J. S. Kirkaldy

The pearlite reaction

A critical appraisal of theory and experiments for both isothermal and forced velocity pearlite is presented. It is concluded for binary systems that both the theoretical models for volume diffusion and boundary diffusion control are well-advanced and adequate for the purposes of experimental test. However, some ambiguity remains in the boundary diffusion model with respect to the thermodynamics of the boundary ”phase” region, so it is still not possible to predict absolute rates of transformation. The theoretical problem for ternary pearlites is also well understood, although rigorous theory seems intractable. A new perturbation procedure for definition of the optimal steady-state spacing is presented and amplified for both isothermal and forced velocity pearlite, and for both volume and boundary diffusion models. In terms of the critical spacing Sc for isothermal pearlite and the spacing at minimum undercooling Sm for forced velocity pearlite the predicted stability points are as follows: {fx2777-1} For isothermal pearlite these perturbation results correspond closely to the state of maximum entropy production rate while for forced velocity pearlite the correspondence is also satisfactory. A detailed analysis of the data leads us to reaffirm the author’s conclusions that the eutectoid reactions in Cu-12 pct Al and some related ternary alloys reported by Asundi and West are controlled by volume diffusion and that the eutectoid reaction in Al-78 Zn reported by Cheetham and Ridley is controlled by boundary diffusion. We conclude further after careful analysis that the pearlite reaction in Fe-0.8 C is controlled for the higher temperatures by volume diffusion of carbon in austenite. We are also led to state that the pearlite transformations in Fe-C-Mn and Fe-C-Ni occur for the most part in a nopartition regime and are therefore controlled by volume diffusion of carbon in austenite, while the transformations in Fe-C-Cr and Fe-C-Mo, being forced by thermodynamics to sustain partition of chromium and molybdenum, are controlled by phase boundary diffusion of the latter elements. nt]mis|M. P. PULS, formerly Postdoctoral Fellow, Department of Metallurgy and Materials Science, McMaster University, Hamilton, Ontario, Canada

Partition of manganese during the proeutectoid ferrite transformation in steel

The late stages of the isothermal proeutectoid ferrite reaction in Fe-C-Mn have been investigated theoretically and experimentally. For the growth of grain-boundary allotriomorphs three temporal regimes must be recognized. In the early regime the grain-size is infinite with respect to the diffusion length of carbon so the growth is parabolic. The middle regime involves the cumulative impingement of the carbon fields from opposite sides of the grains. This regime ends as the carbon activity approaches substantial uniformity through the ferrite and austenite. The final stage involves the extremely slow approach of the manganese towards uniform activity as well. These temporal regimes must be further subdivided into high and low super saturation regions. In the low supersaturation region segregation of manganese must occur at all times, while in the high supersaturation region it must occur significantly only for late times. The growth rates and the diffusion profiles for the third temporal regime have been calculated on a local equilibrium model and compared with the metallographic and microprobe results for alloys within the two regions of supersaturation. The agreement between theory and experiment is in all cases good.

Thermodynamic prediction of the ae3 temperature of steels with additions of Mn, Si, Ni, Cr, Mo, Cu

Published binary phase diagrams and activity data for iron and binary and ternary alloys have been used to evaluate the general linear series expansion of the activity coefficient and the standard free energy changes and these have been employed in turn for the accurate thermodynamic determination of the Ae3 temperature in steels with additions of Mn, Si, Ni, Cr, Mo, and Cu. A computer program accurate for total additions up to 7.0 wt pct and an analytic formula accurate for total additions up to 2.5 wt pct have been developed. The predicted Ae3 temperatures compare favorably with observations on over 200 steels from international compendia. It is demonstrated that existing linear regression formulae are incorrect and that for low alloy additions they should be replaced by linear expressions with carbon concentration dependent coefficients.

Scaling of intragranuiar dendritic microstructure in ingot solidification

Analytic scaling formulas of complete constitutional generality for forced velocity cells and dendrites were in earlier research perfected forin situ steady-state solidification conditions involving binary organic alloys. As a further test, these were used, given the velocity and gradient control parameters, to predict the primary and secondary dendrite arm spacings of unidirectionally cooled Al-Cu alloys for which a large data set is available. Numerical methods were employed to determine the control parameters that exist under unsteady-state ingot solidification conditions according to the Scheil formulation. Primary and secondary arm spacings, corrected empirically for ripening, that by and large agree with the Al-Cu experimental data were obtained, demonstrating that the formulas are adequate for the prediction of dendrite scales in steady and unsteady-state conditions. The predictions have been incorporated into a computer program that displays the time-dependent columnar microstructure and mushy zone in an ingot cross section of an oriented single crystal together with the thermal and liquid-solid distributions.

Solubility product for niobium carbide in austenite

The solubility product for NbC0.87 has been experimentally determined between 950° and 1250 °C by gas equilibration and extraction methods. The new relation log10[pct Nb] [pct C]0.87 = 3.4 − 7920/T on a weight percent basis has been evaluated from the data. This is in good agreement with a compendium of measurements from the literature. The data do not yet justify an analysis which takes account of a variable stoichiometry.

