Grain boundary diffusion and grain boundary phase transition in tungsten in the temperature range of activated sintering

2.1 Measurements of GB diffusivity

Radiotracer diffusion experiments were performed by Lee et al. (39, 40) and all details can be found in those publications. Only the key procedures for required understanding of the GB diffusion experiments are presented here.

The nominal purity of the experimental W sintered samples, used as discs of 10 mm in diameter and a thickness of 2 mm, was 99.98 wt.%. The main impurities were detected to be C (15), Fe (20), Mo (40), and P (20) (the amounts are given in wt.ppm). The samples were polished to a mirror-like finish using mechanical grinding/polishing and pre-annealed at 2400 K for 3 h in vacuum (10⁻⁶ Pa) for recrystallization in order to maintain an average grain size of 150 μm, which is suitable for the GB diffusion experiments in the Harrison’s B-type kinetic regime (42). In addition, the Ni-doped W samples were prepared by annealing a nominally pure W specimen with deposited Ni at 1473 K for 10 h in vacuum. Afterwards, the remaining Ni material was removed from the surface by polishing.

The ⁵⁷Co (half-life of 272 days) and ¹⁸⁵W (half-life of 75 days) radioisotopes were used for investigations of impurity and self-diffusion, respectively. Diffusion annealing treatments were performed below 1673 K for the times which satisfy the Harrison’s B-type kinetics, i.e. (D_vt)^1/2 < 0.3d (43), where D_v is the volume diffusion coefficient, t the annealing time and d the grain size. Simultaneously, the parameter α, α = sδ/2(D_vt)^1/2, was kept less than 0.1. Here s is the corresponding segregation factor (s = 1 for self-diffusion) and δ is the GB width. The diffusional GB width δ was measured to be about 0.5 nm for fcc (44) and bcc (45) metals.

The samples with deposited radioisotopes in question were subjected to the given diffusion annealing treatments and the resulting concentrations as function of the penetration depths were measured by mechanical serial sectioning. Under the B-type kinetic regime of GB diffusion, the so-called triple product P, P = sδD_gb (D_gb is the GB diffusion coefficient), is possible to determine using the Le Claire approximation (46) of the Whipple exact solution (47) of the GB diffusion problem. For this aim, the GB diffusion related branch of the penetration profile is linearized in the coordinates of the layer tracer concentration, C ̄ , against the penetration depth, y, to the power 6/5,

Representative examples for ¹⁸⁵W GB self-diffusion in nominally pure W are shown in Figure 1 and the GB diffusion branches are seen to be linear in the corresponding coordinates. Penetration profiles of a similar quality were measured for W GB self-diffusion in Ni-doped W, too. In order to determine the GB diffusion coefficient, D_gb, the volume diffusion coefficient, D_v, has to be known, as it is seen from Equation (1). The volume diffusion coefficients of Co and W measured at temperatures below 1700 K in W single crystals (48) were used.

Figure 1:

Penetration profiles of ¹⁸⁵W GB diffusion in nominally pure W (38).

2.2 Discontinuous temperature dependence of GB diffusion in W

The determined GB diffusion coefficients of W and Co in nominally pure W and Ni-doped W are shown in Figure 2 as function of the inverse temperature. In the temperature interval of 1370 K–1450 K (0.37–0.4 T_m), the Arrhenius plots show discontinuous GB diffusion behavior in all three cases. The differences in absolute values of the triple products between different cases is explained by a strong segregation of Co in W and enhancement of GB self-diffusion by Ni segregation effect in Ni-doped W. However, at higher or lower temperatures, the GB diffusivities reveal typical linear Arrhenius relationships.

Figure 2:

Arrhenius plots of GB self- and impurity (Co) GB diffusion in sintered W and Ni-doped W (37, 38).

The most striking features of the results of GB diffusion measurements are

1. the discontinuity of the Arrhenius line, which occurs for all three curves with rather wide temperature intervals;

2. the discontinuity is seen at approximately the same mean temperature of 1370 K;

3. the magnitude of this discontinuity, however, is quite different. It is of the order of two to three decades for P^Co and less than one for P^W in nominally pure W. It is quite large again for P^W in Ni-doped W.

