Abstract
In this work we explore the compositions of nonstoichiometric intermediate phases formed by diffusion reactions: a mathematical framework is developed and tested against the specific case of Nb_{3}Sn superconductors. In the first part, the governing equations for the bulk diffusion and interphase 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 that the bulk composition of a compound layer remains unchanged after its initial formation instead of varying with the diffusion reaction system; here the model is modified for the case of grain boundary diffusion. Finally, we apply this model to the Nb_{3}Sn superconductors and propose 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 intermetallics (e.g., Ni–Al_{0.4–0.55}^{3}), etc. For instance, the superconducting Nb_{3}Sn phase, which finds significant applications in the construction of 12–20 T magnets^{4,5,6}, has a composition range of ~17–26 Sn at.%^{7,8}, and its superconducting transition temperature and magnetic field decrease dramatically as Sn content drops^{4,8,9}. The Nb_{3}Sn phase, which is generally fabricated from CuSn and Nb precursors through reactive diffusion processes, is always found to be Snpoor (e.g., 21–24 at.%^{4,9,10}), making composition control one of the primary concerns in Nb_{3}Sn development since 1970s^{10}. Although a large number of previous experiments (e.g.,^{4,9,10,11,12}) 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 ranges – the 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.,^{13,14,15,16,17}), compound formation and instability (e.g.,^{15,16,17}), phase diagram determination (e.g.,^{18}), and interdiffusion coefficient measurement (e.g.,^{19}), 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^{14} for deriving the layer growth kinetics of compounds to calculate their compositions; however, certain assumptions (e.g., steadystate diffusion and firstorder 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.
The model
Let us consider that a nonstoichiometric A_{n}B compound is formed in a system of M–B/A, where M is a third element that does not dissolve in A_{n}B lattice^{20}. The use of the third element M is to decrease the chemical potential of B, so that unwanted highB A–B compounds (e.g., NbSn_{2} and Nb_{6}Sn_{5} in the NbSn system^{7}) that would form in the B/A binary system can be avoided. With the M–B, A_{n}B, and Arich phases denoted as α, β, and γ, respectively, a schematic of the α/β/γ system for a planar geometry is shown in Fig. 1. Let us denote the α/β and β/γ interphase interfaces as I and II, respectively, and the mole fractions, chemical potentials, activities, and diffusion fluxes of B in the β phase at interfaces I and II as X_{I}^{β}, μ_{I}^{β}, a_{I}^{β}, J_{I}^{β}, and X_{II}^{β}, μ_{II}^{β}, a_{II}^{β}, J_{II}^{β}, respectively. The maximum and minimum mole fractions of B in β phase (i.e., A_{n}B compound) from the phase diagram are set as X_{I}^{β,eq} and X_{II}^{β,eq}, respectively. Let us also denote the μ_{B}s and a_{B}s of α and γ as μ_{B}^{α}, a_{B}^{α}, and μ_{B}^{γ}, a_{B}^{γ}, respectively. Let us assume that the solubility of B in γ phase is negligible^{7}. An isothermal cross section of such an M–A–B phase diagram at a certain temperature is shown in Fig. 2. 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 model below can be modified for other cases. Similar to the CuNbSn system, let us assume B is the primary diffusing species in the β phase^{21} and that the α phase can act as an intensive sink for B vacancies in order for it to be an efficient source of B atoms for β layer growth, and that the diffusivity of B in α is high so that the α phase remains homogeneous during the growth of β layer^{11}.
Here we assume that diffusion occurs by vacancy mechanism and the total atomic flux is balanced by the vacancy flux. As discussed in the papers by Svoboda and Fischer et al.^{22,23,24}, the presence of various types of sinks or sources for vacancies may lead to quite different diffusional and conservation laws and equations. For this model, we assume that B vacancies are generated by the reaction at interface II (as will be discussed in detail later), and then diffuse across the β layer to interface I, where they are annihilated by B atoms from α phase (the B source). For the simplicity of the model, we assume that there are no sinks or sources for vacancies in the bulk or grain boundaries of β phase, while the only sink in the system for B vacancies is the α phase. The following model can be modified for cases with other types of sinks or sources for vacancies using the models by Svoboda and Fischer et al.^{22,23,24}.
