THE PHENOMENOLOGICAL MODEL OF NON-EQUILIBRIUM CRYSTALLIZATION AND

STRENGTHENING-PHASE-FORMATION PROCESSES IN THE WELD

V. Mazurovsky1, M.Zinigrad2, A. Zinigrad1, L. Leontiev3, V. Lisin3

1CWC Ltd., Ariel, Israel2Colledge of Judea and Samaria, Ariel, Israel

3Institute of Metallurgy of UD RAS, Ekaterinburg, Russia

Abstract A phenomenological model providing for the prediction of weld

structure based on a modified Schaeffler diagram has been developed. A real empirical diagram constructed for analysis of the structure and phase composition of a weld in its initial state after fusion welding (of any type) has been devised. The model takes into account the nonequilibrium crystallization of metals with any level of alloying and permits calculation of the quantities of each of the phases formed regardless of the concentration level of the alloying components.

IntroductionThe Schaeffler diagram is an important tool for predicting the

constitution of austenitic Cr-Ni steel welds with carbon contents up to 0.12 %. However, it does not allow determination of the composition and volume of the carbide phase. Furthermore, if the carbon content in a weld exceeds 0.12 % (as, for example, in the case of hardfacing), the agreement of predictions with actual data is markedly reduced due to an intense consumption of carbon and carbide-forming elements by the carbide formation process. In addition, an accurate prediction can be complicated either by a lower content (or absence) of Cr and Ni in the weld or by a higher content of Mn (above 4 %). This could be attributed, first, to the fact that the carbide formation process in a weld is not taken into consideration (as mentioned above), and, second, it can be caused by the use of constant empirical coefficients in the equations specifying Cr and Ni equivalents [1]. A third problem relating to microstructure prediction based on the Schaeffler diagram stems from the difficulty in

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determining the volume of a specific phase in multiphase regions. Therefore, the Schaeffler diagram should be modified to provide a more accurate prediction of weld structure by doing the following:

taking into account the carbide formation process; deriving universal equations for calculating Cr and Ni

equivalents, which take into account the nonequilibrium nature of the crystallization of weld metal;

determining the number of phases in multiphase regions of the diagram, as well as the structure and quantity of each phase.

Studies aimed at partially eliminating these shortcomings of the Schaeffler diagram are known. H. Schneider [2] made one of the first attempts, focusing his attention on the influence of such alloying elements as N, Co, Cu, Al, and W on the values of the Cr and Ni equivalents. More recently, R.H. Espy modified the Schaeffler diagram for nitrogen-containing austenitic steels [3]. Y.M. Potak and E.A. Sagalevich devised a fundamentally new type of diagram [4], in which the mutual influence of the alloying components is partially taken into account and it is possible to predict the constitution of weld metal with a carbon content up to 0.3 %. Further refinement of the foregoing diagram [5] greatly expanded the region where constitution of weld metal with various degrees of alloying can be predicted. However, it did not eliminate the problem of predicting the constitution of weld metal with a carbon content greater than 1 % and a high concentration of alloying elements. Several studies have been devoted to more accurately predicting the phase composition in two-phase regions by creating special diagrams [6-8] or modifying the Schaeffler diagram [9]. The studies conducted by D.J. Kotecki merit special attention [6,10-12]. They were devoted to both more accurately predicting the constitution and establishing the martensite boundaries at various levels of alloying with manganese in the steels studied. Very important investigations were carried out by S.A. David, S.S. Babu et al [13-17].

We note that, as a rule, the studies just cited considered individual groups of steels and were intended for particular cases of alloying. Thus, they did not solve the general problem.

The present study is devoted to solving this problem.

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Carbide formation processAs mentioned above, if the carbon concentration exceeds

0.12 %, the volume of carbon and carbide-forming elements consumed in carbide formation can increase markedly. This process is disregarded in the Schaeffler diagram. We believe that it could be the reason for the disparity between the results of matrix microstructure predictions based on the Schaeffler diagram and the actual data for steels with a high carbon content.