4. The unusual sharp decrease of P with decreasing temperature seems to split the Arrhenius line into high- and low-temperature regions. Also, the absolute magnitude of the triple products P is quite different in the three cases.

Table 1 summarizes some results at two characteristic temperatures in the high and low temperature regions for comparison.

The surprising discontinuities in the Arrhenius plots, Figure 2, occur for impurity (Co) as well as for self-diffusion (W) very noticeably at nearly the same temperature. From this it is clear that the cause of this behavior originates from the GB properties of the sintered W material.

Potentially, one may suggest that a GB pre-wetting phase transition induced by residual Ni atoms in sintered W, documented by Glebovski et al. (35), enhances GB diffusivities via formation of quasi-liquid (confined) layers covering high-angle GBs in the material. The discontinuities in GB diffusion induced by pre-melting were indeed observed and reported for Cu–Bi (23) and Ni–Bi (24) alloys. However, one may generally expect that diffusion within melted layers would proceed with a lower activation energy, in an obvious contrast to the present experimental data, Figure 2. Furthermore, from a thermodynamic point of view, the discontinuities (jump-like) of the Arrhenius dependencies are observed at temperatures (about 1100 °C) well below the eutectic temperature for the W–Ni system (1495 °C). Analyzing the W–Ni phase diagram (49), one may hypothetically expect solid state wetting by the NiW₂ or NiW phases below 1060 °C and a transition like solid state wetted–pristine grain boundary. This hypothesis seems to contradict the experimental findings that the effective activation energies below 1100 °C and above 1150 °C for W GB diffusion are very similar, Figure 2. An alternative explanation is required.

Guttmann’s segregation model (50), which includes interactions between segregating atoms in the frame of a regular solution, has been applied by Militzer and Wieting (21) to develop a model for describing interfacial transitions due to self-interactions or mutual interactions if one or different species, respectively, segregate to a boundary. The phase transition results in the occurrence of a miscibility gap of the segregants at the boundary below a critical temperature that depends on the interaction energy and the segregant concentration in the bulk. In its turn, the distribution of segregants, especially in the case of a miscibility gap, influences significantly diffusion, both of the matrix (W) atoms as well as of further impurities (Co in the particular case).

Based on the theoretical framework described above, in a previous work (39) carbon was assumed to be the dominant segregating element in the tungsten GBs and the influence of P was initially neglected for simplicity. If a C segregation-induced GB phase transition occurs, one would expect a comparatively high C GB concentrations at lower temperatures, which decreases discontinuously (step-like) at a certain critical temperature. The transition temperature is the same in all our experiments, because all samples were equally prepared.

4.1 Segregation induced GB phase transition

As was mentioned in the Introduction, there exist numerous reports on GB phase transitions caused by temperature change (18, 25), solute segregation (20, 21, 30), GB sliding (15), and solute induced prewetting at GBs (23, 24). In particular, the latter studies revealed discontinuities (kinks) in GB diffusion at temperatures at which GB structural transformations occur. Those result are likely fingerprints of appearance of a liquid-like phase as was theoretically elaborated by Kikuchi and Cahn (51). In a line with these reasonings, one may suggest that the discontinuity in GB diffusion in W, as described in previous works (39), could be caused by a GB segregation-induced phase transition in sintered W. This interpretation becomes more comprehensive and logical by discussing the AES results based on the following theoretical analysis.

According to Fowler’s binary regular solution theory (52), which includes interactions between segregating atoms in a GB, there is a critical segregation temperature T_c, below which the amount of the segregated element remains practically constant (for an alloy with fixed volume concentration of solute), in a clear contradiction to McLean’s equilibrium segregation theory (53). The solute–solvent interaction is effectively captured by an interaction parameter α, which represents the contribution of the mixing enthalpy H_MA of the two elements, M (solvent, matrix element) and A (solute), in terms of a regular solution model, H_MA = αX_MX_A (29). The case of α > 0 corresponds to a separation tendency and formation of a miscibility gap is possible. A full description of GB thermodynamics has been given in many papers, see e.g. (54], [55], [56). Experimentally, a GB spinodal decomposition was observed, e.g., in Fe–Mn (57) or Ni–Cr–Fe (58) alloys, that contributed to development of GB phase diagrams (59). Note that the case of α = 0 corresponds to an ideal segregation according to McLean’s theory.