In this work let us assume the diffusivity of B in β phase, D, and the molar volume of β phase, V_{m}^{β}, do not vary with X_{B}, in which case the continuity equation in the A_{n}B layer is given by:
According to mass conservation, in a unit time the amount of B transferring across the interface I should equal to that diffusing into the β layer from the interface I, and the amount arriving at the interface II should equal to that transferring across it, i.e., dn/dt_{I} = J_{I}^{β}∙A_{I}, and dn/dt_{II} = J_{II}^{β}∙A_{II}, where A_{I} and A_{II} are the areas of the interfaces I and II, respectively. The molar transport rate dn/dt across an interface equals to r∙A_{int}∙exp(−Q/RT)∙[1−exp(−Δμ/RT)], where r is the transfer rate constant for this interface with the unit of mol/(m^{2}∙s), A_{int} is the interface area, Q is the energy barrier, R is the gas constant, T is the temperature in K, and Δμ is the driving force for atom transfer. For the interface I, Δμ_{I} = μ_{B}^{α}−μ_{I}^{β}. For the interface II, Δμ_{II} = μ_{II}^{β}–μ_{B}^{γ}, and μ_{B}^{γ} = μ_{B}(A−X_{II}^{β,eq} B). With J_{B} = −(D/V_{m})∙(∂X_{B}/∂x), we have:
Eqs. (2) and (3) are the boundary conditions for Eq. (1). Note that X_{B} in α phase, X_{B}^{α}, drops with annealing time as B in α is used for β layer growth, so μ_{B}^{α} drops with t:
where n_{M} and n_{B0} are the moles of M and B in the MB precursor. For those systems without the third element, μ_{B}^{α} is constant, and Eq. (4) is not needed. In addition, since the B atoms diffusing to the interface II are used to form new β layers, we have:
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 γ to β, stress effects need to be considered^{25}.
To simplify Eqs. (2) and (3), we notice that 1−exp[−(μ_{B}^{α}−μ_{I}^{β})/RT] = 1−a_{I}^{β}/a_{B}^{α}, since μ_{B}^{α}−μ_{I}^{β} = RTln(a_{B}^{α}/a_{I}^{β}); similarly, 1−exp[−(μ_{II}^{β}−μ_{B}^{γ})/RT] = 1−a_{B}^{γ}/a_{II}^{β}. Let us also denote D/[V_{m}∙r_{I}∙exp(−Q_{I}/RT)] as φ_{I}, and D/[V_{m}∙r_{II}∙exp(−Q_{II}/RT)] as φ_{II}: clearly φ_{I} and φ_{II} represent the ratios of diffusion rate over interface reaction rates, and have a unit of meter. Then Eqs. (2) and (3) can be respectively written as:
Analytic and Numerical Solutions
Let us first consider two extreme cases. First, for the case that the interface reaction rates are much higher than the diffusion rate across the β layer (i.e., diffusionrate limited), φ_{I} and φ_{II} are near zero; according to Eqs. (2)–(3), , μ_{B}s are continuous at both interfaces, so X_{II}^{β} = X_{II}^{β,eq}. Suppose μ_{B}^{α} and the position of interface I, x_{I}, are both constant with time, then X_{I}^{β} is also constant, and the solutions to Eqs. (1) and (5) are respectively X_{B}(x, t) = X_{I}^{β}−(X_{I}^{β}−X_{II}^{β,eq})∙erf{(x−x_{I})/[2√(Dt)]}/erf(k/2) and l = k√(Dt) for the β layer, where k can be numerically solved from k∙exp(k^{2}/4)∙erf(k/2) = 2/√π∙(X_{I}^{β}−X_{II}^{β,eq})/X_{II}^{β,eq}. For instance, for X_{I}^{β} = 0.26 and X_{II}^{β,eq} = 0.17, k = 0.953. On the other hand, if the interface reaction rates are much lower than the diffusion rate across β (e.g., as the β layer is thin), φ_{I} and φ_{II} are large; according to Eqs. (2) and (3), X_{B} and J_{B} are nearly constant in the entire β layer. Thus, (1−a_{B}/a_{B}^{α})/φ_{I} = (1−a_{B}^{γ}/a_{B})/φ_{II}, from which a_{B} can be calculated. Integration of Eq. (5) gives: l ∝ t, and the prefactor 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 α and β provided. Next, let us consider a compound with a narrow composition range, so that as a Taylor series expansion is performed around X_{II}^{β,eq} for its a(X_{B}) curve, highrank terms can be neglected because X−X_{II}^{β,eq} ≤ (X_{I}^{β,eq}−X_{II}^{β,eq}) is small; that is, a_{X} ≈ a_{B}^{γ} + κ(X−X_{II}^{β,eq}), 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 β composition range is narrow: the X(x) profile of the β 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}^{β}−X_{II}^{β})/l. With these two approximations, we can solve Eqs. (6) and (7) and obtain that:
