. The plot (so-called Arrhenius plot) of ln k versus T^-1 provides straight line with the slope -E_a/R and the intercept with y-axis equal to ln A. Several experiments should be performed in order to obtain the line with insufficient number of points, which enables to determine reliable slope of the Arrhenius plot (please see Fig.18 in Chapter 4 for example of Arrhenius plot).

Sign.	f(α)	g(α) = kt	Name of kinetic function	Rate limiting step		Description
F 1/3	(3/2) (1-α)^1/3	1-(1-α)^2/3	One-third order	Chemical reaction		Chemical process or mechanism non-invoking equations
F 3/4	4 (1-α)^3/4	1-(1-α)^1/4	Three-quarters order
F 3/2	2 (1-α)^3/2	(1-α)^-1/2 -1	One and half order
F 2	(1-α)^2	(1-α)^-1 -1	Second order
F 3	(1/2) (1-α)^3	(1-α)^-2 -1	Third order
P 3/2	(2/3) α ^-1/2	α ^3/2	Mampel power law	Nucleation		Acceleratory rate equations
P 1/2	2 α ^1/2	α ^1/2
P 1/3	3 α ^2/3	α ^1/3
P 1/4	4 α ^3/4	α ^1/4
E 1	α	ln α	Exponential law
A 1, F 1	1-α	-ln(1-α)	KJMA or JMAYK; (Johanson-Mehl-Avrami-Yerofeyev-Kolgomorov equation)	Random nucleation	n =1	Sigmoidal rate equations or random nucleation and subsequent growth
A 3/2	3/2 (1-α)[-ln(1-α)]^1/3	[-ln(1-α)]^2/3	n=3/2
A 2	2 (1-α)[-ln(1-α)]^1/2	[-ln(1-α)]^1/2	n=2
A 3	3 (1-α)[-ln(1-α)]^2/3	[-ln(1-α)]^1/3	n=3
A 4	4 (1-α)[-ln(1-α)]^3/4	[-ln(1-α)]^1/4	n=4
A u	α (1-α)	ln[α/(1-α)]	Prout-Tomkins equation	Branching nuclei
R 1	(1-α)^0	α	Power law	Contracting disc		Phase boundary deceleratory rate equations
R 2	2 (1-α)^1/2	1 -(1-α)^1/2	Contracting cylinder
R 3	3 (1-α)^2/3	1 -(1-α)^1/3	Contracting sphere
D 1	1/2 α	α^2	Parabola law	1D diffusion		Diffusion limited deceleratory reaction
D 2	[-ln(1-α)]^-1	α + (1-α)ln(1-α)	Valensi equation	2D diffusion
D 3	3/2(1-α)^2/3[1-(1-α)^1/3]^-1	[1 -(1-α)^1/3]^2	Jander equation	3D diffusion
D 4	3/2[(1-α)^-1/3-1]^-1		Ginstling-Brounstein equation
D 5	3/2(1-α)^4/3[(1-α)^-1/3-1]^-1		Zhuravlev, Lesokin, Tempelman eq.
D 6	3/2(1+α)^2/3[(1+α)^1/3-1]^-1		anti-Jander eq.
---	3/2 [(1+α)^1/3-1]^-1		anti- Ginstling-Brounstein eq.
---	3/2(1+α)^4/3[(1+α)^-1/3-1]^-1		anti- Zhuravlev, Lesokin, Tempelman eq.

Table 7.

Overview of known kinetic functions representing applied kinetic model for the process.

The non-isothermal method enables to determine the kinetic parameters of the process from one experiment

As was demonstrated in Chapter 4.2.1.

, therefore it is sometime termed as the single run method. The non-isothermal method is based on the assumption that the kinetics of processes should be described by the constitutive formula Eq.104. The function f(α) depends on applied kinetic model (Table 7) and the temperature dependence of the rate constant k(T) is given by the Arrhenius law (Eq.106).