Based upon the thermodynamic model for single element segregation, Guttmann (50) developed Fowler’s theory with respect to ternary systems having two segregating elements (A & B) and applying the regular solution model. In the case of α_AB > 0, co-segregation is promoted by an attractive interaction between the A and B elements, while site competition prevails in the case of α_AB < 0. Based on Guttmann’s work, Militzer et al. (21) calculated the temperature of GB phase transition induced by a site competition between A and B solute atoms (α_AB < 0). The temperature for GB phase transition was arbitrarily estimated for various bulk solubility conditions, as a consequence indicating that the GB phase transition induced segregation absolutely depends on the bulk solubility of the element, in fact on the chemical potential of the element. As a result, above the critical concentration, GB phase transition by formation of a miscibility gap of A, B atoms at GBs is significantly weakened.

According to the Guttmann model (50), the GB concentrations (in at. fractions) of A, X gb A , and B, X gb B , atoms are given by:

and

Here X v A ( X v B ) is the volume concentration of the A (B) atoms and the enrichment factors B_A and B_B are determined as

and

Here Δ G 0 A ( Δ G 0 B ) is the difference between the reference chemical potentials of matrix and A (B) atoms for bulk and GB positions (50) and the interaction parameters α _ij (i, j = A, B, M) correspond to the interactions between atoms i and j in terms of the regular solution model (M denotes here the matrix atoms). The values Δ G 0 A + α AM and Δ G 0 B + α BM correspond to the segregation energies of non-interacting solutes in terms of McLean’s model. Equations (4) and (5) suggest that the co-segregation tendency in terms of separation/site competition is driven by the relative values of the parameters α_AM;, α_BM, and α AB ′ = α AB − α AM − α BM . This model can be extended further accounting for multi-layer segregation (60) and/or different atomic coordination numbers within the bulk and GB layers (61) that will significantly complicate the final expressions like Equations (2)–(5), but the general structure will remain formally similar. For the present discussion and for the sake of simplicity we applied the original Guttmann model.

In Figure 8, the temperature dependencies of the determined GB concentrations of (model) A and B solute atoms are shown. The numerical parameters, Table 2, are chosen somewhat arbitrarily, especially in view of obvious limitations of the simplified Guttmann model, but with a goal to mimic C and P atoms in tungsten, compare Figures 7 and 8. In view of intrinsic limitations of Guttmann’s model, we are not focused on reproducing the measured temperature dependencies of the AES measurements. However, the general trends are qualitatively well reproduced. In fact, the A atoms dominantly segregate at the GBs, the tendency which is almost abruptly switched to dominant B segregation at a critical temperature, Figure 8. This modelling behavior corresponds qualitatively to the experimentally observed behavior of C and P atoms at W GBs, Figure 7.

Figure 8:

Temperature dependence of GB concentrations of the elements A and B (which are considered to mimic C and P, respectively) determined using Guttmann’s model (47), Equations (2)–(5). The volume concentrations of A and B atoms are taken equal to those of C and P atoms in the present W samples ( X v A = 0.23 at.% and X v B = 0.12 at.%). The numerical parameters of the model are listed in Table 2.

In Figure 9 we plot the GB concentrations of atoms A and B against the corresponding volume concentrations and the temperature as it is followed from the simplified Guttmann model. A strong A–B interaction is seen which results in a kind of alternating segregation behavior of the (model) A and B atoms. The numbers are chosen to mimic (not to reproduce) behavior of C and P atoms in nominally pure W. The calculations are performed keeping X v B = 0.12 at.% (Figure 9a and b) and X v A = 0.23 at.% (Figure 9c and d).