where η = φ_{II}a_{B}^{α}/(φ_{I}a_{B}^{α}+κl). Then a_{I}^{β} can be calculated from a_{II}^{β}, and X_{I}^{β} and X_{II}^{β} can be calculated from a_{I}^{β} and a_{II}^{β} using X = X_{II}^{β,eq} +(a_{X} − a_{B}^{γ})/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(a). For simplicity, μ_{B}^{α} of the system is set as μ_{B}(A−X_{I}^{β,eq} B) and is constant (for Nb_{3}Sn systems, this means that Nb_{6}Sn_{5} serves as Sn source), and the other parameters are specified in the figure. The difference between the analytic and numerical solutions is <0.1%, showing that the approximation of linear X(x) is good if the composition range is small (2 at.% in this case). The l(t) result (where t is the annealing time after the incubation period) from the numerical calculations is shown in Fig. 3(b). While the analytic l(t) solution is complicated, some l(t) relations with simple forms can be used as approximations. The most widely used l(t) relation for the case of constant μ_{B}^{α} is l = bt^{m}, in which m = 1 for reactionrate limited and m = 0.5 for diffusionrate limited; however, a defect with this relation is that as l increases from zero, it may shift from reactionrate limited to diffusionrate limited, so m may vary with t. Here a new relation l = q[√(t+τ)−√τ] (where q is a growth constant and τ is a characteristic time) is proposed. Such a relation is consistent with l^{2}/v_{1}+l/v_{2} = t (where v_{1} and v_{2} are constants related to diffusion rate and interface reaction rates, respectively) proposed by previous studies^{14,15}. 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).
Results and 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 β bulk. In such a case, for an α/β/γ system, as μ_{B}^{α} drops with the growth of β layer, X_{B}(x) of the entire β layer should decrease with μ_{B}^{α}, because μ_{B}^{α} ≥ μ_{I}^{β} ≥ μ_{II}^{β} ≥ μ_{B}^{γ}. Finally, one of two cases will occur: either μ_{B}^{α} drops to μ_{B}^{γ} (if A is in excess) so the system ends up with the equilibrium among γ, A–X_{II}^{β,eq} B, and M–X_{1} B (as shown by the shaded region in the isothermal M–A–B phase diagram in Fig. 2), or γ is consumed up and β gets homogenized with time and finally μ_{B}(β) = μ_{B}(α) (as shown by the dashed line in Fig. 2). In either case, β layer 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 CuSn/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 CuSn 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 CuSn/Nb systems with Nb in excess (e.g.,^{4,9,10}), 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_{B}^{γ}), X_{Sn}s of Nb_{3}Sn remain high above X_{II}^{β,eq}, 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)^{21,26,27} and small Nb_{3}Sn grain size (~100 nm). In this case, our model and equilibriumstate analysis apply only to the diffusion zones (i.e., the grain boundaries and the interphase 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 β/γ interface, highB A_{n}B (L_{2} layer) reacts with γ (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 β/γ interphase interface (as shown by green solid arrows), and then along β 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 interplane 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 AonB antisite defects later (e.g., for Nb_{3}Sn, NbonSn antisites are more stable than Sn vacancies^{28}). 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}^{β} of the moment when the reaction frontier sweeps across this point, i.e., the X_{B}(x) of the β layer is simply an accumulation of X_{II}^{β}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 compound. For instance, if bulk diffusivity is high, one method to form highB A_{n}B is increasing the starting B/A ratio so that after long annealing time for homogenization subsequent to the full consumption of A, μ_{B}(α) = μ_{B}(A−X_{I}^{β,eq} B). However, our experiments demonstrate that for compounds with low bulk diffusivity (e.g., Nb_{3}Sn), such an approach does not work; instead, controlling the X_{II}^{β}s while the compounds are growing is the only way. For those compounds with low but nonnegligible bulk diffusivities, their compositions would be between these two extremes.