Most of the transition metal monocarbides form in the B1 (NaCl) structure, fcc metal with carbon occupying the octahedral interstitial sites. The shortest M-M distance is about 30 % greater in the B1 carbide than in the pure metal for the Group IV and V carbides, but drops to less than 10 % greater for the Group VI or VII carbides [18]. At 100 % site occupancy, the stoichiometry of the carbide is MC1.0, though this situation is rarely realized. The concentration and ordering, if any, of the vacancies that result from a nonstoichiometric M-C ratio have a great effect on the thermodynamic, mechanical, electronic, and magnetic properties of the metal carbides; however, the details of these effects are a matter of some debate in the literature, due to the difficulties inherent in synthesizing pure compounds and in measuring the exact details of the crystal structure of a given sample. The metal carbides share many characteristics with the metals themselves, having a plastic deformation like the fcc metals.

Most of the Group IV-V metal carbides conform to the Hägg rules, which were developed empirically to “predict” the structures of the transition metal carbides [19]. The structure adopted by the metal carbide is determined by the ratio of the radius of the nonmetal atom (rx) to that of the metal atom (rm). For r = rx / rm < 0.59, the structures adopted are the simple A1, A2, A3, and hexagonal lattices. For r > 0.59, more complex structures form to prevent the expansion of the lattice – which would have been required for the simple structure to accommodate the large nonmetal atom – from taking the metal atoms beyond the distance for favorable metal-metal interactions. The monocarbides take on an fcc metal lattice with carbon atoms on the octahedral interstitial sites, and carbon occupation of the trigonal prismatic sites in the hcp [20]. Non-Hägg structures are known among the carbides as well, a major example being Cr2C3.

One much-noted feature of the structure of the transition metal

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carbides is that the lattice adopted by the metal in the carbide is never that of the parent metal. That is to say, if the metal has an hcp lattice, its carbide has the metal on an fcc lattice; the fcc parent metal occupies a non-cubic lattice in the daughter carbide; and bcc parent metals have both fcc and hcp lattices available in the daughter carbides. This has been explained using Engel-Brewer theory of metals [21,22], in which the structure adopted by a metal or alloy depends on the s-p electron count [18]. With increasing s-p electron count the metal structure progresses from bcc to hcp to fcc across the transition series. Likewise, the Group IV and V metal carbides MC form in the B1 structure rather than a hexagonal form because the incompletely filled bands of the host metals can accommodate a high ratio of sp-electron-rich carbon to metal. In Group VI the stoichiometry M2C occurs often, while Groups VII and VIII, when they form carbides at all, take on metal-rich stoichiometries M3C and M4C, consistent with an attempt to avoid filling antibonding levels in the metal bands [23].

The nature of the bonding in the monocarbides is a matter of some debate, although all agree that the simple ionic model (M+C- or M-

C+) is not consistent with the properties of the carbides. Ionic materials will not typically slip on the {111} planes due to the strong repulsive (coulombic) interactions across the shear plane in the half-glide position; instead, they will slip on the {110} or {100} planes [20]. However, the exact direction of electron transfer is the subject of some controversy. X-ray photelectron spectroscopy (XPS) and electronegativity considerations suggest simple M --> C electron donation, but the XPS data is questionable due to the possibility for backdonation or screening effects [23]. Furthermore, simple M ‡ C donation would result in ionic compounds such as the alkali and alkaline earth metal carbides. However, these materials are resistors with low optical transparency or reflectivity, and readily hydrolyze to the metal oxide and hydrocarbon. In contrast, the transition metal carbides are conductors with a shiny metallic and colored appearance and are hydrolytically stable. Band occupation suggests C --> M electron transfer. Carbon appears to combine its sp electrons with the metal d bands. Such a donation scheme may be used to explain, if crudely, the trend in melting point maxima, which occurs in Group VI for the metals and Group V for the carbides: the maxima may be associated with the half-filled d shell [23].