Figure 9:

GB concentrations (subscript ‘gb’) of elements A and B (which are considered to mimic C and P, respectively) according to Guttmann’s model (47) as functions of the volume concentrations (subscript ‘v’) and temperature T.

An inspection of Figure 9 allows to distinguish three regions, I, II, and III, of impurity concentrations in crystalline bulk. If X v B is kept constant at 0.12 at.% (Figure 9a and b), a temperature-induced ‘swapping’ of A and B segregations will occur for the volume concentrations of the element A within the region II, 0.13 at.% ≤ X v A ≤ 0.4 at.%. Alternatively, if X v A is kept constant at 0.23 at.% (Figure 9c and d), the temperature-induced ‘swapping’ of A and B segregations will occur for the volume concentrations of the element B belonging to the interval 0.09 at.% ≤   X v B ≤ 0.24 at.%. One recognizes that according to the model, within the concentration interval II, a critical temperature appears which separates two types of GB structures in terms of impurity segregation. This behavior indicates the existence of segregation-induced GB phase transition or complexion transition in the system. The critical temperature of the segregation GB transition is a function of the impurity volume concentrations, i.e. of the corresponding chemical potentials. For the concentrations which correspond to the experimentally observed amounts of C and P in W, the GB segregation transition is ‘severe’ in terms of strong changes of the GB segregations within a limited temperature interval. It is straightforward to suggest that the resulting GB properties, including the GB diffusivities, have to reveal strong discontinuities in the corresponding temperature interval, in full accord with the experimental observations by Lee et al. (40).

Summarizing the present explanation of the GB phase transition in terms of the segregation theory, it is intuitively expected that carbon and phosphorus segregations may be responsible for the GB phase transition in sintered W, resulting in a discontinuity in the W GB diffusion at temperatures where the C enrichment drops below that of P. As clearly confirmed from the AES analysis results of Figure 7, the GB segregation concentrations of C and P follow Guttmann’s theory reasonably well (cf. Figure 8), showing discontinuous changes at about 1000 K. It is important that C and P reveal different segregation tendencies with respect to the GB sites, see Figure 5. While P segregates to the GB core, the C segregation attains its maximum for at more distant, ‘sub-GB’ atomic layers. A discontinuity in the GB self-diffusion behavior at the temperatures where segregation swaps from one dominated by C at sub-GB sites to another dominated by P atoms within the GB core is not unexpected.

However, we have to indicate the existing difference in the temperatures of discontinuities for GB diffusion and C/P segregation evolution. According to Militzer’s model (21) such a difference is probably due to two factors: an inhomogeneity of the impurity concentrations in sintered W and the effect of cooling rate. In particular, the latter point was supported by Drachinskiy et al. (62), which showed that the cooling rate strongly influences the critical temperature for GB segregation in W and transition metals. In this regard, this issue related to the temperature mismatch should be discussed in further works, especially applying advanced thermodynamic modelling of segregation-induced GB phase transition.

The characteristic site competition in W GBs for phosphorus and carbon atoms is obvious from the AES results (Figures 6 and 7) and it is corroborated by the simplified Guttmann model, Figures 8 and 9. The opposite segregation of C and P is in good agreement with the results of Hofmann et al. (63, 64). This is also confirmed by the relationship between the GB concentrations of C and P obtained from the AES analysis, as re-plotted in Figure 10.

Figure 10:

Relation between the GB segregation concentrations of C and P in W.

According to Hoffmann et al. (64), Fe and Ni also reveal strong competitions with both P and especially with C. Since cobalt behaves chemically similarly to these 3d transition elements, we may propose that the carbon segregating atoms influence thermodynamic-as well as kinetic properties of Co at W grain boundaries. As a result, it is believed that GB diffusion of Co in W (with C and P as impurities) underwent a discontinuous temperature dependence at 0.37 T_m (39), as is represented in Figure 2. The temperature interval of the discontinuity in GB diffusivity for Co is notably increased with respect to that for W, Figure 2. This behavior is well explained by the Co–C and Co–P interactions. In fact, the interaction parameters α AC ′ = α AC − α AM − α CM and α AP ′ = α AP − α AM − α PM are positive for both Fe–P and Ni–P and negative for Fe–C and Ni–C, as it was predicted by Hoffmann et al. (64). A similar tendency is expected for Co–P and Co–P interactions, which explains the opposite shifts of low-temperature and high-temperature limits of the discontinuity in Co GB diffusivity with respect to that for W in Figure 2.

