TERMODYNAMICAL ASPECTS OF NUCLEI FORMATION AND STRUCTURE FORMATION OF METALS DURING REDUCTION OF OXIDES

L.Leontiev, S.Shavrin1, G.Maizel and V.Gorbachev2

1 Ural’s Department of Russian Academy of Sciences, S.Kovalevskoi st. 4, 620219, Ekaterinburg, Russia.

2 Scientific-industrial-inculcation enterprise TOREX, Studencheskaya st., 16, 620219, Ekaterinburg Russia.

ABSTRACT

In the work is proposed the thermodynamical description of nuclei formation in the system of finite dimensions, in difference with the classical theories of formation where the system out of the nuclei is infinite.

It is shown that taking into account the finite dimensions of the system allows to determine the mutual arrangement of nuclei (numerous) ensemble, taking into account the phase transformation mechanism peculiarities and it’s kinetic characteristics (diffusion flows) and reaction product structure. Fs an example the analysis of products structure under redaction – oxidation processes in metals oxides is carried out.

  1. Application of classical concepts to formation analysis.

Thermodynamic analysis of formation in the system steam-liquid founded on the works of Gibbs is based on the assumptions range. The main assumption, according to the view point of the most researchers, is the assumption about possibility of properties description of a small drop – nucleus with the help of the conforming parameters of liquid big mass.

As a result more precise solutions of one nucleus appearance taking into account surface properties of a drop and liquid big mass, additional degree of drops freedom etc., that are usually utilized in building kinetic models of some chemical reactions passing (1,2). Important parameters, determining the reaction speed, in kinetic theories are formation activation energy G and critical size (rc) nucleuses concentration. However, the mathematical description of the chemical reaction passing in the way of many nucleuses appearance and growth analysis is based on solving of one nucleus appearance task when the system statement outside the phases separation border is unchangeable.

The correctness of such a transition, from our point of view, needs additional consideration because in practice the not transformed space area surrounding each nucleus is finite and the possibility of its properties change during the given nucleus appearance disregard is not quite obvious. That is why in the present work the attempt is made to analyze the formation in the system having the finite size.

Without harm to the community let us consider the formation of one nucleus in the simplest system steam-liquid. For this purpose let us imagine the closed volume that contains N particles of steam with temperature T and pressure P.

If the steam pressure P exceeds the pressure P0 for the steam that is in balance with the liquid having a flat surface, the formation is thermodynamically permitted and radius r of the nucleus, balanced with the initial steam, will be determined by the difference of P and P0 values. This relation is described by Thomson formula (3)

/ (1)

in which vl – volume on one particle in liquid statement;  - specific surface energy. Though the conclusion of the last relation contains some incorrectnesses, pointed out in the monograph (1), their role is not so significant as to distort the phenomenon essence when utilizing it.

Graphically this relation for T 300 K and P0 = 17,5 mm of mercury column (fig. 1, curve 1) formally shows the conformity between the drop radius and the surrounding steam pressure. In the closed finite system the appearance of the nucleus containing n particles leads to the particles number decrease in the gas phase and accordingly to the pressure decrease by the value P:

The remained steam has the pressure not P but P- P depending on the nucleus size:

P(N-n)=P(N)-An/V / (2)

Where A – constant equal 1/3 mc2 (m- particle mass, c – its speed in gas).

To find the balance condition between the remained steam and drop we use both Thomson formula (1), and (2) which for the convenience of comparison we represent in the following form: P(r)=P-A'r3, because the particles number in the drop n r3 (A'=mc2/3V , V – the system volume).

The graphic solution of the equations (1) and (2) system is presented on fig.1 (accordingly 1 and 2). It can be seen that the balance condition in the finite system may be realized two times – in points A and B on the curve 1. Accordingly in the system containing in the initial statement N particles, nucleuses of two sizes (r1 and r2) may be in balance with the remained steam. This principal difference from classical models determining one nucleus appearance ( with the size rc on fig.1) in the nonfinite system. Nucleus of the r1 size is in unstable balance with the surrounding steam.

It is interesting to determine the system sizes influence (i.e. N) and also the influence of the initial saturation degree on the value and relation of the balanced nucleuses sizes (r1 and r2 ). If N is large ( N = N1 ) the nucleus formation leads to insignificant decrease of the remained steam pressure. In this case the size r1 of the nucleus being in unstable balance insignificantly differs from the classical rс , and in the limit with N is equal to it (fig.2).