The Group IV-V carbides are able to form continuous solid

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solutions with each other over a wide range of compositions, but are only partially miscible with the carbides of the Group VI - VIII metals [18].

On the basis of these data and to pay due regard to the carbide formation process, we use the new concept of a carbide-forming ability (CFA) of an element. It can be described by the following equation:

, (1)

- absolute CFA of i-th element; - atomic radius of i-th element, Ǻ; - number of d electrons in the outermost electron shell of i-th

element.An analysis of equation (1) shows that the CFA increases along

the following sequence: Fe, Mn, Cr, Mo, W, Nb, V, Ta, Ti, Zr, Hf. However, in real alloys, the CFA depends on several factors. The factors influencing the CFA of elements can be divided into two groups:

1) physicochemical factors, which determine the nature of the carbide formation process,

2) technological factors, which influence the carbide formation process by altering parameters from the first group.

Let us examine these groups of factors in somewhat greater detail. The concentration of each carbide-forming element (CFE) and its chemical affinity toward carbon must first be taken into account. The chemical affinity is determined by the magnitude of the change in the Gibbs free energy accompanying the formation of the respective carbide under standard conditions. The value of is uniquely related to the thermodynamic equilibrium constant of the corresponding carbide formation reaction by the familiar relation

. (2)Consequently, corresponds only to the equilibrium concentration of the carbide formed. Since the formation of different carbides of one particular element is possible, the sequence of their formation and the ratio between the quantities of these carbides in the equilibrium state is determined specifically by the value of its affinity toward carbon. Of course, and thus depend on temperature, as is illustrated by the

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presence of particular stable phases on the binary CFE–carbon phase diagram.

However, in a transient welding process, where melting, chemical reactions between the phases, and crystallization take place with high rates, an equilibrium state is rarely achieved. Therefore, the kinetic factors must be taken into account. The latter should include the rate constant of the activation step of the heterogeneous carbide formation reaction, which can be represented in a simplified modification as

, (3)

as well as the diffusion coefficients of carbon ( ) and the CFE ( ).It should be recalled that, in a final analysis, the rate of a

reaction like (3) depends not only on the kinetic constants , , and , but also on the deviation of the reaction from the equilibrium state

and, when liquid phases participate, on the hydrodynamic conditions of the process. The deviation from equilibrium is determined by the ratio between the real concentrations of the reactants in the system under investigation and the value of the equilibrium constant .

The kinetic and thermodynamic constants cited are strongly temperature-dependent, and the hydrodynamic parameters are determined by the thermal and geometric conditions in the weld pool. Therefore, the second group of factors, which influence the CFA’s of elements, i.e., the technological factors, take on special significance. The most important factors in this group are the parameters of the welding process (welding current, voltage, filler material feeding rate, welding speed, electrode dimensions and shape, polarity, etc.), he dimensions of the parts being welded, and the thermal regime of the welding and crystallization processes.

The problem of calculating the composition of weld metal can be solved, in particular, by employing the methods of the mathematical modeling of the interactions of phases with consideration of the factors cited above [24].

In the general case, the CFA of an element can be expressed as:

, (4)

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- actual CFA of i-th element; - a coefficient which takes into account the above factors.

The determination of coefficient is an extremely complicated task. We developed an appropriate method, which permits the determination of . Scrutiny of this method is beyond the scope of this paper.

The quantity of carbon required to form carbides from the i-th CFE can be expressed as follows, wt.  %:

, (5) С % - carbon concentration in weld, wt. %.

