The Effect of Solute Atoms on Grain Boundary Migration: A Solute Pinning Approach

Abstract

The effect of solute atoms on grain boundary migration has been modelled on the basis of the idea that solute atoms will locally perturb the collective re-arrangements of solvent atoms associated with boundary migration. The consequence of such perturbations is cusping of the boundary and corresponding stress concentrations on the solute atoms which will promote thermal activation of these atoms out of the boundary. This thermal activation is considered to be the rate-controlling mechanism in boundary migration. It is demonstrated that the present statistical approach is capable of explaining, in phenomenological terms, the known effects of solute atoms on boundary migration. The experimental results on the effect of copper on boundary migration in aluminium, due to Gordon and Vandermeer, have been well accounted for. INTRODUCTIONIt has been known for a long time that impurity solid solution atoms have a tendency of segregation to grain boundaries, an effect which may strongly retard the grain boundary mobility and thus the kinetics of recrystallization and grain growth in pure metals, even when present in the ppm range. (commonly referred as the dissipation approach). Typical application examples for the force and dissipation approach are respectively [4][5] and [6]. In the force approach the solute drag is estimated by summing the forces that the solute atoms exert on the boundary and in the dissipation approach by evaluating the amount of free energy dissipated due to diffusion when the boundary goes through a volume containing one mole of material. Over the last decades, a lot of efforts have been made to generalize these approaches to a migrating phase boundary into a multi-component system [3,[7][8][9].Indeed in the initial treatment of the solute drag effect by Cahn and by Lücke and Stüwe [10] the equation used for evaluating the solute drag does not apply to phase transformations. It only applies to the migration of grain boundaries, i.e. to one phase materials.The force approach and the dissipation approach have both a sound physical basis and should therefore be equivalent. However the formula for calculating the solute drag for a migrating phase boundary into a multi-component system in steady-state conditions has been a subject of debates over the years and it was only recently that a valid expression has been found with a remarkable amount of empirical insight [11]. The general expression has also been derived in a deductive and completely independent way by applying the principle of maximum dissipation in [12].Even if in the recent years more complex situations than the one treated in the initial works of Cahn [2] and BACKGROUND THEORIESThe where m is the intrinsic boundary mobility (i.e. that corresponding to c = 0). The treatments by Cahn[2] and Lücke and Stüwe [10,23] are derived on the basis of the assumption that the effects on boundary mobility due to the cusp-formation resulting from the solute-boundary interaction can be neglected, the validity of this assumption is discussed belo...

On the Effect of Atoms in Solid Solution on Grain Growth Kinetics

“…In a grain growth situation, the driving pressure is assumed to be well represented by P ¼ 2c GB =r; where c GB is the specific grain boundary energy and r is the grain radius (r ¼ D=2). An expression for the boundary migration rate can be found in the solutepinning analysis referred to above, [10] and a brief summary of some salient ideas in this approach seems necessary in this context. Further, this summary will also include the equations required in the simulations below.…”

“…Table II boundary migration in aluminum by Gordon and Vandermeer [31] and the model analysis of these data by the present authors. [10] The values of the different parameters are reported in Table II under the Case D. It should be noted that in the former analysis, a fitting parameter j was introduced in order to reproduce correctly the variations of the boundary migration rate with the amount of copper obtained by Gordon and Vandermeer. As to estimate the influence of copper on grain growth evolution, this parameter has been reintroduced in Eq.…”

“…The solute-drag theory, however, has not been developed to a level which makes it possible to predict the relevant n-values in this region of solute-boundary instability. However, the new statistical solute-pinning approach to the effect of solute atoms on boundary migration recently developed by the present authors, [10] and extended to the effect of multiple components highly diluted in solid solution, [11] is both physically and mathematically more transparent, which opens for a theoretical analysis of grain growth kinetics in dilute solid solutions.…”

Modeling of grain growth kinetics in complexly alloyed austenite

“…(1), to fit the experimental results, n commonly exceeds 2 corresponding to parabolic growth originally predicted in the classical work [20] for pure metals. Higher values of n are ascribed to the growth retardation due to both phase precipitations (Zener's pinning [18]) and solute atoms of alloying elements or impurities (solute drag [21][22][23]). The corresponding activation energies has no certain physical meaning and strongly vary depending on the composition of the steel and the temperature range under study [2].…”

“…With the mentioned proportionality of activation energy of the grain boundary self-diffusion to AESD kept in mind, it is advisable to employ the former with allowance for its dependence on the chemical composition in modeling the grain growth kinetics. This approach is the first to discard numerous auxiliary parameters conventionally related to the effect of the chemical composition on the solute drag mechanism [21][22][23].…”