It is well known that segregated impurity atoms can drastically reduce the mobility of grain boundaries in pure metals. The phenomenon is considered to be a general effect and is usually referred to simply as “solute drag”. Cahn [1] and Lücke and Stüwe [2] suggested a quantitative treatment of the solute drag, which has been the basis of following works on solute drag. The latter works have mainly focused on extending their approach to a migrating phase boundary into a multi-component system [3-9]. One achievement of the Cahn-Lücke-Stüwe theory (CLS theory) was to demonstrate that the grain boundary velocity is inversely proportional to the solute concentration in the high solute content/low driving force regime.

However this theory presents some major drawbacks. Firstly, while the basic physical idea behind the solute drag theory is principally simple, where the motion of a grain boundary is slowed down by solute atoms which exert a drag force on the boundary, the analytical treatment becomes rather involved: the solute profile around the moving grain boundary must be established by solving Fick’s first law in a moving frame and then the solute drag is determined from this profile. Secondly, few works have tackled the issue of the influence of different types of solute atoms on a moving grain boundary [10,11]. This issue is of industrial relevance because industrial alloys are generally not high purity alloys with only one type of impurity but in most cases made up of different major additions. The last decades have seen the advent of computer aided material science and engineering and so a solute drag approach taking simply into account the effect of different additions should be of great help to achieve more realistic simulations.

Recently a new solute drag model has been proposed based on a solute pinning approach [12]. The grain boundary is pinned by the solute atoms along the boundary, which will induce a local cusping of the boundary at the solute atoms (Figure 1). Stress concentration will arise promoting thermal activation of solute atoms out of the boundary. Compared to the CLS theory, the main benefit of this approach is to be computationally simpler. The present treatment aims to generalize this approach to different types of solute atoms in the high solute content/ low driving force regime. Then let us consider two solutes A and B in a matrix forming an ideal solid solution.

The Equation (19) is actually consistent with an analogous one that can be derived by following Lücke and Detert’s original demonstration of solute drag [14] (the CLS theory has been developed later to overcome the shortcomings of this simple approach but do not change fundamentally the result). Indeed, in their approach it is assumed that a slow moving boundary will drag along its migration a number of solute atoms A and B close to their equilibrium values, which is equal to

and therefore exert a drag pressure

It should be noted that in their demonstration and are not determined by the cusping of the boundary but by the Einstein’s equation

where and are the bulk diffusion coefficient of the solute atoms:

The boundary will move then with the velocity

and therefore,

In their approach, the quantities

and

are respectively the mobility of the boundary when it is fully loaded either with solute atoms of type A or B. Finally, the boundary velocity could be expressed as

i.e. the same formula as the one in Equation (19), although the explicit expressions for the mobility terms are slightly different (see also [12]). The agreement between the solute cusping approach and the most accepted theory on solute drag can be understood by the fact that in the slow driving pressure/high solute content regime the solute cusping approach predicts a boundary which is nearly flat due to the large number of solute atoms pinning it [12], consistent with the original assumptions made by Lücke and Detert [14].

It is also interesting to note that a similar expression, although simpler, for solute drag effects in multi-component alloys was introduced as a phenomenological approach already by Vatne [15], to account for solute effects in the softening model Alsoft [16-18]. Here an effective concentration of solutes as derived from a summation of the solute concentration of the individual alloy elements, weighted by their activation energy for diffusion, is introduced into an equation analogous to Equation (11) [15].

However, although more stringently derived, it should be noted that some limitations also apply to the formula (Equations (19) and (25)). Firstly, interactions between the two types of solute have been neglected. The consequences of their interactions have been studied in [10,11]. Without considering site saturation in the boundary, it has been demonstrated that the solute drag in the presence of solute-solute interactions could either be increased or reduced depending on the nature of their interactions, attractive or repulsive [10]. It has also been proved that co-segregation of solutes competing for the boundary sites but not interacting with each other can lead to a complex behaviour where an impurity addition increases the boundary mobility [11]. Secondly, this formula only apply for the migration of a grain boundary in a dilute solid solution and do not tackle the problem of interphase migration into a multi-component system, which is at the time being out of the reach of this approach.

Despite the limitations mentioned before, and it still remains to be validated against real experiments, it is believed that the establishment of this formula (Equations (19) and (25)) has the potential of more accurate simulations of microstructure evolution (e.g. recrystallisation and sub-grain/grain growth) where solute drag effects are of importance, and thus in computer aided design of industrial processes and alloys. In particular, as compared to simpler approaches, it may be important in alloys where different solute has significantly different diffusion rates and/or boundary segregation tendencies (as expressed by the activation energy)

This research work has been supported by a KMB project (project number: 193179/I40), in Norway. The financial support by the Research Council of Norway and the industrial partners, Hydro Aluminium and Sapa Technology is gratefully acknowledged.