The concentration of carbon dissolved in the matrix is as follows, wt. %:

, (6)

n - the number of CFEs in the weld. The concentration of i-th element consumed to form the j-th ( )-type carbide is

, (7)

- the probability of the formation of j-th carbide; x, y - the stoichiometric coefficients in the j-th carbide chemical formula; - the atomic weight of the i-th CFE; - the atomic weight of carbon.The total concentration of the i-th element consumed to form carbides, wt. %, is

, (8)

k - the number of types of carbides formed from the i-th CFE. Now we can determine the content of the CFE dissolved in the

matrix:. (9)

Therefore, the values of and obtained with allowance for the carbide formation process can be used to determine Cr and Ni

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equivalents.At the same time, now we are able to calculate the quantity of

matrix-strengthening carbide phases. The content of the carbide phase formed by i-th element is given by the following formula:

, (10)

- the content of the carbide phase formed by the i-th element, wt. %;

and - see (5) and (8), respectively.The total content of the carbide phase in a weld is given by the

expression

. (11)

At the same time, we must take into account the consumption of alloying elements in boride and nitride formation. If the Ni content in alloys is over 5 %, formation of the phase will also be taken into account. However, these issues cannot be described within the scope of this paper.

Thus, matrix content (wt. %) in the total volume of the weld metal will be as follows:

. (12)

Now we are able to determine the carbon content and the content of each alloying element in the matrix. For carbon (wt. %) we will obtain:

(13)

The content of the i-th alloying element will be:

(14)

Universal equations for calculating Cr and Ni equivalentsBy analyzing the structures of more than 50 types of weld metal,

the authors determined coefficient variation empirical dependencies for the equations defining Cr ( ) and Ni ( ) equivalents in the Schaeffler diagram [1]. When modified, these equations are as follows:

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, (15)

, (16)

l - number of ferrite-forming elements; - concentration of ferrite-forming i-th element in the matrix, wt %; - function determining empirical coefficient for ferrite-forming i-th element; z - number of austenite-forming elements; - concentration of austenite-forming i-th element in the matrix, wt %; - function determining empirical coefficient for austenite-forming i-th element;

- complex factors defined for the ferrite- and austenite-forming elements based on the structure analysis, as described above.

Phase proportions in multiphase region of the Schaeffler diagramPhase proportions in multi-phase region of the Schaeffler

diagram were found by means of the structure analysis applied to determine the coefficients in equations (15), (16), and taking into account carbide formation process (3). Accordingly, percentage lines for the amount of the second (third) phase were incorporated in the appropriate regions of the diagram. For three-phase regions, algorithms to determine each phase content were developed.

The modified Schaeffler diagram is shown in Fig. 1. The content of the ferrite phase in the ferrite-austenite region is represented by Ferrite Numbers (FN) as recommended by the International Institute of Welding and not by ferrite percentage lines (as at the original Schaeffler diagram). Boundaries of the ferrite-austenite region are still presented (0 %F and 100 %F).

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Fig 1. Modified Schaeffler diagram taking into consideration the effect of carbide formation.

Computer implementation of the model Equations (4)- (14) and the mathematical description of the

Schaeffler [25] diagram served as a base of the mathematical implementation of the model developed by the authors. The model is being implemented for solving two tasks, namely:

i) direct task, i.e., microstructure forecast according to the chemical composition of the weld;

ii) reverse task, i.e., determination of the chemical composition of weld according to the microstructure required.

The reverse task, of course, offers an extremely complicated challenge. A pilot version of the software based on the foregoing

mathematical model has been developed. This computer program is one of the elements of an expert system that the authors use to optimize and develop welding materials and technologies.

Testing of the model

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The software described above was tested by determining over 50 compositions of welds with a known matrix constitution (microstructure) and strengthening phase volume. The data obtained displayed good agreement with the experimental data. The deviation did not exceed 15-20 %.

Conclusions1. An improvement was made to the Schaeffler diagram. This

improvement permits prediction of the constitution of weld metal with consideration of the following factors:

- nonequilibrium crystallization;- the formation of strengthening phases;- mutual influence of the alloying elements.

2. The mathematical equations derived served as the basis of a mathematical model providing for the possibility of determining the number of matrix phases, strengthening phases, and their structure and quantity.

3. Computer implementation and testing of the mathematical model were performed. The results obtained demonstrated the good agreement between the calculated and actual data.

4. The proposed modified diagram makes it possible to predict the constitution of weld metal over a broad range of levels of alloying.

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