Assessment of thermochemical data on steel deoxidation ( • )

It is proposed to develop a method to judge the certainty on the information regarding to deoxidation equilibria of iron melts. To accomplish this objective, thermochemical data was collated and then evaluated. The basic knowledge on deoxidation conditions are framed by the non-ideal Henrian behaviour of diluted solutions of both deoxidizer and oxygen in liquid iron in equilibrium with a pure oxide. Conventional deoxidation reactions were considered at 1,873 K such that in their equilibrium constants, only first order interaction coefficients were considered. The criteria in selecting the most appropriated free energy equation was based on evaluating them under two critical composition points: 1 where they satisfy an oxygen to deoxidizer ratio dictated by its stoichiometry and 2 where oxygen contents at a given amount of deoxidizer reaches a minimum value. These data were plotted on logarithmic scales to appreciate the effects of deoxidizer’s valences. Once such information was classified, under restrictions 1 and 2, previous compositions were related to deoxidizer ́s electronegativities.


1. . R RA AT TI IO ON NA AL LE E O OF F T TH HI IS S W WO OR RK K
Design of mechanical components to satisfy high quality standards demands meticulously controlled processes to such a point where chemical compositions of melts must be satisfied within very narrow limits.This statement becomes yet more valid for steel melts subjected to secondary refining processes such as vacuum ladle metallurgy or electrode remelting processes where assertive compositions are compulsory.These requirements are expected to be met, since the response to further heat treatments, enhance mechanical properties such as fracture toughness or fatigue life among others; which in turn are strongly dependent on the chemical composition and microstructural homogeneity and cleanliness of the resulting metal.
One example of a highly dependent process of chemically restrained stages is that to obtain inclusion shape control by ladle metallurgy.In this process, it is necessary to reach very low sulphur contents in steel melts and thereafter aluminum additions must be such that melts should reach a fully killed stage; so afterward proceeding with calcium treatment.This sequence of events should be considered as a standard practice in a melt shop to a point such that even if air oxidation prevention fails or if effects of slag accretions on refractories affect oxygen potentials significantly.Henceforth, measures should be at hand to trace, prevent and solve this phenomenon instantaneously.The desired fully killed condition of a melt carried out by aluminum additions as well as the calcium treatment are both very critical in the sense that if they are inappropriate, nozzle clogging must be expected due to precipitation of high melting point phases.Thus, through previous assertions one can establish that despite the fact that previous statements are referred to non equilibrium conditions in these processes; their control should be based on mathematical models which include trustable thermochemical data.Indeed, these postulates become much more meaningful when acute compositions of steel melts demand control to levels of parts per million.

2. . T TH HE EO OR RE ET TI IC CA AL L F FR RA AM ME E O OF F R RE EF FE ER RE EN NC CE E
A deoxidation reaction ruled by the Henrian law in mass percent, where solutes are dissolved in pure iron is represented by: The equilibrium constant for reaction (1) is given by: (2) If deoxidation products are pure, then their activity is unitary, so K is reduced to: (3) Additionally, if solutes are infinitely diluted in iron, then a Henrian standard state is given by: (4) Where, h is a Henrian activity coefficient.This equation can be expressed as the deoxidation constant (K'), which is the reciprocal of the former equation, thus (5) By taking the logarithm of the previous equation, it can be found a linear relationship: (6) By assuming real diluted solutions, values of h can be calculated as follows: where, f is commonly referred to as interaction coefficient.
The formal way to represent thermochemical properties of diluted solutions in iron was originally postulated by Wagner [1] .Such proposal is based on Mc Laurin series, which are represented by: (8) As shown in equation (8), the concentration of the species involved in the equilibria are given in mole fractions; however, in practice it is more convenient to express such compositions on a weight percent basis; to do so, we use the following relationship: (9) Substituting back equation (9) into equation (8) yields: ) )  (10) It is interesting to notice that the derivatives in equation (10) represent the so called interaction parameters; and according to the degree of the derivatives, they represent the order of the interaction parameter.Henceforth, by substituting accordingly to such interaction parameters, equation (10) can be represented by: (11) where, f i is the interaction coefficient of the i-th element in the melt, e i J is the first order interaction parameter, r i J is the second order interaction parameter, r i (J, k) is the cross interaction parameter, %J is the deoxidizer in weight percent, %k is a second solute in weight percent.
In view that the scope of this work is to fundamentally address a method to make an evaluation of the simplest thermochemical data, then Wagner's formalism deduced in equation ( 11) is reduced to: (12) Equation ( 12) is known as Wagner's truncated formalism.By disclosing equation (12) in terms of a given deoxidizer and oxygen, we obtain the following expressions: where, M represents a deoxidizer agent.
Therefore, the equilibrium constants for equations (13a) and (13b) are given by: (14) Provided that previous knowledge on thermochemistry of deoxidation is taken into account, a frame of reference is necessary to establish any point of comparison: (1) The primary point in such referral is that deoxidation (in equilibrium) is carried along to a point where oxygen in solution reaches out the minimum content and (2) the second referral is based primarily on calculating an equilibrium such, that reagents and reaction products strictly satisfy a stoichiometric ratio.By considering a starting point, take equation ( 6) and derive it with respect to the amount of deoxidizer M, thus it follows that: (15) Therefore, that equation can be expressed as: (16) where, x and y represent the stoichiometric coefficients.
By including the Henrian 1 wt% standard state in equation (16), it yields: (17) Applying the properties of logarithms to equation (17) and by using the following definitions: and then such equation can be minimized so we obtain: (18a) or, its equivalent (18b) Therefore, if [%M] is expressed in terms of e~2.718281, thus log e ~ 0.434, then the expression for the minimum deoxidizer content is given by: ( ) ) By using the data shown in tables I to VIII, equilibria were calculated by using equation (14).Then these data was plotted as composition changes of [wt% O] vs. [wt% M] at a unit activity of deoxidation product.
By imposing on such plots the stoichiometric line, a unique composition point will be found as an intersection of both equilibria and stoichiometric lines.A second alternative to calculate such point; is by considering that an empirically determined equilibria can be explicitly represented by an expression like: (20) where, k=log K' Therefore, one must look for an explicit function, such that [wt% O] 1wt%Fe = f[M].This second proposal can be accomplished by using Maple TM software to transform the equilibria deoxidation reactions such as equation (20) into: (21) where, λ represent the following parameter: (22) It is important to notice that Lambert's function W is also known as omega function and it is the inverse of the following function: (23) where, w is a complex variable.However, since the correlations of data regarding thermochemical equilibria are real numbers, then only that part of the solution is considered and the complex part is assumed to be zero.