Then what determines the bulk composition as grain boundary diffusion dominates? From Fig. 5, it can be clearly seen that there is a competition deciding the V_{B} fraction in the frontier A_{n}B layer: at t_{2} the reaction across the β/γ interface produces V_{B}s in L_{2} layer, while the diffusion of B along β grain boundaries and α/β interface fills these V_{B}s. Thus, if the diffusion rate is slow relative to the reaction rate at interface II (i.e., φ_{II} is low), a high fraction of V_{B}s would be left behind as the interface II moves ahead, causing low B content; if, on the other hand, the diffusion rate is high relative to the reaction rate at interface II, the A_{n}B layer has enough time to get homogenized with the B source, causing low X_{B} gradient. In this case, the μ_{B} of B source and the reaction rate at interface I together set a upper limit for μ_{B} of β.
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_{B}^{α} and l, while those of the bulk cannot. In such a case, μ_{I}^{β} and μ_{II}^{β} (suppose the diffusivities along the interphase interfaces are large) can still be calculated using our model, provided that the μ(X) relation and D of the β grain boundary (instead of the bulk) are used in all of the equations, and that φ_{I} and φ_{II} are multiplied by a factor of ∑A_{GB}/A_{int} (where ∑A_{GB} is the sum of the cross section areas of the grain boundaries projected to the interphase interfaces), because B diffuses only along β grain boundaries while reactions occur at the entire interfaces. Approximately, ∑A_{GB}/A_{int} ≈ [1−d^{2}/(d + w)^{2}] ≈ 2w/d (where w is the β grain boundary width, and d is the grain size). Apparently, grain growth with annealing time reduces the diffusion rate. According to Eq. (8), a_{II}^{β} is determined by η and a_{B}^{α}, and increases with them, as shown by Fig. 6. Since η = φ_{II}a_{B}^{α}/(φ_{I}a_{B}^{α} + κl) = 1/[φ_{I}/φ_{II} + κl/(φ_{II}a_{B}^{α})], clearly η decreases as φ_{I}/φ_{II} and l increase, and the influence of l (which reflects the X_{II}^{β}x gradient) is mitigated as φ_{II}a_{B}^{α} increases. Thus, to improve X_{II}^{β} of A_{n}B at l = 0, one should increase μ_{B}^{α} and the reaction rate at interface I, and reduce the reaction rate at interface II; meanwhile, to reduce X_{II}^{β}(x) gradient, one should increase φ_{II} (which means improving the diffusion rate or reducing the reaction rate at interface II) and a_{B}^{α}. Apparently, these quantitative conclusions are consistent with the above qualitative analysis.
