The Marker Conservation Law in Multiphase Systems

Introduction

The mathematical description of the diffusion process started in 1855 by Fick. His famous laws were derived based on the heat transfer. The second Fick’s law still can be used to estimate the diffusion process in binary diffusion couples. Later in 1946, Kirkendall along with his student Alice Smigelskas studied the diffusion in Cu–Zn diffusion couple. They proved that a model of diffusion via direct interchange of atoms (Fick’s assumptions) was incorrect and that the movement of interface between the “initially different phases” was a result of interdiffusion [1]. This discovery, forever after known as the “Kirkendall effect”, has supported the idea that atomic diffusion occurs through atom–vacancy exchange [1] due to difference in diffusion coefficients [2].

An entirely new understanding of the diffusion processes in the multicomponent systems started with Darken method, published in 1948 [3] and based on Nernst–Planck, Einstein and Kirkendall works. The model is based on the assumption that the overall diffusion flux is a sum of the diffusion and drift fluxes:

where the drift velocity was common velocity for each component. Interdiffusion proceeds under constraint of almost constant total volume, i.e. we neglect the deviations from Vegard law. This constraint means zero divergence of overall volume flux density [4]:

In one-dimensional closed system with concentration gradient along x-axis it immediately leads to

When the only driving force is concentration gradient the flux is given by:

Below we will concentrate on the kinetic effects that are related to difference of diffusion coefficients, namely the position of the Kirkendall plane. The Kirkendall effect is defined as a movement of lattice from slower diffusant side toward the faster diffusant side with some drift velocity. This velocity is common for all components and measured by inert markers frozen within moving lattice. Thus from eqs (2) and (4) we get:

where Ni denotes the molar fraction, Ji=−Di∂ci∂x+ci∑j=1rDj∂Nj∂x. Thus the overall flux, eq. (4) equals

The Kirkendall effect can be estimated from the known drift velocity, eq. (5). It is worth to notice that the drift velocity at K-plane position can be defined as

In the literature exists few methods which allow to estimate the position of the K-plane. One of the first methods was described by Cornet and Calais [5]. The method is based on the drift velocity curve. The position of the Kirkendall plane, xK, is defined by xK=υdrift2tk and graphically determined from the intersection of the drift velocity curve, eq. (5), with the υ=x/2tk plot. Second method is called the trajectory method. In this method, the xK position is calculated from the drift velocity by the integrations split into the time steps during the diffusion process:

where Δt is the time step Δt=ti+1−ti. After each time interval a new position of the Kirkendall plane is calculated. This method confirms, what is generally accepted, that the movement of the Kirkendall plane relative to the initial contact is given by parabolic dependence. This method is numerically effective and allows estimating the position of the Kirkendall plane for any time (from the parabolic dependence); however, its use is limited to one plane only. The last one is described by marker conservation law:

where k denotes the initial concentration of markers (defined by Dirac function).

In multiphase diffusion couples, the Kirkendall velocity is usually discontinuous at moving interfaces and moving interface is an extremely effective source/sink of vacancies. Experimentally, van Dal et al. [6] observed double Kirkendall planes in different phases of a multiphase Ni–Ti couple. The Ni–Ti system represents a technologically important class of order intermetallics with the B2 lattice. This is the shape memory alloy that combines two very unique properties, pseudoelasticity and the shape memory effect. Due to its biocapability Ni–Ti alloys are widely applied in medical treatments, e.g. stents. The knowledge of the diffusion kinetics in Ni–Ti is important for the long-time stability of structures and the improvement of the respective thermomechanical treatment [7].

In our last paper, the Kirkendall effect in single phase was discussed. The new method named entropy density method was presented [8]. The purpose of this paper is to use the entropy density method (marker conservation law) to calculate the Kirkendall plane positions in multiphase Ni–Ti alloy. The results are compared with experiments.

