A model for the compositions of non-stoichiometric intermediate phases formed by diffusion reactions, and its application to Nb3Sn superconductors

In this work we explore the compositions of non-stoichiometric intermediate phases formed by diffusion reactions: a mathematical framework is developed and tested against the specific case of Nb3Sn superconductors. In the first part, the governing equations for the bulk diffusion and inter-phase interface reactions during the growth of a compound are derived, numerical solutions to which give both the composition profile and growth rate of the compound layer. The analytic solutions are obtained with certain approximations made. In the second part, we explain an effect that the composition characteristics of compounds can be quite different depending on whether it is the bulk diffusion or grain boundary diffusion that dominates in the compounds, and that frozen bulk diffusion leads to unique composition characteristics quite distinct from equilibrium expectations; then the model is modified for the case of grain boundary diffusion. Finally, we apply this model to the Nb3Sn superconductors and propose the approaches to control their compositions.


Introduction
Intermediate phases with finite composition ranges represent a large class of materials, and their compositions may influence their performance in application, as demonstrated in a variety of materials, such as electrical conductivity of oxides (e.g., TiO 2-y 1 ), electromagnetic properties of superconductors (e.g., Nb 3 Sn and YBa 2 Cu 3 O 7-y 2 ), and mechanical properties of some since 1970s 10 . Although a large number of previous experiments (e.g., 4,[8][9][10][11] ) have uncovered some factors that influence the Sn content, it is still a puzzle what fundamentally determines the Nb 3 Sn composition. This work aims to fill that gap. Here it is worth mentioning that the composition interval of a compound layer does not necessarily coincide with its equilibrium phase field rangesthe former can be narrower (e.g., the Nb 3 Sn example above) if the interphase interface reaction rates are slow relative to the diffusion rate across the compound, which results in discontinuities in chemical potentials at the interfaces.
There have been numerous studies regarding diffusion reaction processes, most of which have focused on layer growth kinetics (e.g., [12][13][14][15][16], compound formation and instability (e.g., [14][15][16] ), phase diagram determination (e.g., 17 ), and interdiffusion coefficient measurement (e.g., 18 ), while a systematic model exploring how to control compound composition is still lacking. We find it indeed possible to modify the model developed by Gosele and Tu 13 for deriving the layer growth kinetics of compounds to calculate their compositions; however, certain assumptions (e.g., steady-state diffusion and first-order interface reaction rates) that the model was based on may limit the accuracy of the composition results. In this work, we aim to develop a rigorous, systematic mathematical framework for the compositions of intermediate phases.

Results
Let us consider that an A n B compound with a B content range of X l -X u (X l < X u <1) is formed in a system of M-B/A, where M is a third element that does not dissolve in A n B lattice 19 , and the solubility of B in A is negligible 6 . This is the case we see for the Nb 3 Sn example above (for which A stands for Nb, B for Sn, and M for Cu), but the work below can be modified for other cases. An isothermal cross section of such an M-A-B phase diagram at a certain temperature is shown in Fig. 1. The use of the third element M is to decrease the chemical potential of B, so that unwanted high-B A-B compounds (e.g., NbSn 2 and Nb 6 Sn 5 6 ) that would form in the B/A binary system can be avoided. Similar to the Cu-Nb-Sn system, let us assume B is the primary diffusing species in the A n B layer 20 and that the diffusivity of B in M-B is high 10 .
A schematic of the M-B/A n B/A system for a planar geometry is shown in Fig. 2 Eqs. (2) and (3) Eqs. (1)-(5) are the governing equations for the system set up above, solutions to which give both the X B (x, t) and the l(t) of a growing A n B layer. It should be noted that for the systems with large volume expansion associated with transformation from A to A n B, stress effects need to be considered 21 .
To simplify Eqs.
For now, let us consider two extreme cases.
First, for the case that the interface reaction rates are much higher than the diffusion rate across the A n B layer (i.e., diffusion-rate limited), φ I and φ II are near zero; according to Eqs.  1) and (5) are respectively from which a B can be calculated. Integration of Eq. (5) gives: l ∝ t, and the pre-factor depends on the interface reaction rates.
For a general case between these two extremes, the equations can only be solved with the μ(X) relations of M-B and A n B provided. Next, let us consider a compound with a narrow composition range, so that as a Taylor series expansion is performed around X l for its a(X B ) curve, high-rank terms can be neglected because |X- where κ is the linear coefficient of the a(X) curve. Given the complex boundary conditions for Eq. (1), to obtain the analytic solutions we introduce a second approximation if the A n B composition range is narrow: the X(x) profile of the A n B layer is linear so that at a certain time J is constant with x, such that -(∂X B /∂x)| I ≈ -(∂X B /∂x)| II ≈(X I -X II )/l. With these two approximations, we can solve Eqs. (6)-(7) and obtain that: where η=φ II a s /(φ I a s +κl). Then a I can be calculated from a II , and X I and X II can be calculated from a I and a II using X =X l +(a X -a l )/κ.
To verify the results, the equations are solved for a hypothetical system analytically and numerically, with and without the assumption that X(x) is linear, respectively. The obtained composition profiles are shown in Fig. 3  are constants related to diffusion rate and interface reaction rates, respectively) proposed by previous studies 13,14 . This relation overcomes the above problem because as t << τ, l =[q/(2τ)]•t and as t >> τ, l=q√t. As can be seen from Fig. 3 (b), a better fit to the numerical l(t) curve in the whole range is achieved by l=q(√(t+τ)-√τ).