3. . L LI IT TE ER RA AT TU UR RE E S SU UR RV VE EY Y
Thermochemical data on single deoxidation of iron melts were collected from several sources [2-36] .In cases where deoxidizing elements form more than a single solid deoxidation product a single equilibrium was considered for each product along with its own interaction parameters.From all the elements considered, two of them show different oxidation states, namely caesium and titanium.Henceforth, where information about these equilibria was available, it was incorporated to a table as additional columns.The Henrian behaviour of infinitely dissolved oxygen and deoxidizer in iron in equilibrium with their oxides with unitary activity are listed in tables I to XI.

4. . R RE ES SU UL LT TS S A AN ND D D DI IS SC CU US SS SI IO ON N
The thermochemistry of iron deoxidation is explored by using several sources of information on deoxidizers, oxygen and deoxidation products in equilibria with iron.Data to carry out this task is considered as a trustful source and these are shown in tables I to XI.
Deoxidation equilibria were calculated by assuming the following: real Henrian behaviour of dissolved species in iron by considering first order interaction parameters, pure oxides or unity activity of deoxidation products and a constant temperature, 1,873 K.
The major variations of deoxidation equilibria are expected to be due to: (i) Change in free energy of formation that will convey to values of log K and (ii) magnitude of first order interaction coefficients.Thus variations of type (i) will be identified as a parallelism between two Henrian behaviours where they may be either real or ideal.And, the type (ii) will be traced as the intensity of inflections of non ideal Henrian behaviours.The latter ones are expected to manifest themselves around the minimum oxygen contents and their intensities will be traced as a more pronounced departure from the ideal Henrian behaviour.Therefore, a more negative and higher value of e O M (hence e M O ) would render curlier oxygen-deoxidizer distributions.
It is important to mention that to appreciate those effects in a given deoxidation equilibrium; the deoxidizing agent as well as the oxygen content are both plotted on logarithmic scales.These results are shown in figures 1-12.Subsequently, deoxidation equilibria are solely represented by the critical composition of ratios found where the stoichiometry F Fi ig gu ur re e 1. .Deoxidation with aluminium on different steel melts at 1,873 K.  is satisfied and where the lowest oxygen content is evaluated.However, it must be established that these ratios do not represent a linear composition between them, instead as shown in figure 1, which corresponds to aluminium deoxidation, for several steel melts, minima composition ratios as described earlier are located around the major inflection of the curve.Thus, while figure 1 shows effects of several activity coefficients which corresponds to various steel melts, figure 2 shows data from table I that includes several values for log K as well as different values for interaction coefficients.It is interesting to note that in both figure 1 and figure 2, composition ratios related to stoichiometric points are less dispersed than those representing the minima oxygen contents.At first sight, deoxidation equilibria shown in figures 2 to 12 can be categorized according to their degree of dispersion.Then, three major groups can comprise the following deoxidizing agents can be identified, namely: 1) Ca, Mg, 2)Al, Zr, Si, Mn and 3) Cr, V, Ti and La.
As a first estimation as to why data shows such inconsistencies, may be attributable to: (1) accuracy of chemical assays in respect to indexes of detection of certain species.Under this premise, one can appreciate that tracing calcium and magnesium in equilibrium with oxygen, all of them infinitely dissolved in iron, it is technically more difficult than tracing chromium or vanadium where equilibria contents are much higher in melts.(2) Incongruities related to basic equilibria data such as log (K) and F Fi ig gu ur e 3 .Comparison of equilibria data on steel deoxidation with calcium at 1,873 K.If deoxidation of iron is considered as one set of equivalent equilibria that are related by the ability of a deoxidizer to react with oxygen, then these critical compositional ratios can be conveniently rearranged, according to their oxidation state.Hence, three major groups of equilibria are considered.These would involve oxides constituted of: bivalent, trivalent or tetravalent deoxidizing elements.Such set of equilibria plotted as weight percents of deoxidizer vs oxygen on logarithmic scales show that stoichiometric ratios are more congruent among themselves than the ones representing minimum oxygen contents.Such results are shown in figures 13 to 15.