So, for each saturation degree in the system there is threshold value of the particles number N below which the formation is principally impossible. These are the unbalanced systems to which we can apply the term “ metastable system”.

The same regulations can be watched in changing of the saturation degree (pressure P). On fig.3 it is seen that with the P decrease the balanced nucleuses sizes values draw together under P = P3 , r = r2 , and under P < P3 the formation is impossible because the drop appearance leads to the statement when the steam pressure in the system is always less than the balanced one for the nucleus of any size. Thus, the nucleus formation in the system is possible only under the pressure exceeding P3.

So, the system statement change while the second phase unit appearance can principally change the formation character. Next paragraph is devoted to the quantitative assessment of this factor.

  1. The finite system free energy change.

As in the previous paragraph the energetic equations building we fulfill for the simplest system steam – liquid. The steam statement we describe by the equations of the ideal gas statement. So, the chemical potential of 1 mole of one-component ideal gas (thermodynamic potential of Gibbs) has the form

G(T,P)=G(T,P0)+kTln(P/P0) / (3)

where P0 - gas initial pressure. Similarly the free energy on one particle of gas will be:

g(T,P)=g(T,P0)+kTln(P/P0) / (4)

Let us consider the steam containing N particles on each of which there is free energy gv .Thus, the system initial free energy constitutes G1 =Ngv . If hypothetically one steam particle segregates into liquid statement the system energy change will constitute

G1=(N-1)gv+gl-Ngv+ws1 / (5)

where gl - particle energy in liquid statement; ws1 - work of formation of phases separation surface. Relation (5) may be brought to the form G1=(gl-gv+ws1) .

After this act each particle free energy in the steam will constitute

gv1=gv+kT ln(N-1)/N

and while lim ln x = x – 1 than

gv1=gv - kT /N / (6)

After transition of the second particle into liquid statement the common change of the system free energy is equal to

G2=(gl-gv)+(gl-gv1)+ws2

Where the first member characterizes the energy gain as a result of the first particle transition into liquid, the second member – accordingly second particle.

Consequently changing the stated approach can be received the change of the free energy of the system containing a nucleus of n particles:

Gn=n(gl-gv)+kTn2/2N+n2/3 / (7)

The latter relation is different from the received one for nucleus formed in the nonfinite system in the member kTn2/2N .

Let us analyze its role in nucleus formation parameters calculation.

The function graphic (7) has two extremums one of which is maximum and the other one is minimum. The position of both extremal values determining accordingly the size of the stable and unstable nucleus being in balance with the surrounding system . The first maximum value weakly depends on the saturation degree and on particles number in the system. Both these relations are much remarkable for the second extremum determining the size of the stable nucleus (fig.4,5).

Utilizing of this energetic balance taking into account the tense statement energy, solid phase mechanism and also crystal-chemical peculiarities of the reagent and product crystals allowed to formulate the approach to the calculation determination of the reaction products structures. As an example the transformations in ferric oxides from the highest to the lowest till metal.

  1. Structure of the product of reduction reaction of Fe2O3 .

Let us assume that on the initial oxide surface, reacting with the gas-reducer, there are Fe3O4 nucleuses distributed in random order. While removing oxygen ions from the surface on it some excessive concentration of iron ions is supported. The value of it is always higher than in the areas directly surrounding the nucleuses (fig.6, a). The size of these areas ( B ) is determined by the relation of anions removal from the surface and mass transfer of cations to the growing element of the new phase.

As was already mentioned, new nucleuses may be formed only outside these areas and the growth of the old ones goes independently from each other till that moment when the areas B, conforming to the adjoining nucleuses, touch each other (fig.6, b). In this case the growth speed along the touch lines is decelerated. And along other directions, to the more distant nucleuses, remains the same. Such a change of the growth speed leads to ordering of the growing elements of reaction product position on the surface of the reagent.

The first stage of growth stops when all the surface is filled with the nucleuses and areas B. After that the second growth stage begins when the excessive concentration of cations on the surface is less than it is necessary for new nucleus formation. Consequently new nucleuses are not formed. In difference from the first stage the growth speed remains constant right to the nucleuses touch after what the reaction product structure has the form represented on fig.6,c.

Before analysis of the product structure during the further reaction passing let us consider the character of development of the interphase borders on the example of two nucleuses (fig.7,a). In case of low temperature reduction, when the determining role in mass transfer plays the diffusion along the phases separation border, the borders shape determines both process development character in space. If we suppose that the new phase Fe3O4 growth is determined by the separation borders saturation with cations than the two nucleuses coming close goes as long as the part of the interphase border, perpendicular to the growth direction exists (fig.7,a). However this part with the process passing decreases and disappears when the surface, reacting with the gas-reducer, and the interphase border are in the same plane (fig.7). The nucleuses coming close in this case stops and the further growth goes into the crystal depth (fig.7, c).