Thus, carbon atoms compete for GB sites with P (and with most of other segregating elemental impurities in the sintered W GBs) giving rise to a segregation GB phase transition at a certain critical temperature. Although there is a temperature difference between the results of GB diffusion measurements and AES analysis with respect to the critical temperature, we propose to explain the discontinuity in the temperature dependence of the W GB diffusion in terms of a sequence of GB phase transitions. In fact, at temperatures below 0.37 T_m, carbon atoms enrich highly the W GBs, whereas in the high-temperature range above 0.4 T_m, W GBs are characterized by strong P segregation and a low C coverage. At about the critical temperature, corresponding to the discontinuous GB diffusion behavior, a miscibility gap is formed with co-existing of low- and high-C regions. Thus, we suggest that the C segregation is expected to induce GB phase transition by 2D spinodal decomposition in sintered W GB. That behavior is accompanied by alternating evolution of P segregation/desegregation. As a result, at the critical temperature, carbon atoms increasingly leave probably the interstitial positions of W atoms in GBs, resulting in a sharp increase in GB diffusivity. However, the phase transition by the carbon atoms does not affect the diffusion mechanism as vacancy-mediated one (40).

4.2 Impact of Ni alloying on GB diffusion/GB phase transition and activated sintering

In order to elucidate the impact of Ni alloying, we will compare the GB diffusion coefficients measured in of W and Ni-doped W at two temperatures, 1327 K and 827 K, corresponding to the high- and low-temperature regions, Table 1. As is seen, the GB diffusivity (triple product) of W atoms in W–Ni is enhanced with respect to that in nominally pure W by a factor of 31 (489) at a low (high) temperature. The enhancement of P W − Ni W with respect to P W W can be explained considering the results of AES investigation. The triple products for W can formally be seen as double products of the corresponding diffusion coefficients and the GB width, since the segregation factor for W can safely be assumed to be unity (at least not strongly deviates from unity). The Ni alloying does probably not affect the diffusional GB width (or changes it by less than a factor of, say, two). Thus, the above-mentioned enhancement is attributed to the Ni alloying induced changes of D W − Ni W . Now, suggesting that Ni, similar to Co, is partially interstitially dissolved in W GBs and diffuses interstitially, formation of W–Ni dumbbells can be assumed. Rotational and dissociative jumps of these dumbbells enlarge the mobility of W atoms considerably. The probability of dumbbell formation increases with the Ni-content in the GB, which is influenced by site competition with C. This explanation agrees with the earlier reasonings (40). It is instructive to relate the process of activated sintering to W GB diffusion which is enhanced by the formation of rapidly mobile W–Ni atom pairs.

Summarizing the above interpretation, we can discuss the mechanism of activated sintering in W in terms of discontinuous self-diffusion and carbon-induced GB phase transition. It is well known that a small amount of Ni addition significantly improves the densification process of W powder compacts at low temperatures of about 1473 K (4, 5). As was described in the Introduction, the mechanism of W activated sintering is not clarified yet. Here, accounting for the developed understanding of GB properties including the structure, dynamics and atomic diffusion, we will propose new insights. W self-diffusion in the Ni-alloyed W GBs is enhanced with respect to that in nominally pure W by a factor of 100 or even more, Figure 4. According to Hoffmann et al. (64), Ni atoms have a strong affinity with W atoms and a strong repulsive interaction with carbon atoms that enhances W GB self-diffusion, especially in the state with a low-C coverage. Note that the discontinuous transition of the GB diffusivity occurs around the temperature of the activated sintering regardless of the additive element.