Discussion
Before discussing the application of this model to a specific material system, it must be pointed out that all of the analysis and calculations above are for the case that B diffuses through A n B bulk. In such a case, for an M-B/A n B/A system, as μ s drops with the growth of A n B layer,  Fig. 1). In either case, A n B eventually reaches homogeneity.
However, we find that the composition could be different for a compound in which the bulk diffusion is low while grain boundary diffusion dominates. One such example is Nb 3 Sn, the composition of which displays some extraordinary features. As an illustration, the X Sn s of a Cu-Sn/Nb 3 Sn/Nb diffusion reaction couple after various annealing times are shown in Fig. 4.
Clearly, as the X Sn (and µ Sn ) of Cu-Sn drop with time, the X Sn s of Nb 3 Sn do not drop accordingly; instead, they more or less remain constant with time. In addition, from 320 hours to 600 hours, although Nb has been fully consumed, the X Sn of Nb 3 Sn does not homogenize (i.e., the X Sn gradient does not decrease) with time. In many other studies on Cu-Sn/Nb systems with Nb in excess (e.g., 4,[8][9] ), even after extended annealing times after the Nb 3 Sn layers have finished growing (which indicates that the Sn sources have been depleted, i.e., µ Sn s have dropped to µ l ), X Sn s of Nb 3 Sn remain high above X l , without dropping with annealing time.
The reason for these peculiarities is that grain boundary diffusion in Nb 3 Sn dominates due to extremely low bulk diffusivity (e.g., lower than 10 -23 m 2 /s at 650 °C) 20,22,23 and small Nb 3 Sn grain size (~100 nm). In this case, our model and equilibrium-state analysis apply only to the diffusion zones (i.e., the grain boundaries and the inter-phase interfaces) instead of the bulk. To clarify this point more clearly, a schematic of the diffusion reaction process is shown in Fig. 5.
At time t 1 , at the A n B/A interface, high-B A n B (L 2 layer) reacts with A (L 3 layer) to form some new A n B cells, leaving B vacancies (noted as V B s) in L 2 layer (time t 2 ). If bulk diffusivity is high, V B s simply diffuse through bulk (e.g., from L 2 to L 1 , as shown by grey dotted arrows) to the B source. If bulk diffusion is frozen, the V B s diffuse first along A n B/A inter-phase interface (as shown by green solid arrows), and then along A n B grain boundaries to the B source. This process continues until this L 3 layer entirely becomes A n B (time t 3 ), so the reaction frontier moves ahead to L 3 /L 4 , while the L 2 /L 3 interface now becomes an inter-plane inside A n B lattice. If bulk diffusion is frozen, the V B s in the L 2 layer that have not diffused to B source will be frozen in this layer forever, and will perhaps transform to A-on-B anti-site defects later (e.g., for Nb 3 Sn, Nb-on-Sn anti-sites are more stable than Sn vacancies 24 ). Since these point defects determine the A n B composition, the X B in this L 2 layer cannot change anymore regardless of μ B variations in grain boundaries. That is to say, X B of any point is just the X II of the moment when the reaction frontier sweeps across this point, i.e., the X B (x) of an A n B layer is simply an accumulation of X II s with l increase. Returning to Fig. 3 (a), the dashed lines display the evolution of X B (x) with l increase for bulk diffusion, while that for grain boundary diffusion is shown by the solid lines.
Since the energy dispersive spectroscopy (EDS) attached to scanning electron microscopes (SEM) that is used to measure the compositions typically has a spatial resolution of 0.5-2 μm, and thus mainly reflects the bulk composition, the composition characteristics of Nb 3 Sn layers as described above can be explained. It should be noted that knowledge of the difference between bulk diffusion and grain boundary diffusion is important in controlling the final composition of a Next, we will modify the above model for the case of grain boundary diffusion for quantitative analysis. As pointed out earlier, the chemical potentials of grain boundaries can change with μ s and l, while those of the bulk cannot. In such a case, µ I and µ II (suppose the diffusivities along the inter-phase interfaces are large) can still be calculated using our model, width, and d is the grain size). Apparently, grain growth with annealing time reduces the diffusion rate. According to Eq. (8), a II is determined by η and a s , and increases with them, as shown by Fig. 6. Since η=φ II a s /(φ I a s +κl)= 1/[φ I /φ II +κl/(φ II a s )], clearly η decreases as φ I /φ II and l increase, and the influence of l (which reflects the X II -x gradient) is mitigated as φ II a s increases.
Thus, to improve X II of A n B at l=0, one should increase μ s and the reaction rate at interface I, and reduce the reaction rate at interface II; meanwhile, to reduce X II (x) gradient, one should increase φ II (which means improving the diffusion rate or reducing the reaction rate at interface II) and a s .
Apparently, these quantitative conclusions are consistent with the above qualitative analysis.   Fig. 7. Besides, we can also infer that the Sn transfer rate at the Cu-Sn/Nb 3 Sn interface must be much faster than that at the Nb 3 Sn/Nb interface, so μ Sn discontinuity across the interface I is small. These explain why low-Sn Cu-Sn can lead to the formation of high-Sn Nb 3 Sn. It is worth mentioning that from Fig. 7, it is clear that the Taylor series for the true a(X) relation of Nb 3 Sn have more high-rank terms than a(X) ≈ a l + κ(X-X l ); however, our numerical calculations show that adding high-rank terms to the a(X) relation does not lead to different conclusions regarding the influences of a s , φ I , φ II , and l on X II . Thus, the above qualitative and quantitative analysis still applies.
In summary, a mathematical framework is developed to describe the compositions and layer

Methods
For the Cu-Sn/Nb 3 Sn/Nb diffusion reaction couples that were used for Sn content measurements (the results of which are shown in Fig. 4), the initial composition of the precursor Institute of Technology was used for calibrating the Sn content of the samples. The standard deviation in the measurements was found to be about ± 0.5 at.%. Parameters: X l =0.24, X u =0.26, a s /a l =2