Such a process is quasystationary because for the phases separation borders motion the same quantity of iron ions are spent as it is formed on the oxide surface bordering with the gas phase.

Actually, if the excessive ions quantity exceeds the necessary one for the separation border saturation its motion speed will exceed the speed of motion of the border oxide – gas and the canal between the nucleuses decreases (fig.8, a). It decreases till the moment when the balance between appearing and needed cations is achieved. On the contrary, if little excessive iron ions are formed the canal widens (fig.8, b). The achieved in this case statement of the dynamic balance allows to carry out the approbation of the stated model of appearance and growth of nucleuses by comparison of the calculated process activation energy value with the experimental one. In the basis of the calculation lies the solving of the border diffusion task that allows to tie together the depths of the substance, diffusing on the border and volume, penetration with time and temperature. For this purpose according to the work we bring the differential equations system

c/=D2c/x2; c=1 at y=0 and x=0; c=0 at y=
ac/= a D12c/x2+ Dc/x at x=0,
c/x=0 at x=; c=0 at t=0. / (8)

Here and further the following designations are introduced: l1 and l – accordingly diffusing substance penetration depth along the border and on grain volume; c – substance concentration; D and D1 - ratios of volume and border diffusion accordingly; a – border thickness.

The equations system solution is described in detail by the authors (5) and temperature relation of the substance penetration depth along the border has the form

l1(T)=A exp((1/2Q-Q1)/2RT)

where Q and Q1 - accordingly activation energies of volume and border diffusion; A – constant depending on time and border geometry. It should be mentioned that the task of excessive cations diffusion from the ferric oxide surface into the phases separation border has essential peculiarities: first, it is the axial symmetry of the diffusion profile; second, inequality of the flows in the volume on different sides from the border. That is why the more correct task solution should contain the function of at least three variables

l1(T)=A'(Q1,Qg,Qm)

Where Q and Q1 - activation energies of iron ions volume diffusion into Fe2O3 and Fe3O4 accordingly. However, as we consider only the transfer Fe2O3 -- Fe3O4 we can neglect the suctionof cations from the separation border into new phase ( otherwise it would mean also the reduction of Fe3O4 till FeO). That is why in the first approximation we will use the relation l1 (T) in the form

l1(T)=A' exp((1/2Qg-Q1)/2RT) / (9)

which allows to reveal the unambiguous bond l1(T) with the speed of the low temperature reduction. Assuming that the latter one is determined by the phases separation border saturation speed in the given flat task (fig.9)

=al1l1/=A'' exp((1/2Qg-Q1)/RT) / (10)

where A = aA'A'/.

The relation (10) we represent in the form

=A'' exp(-Eef/RT) / (11)

where Eef = 1/2Qg-Q1 and we will try to assess the value of the “seeming” activation energy from the analysis of cations penetration depth experimental temperature relation on the phases separation border. Because the latter one is directly connected with the stationary distance between canals than according to (9) and with the help of the temperature relation data l1(T) we will determine using (11) the value of Eef30kkal/mol

T, oC / 400 / 600 / 700 / 800
l1, mkm / 410-4 / 1 / 2 / 3

Conformity of this value to the value of “seeming” energy of activation, received from direct experiments on ferric oxide reduction kinetics (6,7) (Eef=25-30 kkal/mol), allows to make conclusion about the determining role of the iron ions diffusion flows in the regulations of reaction product growth. In this case activation energy is a linear function of the conforming parameters of border and volume diffusion (Qg and Q1 ). However the product structure, in this case Fe3O4 with the system of canals directed to the phases separation border, is determined also by the regulations of formation in the surface layer of ferric oxide (9,10).


So, utilizing of the proposed thermodynamic approach to the analysis of formation in finite systems along with taking into account the details and peculiarities of the phase transition mechanism allows, in the first approximation, to determine the products structure in reactions with the participation of solid phases. The further works are devoted to the analysis of the structures of other reactions connected with the metals reduction.

Literature.

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  6. Nabi G., Lu W.-K. Reduction kinetics of hematite to magnetite in hydrogen-water vapor mixtures.- Trans. Met. Soc. AIME, 1968, vol. 242, No 12
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