However; is worth to mention that stoichiometric ratios as expected; show a characteristic slope for each group of data.On the contrary, information related to minimum oxygen contents appear as a wide band of data without regard the valence considered in each group.In fact, the major incongruities are those related to the most stable lanthanide species, i.e.La/La 2 O 3 F Fi ig gu ur re e 5 5. .Comparison of equilibria data on steel deoxidation with chromium at 1,873 K.    and Ce/Ce 2 O 3 , see figure 15.It is also worth noting that in the case of M 2 O 3 oxides, both the stoichiometric line and the lines defining the minimum oxygen content converge to points that satisfy the condition defined by the difference of electronegativities equal to zero; such convergence is not observed in MO 2 nor MO oxide systems.
A global deoxidation behaviour, which includes most of the deoxidizers is shown in figure 15.On this figure, both stoichiometric and minimum oxygen contents as bands of data are shown.If the electronegativity concept is applied to a deoxidizer with respect to oxygen to form an oxide, then this periodic potentiality can be used as a guideline to reframe critical composition ratios previously cited.By selecting the most congruent thermochemical data to satisfy this chemical ability, better correlations are found for both stoichiometric and minimum composition ratios.However, it should be pointed out that both sets of critical ratios still manifest themselves as bands of data (Fig. 15).Strictly speaking, electronegativities vs minimum oxygen and stoichiometric ratios are not yet the best way of plotting these data.
If the analysis on thermochemical data is carried out, it can be found that inaccuracies are, as expected, related to chemical factors such as: iron deoxidizing agent and crucible impurities; composition of reaction products and of course a time period to reach actual equilibrium conditions.It is worth mentioning that fewer incongruities are found on deoxidation equilibria of iron melts since the advent of solid electrolyte cells.By using these cells, true oxygen dissolved in iron instead of total oxygen contents is traced.Therefore, provided melts are homogeneous, oxygen contents and temperatures monitored by these cells are determined in situ.It is important to recall that these experimental findings jointly with deoxidizing assays are compelled to satisfy theoretical grounds or vice versa.
On the other hand, despite the fact that most equilibria on deoxidation take into account deviations from the ideal Henrian behaviour, due to interrelated effects among free energies of formation or equilibrium constants and interaction parameters, small changes in them lead to significative variations of those results predicted by equation (14).To bring in a closer assessment on F Fi ig gu ur re e 9 9. .Comparison of equilibria data on steel deoxidation with silicon at 1,873 K.   deviations observed in a real Henrian equilibrium, one can observe that departures from ideality occur at composition ratios driven towards the minimum oxygen content, while at ratios closer to the stoichiometric one, both real and ideal Henrian behaviours have essentially equal ratios.The latter finding indicates to us that a real equilibria represented by composition ratios of oxygen vs. deoxidizer plotted as logarithms show both a lineal and non-lineal behaviours.
Another factor which sensibly induces deviations of composition ratios to unreasonable values of equilibria is that shown by deoxidizing agents with several oxidation states.This is expected since equilibrium conditions may be affected by more than one stable oxide.Among these one can consider Ce and Ti, where Ce 2 O 3 and CeO 2 and Ti 2 O 3 and TiO 2 may coexist.
By carrying out an analysis on pure chemical basis one can find that the degree of ionic bonding between two atoms is defined from the difference in affinity of the species for electrons.Thus, the larger this difference the stronger will be the tendency for electron exchange.In other words, if the chemical affinity is considered as a measure by which an electron can be removed from an atom to form an ion, such statement should be understood as the definition of electronegativity.Under these principles, Pauling assigned a scale of electronegativities between two elements, which has been used to account for bond enthalpies, percentage of ionic character of the bond and type of oxide (basic or acidic).In this sense, electronegativity has to be defined as the power of an atom to attract electrons by itself when it is combined in a compound.

2 2 .
.1 1. .M Mi in ni im mu um m o ox xy yg ge en n c co on nt te en nt t

T
Ta ab bl le e I I. Thermodynamic data for the [Al] -[O] equilibrium in iron melts at 1,873 K Tabla I. Datos termodinámicos para el equilibrio [Al] -[O] en hierro fundido a 1

573 22 T 28 T
Ta ab bl le e I IV V. Thermodynamic data for the [Cr] -[O] equilibrium in iron melts at 1,873 K Tabla IV.Datos termodinámicos para el equilibrio [Cr] -[O] en hierro fundido a 1Ta ab bl le e V V. Thermodynamic data for the [La] -[O] equilibrium in iron melts at 1,873 K Tabla V. Datos termodinámicos para el equilibrio [La] -[O] en hierro fundido a 1