Competition between Chemical and Gravity Forces in Binary Alloys

Introduction

The sedimentation process is widely known as the transport phenomena of macroscopic solutes induced by a gravitational or centrifugal field in a liquid solvent [1]. Almost all sedimentation processes have been described by the Lamm equation proposed in 1929 [2]. The fundamental idea of Lamm is that the driving force between the centrifugal field acting on the macro-solute is given by the difference between the centrifugal force and the buoyant force caused by the surrounding liquid solvent [1]. In 1996, Mashimo et al. developed an ultracentrifuge apparatus that can generate an acceleration field up to 10⁶ G for a long time at high temperatures [3]. The second-generation ultracentrifuge was newly developed, in 2001 [4]. Using this apparatus he studied the sedimentation of the substitutional solute atoms of different elements [5, 6, 7], e. g. ultracentrifuge experiments at high temperature on elemental selenium to examine the sedimentation of isotope atoms in liquid matter [8].

The postulate of the quasi-equilibrium vacancies indicates that the high current densities, stresses, etc. do not affect vacancy concentration, which depends only on the temperature, e. g. due to the unlimited power of vacancy sinks-sources. Thus, it is well known how to provide volume conservation namely, lattice drift (Kirkendall effect) caused by the vacancy flux divergence leading to dislocation climb and subsequently to the construction of extra planes in the accumulation region and dismantling atomic planes in the depleting region [9, 10].

We will consider the problem of one-dimensional transport in two-component mixture induced by the gravity (centrifugal) field. The external force per unit mass Fiext in the case of artificial gravitation (e. g., centrifuge) is equal to [11]:

where r is a radius and ω the angular speed and Mi is the molar mass. Now, we consider a case of steady-state sedimentation, controlled by mass diffusion, for which the quasi-stationary assumption holds. Thus from the momentum equation it follows that:

where: p is the pressure and ρ denote the overall mass density of the mixture.

The main equations for interdiffusion fluxes in the bi-velocity form (include the Darken drift of the material [12]) are [13, 14]:

where cA is the molar fraction of the A component, BA its mobility and p pressure field. Here FBext has different interpretations for different external driving forces. For example [15], in the case of electromigration, FBext=ZBeρj (ZB is a partial effective charge of B-ions under electron wind), in case of thermomigration FBext=−1TQBgradT, where QB is a partial heat of transport for B-atoms. In the case of artificial gravitation (e. g., centrifuge), FBext=MBg=MBω2r.

For simplification assuming that the molar volumes are constant and then addition of eqs (3) and (4) gives:

Condition (5) in our one-dimensional problem with zero fluxes and gradients at infinities gives:

Substitution of eq. (6) into eqs (3) and (4) gives:

and, for conservation of mass:

On the other hand, the momentum equation determines the rate of pressure changes according to eq. (2). Combining eqs (2) and (8), one gets:

Equation (9) gives self-consistent coupled equations of diffusion type for atomic fraction. An exact solution can be obtained by using implicit difference methods for differential functional equations. A numeric solution should give the time evolution of the spatial separation of components.

In these studies, we present the first experimental results of the growth of Cu₆Sn₅ and Cu₃Sn during the sedimentation process under a strong gravitational field of 779,000 G at 200 °C.

Sedimentation during binary IMC growth

Experiments

Pure (99.99 %) Cu and Sn specimens were used during the experiment. The first step of the experiment was fabrication of Cu₆Sn₅ and Cu₃Sn layers using diffusion experiments. Thus, the thin foils, each ~2 mm thick, were inserted into the furnace to implement the diffusion couple experiment. Diffusion annealing was carried out for 1 month (720 h) at 200 °C in argon atmosphere. After that, the growth of the two intermetallic layers is schematically presented in Figure 1. The cross-sectional image of the Cu-Sn system before sedimentation process is presented in Figure 2(a). The thickness of the layers was approximately XCu3Sn≈10 and XCu6Sn5≈9 µm. Thus, the samples were fixed into a titanium alloy rotor with an outside diameter of 80 mm and the distances from the rotor axis for the specimen were 35.5 mm.