If the constant heating rate (CHR methods) Θ is applied, i.e. dT=Θ dt, the combination of Eqs.106 and 107 leads to the relationship [73-75,87]:

$$\frac{d\alpha }{f(\alpha )}=\frac{A}{\Theta }\cdot \exp \left(-\frac{E_a}{RT}\right)dT$$E108

The integration of Eq.108 yields to the formula:

$$g(\alpha )\cong \frac{A{E}_a}{\Theta R}\cdot p\left(x\right)$$E109

where the function g(α) represents the kinetic function and A, E_a, R and Θ are the temperature independent parameters. The determination and accurate calculation of the p(x) function is a marginal problem in the non-isothermal kinetic [64]. The Coats and Redfern approximation of the p(x) function is usually used [76].

$$\ln \left(\frac{g(\alpha )}{T^2}\right)=\left[\ln \frac{AR}{\Theta E_a}\left(1-\frac{2RT}{E_a}\right)\right]-\frac{E_a}{R}\cdot \frac{1}{T}\cong \ln \left(\frac{AR}{\Theta E}\right)-\frac{E_a}{R}\cdot \frac{1}{T}$$E110

A straight line was obtained by plotting ln[g(α)/T²] versus reciprocal temperature (T^-1) for correct mathematical model g(α) of the reaction mechanism. The overall activation energy and pre-exponential factor were calculated from the slope and the intercept with y-axis of the plot, respectively.

The expression of p(x) according to Schlomilch can be written as [77,456]:

$$p(x)=\exp \left(-\frac{E_a}{RT}\right)\left(\frac{1}{x}\right)\left(\frac{1}{x+2}\right)$$E111

6.2. Model free methods

The Kissinger`s kinetic approach is simple and often applied kinetic equation [78,79]:

$$\ln \left[\frac{\Theta }{T_m^2}\right]=\ln \left[\frac{AR}{E_a}n{(1-{\alpha }_m)}^{n-1}\right]-\frac{E_a}{R{T}_m}=const.-\frac{E_a}{R{T}_m}$$E112

where T_m is the peak temperature measured under applied heating rate Θ, n is the empirical reaction order (kinetic exponent), α_m is the fractional conversion reached for the temperature T_m and R is the universal gas constant. The plot of ln (AR/ E_a) versus T_m^-1 was fitted by the straight line with the slope equal to-E_a/R whereas the intercept yielded to the constant term of Eq.112.

The peak methods approximate the fixed state of reaction to the stage at which the maximum rate of the process is achieved, i.e. to the peak of thermoanalytical curve. It must be pointed out that the Kissinger approach can be used for the rate (differential)-isoconversion (Friedman or FR-[80]) type as well as for the p(y)-isoconversion methods in principle, but the method’s popularity place the Eq.112 into a group of maximum rate methods [81].

6.3. Estimation of the mechanism of process

The mechanism was estimated from the shape of DTG peak via the value of kinetic exponent (n) which was related to the empirical order of reaction [78,82] The exponent can be calculated from the equation [83,84]:

$$n=\frac{2.5R{T}_m^2}{w_{1/2}{E}_a}$$E113

where w_1/2 is the half-width (width at a half high) of peak. The value of kinetic exponent is typical for various mechanisms of investigated process [85].

Based on the foundation that the peak asymmetry increases with decreasing value of n Kissinger [78] has proposed the method that uses the shape index (SI) of TA curve peak that can be analytically expressed as:

$$SI=\frac{{\left({\partial }^{2}N/\partial t^2\right)}_{T_1}}{{\left({\partial }^{2}N/\partial t^2\right)}_{T_2}}$$E114

where N denotes the value of measured property, t is the time, T₁ and T₂ are the first (frontal) and second (terminal) inflection point, respectively. The shape index is only the function of reaction order.

The value of kinetic exponent can be then calculated according to the first [78] and the second Kissinger approach [76]:

$$n_{\text{K}1}=1.26S{I}^{1/2}$$E115

$$n_{\text{K}2}=1.88SI$$E116

6.4. Thermodynamic consideration

The correlation between kinetic and thermodynamic parameters of the investigated process results from the combination of Arrhenius with Eyring or Wertera and Zenera laws related to the temperature dependence of rate constant (k(T)) [86]:

$$\begin{array}{l}k(T)=A\exp \left[-\frac{E_a}{RT}\right]=\frac{k_{B}T}{h}\exp \left[\frac{\Delta S^{\#}}{R}\right]\exp \left[-\frac{\Delta H^{\#}}{RT}\right]\\ =\nu \exp \left[-\frac{\Delta G^{\#}}{RT}\right]=\nu K^{\#}\end{array}$$E117

where k_B, h and ν=k_BT/h are the Boltzmann, the Plank constant and the vibration frequency, respectively. The thermodynamic parameters of activated complex, including free energy (ΔG^#), enthalpy (ΔH^#) and entropy (ΔS^#) of the process were calculated using Eyring equations [86,87,88]:

$$\Delta H^{\#}=E_{a,\alpha }-R{T}_{\alpha }$$E118

$$\Delta S^{\#}=R\left[\ln \left(\frac{h{A}_{\alpha }}{k_B{T}_{\alpha }}\right)-1\right]$$E119

$$\Delta G^{\#}=\Delta H^{\#}-T_{\alpha }\Delta S^{\#}$$E120

The thermodynamic parameters of activated complex are often calculated using the peak temperature T_m so that the value of ΔG^#, ΔH^# and S^# is related to the highest rate of process.