Kinetics of the pearlite reaction in Fe−C−Cr

Velocity and spacing measurements on the pearlite reaction in three eutectoid alloys of Cr (0.4, 0.9 and 1.8 wt pct) have been recorded and compared with growth and spacing equations based on a variety of thermodynamic and kinetic models. At high temperatures the reaction is controlled by phase boundary diffusion of Cr while at lower temperatures, a local equilibrium no-partition model with kinetic control by carbon diffusion best represents the data. At intermediate temperatures there is a smooth transition rather than a sharp break betwewn the two regimes. There is a discrepancy between theory and experiment on the slope of the spacingvs inverse under cooling plot which can be removed by assuming that the ferrite-cementite epitaxy improves with the addition of chromium.

Homogenization by diffusion

The problem of removal of microsegregation by solid-state diffusional processes is reviewed. For most cases of practical interest, volume diffusion coefficients as measured in laboratory couples determine homogenization rates. If the initial distribution of solute is sufficiently regular and well-characterized, the decay of microsegregation can be predicted with good precision. Often, however, order-of-magnitude estimates are all that is required, and these may be made from less complete information. The interactions of diffusing solutes with crystalline defects and with other solutes, are of practical as well as academic interest: in particular, multi-component interactions can significantly influence the course of homogenization.

Thermodynamics and phase equilibria for the Fe−C−Cr system in the vicinity of the eutectoid temperature

The low chromium Fe−C−Cr phase equilibria pertinent to the eutectoid transformation of austenite have been modeled and calculated from published thermodynamic information and verified through a series of selected equilibration experiments. In particular, α-γ, α-cm and γ-cm partition coefficients have been measured within the range 700 to 770°C and compared with austenite and carbide models proposed by Hillert and coworkers. While the theoretical-empirical closure is not perfect, the models are proven to be completely adequate for incorporation within the kinetic description of the corresponding ternary pearlite reaction.

Transition between internal and external nitridation of Ni-Ti alloys

A series of Ni-Ti alloys ranging in composition from 0.1 to 5 wt pct Ti were annealed in nitrogen gas or a nitrogen/argon gas mixture between 800 °C and 1020 °C. The evolution of surface and subscale structures, along with the diffusion profile of Ti in Ni, were investigated using scanning electron microscopy and energy dispersive X-ray analysis (EDX), respectively. A strong extrusion of Ni to accommodate the excess volume of internal TiN precipitation was observed between 0.5 and 1.0 wt pct Ti at 1020 °C, where a continuous superficial layer of stoichiometric TiN begins to form. A finite difference computational algorithm was developed based upon a ternary model of simultaneous diffusion and precipitation, which generates the concentration profile of Ti in Ni and the particle distribution of TiN and subsumes a transition from internal to external nitridation. Because there is a dearth of independent thermodynamic and kinetic data on this system, we were forced to use parameters established by a selected minimal set of our own experiments to predict outcomes for the main body of experimental work, thereby obtaining satisfactory closure between theory and experiment.

Entropy criteria applied to pattern selection in systems with free boundaries

The steady state differential or integral equations which describe patterned dissipative structures, typically to be identified with first order phase transformation morphologies like isothermal pearlites, are invariably degenerate in one or more order parameters (the lamellar spacing in the pearlite case). It is often observed that a different pattern is attained at the steady state for each initial condition (the hysteresis or metastable case). Alternatively, boundary perturbations and internal fluctuations during transition up to, or at the steady state, destroy the path coherence. In this case a statistical ensemble of imperfect patterns often emerges which represents a fluctuating but recognizably patterned and unique average steady state. It is cases like cellular, lamellar pearlite, involving an assembly of individual cell patterns which are regularly perturbed by local fluctuation and growth processes, which concern us here. Such weakly fluctuating nonlinear steady state ensembles can be arranged in a thought experiment so as to evolve as subsystems linking two very large mass-energy reservoirs in isolation. Operating on this discontinuous thermodynamic ideal, Onsager’s principle of maximum path probability for isolated systems, which we interpret as a minimal time correlation function connecting subsystem and baths, identifies the stable steady state at a parametric minimum or maximum (or both) in the dissipation rate. This nonlinear principle is independent of the Principle of Minimum Dissipation which is applicable in the linear regime of irreversible thermodynamics. The statistical argument is equivalent to the weak requirement that the isolated system entropy as a function of time be differentiable to the second order despite the macroscopic pattern fluctuations which occur in the subsystem. This differentiability condition is taken for granted in classical stability theory based on the 2nd Law. The optimal principle as applied to isothermal and forced velocity pearlites (in this case maximal) possesses a Le Chatelier (perturbation) Principle which can be formulated exactlyvia Langer’s conjecture that “each lamella must grow in a direction which is perpendicular to the solidification front”. This is the first example of such an equivalence to be experimentally and theoretically recognized in nonlinear irreversible thermodynamics. A further application to binary solidification cells is reviewed. In this case the optimum in the dissipation is a minimum and the closure between theory and experiment is excellent. Other applications in thermal-hydraulics, biology, and solid state physics are briefy described.