Figure 1:

Schematic view of the sedimentation process in the intermetallic Cu-Sn system.

Figure 2:

Cross-sectional image of the Cu-Sn system: (a) the initial sample for the sedimentation process – annealing at 200 °C for 720 h, (b) after sedimentation at 200 °C for 24 h.

The ultracentrifuge experiments were performed using the turbine motor-type ultracentrifuge apparatus in Japan – Kumamoto University, at a temperature of 200 °C for 24 h, which was below the melting point of the starting material. The system consisted of a turbine motor with an oil-floating bearing and a specimen rotor. The turbine motor was driven by hot compressed air supplied by a screw compressor and a combustion system. The details of the apparatus are described in Ref. [3]. The rotation rate was 140,000 rpm. For the ultracentrifuged specimen, the thickness of the layers was estimated by using an optical microscope. It is shown that the ultracentrifuge force causes the change in the layer thickness, mainly XCu3Sn≈16 and XCu6Sn5≈7 microns, as shown in Figure 2(b).

Model and results

Let us consider an alloy where two intermediate compounds form, e. g. Cu-Sn system. In such a case, the sedimentation (centrifugal force) influences the thickness of the formed intermetallic layers. The main equations for balance the interdiffusion fluxes for A and B components are [16]:

where X•=dXj/dt is the interface velocity, JB,jR denotes the flux of the B component on the right of the j-th phase.

The diffusional fluxes through the intermetallic phases in the case of intermediate compounds can be expressed using Wagner’s diffusion coefficient [15]:

where the term \vintci,jLci,jRDi,jdc is called the Wagner’s integral diffusion coefficient [17] and is often written as:

Dˉi,jΔci,j is an effective diffusion coefficient averaged over the phase.

In the case of the sedimentation process, the overall flux can be derived from eq. (7):

Equations (10) and (14) allow us to calculate the change of the thickness of the layers in time.

The calculations were divided into two parts: (1) the average diffusion coefficient (Wagner’s diffusivity) of the phase was determined by the inverse method from the experiment and (2) the calculations of the sedimentation process.

The inverse method based on the eqs (10) and (11) for both Cu3Sn and Cu6Sn5 phase. When the initial as well as after the experiment data are known – in this case the initial composition and thickness after the experiment, than the diffusion coefficients can be determined. Thus, the minimization of the least squares metric can be used. The calculations are finished when the required error is achieved.

The diffusion coefficient was estimated from the known kinetics of the process by the inverse method, Figure 3. The sample was annealed for 1 month at 200 °C. The kinetics of the Cu₆Sn₅ and Cu₃Sn growth is presented in Figure 4 the obtain data are presented in Table 1.

Figure 3:

Kinetics of the Cu₆Sn₅ and Cu₃Sn growth during diffusion annealing at 200 °C after 720 h

Figure 4:

Kinetics of the Cu₆Sn₅ and Cu₃Sn growth after the sedimentation process at 200 °C after 24 h.

The data presented in Table 1 were used later in sedimentation calculations. The initial thickness of the Cu₆Sn₅ and Cu₃Sn layers was approximately 10 µm (from experiments). The result of the sedimentation is presented in Figure 4.

During the sedimentation process, the calculated kinetics are linear, Figure 4. The thickness of Cu₃Sn grows, however, the Cu₆Sn₅ phase is now consumed.

Summary

The phenomenological model of the sedimentation process in multiphase binary Cu-Sn system was formulated. Mass transfer occurs under the combined mechanisms of the Kirkendall effect, backstress and external force (sedimentation). We have shown the self-consistent coupled equations of diffusion type for atomic fraction. Exact solution can be obtained based on implicit difference methods for differential functional equations. The Wagner’s diffusion coefficient was determined from the preliminary results – the annealing process by the inverse method.

For the first time the sedimentation process in a multiphase system was presented experimentally. The results of the thickness of the Cu₃Sn and Cu₆Sn₅ phases were compared with simulations. It was presented that the sedimentation process can be used for quick generation of the Cu₃Sn phase – the kinetics are much higher than in the Cu₆Sn₅ phase.