6.5. Calculation of theoretical value of activation energy

The approach is known as the “Congruent dissociative vaporization mechanism” (CDV). In the case of a solid compound S decomposed into gaseous products A and B with simultaneous condensation of low-volatility of species A, that is [89,369]:

$$S(s)\to aA(g)\downarrow +bB(g)$$E121

The theoretical value of activation energy E_a^T should be different for equimolar (in the absence of gaseous product B in the reactor atmosphere) and isobaric (in the presence of the excess of gaseous product B) modes of decomposition. The relations for the equimolar (E_a^Te) mode are:

$$E_a^{Te}=\frac{{\Delta }_{rc}{H}^{\circ }(T)}{\nu }=\frac{{\Delta }_r{H}^{\circ }(T)}{a+b}$$E122

and for the isobaric mode (E_a^Ti) they are:

$$E_a^{Ti}=\frac{{\Delta }_{rc}{H}^{\circ }(T)}{\nu -b}=\frac{{\Delta }_r{H}^{\circ }(T)}{a}$$E123

$${\Delta }_r{H}^{\circ }(T)={\displaystyle \sum {\nu }_i}{\Delta }_{f}H{}_i{}^{\circ }$$E124

ν denotes total number of moles of gaseous product (a+b) and Δ_rH° is the reaction enthalpy for given temperature (Eq.124). The temperature dependence of enthalpy is given by Eq.21 in Chapter 4. In both cases, the E_a^T parameter corresponds to the specific enthalpy, i.e. the enthalpy of the decomposition reaction reduced to one mole of primary products without including components present in excess.

In order to take into account the partial transfer of energy released in the condensation of low-volatility product A to the reactant, the calculations of enthalpy of decomposition reaction 121 require an additional term (τ a Δ_cH°(A,T)) where the coefficient τ corresponds to the fraction of condensation energy transferred to the reactant at the interface. The reaction enthalpy is then given by the relation:

$$\begin{array}{l}{\Delta }_{rc}{H}^{\circ }(T)=a{\Delta }_f{H}^{\circ }(A,T)+b{\Delta }_f{H}^{\circ }(B,T)\\ -c{\Delta }_f{H}^{\circ }(C,T)+\tau a{\Delta }_c{H}^{\circ }(A,T)\end{array}$$E125

For the majority of substances, the condition τ=0.5 can be applied [89,90].

An essential difference between the CDV mechanism and the Arrhenius activation mechanism is that during the interface reactions, a proportion of the energy released on condensation of a non-volatile product is transferred to the solid reactant, reducing the energy barrier for further reactant volatilization. Thus, ‘recycled’ energy is responsible for the autocatalytic behaviour, justifying the following important generalizations [70]:

• Models identifying the preferred occurrence of chemical change at reactant/product contact interfaces with (uncharacterized, qualitative) ‘strain’, ‘catalysis of by product’, etc., can now be discarded as providing no insights into the reaction controls and mechanisms.

• Autocatalytic behaviour, resulting from the redistribution of product condensation energy, occurs when the decomposition proceeds at a reactant/product contact interface. Reactions yielding non-condensed (non-volatile) product transfer no energy, so that τ=0.

• Because the energy transfer is responsible for the autocatalysis, the variations in τ, found for a number of diverse substances, are identified with supersaturation of the non-volatile component rather than the chemical properties of the different original reactants.

The simplest presumption for the energy redistribution at an interface is that the condensation energy is shared equally between the reactant and the solid products, which is expressed as τ=0.5. The deviations, where Δ_cH° is distributed unequally between the solid reactant and the product phases in the ratio τ/(1-τ), are ascribed to the degree of supersaturation of the non-volatile vapor.

The Arrhenius model is often represented by the familiar graph of energy variations as the reaction progresses by the “Advance along the Reaction Coordinate”. This shows an initial rise to a maximum value, to form the ‘transition complex’, being followed by a decline thereafter. The activation energy is then the energy required for forming the ‘activated’ transition complex in an assumed “rate-determining step”. However, in CDV theory, the value of parameter E represents the vaporization enthalpy [70].