The binary A–B multiphase systems with low non-stoichiometry [9]

The volume produced due to the reaction (unbalanced fluxes) at the moving boundaries of j-phase follows from Leibnitz theorem and is given by (e.g. for component B)

where cB,jR and cB,jL denote the concentrations at the right (R) and left (L) boundaries of j-phase. JB,jR and JB,jL denote the overall fluxes of component B in j-phase and Xj is the position of the jth phase boundary. The overall flux can be written in its usual form by Fick first law:

Finally, to calculate the growth of the particular intermetallic phase the equilibrium molar fractions (from the phase diagram) can be introduced and then eq. (11) should be solved for each boundary, mainly:

where ΔXj denotes the thickness of the jth phase.

Integral diffusivity

In this paper we use the Wagner integral diffusion coefficient (integral interdiffusivity) in a metallic binary compound in which reactive diffusion occurs. This diffusivity can be used when (1) the reactions are restricted to the two planar phase boundaries where the para-equilibrium conditions are held [10], (2) the binary compound is formed, (3) the reaction product shows low non-stoichiometry and steady-state approximation is valid, (4) the terminal phases (substrates A and B) are the pure elements and (5) the diffusivities vary within the narrow homogeneity range of the growing compound.

By combining diffusion fluxes with the mass balances at interfaces the velocity of interface is given by

where diffusivity is expressed by the interdiffusion coefficient; Ni,j*=Ni,j/〈ci,j〉 denotes the molar ratio of i-component in its sublattice (because we consider compounds showing low non-stoichiometry, the average concentrations are composition invariant: ci,j≅const). From eq. (14) the kinetic equation can be obtained by integrating the flux over the whole growing layer:

where ∫Xj(L)Xj(R)dx=Xj(R)−Xj(L)=ΔXj(t) is the thickness of the jth layer. The above equation can be rewritten in its usual form:

where kjp=∫eqNi,j(R)eqNi,j(L)D˜j dNi,j=D˜jInt is the Wagner integral diffusivity.

Kirkendall effect in binary multiphase systems

It is well known how to determine the position of the Kirkendall plane in single phase. We can use several methods [8] like (1) velocity curve method, (2) the trajectory method and (3) entropy density approximation (markers conservation law). However, all of the above methods are strongly based on the definition of the drift velocity (the difference in intrinsic diffusion coefficients of the elements).

This method most often cannot be used in multiphase binary alloys where the integral diffusion coefficient is used (the same coefficient for both components). Moreover, due to the low non-stoichiometry the drift velocity is almost constant. The first challenge in Kirkendall plane calculation is to determine the drift velocity. It can be determined in terms of integral diffusivity, eq. (15), in binary multiphase system as

In the Results section we will describe two methods (1) proposed by Cornet and Calais [5] later rediscovered by van Dal et al. [6] of calculating the Kirkendall plane position in terms of curve method and (2) the marker conservation law method in Ni–Ti multiphase binary alloy. The phase diagram used in computation is presented in Figure 1.

Figure 1:

The Ni–Ti phase diagram from FactSage program.

Results

To calculate the Kirkendall plane position the velocity curve method will be used and applied to Ni–Ti binary system at two different temperatures and two different times. The Ni–Ti system at 850°C and 900°C has several intermetallic phases, Figure 1. Three of them (TiNi₃, Ti–Ni and Ti₂Ni) are considered here. The data used in simulations are shown in Table 1. The effective (integrated) diffusion coefficients were approximated from the experimental data.

For proposed model evaluation, the concentration profile during the reactive diffusion in Ti–Ni system was investigated experimentally. Samples of pure nickel and titanium, 99.99% purity, were used for the experiments. Specimens with dimensions 20 × 10 mm were cut from nickel foil 2 mm thick. The substrates were cleaned and dried prior to the experiment, then were suspended in the center of the reactor. The processing time was 100 and 150 h. After this time the sample was cooled. Then the samples were removed, cleaned thoroughly in cold water and dried. The transverse sections of the selected specimens were observed in an optical microscope and the thickness of the various layers were measured using a Nikon microscope EPIPHOT 300. The layer compositions were evaluated using scanning electron microscope (SEM), the phase compositions were determined by energy-dispersive X-ray spectroscopy (EDX), Figure 2.

Figure 2:

The SEM microphotography of diffusion in Ni–Ti system after 200 h at 1,123 K.