Next let us compare this model with the example of Nb_{3}Sn. It has been well established from experimental work that there are mainly two factors that can significantly influence the Sn content of Nb_{3}Sn in a CuSn/Nb_{3}Sn/Nb diffusion reaction couple: heat treatment temperature and CuSn source. The heat treatment temperature can simultaneously influence multiple factors of Eq. (8), such as a_{B}^{α}, D, and reaction rates at both interfaces, etc. Thus, the explanation of the influence of temperature on Sn contents using this theory requires knowledge of the quantitative variations of these factors with temperature. For the other factor, CuSn source, the diffusion reaction couples can be classified into two types based on the CuSn source: the type I uses bronze (with Sn content in CuSn typically below 9 at.%) as Sn source, and the type II uses highSn CuSn (e.g., Cu25 at.% Sn). Previous measurements^{4,9,10,11,29} show that both types of samples have Sn contents above 24 at.% for the Nb_{3}Sn layer next to the CuSn source; however, they have distinct Sn content gradients as the Nb_{3}Sn layers grow thicker: the type I generally has Sn content gradients above 3 at.%/μm^{29}, while those of the type II are below 0.5 at.%/μm^{4,9,10}. Such a difference in the Sn content gradients leads to distinct grain morphologies and superconducting properties. The different X_{Sn} gradients in the two types of samples with different CuSn sources can be easily explained by our theory above: according to Eq. (8), increased μ_{B}^{α} can decrease X_{Sn} gradients. It may also need further investigation regarding whether CuSn source can also influence diffusion rates in Nb_{3}Sn layer (e.g., via thermodynamic factor), because greater D leads to greater φ_{II}, which helps decreasing X_{Sn} gradients. As to the phenomenon that different wires have similar X_{Sn} in the Nb_{3}Sn layer next to the CuSn source, the relation between μ_{Sn}(CuSn) and μ_{Sn}(NbX_{Sn} Sn) is required. The CuSn system has been well studied, and the phase diagram calculated by the CALPHAD technique using the thermodynamic parameters given by ref. 30 is well consistent with the experimentally measured diagram^{31}. Thus, the parameters from ref. 30 are used to calculate μ_{Sn} of CuSn, which is shown in Fig. 7. On the other hand, although thermodynamic data of NbSn system were proposed by refs. 30 and 32, in these studies Nb_{3}Sn was treated as a line compound. However, some information about μ_{Sn} of Nb_{3}Sn can be inferred from its relation with μ_{Sn} of CuSn: since Cu7 at.% Sn leads to the formation of Nb24 at.% Sn near the CuSn source^{29}, we have μ_{Sn}(Cu7 at.% Sn) ≥ μ_{Sn}(Nb24 at.% Sn). Thus, the expected approximate μ_{Sn}(NbX_{Sn} Sn) curve in a power function is shown in Fig. 7. Furthermore, we can also infer that the Sn transfer rate at the CuSn/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 lowSn CuSn can lead to the formation of highSn 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 highrank terms than a(X)≈a_{B}^{γ} + κ(X−X_{II}^{β,eq}); however, our numerical calculations show that adding highrank terms to the a(X) relation does not lead to different conclusions regarding the influences of a_{B}^{α}, φ_{I}, φ_{II}, and l on X_{II}^{β}. Thus, the above qualitative and quantitative analysis still applies.
In summary, a mathematical framework is developed to describe the compositions and layer growth rates of nonstoichiometric intermediate phases formed by diffusion reactions. The governing equations are derived and analytic solutions are given for compounds with narrow composition ranges under certain approximations. We also modify our model for compounds in which bulk diffusion is frozen, of which the bulk is not in equilibrium with the rest of the system. Based on this model, the factors that fundamentally determine the compositions of nonstoichiometric compounds formed by diffusion reactions are found and approaches to control the compositions are proposed.
Methods
For the CuSn/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 CuSn alloy was Cu12 at.% Sn. The samples were reacted at 650 °C for 65 h, 130 h, 320 h, and 600 h. Then the surfaces of the samples were polished to 0.05 μm and the compositions were measured using an EDS system attached to an SEM. An accelerating voltage of 15 kV was used for the quantitative line scans. A standard Nb25 at.% Sn bulk sample provided by Dr. Goldacker from Karlsruhe 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.%.
Additional Information
How to cite this article: Xu, X. and Sumption, M. D. A model for the compositions of nonstoichiometric intermediate phases formed by diffusion reactions, and its application to Nb_{3}Sn superconductors. Sci. Rep. 6, 19096; doi: 10.1038/srep19096 (2016).
References
 1.
Kim, M., Baek, S., Paik, U., Nam, S. & Byun, J. Electrical conductivity and oxygen diffusion in nonstoichiometric TiO_{2x}. J. Korean Phys. Soc. 32, S1127–S1130 (1998).