The results of the experiments compared with the calculations are presented in Figures 3 and 4. The points represent the experimental data and the straight lines show calculations. During the calculations the thickness of the intermetalic phases was also determined.

Figure 3:

(a) The concentration profile after the diffusional annealing in Ni–Ti diffusion couple after 100 h at 1,123 K. Dots show the experimental results and lines are simulations; (b) vertical lines represent the experimentally determined position of Kirkendall planes, the dotted line denotes the calculated drift velocity.

Figure 4:

(a) The concentration profile after the diffusional annealing in Ni–Ti diffusion couple after 150 h at 1,173 K. Dots show the experimental results and lines are simulations; (b) vertical lines represent the experimentally determined position of Kirkendall planes, the dotted line denotes the calculated drift velocity.

During the experiments the Kirkendall plane positions were also determined (the vertical line inside TiNi₃ and Ti–Ni phases in Ti₂Ni the Kirkendall plane was not observed).

Simulation of the Kirkendall plane positions

The marker conservation law

When the kinetic of the process is known (the thickness of each intermetallic phase), Figure 5, the Kirkendall plane position can be calculated by marker conservation law, mainly:

(18)∂k∂t+divkυdrift=0

Figure 5:

The kinetic of the intermetallic phase formation in Ni–Ti alloy at 1,123 K.

where k denotes the density of markers. At the beginning the k should be assumed as Dirac function:

(19)k={+∞,x=00,x≠0

and υdrift denotes the drift velocity, which can be approximated by eq. (17) (note that the drift velocity is only time function here: υdrift=υdriftt).

In Figure 6 the evolution of the marker density for different times in Ni–Ti phase is shown. At the beginning the phase thickness was 0 and the marker was at the boundary of pure Ni and Ti. After some time the marker moved from the boundary to the interior of the formed Ni–Ti phase. The final position of the marker shows the maximum on the marker density versus position graph (Figure 6).

Figure 6:

The marker density (Kirkendall plane position) in Ni–Ti intermetallic phase at 1,123 K vs final thickness of the phase.

The final position of the Kirkendall plane positions compared with experiments is presented in Table 2.

Table 2:

The comparison of the Kirkendall plane positions in Ni–Ti alloy at 1,123 and 1,173 K from experiments and calculations (the position is calculated with respect to the phase boundary).

		Experiment	Velocity curve method	Marker conservation law
850°C	Phase:
	TiNi3	0.75 µm	0.7 µm	3.5 µm
	Ti–Ni	63 µm	3.37 µm	6.3 µm
	Ti2Ni	0 µm	0 µm	4.3 µm
900°C	Phase:
	TiNi3	0.05 µm	0.03 µm	7.6 µm
	Ti–Ni	10.24 µm	4.31 µm	13.2 µm
	Ti2Ni	0 µm	0 µm	9.2 µm

The results presented in Table 2 shows that the marker conservation law can be used in calculation the Kirkendall plane position in multiphase system. In both cases, in Ti–Ni phase, the entropy density gave better results of the plane position than the velocity curve method. However, in TiNi₃ intermetallic phase better results are calculated by velocity curve method. Only marker conservation (entropy density) method can be used to approximate the position of the Kirkendall plane in Ti₂Ni phase. Experimentally, in this phase the markers do not exist; however, the reason can be the ThO₂ application procedure or during early stages of diffusion process first two phases are nucleating earlier, thus consuming all ThO₂ particles in the system.

Conclusions

In this paper the bivelocity method and the effective use of the volume continuity equation were presented. It was shown that the volume continuity equation allows defining the material drift velocity and can be used in practical computations. Presented method allows quantitative description of the diffusion controlled processes, the diffusional coatings formation, the life time of the material, i.e. the critical consumption of the reacting element. The resulting system of physical laws, initial and boundary conditions (i.e. the initial boundary value problem) allows using the bivelocity method in a case of non-ideal systems.

The method allows also to determine the Kirkendall plane positions in multiphase binary systems. The drift velocity can be calculated by the means of constant integral Wagner diffusion coefficients. The method gives satisfactionary comparison with experimental results.