 2.
Park, S. I., Tsuei, C. C. & Tu, K. N. Effect of oxygen deficiency on the normal and superconducting properties of YBa_{2}Cu_{3}O_{7δ}. Phys. Rev. B 37, 2305–2308 (1988).
 3.
Noebe, R. D., Bowman, R. R. & Nathal, M. V. Physical and mechanical properties of the B2 compound NiAl. Int. Mater. Rev. 38, 193–232 (1993).
 4.
Godeke, A. Performance boundaries in Nb_{3}Sn superconductors (Ph.D. thesis) 89–104 (University of Twente, 2005).
 5.
Xu, X., Sumption, M. D., Bhartiya, S., Peng, X. & Collings, E. W. Critical current densities and microstructures in rodintube and tube type Nb_{3}Sn strands – present status and prospects for improvement. Supercond. Sci. Technol. 26, 075015 (2013).
 6.
Xu, X., Sumption, M. D. & Peng, X. Internally oxidized Nb_{3}Sn strands with fine grain size and high critical current density. Adv. Mater. 27, 1346–1350 (2015).
 7.
Charlesworth, J. P., Macphail, I. & Madsen, P. E. Experimental work on the niobiumtin constitution diagram and related studies. J. Mater. Sci. 5, 580–603 (1970).
 8.
Zhou, J. et al. Evidence that the upper critical field of Nb_{3}Sn is independent of whether it is cubic or tetragonal. Appl. Phys. Lett. 99, 122507 (2011).
 9.
Lee, P. J., Fischer, C. M., Naus, M. T., Squitieri, A. A. & Larbalestier, D. C. The microstructure and microchemistry of high critical current Nb3Sn strands manufactured by the bronze, internalSn and PIT techniques. IEEE Trans. Appl. Supercond. 13, 3422–3425 (2003).
 10.
Peng, X. et al. Composition profiles and upper critical field measurement of internalSn and tubetype conductors. IEEE Trans. Appl. Supercon. 17, 2668–2671 (2007).
 11.
Suenaga, M. In Superconductor Materials Science: Metallurgy, Fabrication, and Applications (eds Foner, S. & Schwartz, B. B.) 201–274 (Plenum, 1981).
 12.
Xu, X., Collings, E. W., Sumption, M. D., Kovacs, C. & Peng, X. The effects of Ti addition and high Cu/Sn ratio on tube type (Nb, Ta)_{3}Sn strands, and a new type of strand designed to reduce unreacted Nb ratio. IEEE Trans. Appl. Supercond. 24, 6000904 (2014).
 13.
Farrell, H. H., Gilmer, G. H. & Suenaga, M. J. Grain boundary diffusion and growth of intermetallic layers: Nb_{3}Sn. J. Appl. Phys. 45, 4025–4035 (1974).
 14.
Gosele, U. & Tu, K. N. Growth kinetics of planar binary diffusion couples: “Thinfilm case” versus “Bulk cases”. J. Appl. Phys. 53, 3252–3260 (1982).
 15.
Debkov, V. I. in Reaction Difusion and Solid State Chemical Kinetics (ed. Debkov, V. I.) 3–19 (IPMS, 2002).
 16.
Gusak, A. M. et al. In Diffusioncontrolled Solid State Reactions: In Alloys, Thin Films and Nanosystems (eds Gusak, A. M. et al.) 99–133 (WileyVCH, 2010).
 17.
Tu, K. N. & Gusak & A. M. In Kinetics in Nanoscale Materials (eds Tu, K. N. & Gusak, A. M.) 187–193 (WileyVCH, 2014).
 18.
Kodentsov, A. A., Bastin, G. F. & van Loo, F. J. J. The diffusion couple technique in phase diagram determination. J. Alloy Compd. 320, 207–217 (2001).
 19.
Gong, W., Zhang, L., Wei, H. & Zhou, C. Phase equilibria, diffusion growth and diffusivities in NiAlPt system using Pt/βNiAl diffusion couples. Prog. Nat. Sci. 21, 221–226 (2011).
 20.
Sandim, M. J. R. et al. Grain boundary segregation in a bronzeroute Nb_{3}Sn superconducting wire studied by atom probe tomography. Supercond. Sci. Technol. 26, 055008 (2013).
 21.
Laurila, T., Vuorinen, V., Kumar, A. K. & Paul, A. Diffusion and growth mechanism of Nb_{3}Sn superconductor grown by bronze technique. Appl. Phys. Lett. 96, 231910 (2010).
 22.
Svoboda, J., Fischer, F. D., Fratzl, P. & Kroupa, A. Diffusion in multicomponent systems with no or dense sources and sinks for vacancies. Acta Mater. 50, 1369–1381 (2002).
 23.
Svoboda, J., Fischer, F. D. & Fratzl, P. Diffusion and creep in multicomponent alloys with nonideal sources and sinks for vacancies. Acta Mater. 54, 3043–3053 (2006).
 24.
Fischer, F. D. & Svoboda, J. Substitutional diffusion in multicomponent solids with nonideal sources and sinks for vacancies. Acta Mater. 58, 2698–2707 (2010).
 25.
Cui, Z. W., Gao, F. & Qu, J. M. Interfacereaction controlled diffusion in binary solids with applications to lithiation of silicon in lithiumion batteries. J. Mech. Phys. Solids. 61, 293–310 (2013).
 26.
Bochvar, A. A. et al. Diffusion of tin during growth of the Nb_{3}Sn layer. Met. Sci. Heat. Treat. 22, 904–907 (1980).
 27.
Müller, H. & Schneider, T. H. Heat treatment of Nb_{3}Sn conductors. Cryogenics 48, 323–330 (2008).
 28.
Besson, R., Guyot, S. & Legris, A. Atomicscale study of diffusion in A15 Nb_{3}Sn. Phys. Rev. B 75, 054105 (2007).
 29.
Abächerli, V. et al. The influence of Ti doping methods on the high field performance of (Nb, Ta, Ti)_{3}Sn multifilamentary wires using Osprey bronze. IEEE Trans. Appl. Supercon. 15, 3482–3485 (2005).
 30.
Li, M., Du, Z., Guo, C. & Li, C. Thermodynamic optimization of the CuSn and CuNbSn systems. J. Alloys Compd. 477, 104–117 (2009).
 31.
Hansen, M. & Anderko, R. P. In Constitution of binary alloys (eds Hansen, M. & Anderko, R. P.) 634 (McGrawHill, 1958).
 32.
Toffolon, C., Servant, C., Gachon, J. C. & Sundman, B. Reassessment of the NbSn system. J. Phase Equilib. 23, 134–139 (2002).
Acknowledgements
The authors thank S. Dregia and J. Morral for useful discussions, and X. Peng and Hyper Tech Research Inc. for providing Nb_{3}Sn samples for analysis. The work is funded by the US Department of Energy, Division of High Energy Physics, under an SBIR program.
Author information
Affiliations
Department of Materials Science and Engineering, the Ohio State University, Columbus, OH 43210 USA
 X. Xu
 & M. D. Sumption
Authors
Search for X. Xu in:
Search for M. D. Sumption in:
Contributions
X.X. initiated this study, developed the model, and wrote the manuscript. M.D.S. supported this work, discussed the results, and reviewed the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to X. Xu.
Rights and permissions
This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
About this article
Further reading

Effect of Zn addition and Ti doping position on the diffusion reaction of internal tin Nb3Sn conductors
Superconductor Science and Technology (2019)

Ternary Nb3Sn superconductors with artificial pinning centers and high upper critical fields
Superconductor Science and Technology (2019)

Atomicscale analyses of Nb3Sn on Nb prepared by vapor diffusion for superconducting radiofrequency cavity applications: a correlative study
Superconductor Science and Technology (2019)

The effects of Mg doping on the microstructure and transport properties of internal tinprocessed brass matrix Nb3Sn superconductors
Superconductor Science and Technology (2019)

Effects of inhomogeneities on pinning force scaling in Nb3Sn wires
Superconductor Science and Technology (2018)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.